Mod Two Homology and Cohomology (Jean Claude)

Mod Two Homology and Cohomology

Book Project

Jean-Claude HAUSMANNUniversity of Geneva, Switzerland

September 1, 2013

Contents

Introduction 7

Chapter 1. Simplicial (co)homology 91. Simplicial complexes 92. Definitions of simplicial (co)homology 143. Kronecker pairs 164. First computations 214.1. Reduction to components 214.2. 0-dimensional (co)homology 214.3. Pseudomanifolds 224.4. Poincare series and polynomials 234.5. (Co)homology of a cone 234.6. The Euler characteristic 254.7. Surfaces 255. The homomorphism induced by a simplicial map 286. Exact sequences 337. Relative (co)homology 388. Mayer-Vietoris sequences 439. Appendix A: an acyclic carrier result 4410. Appendix B: ordered simplicial (co)homology 4511. Exercises for Chapter 1 49

Chapter 2. Singular and cellular (co)homologies 5312. Singular (co)homology 5312.1. Definitions 5312.2. Relative singular (co)homology 5912.3. The homotopy property 6412.4. Excision 6512.5. Well cofibrant pairs 6812.6. Mayer-Vietoris sequences 7513. Spheres, disks, degree 7614. Classical applications of the mod 2-(co)homology 8115. CW-complexes 8316. Cellular (co)homology 8717. Isomorphisms between simplicial and singular (co)homology 9318. CW-approximations 9619. Eilenberg-McLane spaces 10120. Generalized cohomology theories 10521. Exercises for Chapter 2 106

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4 CONTENTS

Chapter 3. Products 10922. The cup product 10922.1. The cup product in simplicial cohomology 10922.2. The cup product in singular cohomology 11223. Examples 11424. Two-fold coverings 11824.1. H1, fundamental group and 2-fold coverings 11824.2. The characteristic class 12024.3. The transfer exact sequence of a 2-fold covering 12224.4. The cohomology ring of RPn 12425. Nilpotency, Lusternik-Schnirelmann categories and topological

complexity 12426. The cap product 12727. The cross product and the Kunneth theorem 13128. Some applications of the Kunneth theorem 13928.1. Poincare series and Euler characteristic of a product 13928.2. Slices 13928.3. The cohomology ring of a product of spheres 14028.4. Smash products and joins 14128.5. The theorem of Leray-Hirsch 14428.6. The Thom isomorphism 15128.7. Bundles over spheres 15928.8. The face space of a simplicial complex 16328.9. Continuous multiplications on K(Z2,m) 16429. Exercises for Chapter 3 166

Chapter 4. Poincare Duality 16930. Algebraic topology and manifolds 16931. Poincare Duality in polyhedral homology manifolds 17032. Other forms of Poincare Duality 17832.1. Relative manifolds 17832.2. Manifolds with boundary 18132.3. The intersection form 18332.4. Non degeneracy of the cup product 18432.5. Alexander Duality 18533. Poincare duality and submanifolds 18633.1. The Poincare dual of a submanifold 18633.2. The Gysin Homomorphism 18933.3. Intersections of submanifolds 19133.4. The linking number 19434. Exercises for Chapter 4 198

Chapter 5. Projective spaces 20135. The cohomology ring of projective spaces - Hopf bundles 20136. Applications 20536.1. The Borsuk-Ulam theorem 20536.2. Non-singular and axial maps 20637. The Hopf invariant 20937.1. Definition 209

CONTENTS 5

37.2. The Hopf invariant and continuous multiplications 21037.3. Dimension restrictions 21237.4. The Hopf invariant and linking numbers 21338. Exercises for Chapter 5 215

Chapter 6. Equivariant cohomology 21739. Spaces with involution 21740. The general case 22941. Localization theorems and Smith theory 23742. Equivariant cross products and Kunneth theorems 24243. Equivariant bundles and Euler classes 24944. Equivariant Morse-Bott Theory 259

Chapter 7. The Steenrod squares 26945. Cohomology operations 26946. Properties of the Steenrod squares 27347. Construction of the Steenrod squares 27548. The Adem relations 28049. The Steenrod algebra 28650. Applications 291

Chapter 8. Stiefel-Whitney classes 29351. Trivializations and structures on vector bundles 29352. The class w1 – Orientability 29953. The class w2 – Spin structures 30254. Definition and properties of the Stiefel-Whitney classes 30755. Real flag manifolds 30955.1. Definitions and Morse theory 31055.2. Cohomology rings 31455.3. Schubert cells and Stiefel-Whitney classes 32156. Splitting principles 32957. Complex flag manifolds 33358. The Wu formula 33858.1. Wu’s classes and formula 33858.2. Orientability and spin structures 34158.3. Applications to 3-manifolds 34458.4. The universal class for double points 34559. Thom’s theorems 35059.1. Representing homology classes by manifolds 35059.2. Cobordism and Stiefel-Whitney numbers 353

Chapter 9. Miscellaneous applications and developments 35760. Actions with scattered or discrete fixed point sets 35761. Conjugation spaces 36062. Chain and polygon spaces 36662.1. Definitions and basic properties 36662.2. Equivariant cohomology 37062.3. Non-equivariant cohomology 37862.4. The inverse problem 38362.5. Spatial polygon spaces and conjugation spaces 386

6 CONTENTS

63. Equivariant characteristic classes 38764. The equivariant cohomology of certain homogeneous spaces 39265. The Kervaire invariant 399

Bibliography 413

Index 421

Introduction

The homology mod 2 first occurred in 1908 in a paper of Tietze [192] (see also[38, pp. 41–42]). Till around 1935, its use was limited to providing Poincare dualityfor closed manifolds (even non-orientable), first obtained by Veblen and Alexanderin 1912 [196]. A main consequence is that the Euler characteristic of any closedodd-dimensional manifold vanishes. The discoveries of the Stiefel-Whitney classesin 1936–38 and of the Steenrod squares in 1947–50 gave the cohomology mod 2 itsstatus of a major tool in algebraic topology, providing for instance the theory ofspin structures and Thom’s work on the cobordism ring.

These notes are an introduction, at graduate student’s level, of the (co)homologymod 2 (there will be essentially no other). They include classical applications(Brouwer fixed point theorem, Poincare duality, Borsuk-Ulam theorem, Smith the-ory, etc) and less classical ones (face spaces, topological complexity, equivariantMorse theory, etc). The cohomology of flag manifolds is treated in details, in-cluding for Grassmannians the relationship between Stiefel-Whitney classes andSchubert calculus. Some original applications are given in Chapter 9.

Our approach is different than that of classical textbooks, in which the (co)ho-mology mod 2 is just a particular case of the (co)homology with arbitrary coeffi-cients. Also, most authors start with a full account of homology before approachingcohomology. In these notes, (co)homology mod 2 is treated as a subject by itselfand we start with cohomology and homology together from the beginning. Theadvantage of this approach is the following.

• The definition of a (co)chain is simple and intuitive: an (say, simplicial)m-cochain is a set of m-simplexes; an m-chain is a finite set of m-simplexes.The concept of cochain is simpler than that of chain (one less word in thedefinition...), more flexible and somehow more natural. We thus tend toconsider cohomology as the main concept and homology as a (useful) toolfor some arguments.• Working with Z2 and its standard linear algebra is much simpler than

working with Z. For instance, the Kronecker pairing has an intuitivegeometric interpretation occurring at the beginning and making in anelementary way the cohomology as the dual of the homology. Also, com-putations, like the homology of surfaces, are quite easy and come early inthe exposition.• The absence of sign and orientation considerations is an enormous techni-

cal simplification (even of importance in computer algorithms computinghomology). With much lighter computations and technicalities, the ideasof proofs are more apparent.

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8 INTRODUCTION

We hope that these notes will be, for students and teachers, a complement orcompanion to textbooks like those of A. Hatcher [80] or J. Munkres [152]. Fromour teaching experience, starting with the mod 2 (co)homology is a great help tograsp the ideas of the subject. The technical difficulties of signs and orientationsfor finer theories, like integral (co)homology, may then be introduced afterwards,as an adaptation of the intuitive mod 2 (co)homology.

Not in this book. The following tools are not used in these notes.

• Augmented (co)chain complexes. The reduced cohomology H∗(X) is de-fined as coker (H∗(pt)→ H∗(X)) for the unique map X → pt.• Simplicial approximation.• Spectral sequences (except in the proof of Proposition 40.28).

Also, we do not use any advanced homotopy tool, like spectra, completions, etc.Because of this, some prominent problems using the cohomology mod 2 are onlybriefly surveyed: the work by Adams on the Hopf-invariant-one problem (p.292),the Sullivan’s conjecture (pp. 202 and 292) and the Kervaire invariant (§ 65).

Prerequisites. The reader is assumed to have some familiarity with the followingsubjects:

• general point set topology (compactness, connectedness, etc).• elementary language of categories and functors.• simple techniques of exact sequences, like the five lemma.• elementary facts about fundamental groups, coverings and higher homo-

topy groups are sometimes used.

Acknowledgments: A special thank is due to Volker Puppe who provided severalvaluable suggestions and simplifications. Michel Zisman, Pierre de la Harpe andSamuel Tinguely have carefully read several sections of these notes. The author isalso grateful for useful comments to Jim Davis, Matthias Franz, Rebecca Goldin,Andre Haefliger, Tara Holm, Allen Knutson, Jerome Scherer, Dirk Schuetz, AndrasSzenes, Vladimir Turaev and Claude Weber.

CHAPTER 1

Simplicial (co)homology

Simplicial homology was invented by H. Poincare in 1899 [159] and its mod 2-version, presented in this chapter, was introduced in 1908 by H. Tietze [192]. It isthe simplest homology theory to understand and, for finite complexes, it may becomputed algorithmically. The mod 2-version permits rapid computations on easybut non-trivial examples, like spheres and surfaces (see § 4).

Simplicial (co)homology is defined for a simplicial complex, but is an invariantof the homotopy type of its geometric realization (this result will be obtained indifferent ways using singular homology: see § 17). The first section of this chapterintroduces classical techniques of (abstract) simplicial complexes. Since simplicialhomology has been the only existing (co)homology theory till the 1930’s, simplicialcomplexes played a predominant role in algebraic topology during the first thirdof the XXth century (see the introduction of Chapter 30). Later developments of(co)homology theories, defined directly for topological spaces, made this combina-torial approach less crucial. However, simplicial complexes remain an efficient wayto construct topological spaces, also largely used in computer science.

1. Simplicial complexes

In this section we fix the notation and recall classical facts about (abstract)simplicial complexes. For more details, see [175, Chapter 3].

A simplicial complex K consists of

• a set V (K), the set of vertices of K.• a set S(K) of finite non-empty subsets of V (K) which is closed under

inclusion: if σ ∈ S(K) and τ ⊂ σ, then τ ∈ S(K). We require thatv ∈ S(K) for all v ∈ V (K).

An element σ of S(K) is called a simplex of K (“simplexes” and “simplices” areadmitted as plural of “simplex”; we shall use “simplexes”, in analogy with “com-plexes”). If ♯(σ) = m+1, we say that σ is of dimension m or that σ is a m-simplex.The set of m-simplexes of K is denoted by Sm(K). The set S0(K) of 0-simplexes isin bijection with V (K), and we usually identify v ∈ V (K) with v ∈ S0(K). Wesay that K is of dimension ≤ n if Sm(K) = ∅ for m > n, and that K is of dimen-sion n (or n-dimensional) if it is of dimension ≤ n but not of dimension ≤ n− 1.A simplicial complex of dimension ≤ 1 is called a simplicial graph. A simplicialcomplex K is called finite if V (K) is a finite set.

If σ ∈ S(K) and τ ⊂ σ, we say that τ is a face of σ. As S(K) is closed underinclusion, it is determined by it subset Smax(K) of maximal simplexes (if K is finitedimensional). A subcomplex L of K is a simplicial complex such that V (L) ⊂ V (K)and S(L) ⊂ S(K). If S ⊂ S(K) we denote by S the subcomplex generated by S,

9

10 1. SIMPLICIAL (CO)HOMOLOGY

i.e. the smallest subcomplex of K such that S ⊂ S(S). The m-skeleton Km of Kis the subcomplex of K generated by the union of Sk(K) for k ≤ m.

Let σ ∈ S(K). We denote by σ the subcomplex of K formed by σ and all its

faces (σ in the above notation). The subcomplex σ of σ generated by the properfaces of σ is called the boundary of σ.

1.1. Geometric realization. The geometric realization |K| of a simplicial com-plex K is, as a set, defined by

|K| := µ : V (K)→ [0, 1]∣∣∑

v∈V (K) µ(v) = 1 and µ−1((0, 1]) ∈ S(K) .We see that |K| is the set of probability measures on V (K) which are supportedby the simplexes. There is a distance on |K| defined by

d(µ, ν) =

√ ∑

v∈V (K)

[µ(v)− ν(v)]2

which defines the metric topology on |K|. The set |K| with the metric topology isdenoted by |K|d. For instance, if σ ∈ Sm(K), then |σ|d is isometric to the standardEuclidean simplex ∆m = (x0, . . . , xm) ∈ Rm+1 | xi ≥ 0 and

∑xi = 1.

However, a more used topology for |K| is the weak topology, for which A ⊂ |K|is closed if and only if A ∩ |σ|d is closed in |σ|d for all σ ∈ S(K). The notation|K| stands for the set |K| endowed with the weak topology. A map f from |K|to a topological space X is then continuous if and only if its restriction to |σ|d iscontinuous for each σ ∈ S(K). In particular, the identity |K| → |K|d is continuous,which implies that |K| is Hausdorff. The weak and the metric topology coincide ifand only if K is locally finite, that is each vertex is contained in a finite numberof simplexes. When K is not locally finite, |K| is not metrizable (see e.g. [175,Theorem 3.2.8]).

When a simplicial complex K is locally finite, has countably many vertices andis finite dimensional, it admits a Euclidean realization, i.e. an embedding of |K|into some Euclidean space RN which is piecewise affine. A map f : |K| → RN ispiecewise affine if, for each σ ∈ S(K), the restriction of f to |σ| is an affine map.Thus, for each simplex σ, the image of |σ| is an affine simplex of RN . If dimK ≤ n,such a realization exists in R2n+1 (see e.g. [175, Theorem 3.3.9]).

If σ ∈ S(K) then |σ| ⊂ |K|. We call |σ| the geometric simplex associated to σ.Its boundary is |σ|. The space |σ| − |σ| is the geometric open simplex associated toσ. Observe that |K| is the disjoint union of its geometric open simplexes.

There is a natural injection i : V (K) → |K| sending v to the Dirac measurewith value 1 on v. We usually identify v with i(v), seeing a simplex v as a pointof |K| (a geometric vertex). In this way, a point µ ∈ |K| may be expressed as aconvex combination of (geometric) vertices:

(1.2) µ =∑

v∈V (K)

µ(v)v .

1.3. Let K and L be simplicial complexes. Their join is the simplicial complexK ∗ L defined by

(1) V (K ∗ L) = V (K) ∪V (L).(2) S(K ∗ L) = S(K) ∪ S(L) ∪ σ ∪ τ | σ ∈ S(K) and τ ∈ S(L).

Observe that, if σ ∈ Sr(K) and τ ∈ Ss(L), then σ ∪ τ ∈ Sr+s+1(K ∗ L). Also,σ ∪ τ = σ ∗ τ , the topological join of two spaces (see p. 144).

1. SIMPLICIAL COMPLEXES 11

1.4. Stars, links, etc. Let K be a simplicial complex and σ ∈ S(K). The starSt(σ) of σ is the subcomplex of K generated by all the simplexes containing σ. Thelink Lk(σ) of σ is the subcomplex of K formed by the simplexes τ ∈ S(K) suchthat τ ∩ σ = ∅ and τ ∪ σ ∈ S(K). Thus, Lk(σ) is a subcomplex of St(σ) and

St(σ) = σ ∗ Lk(σ) .

More generally, if L is a subcomplex of K, the star St(L) of L is the subcomplexof K generated by all the simplexes containing a simplex of L. The link Lk(L) ofL is the subcomplex of K formed by the simplexes τ ∈ S(St(L))− S(L). One hasSt(L) = L ∗ Lk(L). The open star Ost(L) of L is the open neighbourhood of |L| in|K| defined by

Ost(L) = µ ∈ |K| | µ(v) > 0 if v ∈ V (L) .This is the interior of |St(L)| in |K|.

1.5. Simplicial maps. Let K and L be two simplicial complexes. A simplicialmap f : K → L is a map f : V (K)→ V (L) such that f(σ) ∈ S(L) if σ ∈ S(K), i.e.the image of a simplex of K is a simplex of L. Simplicial complexes and simplicialmaps form a category, the simplicial category, denoted by Simp.

A simplicial map f : K → L induces a continuous map |f | : |K| → |L| defined,for w ∈ V (L), by

|f |(µ)(w) =∑

v∈f−1(w)

µ(v) .

In other words, |f |(µ) is the pushforward of the probability measure µ on |L|. Thegeometric realization is thus a covariant functor from the simplicial category Simpto the topological category Top of topological spaces and continuous maps.

1.6. Components. Let K be a simplicial complex. We define an equivalencerelation on V (K) by saying that v ∼ v′ if there exists x0, . . . , xm ∈ V (K) withx0 = v, xm = v′ and xi, xi+1 ∈ S(K). A maximal subcomplex L of K such thatV (L) is an equivalence class is called a component of K. The set of componentsof K is denoted by π0(K). As the vertices of a simplex are all equivalent, K isthe disjoint union of its components and π0(K) is in bijection with V (K)/ ∼. Therelationship with π0(|K|), the set of (path)-components of the topological space |K|is given in Lemma 1.7 below

Lemma 1.7. The natural injection j : V (K) → |K| descends to a bijection

j : π0(K)≈→ π0(|K|).

Proof. The definition of the relation ∼ makes clear that j descends to amap j : π0(K) → π0(|K)|. Any point of |K| is joinable by a continuous pathto some vertex j(v). Hence, j is surjective. To check the injectivity of j, letv, v′ ∈ V (K) with j(v) = j(v′). There exists then a continuous path c : [0, 1]→ |K|with c(0) = j(v) and c(1) = j(v′). Consider the open cover Ost(w) | w ∈ V (K) of|K|. By compactness of [0, 1], there exists n ∈ N and vertices v0, . . . , vn−1 ∈ V (K)such that c([k/n, (k + 1)/n]) ⊂ Ost(vk) for all k = 0, . . . , n − 1. As c(0) = j(v)and c(1) = j(v′), one deduces that v0 = v and vn−1 = v′. For 0 < k ≤ n − 1,one has c(k/n) ∈ Ost(vk−1) ∩Ost(vk). This implies that vk−1, vk ∈ S(K) for allk = 1, . . . , n− 1, proving that v ∼ v′.

A simplicial complex is called connected if it is either empty or has one compo-nent. Note that |K| is locally path-connected for any simplicial complex K. Indeed,


any point has a neighborhood of the form |St(v)| for some vertex v, and |St(v)| path-connected. Therefore, |K| is path-connected if and only if |K| is connected. UsingLemma 1.7, this proves the following lemma.

Lemma 1.8. Let K be a simplicial complex. Then K is connected if and onlyif |K| is a connected space.

Finally, we note the functoriality of π0. Let f : K → L be a simplicial map. Ifv ∼ v′ for v, v′ ∈ V (K), then f(v) ∼ f(v′), so f descends to a map π0f : π0(K)→π0(L). If f : K → L and g : L → M are two simplicial maps, then π0(gf) =π0gπ0f . Also, π0idK = idπ0(K). Thus, π0 is a covariant functor from the simplicialcategory Simp to the category Set of sets and maps.

1.9. Simplicial order. A simplicial order on a simplicial complex L is a partialorder ≤ on V (L) such that each simplex is totally ordered. For example, a totalorder on V (L), as in examples where vertices are labelled by integers, is a simpli-cial order. A simplicial order always exists, as a consequence of the well-orderingtheorem.

1.10. Triangulations. A triangulation of a topological space X is a homeo-morphism h : |K| → X , where K is a simplicial complex. A topological space istriangulable if it admits a triangulation. It will be useful to have a good process totriangulate some subspaces of Rn. A compact subspace A of Rn is a convex cell ifit is the set of solutions of families of affine equations and inequalities

fi(x) = 0, i = 1 . . . r and gj(x) ≥ 0, j = 1 . . . s .

A face B of A is a convex cell obtained by replacing some of the inequalities gj ≥ 0by equations gj = 0. The dimension of B is the dimension of the smallest affinesubspace of Rn containing B. A vertex of A is a cell of dimension 0. By inductionon the dimension, one proves that a convex cell is the convex hull of its vertices(see e.g. [135, Theorem 5.2.2]).

A convex-cell complex P is a finite union of convex cells in Rn such that:

(i) if A is a cell of P , so are the faces of A;(ii) the intersection of two cells of P is a common face of each of them.

The dimension of P is the maximal dimension of a cell of P . The r-skeleton P r

is the subcomplex formed by the cells of dimension ≤ r. The 0-skeleton coincideswith the set V (P ) of vertices of P .

A partial order ≤ on V (P ) is an affine order for P if any subset R ∈ V (P )formed by affinely independent points is totally ordered. For instance, a total orderon V (P ) is an affine order. The following lemma is a variant of [102, Lemma 1.4].

Lemma 1.11. Let P be a convex-cell complex. An affine order ≤ for P de-

termines a triangulation h≤ : |L≤| ≈−→ P , where L≤ is a simplicial complex withV (L≤) = V (P ). The homeomorphism h≤ is piecewise affine and ≤ is a simplicialorder on L≤.

Proof. The order ≤ being chosen, we drop it from the notations. For eachsubcomplex Q of P , we shall construct a simplicial complex L(Q) and a piecewiseaffine homeomorphism hQ : |L(Q)| → Q such that,

(i) V (L(Q)) = V (Q);(ii) if Q′ ⊂ Q, then L(Q′) ⊂ L(Q) and hQ′ is the restriction of hQ to |L(Q′)|.

1. SIMPLICIAL COMPLEXES 13

The case Q = P will prove the lemma. The construction is by induction on thedimension of Q, setting L(Q) = Q and hQ = id if dimQ = 0.

Suppose that L(Q) and hQ have been constructed, satisfying (i) and (ii) above,for each subcomplex Q of P of dimension ≤ k − 1. Let A be a k-cell of K withminimal vertex a. Then A is the topological cone, with cone-vertex a, of theunion B of faces of A not containing a. The triangulation hB : |L(B)| → |B| beingconstructed by induction hypothesis, define L(A) to be the join L(B) ∗ a and hAto be the unique piecewise affine extension of hB. Observe that, if C is a face ofA, then hC is the restriction to L(C) of hA. Therefore, this process may be usedfor each k-cell of P to construct hQ : |L(Q)| → Q for each subcomplex Q of P withdimQ ≤ k.

1.12. Subdivisions. Let Z be a set and A be a family of subsets of Z. Asimplicial complex L such that

(a) V (L) ⊂ Z;(b) for each σ ∈ S(L) there exists A ∈ A such that σ ⊂ A;

is called a (Z,A)-simplicial complex, or a Z-simplicial complex supported by A.Let K be a simplicial complex. Let N be a (|K|,GS(K))-simplicial complex,

where

GS(K) = |σ| | σ ∈ S(K)is the family of geometric simplexes of K. A continuous map j : |N | → |K| isassociated to N , defined by

j(µ) =∑

w∈V (N)

µ(w)w .

In other word, j is the piecewise affine map sending each vertex of N to to the cor-responding point of |K|. A subdivision of a simplicial complex K is a (|K|,GS(K))-simplicial complex N for which the associated map j : |N | → |K| is a homeomor-phism (in other words, j is a triangulation of |K|).

Let N be a (|K|,GS(K))-simplicial complex for a simplicial complex K. If Lis a subcomplex of K, then

NL = σ ∈ S(N)|σ ⊂ |L|is a (|L|,GS(L))-simplicial complex. Its associated map jL : |NL| → |L| is therestriction of j to |L|. The following lemma is useful to recognise a subdivision(compare [175, Ch. 3, Sec. 3, Th. 4]).

Lemma 1.13. Let N be a (|K|,GS(K))-simplicial complex. Then N is a sub-division of K if and only if, for each τ ∈ S(K), the simplicial complex Nτ is finiteand jτ : |Nτ | → |τ | is bijective.

Proof. If N is a subdivision of K, then jτ is bijective since j is a homeomor-phism. Also, |Nτ | = j−1(|τ |) is compact, so Nτ is finite.

Conversely, The fact that jτ is bijective for each τ ∈ S(K) implies that thecontinuous map j is bijective. If Nτ is finite, then jτ is a continuous bijectionbetween compact spaces, hence a homeomorphism. This implies that the map j−1,restricted to each geometric simplex, is continuous. Therefore, j−1 is continuoussince K is endowed with the weak topology.

Seeing V (K) as a subset of |K|, we get the following corollary.


Corollary 1.14. Let N be a subdivision of K. Then V (K) ⊂ V (N).

A useful systematic subdivision process is the barycentric subdivision. Letσ ∈ Sm(K) be a m-simplex of a simplicial complex K. The barycenter σ ∈ |K| ofσ is defined by

σ =1

m+ 1

∑

v∈σ

v .

The barycentric subdivision K ′ of K is the (|K|,GS(K))-simplicial complex where

• V (K ′) = σ ∈ |K| | σ ∈ S(K);• σ0, . . . , σm ∈ Sm(K ′) whenever σ0 ⊂ · · · ⊂ σm (σi 6= σj if i 6= j).

Using Lemma 1.13, the reader can check that K ′ is a subdivision of K. Observethat the partial order “≤” defined by

(1.15) σ ≤ τ ⇐⇒ σ ⊂ τis a simplicial order on K ′.

2. Definitions of simplicial (co)homology

Let K be a simplicial complex. In this section, we give the definitions of thehomology H∗(K) and cohomology H∗(K) of K under the various and peculiarforms available when the coefficients are in the field Z2 = 0, 1.Definitions I (subset definitions):

(a) A m-cochain is a subset of Sm(K).(b) A m-chain is a finite subset of Sm(K).

The set of m-cochains of K is denoted by Cm(K) and that of m-chains byCm(K). By identifying σ ∈ Sm(K) with the singleton σ, we see Sm(K) as asubset of both Cm(K) and Cm(K). Each subset A of Sm(K) is determined by itscharacteristic function χA : Sm(K)→ Z2, defined by

χA(σ) =

1 if σ ∈ A0 otherwise.

This gives a bijection between subsets of Sm(K) and functions from Sm(K) to Z2.We see such a function as a colouring (0 = white and 1 = black). The following“colouring definition” is equivalent to the subset definition:

Definitions II (colouring definitions):(a) A m-cochain is a function a : Sm(K)→ Z2.(b) A m-chain is a function α : Sm(K)→ Z2 with finite support.

The colouring definition is used in low-dimentional graphical examples to draw(co)chains in black (bold lines for 1-(co)chains).

Definitions II endow Cm(K) and Cm(K) with a structure of a Z2-vector space.The singletons provide a basis of Cm(K), in bijection with Sm(K). Thus, DefinitionII.b is equivalent to

Definition III: Cm(K) is the Z2-vector space with basis Sm(K):

Cm(K) =⊕

σ∈Sm(K)

Z2 σ .

We shall pass from one of Definitions I, II or III to another without notice;the context usually prevents ambiguity. We consider C∗(K) = ⊕m∈NCm(K) and

2. DEFINITIONS OF SIMPLICIAL (CO)HOMOLOGY 15

C∗(K) = ⊕m∈NCm(K) as graded Z2-vector spaces. The convention C−1(K) =

C−1(K) = 0 is useful.We now define the Kronecker pairing on (co)chains

Cm(K)× Cm(K)〈 , 〉−→ Z2

by the equivalent formulae

(2.1)

〈a, α〉 = ♯(a ∩ α) (mod 2) using Definitions I.a and I.b

=∑

σ∈α a(σ) using Definitions I.a and II.b

=∑

σ∈Sm(K) a(σ)α(σ) using Definitions II.a and II.b .

Lemma 2.2. The Kronecker pairing is bilinear and the map a 7→ 〈a, 〉 is anisomorphism between Cm(K) and Cm(K)♯ = hom(Cm(K),Z2).

Proof. The bilinearity is obvious from the third line of Equations (2.1). Let0 6= a ∈ Cm(K). This means that, as a subset of Sm(K), a is not empty. If σ ∈ a,then 〈a, σ〉 6= 0, which proves the injectivity of a 7→ 〈a, 〉. As for its surjectivity, leth ∈ hom(Cm(K),Z2). Using the inclusion Sm(K) → Cm(K) given by τ 7→ τ,define

a = τ ∈ Sm(K) | h(τ) = 1 .For each σ ∈ Sm(K) the equation h(σ) = 〈a, σ〉 holds true. As Sm(K) is a basis ofCm(K), this implies that h = 〈a, 〉.

We now define the boundary and coboundary operators. The boundary operator∂ : Cm(K)→ Cm−1(K) is the Z2-linear map defined by

(2.3) ∂(σ) = (m− 1)-faces of σ = Sm−1(σ) , σ ∈ Sm(K) .

Formula (2.3) is written in the language of Definition I.b. Using Definition III, weget

(2.4) ∂(σ) =∑

τ∈Sm−1(σ)

τ .

The coboundary operator δ : Cm(K)→ Cm+1(K) is defined by the equation

(2.5) 〈δa, α〉 = 〈a, ∂α〉 .The last equation indeed defines δ by Lemma 2.2 and δ may be seen as the Kroneckeradjoint of ∂. In particular, if σ ∈ Sm(K) and τ ∈ Sm−1(K) then

(2.6) τ ∈ ∂(σ) ⇔ τ ⊂ σ ⇔ σ ∈ δ(τ) .The first equivalence determines the operator ∂ since Sm(K) is a basis for Cm(K).The second equivalence determines δ if Sm−1(K) is finite. Note that the definitionof δ may also be given as follows: if a ∈ Cm(K), then

δ(a) = τ ∈ Sm+1(K) | ♯ (a ∩ ∂(τ)) is odd .Let σ ∈ Sm(K). Each τ ∈ Sm−2(K) with τ ⊂ σ belongs to the boundary of

exactly two (m−1)-simplexes of σ. Using Equation (2.4), this implies that ∂∂ = 0.By Equation (2.5) and Lemma 2.2, we get δδ = 0. We define the Z2-vector spaces

• Zm(K) = ker(∂ : Cm(K)→ Cm−1(K)), the m-cycles of K.• Bm(K) = image (∂ : Cm+1(K)→ Cm(K)), the m-boundaries of K.• Zm(K) = ker(δ : Cm(K)→ Cm+1(K)), the m-cocycles of K.


• Bm(K) = image (δ : Cm−1(K)→ Cm(K)), the m-coboundaries of K.

For example, Figure 1 shows a triangulationK of the plane, with V (K) = Z×Z.The bold line is a cocycle a which is a coboundary: a = δB, with B = (m,n) |(m,n) ∈ V (K) and m ≤ 0, drawn in bold dots.

0

a

Figure 1.

Since ∂∂ = 0 and δδ = 0, one has Bm(K) ⊂ Zm(K) and Bm(K) ⊂ Zm(K).We form the quotient vector spaces

• Hm(K) = Zm(K)/Bm(K), the mth-homology vector space of K.• Hm(K) = Zm(K)/Bm(K), the mth-cohomology vector space of K.

As for the (co)chains, the notationsH∗(K) = ⊕m∈NHm(K) andH∗(K) = ⊕m∈NHm(K)

stand for the (co)homology seen as graded Z2-vector spaces. By convention,H−1(K) =H−1(K) = 0. Also, the homology and the cohomology are in duality via the Kro-necker pairing:

Proposition 2.7 (Kronecker duality). The Kronecker pairing on (co)chainsinduces a bilinear map

Hm(K)×Hm(K)〈 , 〉−−→ Z2 .

Moreover, the correspondence a 7→ 〈a, 〉 is an isomorphism

Hm(K)k−−−→≈

hom(Hm(K),Z2) .

Proof. Instead of giving a direct proof, which the reader may do as an exercise,we will take advantage of the more general setting of Kronecker pairs, developed inthe next section. In this way, Proposition 2.7 follows from Proposition 3.7.

3. Kronecker pairs

All the vector spaces in this section are over an arbitrary fixed field F. Thedual of a vector space V is denoted by V ♯.

A chain complex is a pair (C∗, ∂), where

• C∗ is a graded vector space C∗ =⊕

m∈NCm. We add the convention that

C−1 = 0.• ∂ : C∗ → C∗ is a linear map of degree −1, i.e. ∂(Cm) ⊂ Cm−1, satisfying∂∂ = 0. The operator ∂ is called the boundary of the chain complex.

A cochain complex is a pair (C∗, δ), where

3. KRONECKER PAIRS 17

• C∗ is a graded vector space C∗ =⊕

m∈NCm. We add the convention that

C−1 = 0.• δ : C∗ → C∗ is a linear map of degree +1, i.e. ∂(Cm) ⊂ Cm+1, satisfyingδδ = 0. The operator δ is called the coboundary of the cochain complex.

A Kronecker pair consists of three items:

(a) a chain complex (C∗, ∂).(b) a cochain complex (C∗, δ).(c) a bilinear map

Cm × Cm〈 , 〉−→ F

satisfying the equation

(3.1) 〈δa, α〉 = 〈a, ∂α〉 .for all a ∈ Cm and α ∈ Cm+1 and all m ∈ N. Moreover, we require thatthe map k : Cm → C♯m, given by k(a) = 〈a, 〉, is an isomorphism.

Example 3.2. Let K be a simplicial complex. Its simplicial (co)chain com-plexes (C∗(K), δ), (C∗(K), ∂), together with the pairing 〈 , 〉 of § 2 is a Kroneckerpair, with F = Z2, as seen in Lemma 2.2 and Equation (3.1).

Example 3.3. Let (C∗, ∂) be a chain complex. One can define a cochaincomplex (C∗, δ) by Cm = C♯m and δ = ∂♯ and then get a bilinear map (pairing) 〈,〉by the evaluation: 〈a, α〉 = a(α). These constitute a Kronecker pair. Actually, viathe map k, any Kronecker pair is isomorphic to this one. The reader may use thisfact to produce alternative proofs of the results of this section.

We first observe that, in a Kronecker pair, chains and cochains mutually deter-mine each other:

Lemma 3.4. Let((C∗, δ), (C∗, ∂), 〈 , 〉

)be a Kronecker pair.

(a) Let a, a′ ∈ Cm. Suppose that 〈a, α〉 = 〈a′, α〉 for all α ∈ Cm. Then a = a′.(b) Let α, α′ ∈ Cm. Suppose that 〈a, α〉 = 〈a, α′〉 for all a ∈ Cm. Then

α = α′.(c) Let Sm be a basis for Cm and let f : Sm → F be a map. Then, there is a

unique a ∈ Cm such that 〈a, σ〉 = f(σ) for all σ ∈ Sm.

Proof. In Point (a), the hypotheses imply that k(a) = k(a′). As k is injective,this shows that a = a′.

In Point (b), suppose that α 6= α′. Let A ∈ (Cm)♯ such that A(α − α′) 6= 0.Then, 〈a, α〉 6= 〈a, α′〉 for a = k−1(A) ∈ Cm.

Finally, the condition a(σ) = f(σ) for all σ ∈ Sm defines a unique a ∈ C♯m anda = k−1(a).

As is § 2, we consider the Z2-vector spaces

• Zm = ker(∂ : Cm → Cm−1), the m-cycles (of C∗).• Bm = image (∂ : Cm+1 → Cm), the m-boundaries.• Zm = ker(δ : Cm → Cm+1), the m-cocycles.• Bm = image (δ : Cm−1 → Cm), the m-coboundaries.

Since ∂∂ = 0 and δδ = 0, one has Bm ⊂ Zm and Bm ⊂ Zm. We form thequotient vector spaces

• Hm = Zm/Bm, the mth-homology group (or vector space).


• Hm = Zm/Bm, the mth-cohomology group (or vector space).

We consider the (co)homology as graded vector spaces: H∗ = ⊕m∈NHm and H∗ =⊕m∈NH

m.The cocycles and coboundaries may be detected by the pairing:

Lemma 3.5. Let a ∈ Cm. Then

(i) a ∈ Zm if and only if 〈a,Bm〉 = 0.(ii) a ∈ Bm if and only if 〈a, Zm〉 = 0.

Proof. Part (i) directly follows from Equation (3.1) and the fact that k isinjective. Also, if a ∈ Bm, Equation (3.1) implies that 〈a, Zm〉 = 0. It remains toprove the converse (this is the only place in this lemma where we need vector spacesover a field instead just module over a ring). We consider the exact sequence

(3.6) 0→ Zm → Cm → Bm−1∂−→ 0 .

Let a ∈ Cm such that 〈a, Zm〉 = 0. By (3.6), there exists a1 ∈ B♯m−1 such that〈a, 〉 = a1∂. As we are dealing with vector spaces, Bm−1 is a direct summand

of Cm−1. We can thus extend a1 to a2 ∈ C♯m−1. As k is surjective, there exists

a3 ∈ Cm−1 such that 〈a3, 〉 = a2. For all α ∈ Cm, one then has

〈δa3, α〉 = 〈a3, ∂α〉 = a2(∂α) = a1(∂α) = 〈a, α〉 .As k is injective this implies that a = δa3 ∈ Bm.

Let us restrict the pairing 〈 , 〉 to Zm × Zm. Formula (3.1) implies that

〈Zm, Bm〉 = 〈Bm, Zm〉 = 0 .

Hence, the pairing descends to a bilinear map Hm × Hm〈 , 〉−→ F, giving rise to a

linear map k : Hm → H♯m, called the Kronecker pairing on (co)homology. We see

H∗ and H∗ as (co)chain complexes by setting ∂ = 0 and δ = 0.

Proposition 3.7. (H∗, H∗, 〈 , 〉) is a Kronecker pair.

Proof. Equation (3.1) holds trivially since ∂ and δ both vanish. It remainsto show that k : Hm → H♯

m is bijective.Let a0 ∈ H♯

m. Pre-composing a0 with the projection Zm →→ Hm producesa1 ∈ Z♯m. As Zm is a direct summand in Cm, one can extend a1 to a2 ∈ C♯m.Since (C∗, C

∗, 〈 , 〉) is a Kronecker pair, there exists a ∈ Cm such that 〈a, 〉 = a2.The cochain a satisfies 〈a,Bm〉 = a2(Bm) = 0 which, by Lemma 3.5, implies thata ∈ Zm. The cohomology class [a] ∈ Hm of a then satisfies 〈[a], 〉 = a0. Thus, k issurjective.

For the injectivity of k, let b ∈ Hm with 〈b,Hm〉 = 0. Represent b by b ∈ Zm,

which then satisfies 〈b, Zm〉 = 0. By Lemma 3.5, b ∈ Bm and thus b = 0.

Let (C∗, ∂) and (C∗, ∂) be two chain complexes. A map ϕ : C∗ → C∗ is amorphism of chain complexes or a chain map if it is linear map of degree 0 (i.e.ϕ(Cm) ⊂ Cm) such that ϕ∂ = ∂ϕ. This implies that ϕ(Zm) ⊂ Zm and ϕ(Bm) ⊂Bm. Hence, ϕ induces a linear map H∗ϕ : Hm → Hm for all m.

In the same way, let (C∗, δ) and (C∗, δ) be two cochain complexes. A linearmap φ : C∗ → C∗ of degree 0 is a morphism of cochain complexes or a cochain mapif φ δ = δφ. Hence, φ induces a linear map H∗φ : Hm → Hm for all m.

3. KRONECKER PAIRS 19

Let P = (C∗, ∂, C∗, δ, 〈 , 〉) and P = (C∗, ∂, C

∗, δ, 〈 , 〉−) be two Kronecker pairs.A morphism of Kronecker pairs, from P to P, consists of a pair (ϕ, φ) whereϕ : C∗ → C∗ is a morphism of chain complexes and φ : C∗ → C∗ is a morphism ofcochain complexes such that

(3.8) 〈a, ϕ(α)〉− = 〈φ(a), α〉 .Using the isomorphisms k and k, Equation (3.8) is equivalent to the commutativityof the diagram

(3.9)

C∗

k≈

φ // C∗

k≈

C♯∗ϕ♯

// C♯∗

.

Lemma 3.10. Let P and P be Kronecker pairs as above. Let ϕ : C∗ → C∗ bea morphism of chain complex. Define φ : C∗ → C∗ by Equation (3.8) (or Dia-gram (3.9)). Then the pair (ϕ, φ) is a morphism of Kronecker pairs.

Proof. Obviously, φ is a linear map of degree 0 and Equation (3.8) is satisfied.It remains to show that φ is a morphism of cochain-complexes. But, if b ∈ Cm(K)and α ∈ Cm+1(K), one has

〈δφ(b), α〉 = 〈φ(b), ∂α〉 = 〈b, ϕ(∂α)〉− = 〈b, ∂ϕ(α)〉−= 〈δb, ϕ(α)〉− = 〈φ(δb), α〉 ,

which proves that δφ(b) = φ(δb).

A morphism (ϕ, φ) of Kronecker pairs determines a morphism of Kroneckerpairs (H∗ϕ,H

∗φ) from (H∗, H∗, 〈 , 〉) to (H∗, H

∗, 〈 , 〉−). This process is functorial:

Lemma 3.11. Let (ϕ1, φ1) be a morphism of Kronecker pairs from P to P and

let (ϕ2, φ2) be a morphism of Kronecker pairs from P to P. Then

(H∗ϕ2H∗ϕ1, H∗φ1H

∗φ2) = (H∗(ϕ2ϕ1), H∗(φ2 φ1))

Proof. That H∗ϕ2H∗ϕ1 = H∗(ϕ2 ϕ1) is a tautology. For the cohomologyequality, we use that

〈H∗φ1 H∗φ2(a), α〉 = 〈H∗φ2(a), H∗ϕ1(α)〉 = 〈a,H∗ϕ2 H∗ϕ1(α)〉

= 〈a,H∗(ϕ2 ϕ1)(α)〉 = 〈H∗(φ2 φ1))(a), α〉holds for all a ∈ H∗ and all α ∈ H∗.

We finish this section with some technical results which will be used later.

Lemma 3.12. Let f : U → V and g : V →W be two linear maps between vectorspaces. Then, the sequence

(3.13) Uf−→ V

g−→W

is exact at V if and only if the sequence

(3.14) U ♯f♯

←− V ♯ g♯

←−W ♯

is exact at V ♯.


Proof. As f ♯g♯ = (gf)♯, then f ♯g♯ = 0 if and only if gf = 0.On the other hand, suppose that ker g ⊂ image f . We shall prove that ker f ♯ ⊂

image g♯. Indeed, let a ∈ ker f ♯. Then, a(image f) = 0 and, using the inclusionker g ⊂ image f , we deduce that a(ker g) = 0. Therefore, a descends to a linearmap a : V/ ker g → F. The quotient space V/ ker g injects into W , so there existsb ∈ W ♯ such that a = bg = g♯(b), proving that a ∈ image g♯.

Finally, suppose that ker g 6⊂ image f . Then there exists a ∈ V ♯ such thata(image f) = 0, i.e., a ∈ ker f ♯, and a(ker g) 6= 0, i.e. a /∈ image g♯. This provesthat ker f ♯ 6⊂ image g♯.

Lemma 3.15. Let (ϕ, φ) be a morphism of Kronecker pairs from P = (C∗, ∂, C∗, δ, 〈 , 〉)

to P = (C∗, ∂, C∗, δ, 〈 , 〉−). Then the pairings 〈 , 〉 and 〈 , 〉− induce bilinear maps

cokerφ× kerϕ〈,〉−→ F and kerφ× cokerϕ

〈,〉−−−−→ F

such that the induced linear maps

cokerφk−→ (kerϕ)♯ and kerφ

k−→ (cokerϕ)♯

are isomorphisms.

Proof. Equation (3.8) implies that 〈φ(C∗), kerϕ〉 = 0 and 〈kerφ, ϕ(C∗)〉− =0, whence the induced pairings. Consider the exact sequence

0 // kerϕ // C∗ϕ // C∗ // cokerϕ // 0 .

By Lemma 3.12, passing to the dual preserves exactness. Using Diagram (3.9), onegets a commutative diagram

0 (kerϕ)♯oo C♯∗oo C♯∗

ϕ♯

oo (cokerϕ)♯oo 0oo

0 cokerφoo

k

OO

H∗oo

k≈

OO

C∗φoo

k≈

OO

kerφoo

k

OO

0oo

.

By a chasing diagram argument, the two extreme up-arrows are bijective (one canalso invoke the famous five-lemma: see e.g. [175, Ch.4, Sec.5, Lemma 11]).

Corollary 3.16. Let (ϕ, φ) be a morphism of Kronecker pairs from(C∗, ∂, C

∗, δ, 〈 , 〉) to (C∗, ∂, C∗, δ, 〈 , 〉−). Then the pairings 〈 , 〉 and 〈 , 〉− on (co)homo-

logy induce bilinear maps

cokerH∗φ× kerH∗ϕ〈,〉−→ F and kerH∗φ× cokerH∗ϕ

〈,〉−−−−→ F

such that the induced linear maps

cokerH∗φk−→ (kerH∗ϕ)♯ and kerH∗φ

k−→ (cokerH∗ϕ)♯

are isomorphisms.

Proof. The morphism (φ, ϕ) induces a morphism of Kronecker pairs (H∗φ,H∗ϕ)from (H∗, H∗, 〈 , 〉) to (H∗, H∗, 〈 , 〉−). Corollary 3.16 follows then from Lemma 3.15applied to (H∗φ,H∗ϕ).

Corollary 3.16 implies the following

4. FIRST COMPUTATIONS 21

Corollary 3.17. Let (ϕ, φ) be a morphism of Kronecker pairs from (C∗, ∂, C∗, δ, 〈 , 〉)

to (C∗, ∂, C∗, δ, 〈 , 〉−). Then

(a) H∗φ is surjective if and only if H∗ϕ is injective.(b) H∗φ is injective if and only if H∗ϕ is surjective.(c) H∗φ is bijective if and only if H∗ϕ is bijective.

4. First computations

4.1. Reduction to components. Let K be a simplicial complex. We haveseen in 1.6 that K is the disjoint union of its components, whose set is denoted byπ0(K). Therefore, Sm(K) =

∐L∈π0(K) Sm(L) which, by Definition III, p. 14, gives

a canonical isomorphism⊕

L∈π0(K)

Cm(L)≈→ Cm(K) .

This direct sum decomposition commutes with the boundary operators, giving acanonical isomorphism

(4.1)⊕

L∈π0(K)

H∗(L)≈→ H∗(K) .

As for the cohomology, seeing a m-cochain as a map α : Sm(K) → Z2 (DefinitionII, p. 14) the restrictions of α to Sm(L) for all L ∈ π0(K) gives an isomorphism

Cm(K)≈−→

∏

L∈π0(K)

Cm(L)

commuting with the coboundary operators. This gives an isomorphism

(4.2) H∗(K)≈−→

∏

L∈π0(K)

H∗(L) .

The isomorphisms of (4.1) and (4.2) permit us to reduce (co)homology compu-tations to connected simplicial complexes. They are of course compatible with theKronecker duality (Proposition 2.7). A formulation of these isomorphisms usingsimplicial maps is given in Proposition 5.6.

4.2. 0-dimensional (co)homology. Let K be a simplicial complex. Theunit cochain 1 ∈ C0(K) is defined by 1 = S0(K), using the subset definition. Inthe language of colouring, one has 1(v) = 1 for all v ∈ V (K) = S0(K), that is allvertices are black. If β = v, w ∈ S1(K), then

〈δ1, β〉 = 〈1, ∂β〉 = 1(v) + 1(w) = 0 ,

which proves that δ(1) = 0 by Lemma 2.2. Hence, 1 is a cocycle, whose cohomologyclass is again denoted by 1 ∈ H0(K).

Proposition 4.3. Let K be a non-empty connected simplicial complex. Then,

(i) H0(K) = Z2, generated by 1 which is the only non-vanishing 0-cocycle.(ii) H0(K) = Z2. Any 0-chain α is a cycle, which represents the non-zero

element of H0(K) if and only if ♯α is odd.


Proof. If K is non-empty the unit cochain does not vanish. As C−1(K) = 0,this implies that 1 6= 0 in H0(K).

Let a ∈ C0(K) with a 6= 0,1. Then there exists v, v′ ∈ V (K) with a(v) 6= a(v′).Since K is connected, there exists x0, . . . , xm ∈ V (K) with x0 = v, xm = v′ andxi, xi+1 ∈ S(K). Therefore, there exists 0 ≤ k < m with a(xk) 6= a(xk+1). Thisimplies that xk, xk+1 ∈ δa, proving that δa 6= 0. We have thus proved (i).

Now, H0(K) = Z2 since H0(K) ≈ H0(K)♯. Any α ∈ C0(K) is a cycle sinceC−1(K) = 0. It represents the non-zero homology class if and only if 〈1, α〉 = 1,that is if and only if ♯α is odd.

Corollary 4.4. Let K be a simplicial complex. Then H0(K) ≈ Zπ0(K)2 .

Here, Zπ0(K)2 denotes the set of maps from π0(K) to Z2. The isomorphism of

Corollary 4.4 is natural for simplicial maps (see Corollary 5.9).

Proof. By Proposition 4.3 and its proof, H0(K) = Z0(K) is the set of mapsfrom V (K) to Z2 which are constant on each component. Such a map is determinedby a map from π0(K) to Z2 and conversely.

4.3. Pseudomanifolds. A n-dimensional pseudomanifold is a simplicial com-plex M such that

(a) every simplex of M is contained in a n-simplex of M .(b) every (n− 1)-simplex of M is a face of exactly two n-simplexes of M .(c) for any σ, σ′ ∈ Sn(M), there exists a sequence σ = σ0, . . . , σm = σ′ of

n-simplexes such that σi and σi+1 have an (n − 1)-face in common fori ≤ 1 < n.

Examples 4.5. (1) Let m be an integer with m ≥ 3. The polygon Pm is the1-dimensional pseudomanifold for which V (Pm) = 0, 1, . . . ,m − 1 = Z/mZ andS1(Pm) = k, k + 1 | k ∈ V (Pm). It can be visualized in the complex plane asthe equilateral m-gon whose vertices are the mth roots of the unity.

(2) Consider the triangulation of S2 given by an icosahedron. Choose one pairof antipodal vertices and identify them in a single point. This gives a quotientsimplicial complex K which is a 2-dimensional pseudomanifold. Observe that |K|is not a topological manifold.

Pseudomanifolds have been introduced in 1911 by L.E.J. Brouwer [21, p. 477],for his work on the degree and on the invariance of the dimension. They are alsocalled an n-circuit in the literature. Proposition 4.6 below and its proof, togetherwith Proposition 4.3, shows how n-dimensional pseudomanifolds satisfy Poincareduality in dimensions 0 and n.

Let M be a finite n-dimensional pseudomanifold. The n-chain [M ] = Sn(M) ∈Cn(M) is called the fundamental cycle of M (it is a cycle by Point (b) of theabove definition). Its homology class, also denoted by [M ] ∈ Hn(M) is called thefundamental class of M .

Proposition 4.6. Let M be a finite non-empty n-dimensional pseudomanifold.Then,

(i) Hn(M) = Z2, generated by [M ] which is the only non-vanishing n-cycle.(ii) Hn(M) = Z2. Any n-cochain a is a cocycle, and [a] 6= 0 in Hn(M) if and

only if ♯a is odd.


Proof. We define a simplicial graph L with V (L) = Sn(M) by setting σ, σ′ ∈S1(L) if and only if σ and σ′ have an (n − 1)-face in common. The identificationSn(M) = V (L) produces isomorphisms

(4.7) Fn : Cn(M)≈−→ C0(L) and Fn : Cn(M)

≈−→ C0(L) .

(As M is finite, so is L and C∗(L) is equal to C∗(L), using Definition II, p. 14.)On the other hand, by Point (b) of the definition of a pseudomanifold, one gets a

bijection F : Sn−1(M)≈−→ S1(L). It gives rise to isomorphisms

(4.8) Fn−1 : Cn−1(M)≈−→ C1(L) and Fn−1 : Cn−1(M)

≈−→ C1(L) .

The isomorphisms of (4.7) and (4.8) satisfy

Fn−1∂ = δFn and ∂ Fn−1 = Fnδ .

Since Cn+1(M) = 0 by Point (a) of the definition of a pseudomanifold, the aboveisomorphisms give rise to isomorphisms

F∗ : Hn(M)≈−→ H0(L) and F ∗ : Hn(M)

≈−→ H0(L)

with F∗([M ]) = 1. By Point (c) of the definition of a pseudomanifold, the graph Lis connected. Therefore, Proposition 4.6 follows from Proposition 4.3.

The proof of Proposition 4.6 actually gives the following result.

Proposition 4.9. Let M be a finite non-empty simplicial complex satisfyingConditions (a) and (b) of the definition of a n-dimensional pseudomanifold. Then,M is a pseudomanifold if and only if Hn(M) = Z2.

4.4. Poincare series and polynomials. A graded Z2-vector space A∗ =⊕i∈N

Ai is of finite type if Ai is finite dimensional for all i ∈ N. In this case, thePoincare series of A∗ is the formal power series defined by

Pt(A∗) =∑

i∈N

dimAiti ∈ N[[t]].

When dimA∗ < ∞, the series Pt(A∗) is a polynomial, also called the Poincarepolynomial of A∗.

A simplicial complex K is of finite (co)-homology type if H∗(K) (or, equiva-lently, H∗(K)) is of finite type. In this case, the Poincare series of K is that ofH∗(K). The (co)-homology of a simplicial complex of finite (co)-homology type is,up to isomorphism, determined by its Poincare series, which is often the shortestway to describe it. The number dimHm(K) is called the m-th Betti number of K.The vector space C∗(K) is endowed with the basis S(K) for which the matrix of theboundary operator is given explicitely. Thus, the Betti numbers may be effectivelycomputed by standard algorithms of linear algebra.

4.5. (Co)homology of a cone. The simplest non-empty simplicial complexis a point whose (co)-homology is obviously

(4.10) Hm(pt) ≈ Hm(pt) ≈

0 if m > 0

Z2 if m = 0 .

In terms of Poincare polynomial: Pt(pt) = 1. Let L be a simplicial complex. Thecone on L is the simplicial complex CL defined by V (CL) = V (L) ∪ ∞ and

Sm(CL) = Sm(L) ∪ σ ∪ ∞ | σ ∈ Sm−1(L) .


Note that CL is the join CL ≈ L ∗ ∞.

Proposition 4.11. The cone CL on a simplicial complex L has its (co)homologyisomorphic to that of a point. In other words, Pt(CL) = 1.

Proof. By Kronecker duality, it is enough to prove the result on homology.The cone CL is obviously connected and non-empty (it contains ∞), so H0(CL) =Z2.

Define a linear map D : Cm(CL)→ Cm+1(CL) by setting, for σ ∈ Sm(CL):

D(σ) =

σ ∪ ∞ if ∞ /∈ σ0 if ∞ ∈ σ .

Hence, DD = 0. If ∞ /∈ σ, the formula

(4.12) ∂D(σ) = D(∂σ) + σ

holds true in Cm(CL) (and has a clear geometrical interpretation). Suppose that∞ ∈ σ and dimσ ≥ 1. Then σ = D(τ) with τ = σ − ∞. Using Formula (4.12)and that DD = 0, one has

D(∂σ) + σ = D(∂D(τ)) + σ = D(D(∂τ) + τ) +D(τ) = 0 .

Therefore, Formula (4.12) holds also true if ∞ ∈ σ, provided dimσ ≥ 1. Thisproves that

(4.13) ∂D(α) = D(∂α) + α for all α ∈ Cm(CL) with m ≥ 1 .

Now, if α ∈ Cm(CL) satisfies ∂α = 0, Formula (4.13) implies that α = ∂D(α),which proves that Hm(CL) = 0 if m ≥ 1.

As an application of Proposition 4.11, let A be a set. The full complex FA on Ais the simplicial complex for which V (FA) = A and S(FA) is the family of all finitenon-empty subsets of A. If A is finite and non-empty, then FA is isomorphic to asimplex of dimension ♯A − 1. Denote by FA the subcomplex of FA generated bythe proper (i.e. 6= A) subsets of A. For instance, FA = FA if A is infinite.

Corollary 4.14. Let A be a non-empty set. Then

(i) FA has its (co)homology isomorphic to that of a point, i.e. Pt(FA) = 1.

(ii) If 3 ≤ ♯A ≤ ∞, then Pt(FA) = 1 + t♯A−1.

(iii) If ♯A = 2, then Pt(FA) = 2.

Proof. As A is not empty, FA is isomorphic to the cone over FA deprived ofone of its elements. Point (i) then follows from Proposition 4.11. Let n = ♯A − 1.The chain complex of FA looks like a sequence

0→ Cn(FA)∂n−→ Cn−1(FA)

∂n−1−−−→ · · · → C0(FA)→ 0 ,

which, by (i), is exact except at C0(FA). One has Cn(FA) = Z2, generated by the

A ∈ Sn(FA). Hence, ker ∂n−1 ≈ Z2. As the chain complex C∗(FA) is the same as

that of FA with Cn replaced by 0, this proves (ii). If ♯A = 2, then FA consists oftwo 0-simplexes and Point (iii) follows from (4.10) and (4.1).


4.6. The Euler characteristic. Let K be a finite simplicial complex. ItsEuler characteristic χ(K) is defined as

χ(K) =∑

m∈N

(−1)m ♯Sm(K) ∈ Z .

Proposition 4.15. Let K be a finite simplicial complex. Then

χ(K) =∑

m∈N

(−1)m dimHm(K) =∑

m

(−1)m dimHm(K) .

As in the definition of the Poincare polynomial, the number dimHm(K) isthe dimension of Hm(K) as a Z2-vector space. In other words, dimHm(K) is them-th Betti number of K. Proposition 4.15 holds true for the (co)homology withcoefficients in any field F, though the Betti numbers depend individually on F.

Proof. By Kronecker duality, only the first equality requires a proof. Letcm, zm, bm and hm be the dimensions of Cm(K), Zm(K), Bm(K) and Hm(K). El-ementary linear algebra gives the equalities

cm = zm + bm−1

zm = bm + hm .

We deduce that

χ(K) =∑

m∈N

(−1)mcm =∑

m∈N

(−1)mhm +∑

m∈N

(−1)mbm +∑

m∈N

(−1)mbm−1 .

As b−1 = 0, the last two sums cancels each other, proving Proposition 4.15.

Corollary 4.16. Let K be a finite simplicial complex. Then

χ(K) = Pt(K)t=−1.

The following additive formula for the Euler characteristic is useful.

Lemma 4.17. Let K be a simplicial complex. Let K1 and K2 be two subcom-plexes of K such that K = K1 ∪K2. Then,

χ(K) = χ(K1) + χ(K2)− χ(K1 ∩K2) .

Proof. The formula follows directly from the equations Sm(K) = Sm(K1) ∪Sm(K2) and Sm(K1 ∩K2) = Sm(K1) ∩ Sm(K2).

4.7. Surfaces. A surface is a manifold of dimension 2. In this section, we giveexamples of triangulations of surfaces and compute their (co)homology. Strictlyspeaking, the results would hold only for the given triangulations, but we allow usto formulate them in more general terms. For this, we somehow admit that

• a connected surface is a pseudomanifold of dimension 2. This will beestablished rigorously in Corollary 31.7 but the reader may find a proofas an exercise and this is easy to check for the particular triangulationsgiven below.• up to isomorphism, the (co)homology of a simplicial complex K depends

only of the homotopy type of |K|. This will be proved in § 17. In partic-ular, the Euler characteristic of two triangulations of a surface coincide.

The 2-sphere. The 2-sphere S2 being homeomorphic to the boundary of a3-simplex, it follows from Corollary 4.14 that:

Pt(S2) = 1 + t2 .


0

1

23

4

5

1

2

3

45

a

Figure 2. A triangulation of RP 2

The projective plane. The projective plane RP 2 is the quotient of S2 by theantipodal map. The triangulation of S2 as a regular icosahedron being invariantunder the antipodal map, it gives a triangulation of RP 2 given in Figure 2. Notethat the border edges appear twice, showing as expected that RP 2 is the quotientof a 2-disk modulo the antipodal involution on its boundary.

Being a quotient of an icosahedron, the triangulation of Figure 2 has 6 vertices,15 edges and 10 facets, thus χ(RP 2) = 1. Using that RP 2 is a connected 2-dimensional pseudomanifold, we deduce that

(4.18) Pt(RP2) = 1 + t+ t2 .

To identify the generators of H1(RP 2) ≈ Z2 and H1(RP 2), we define

(4.19) a = α =1, 2, 2, 3, 3, 4, 4, 5, 5, 1

⊂ S1(RP

2) .

We see a ∈ C1(RP 2) and α ∈ C1(RP 2). The cochain a is drawn in bold on Figure 2,where it looks as the set of border edges, since each of its edges appears twice onthe figure. It is easy to check that δ(a) = 0 and ∂(α) = 0. As ♯α = 5 is odd,one has 〈a, α〉 = 1, showing that a is the generator of H1(RP 2) = Z2 and α is thegenerator of H1(RP 2) = Z2.

The 2-torus. The 2-torus T 2 = S1 × S1 is the quotient of a square whoseopposite sides are identified. A triangulation of T 2 is described (in two copies) inFigure 3. This triangulation has 9 vertices, 27 edges and 18 facets, which impliesthat χ(T 2) = 0. Since T 2 is a connected 2-dimensional pseudomanifold, we deducethat

Pt(T2) = (1 + t)2 .

In Figure 3 are drawn two chains α, β ∈ C1(T2) given by

α =3, 8, 8, 9, 9, 3

and β =

5, 7, 7, 9, 9, 5

.

We also drew two cochains a, b ∈ C1(T 2) defined as

a =4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 4

and

b =2, 3, 3, 6, 6, 8, 8, 7, 7, 9, 9, 2

.

One checks that ∂α = ∂β = 0 and that δa = δb = 0. Therefore, they representclasses a, b ∈ H1(T 2) and α, β ∈ H1(T

2). The equalities

〈a, α〉 = 1 , 〈a, β〉 = 0 , 〈b, α〉 = 0 , 〈b, β〉 = 1


1 2 3 1

4

5

1 2 3 1

4

5

6

7

8

9

a

α

1 2 3 1

4

5

1 2 3 1

4

5

6

7

8

9

b

β

Figure 3. Two copies of a triangulation of the 2-torus T 2, showinggenerators of H1(T 2) and H1(T

2)

imply that a, b is a basis of H1(T 2) and α, β is a basis of H1(T2).

If we consider a and b as 1-chains (call them a and b), we also have ∂a = ∂b = 0.Note that

〈a, b〉 = 1 , 〈a, a〉 = 0 , 〈b, b〉 = 0 , 〈b, a〉 = 1

This proves that a = β and b = α in H1(T2).

The Klein bottle. A triangulation of the Klein bottle K is pictured in Fig-ure 4. As the 2-torus, the Klein bottle is the quotient of a square with opposite sideidentified, one of these identifications “reversing the orientation”. One checks thatχ(K) = 0. Since K is a connected 2-dimensional pseudomanifold, the (co)homologyof K is abstractly isomorphic to that of T 2:

Pt(K) = (1 + t)2

(In Chapter 3, H∗(T 2) and H∗(K) will be distinguished by their cup product: seep. 117). In Figure 4 the dot lines show two 1-chains α, β ∈ C1(K) given by

(4.20) α =3, 8, 8, 9, 9, 3

and β =

5, 7, 7, 9, 9, 5

.

The bold lines describe two 1-cochains a, b ∈ C1(K) defined as

(4.21) a =4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 5

and

(4.22) b =2, 3, 3, 6, 6, 8, 8, 7, 7, 9, 9, 2

.

One checks that ∂α = ∂β = 0 and that δa = δb = 0. Therefore, they representclasses a, b ∈ H1(K) and α, β ∈ H1(K). The equalities

〈a, α〉 = 1 , 〈a, β〉 = 1 , 〈b, α〉 = 0 , 〈b, β〉 = 1

imply that a, b is a basis of H1(K) and α, β is a basis of H1(K).

As in the case of T 2, we may regard a and b as 1-chains (call them a and b).

Here ∂b = 0 but ∂a = 4+ 5 6= 0.Other surfaces. Let K1 and K2 be two simplicial complexes such that |K1|

and |K2| are surfaces. A simplicial complex L with |L| homeomorphic to the con-nected sum |K1|♯|K2| may be obtained in the following way: choose 2-simplexesσ1 ∈ K1 and σ2 ∈ K2. Let Li = Ki − σi and let L be obtained by taking thedisjoint union of L1 and L2 and identifying σ1 with σ2. Thus, L = L1 ∪ L2 andL0 = L1 ∩ L2 is isomorphic to the boundary of a 2-simplex.


1 2 3 1

4

5

1 2 3 1

5

4

6

7

8

9

a

α

1 2 3 1

4

5

1 2 3 1

5

4

6

7

8

9

b

β

Figure 4. Two copies of a triangulation of the Klein bottle K,showing generators of H1(K) and H1(K)

By Lemma 4.17, one has

χ(L) = χ(L1) + χ(L2)− χ(L0)

= χ(K1)− 1 + χ(K2)− 1− 0

= χ(K1) + χ(K2)− 2 .(4.22)

The orientable surface Σg of genus g is defined as the connected sum of g copies ofT 2. By Formula (4.22), one has

(4.23) χ(Σg) = 2− 2g .

As Σg is a 2-dimensional connected pseudomanifold, one has

Pt(Σg) = 1 + 2gt+ t2 .

The nonorientable surface Σg of genus g is defined as the connected sum of gcopies of RP 2. For instance, Σ1 = RP 2 and Σ2 is the Klein bottle. Formula (4.22)implies

(4.24) χ(Σg) = 2− g .As Σg is a 2-dimensional connected pseudomanifold, one has

Pt(Σg) = 1 + gt+ t2 .

5. The homomorphism induced by a simplicial map

Let f : K → L be a simplicial map between the simplicial complexes K andL. Recall that f is given by a map f : V (K) → V (L) such that f(σ) ∈ S(L) ifσ ∈ S(K), i.e. the image of a m-simplex of K is an n-simplex of L with n ≤ m.We define C∗f : C∗(K) → C∗(L) as the degree 0 linear map such that, for allσ ∈ Sm(K), one has

(5.1) C∗f(σ) =

f(σ) if f(σ) ∈ Sm(L) (i.e. if f|σ is injective)

0 otherwise.

We also define C∗f : C∗(L)→ C∗(K) by setting, for a ∈ Cm(L),

(5.2) C∗f(a) =σ ∈ Sm(K) | f(σ) ∈ a

.

In the following lemma, we use the same notation for the (co)boundary operators∂ and δ and the Kronecker product 〈 , 〉, both for K of for L.

5. THE HOMOMORPHISM INDUCED BY A SIMPLICIAL MAP 29

Lemma 5.3. Let f : K → L be a simplicial map. Then

(a) C∗f ∂ = ∂C∗f .(b) δC∗f = C∗f δ.(c) 〈C∗f(b), α〉 = 〈b, C∗f(α)〉 for all b ∈ C∗(L) and all α ∈ C∗(K).

In other words, the couple (C∗f, C∗f) is a morphism of Kronecker pairs.

Proof. To prove (a), let σ ∈ Sm(K). If f restricted to σ is injective, it isstraightforward that C∗f ∂(σ) = ∂C∗f(σ). Otherwise, we have to show thatC∗f ∂(σ) = 0. Let us label the vertices v0, v1, . . . , vm of σ in such a way thatf(v0) = f(v1). Then, C∗f ∂(σ) is a sum of two terms C∗f ∂(σ) = C∗f(τ0) +C∗f(τ1), where τ0 = v1, v2, . . . , vm and τ1 = v0, v2, . . . , vm. As C∗f(τ0) =C∗f(τ1), one has C∗f ∂(σ) = 0. Thus, Point (a) is established. Point (c) canbe easily deduced from Definitions (5.1) and (5.2), taking for α a simplex of K.Point (b) then follows from Points (a) and (c), using Lemma 3.10 and its proof.

By Lemma 5.3 and Proposition 3.7, the couple (C∗f, C∗f) determines linear

maps of degree zero

H∗f : H∗(K)→ H∗(L) and H∗f : H∗(L)→ H∗(K)

such that

(5.4) 〈H∗f(a), α〉 = 〈a,H∗f(α)〉 for all a ∈ H∗(L) and α ∈ H∗(K) .

Lemma 5.5 (Functoriality). Let f : K → L and g : L→M be simplicial maps.Then H∗(gf) = H∗g H∗f and H∗(gf) = H∗f H∗g. Also H∗idK = idH∗(K)

and H∗idK = idH∗(K)

In other words, H∗ and H∗ are functors from the simplicial category Simp tothe category GrV of graded vector spaces and degree 0 linear maps. The cohomol-ogy is contravariant and the homology is covariant.

Proof. For σ ∈ S(K), the formula C∗(gf)(σ) = C∗g C∗f(σ) follows di-rectly from Definition (5.1). Therefore C∗(gf) = C∗g C∗f and then H∗(gf) =H∗g H∗f . The corresponding formulae for cochains and cohomology follow fromPoint (c) of Lemma 5.3. The formulae for the idK is obvious.

Simplicial maps and components. Let K be a simplicial complex. For eachcomponent L ∈ π0(K) of K, the inclusion iL : L → K is a simplicial map. Theresults of § 4.1 may be strengthend as follows.

Proposition 5.6. Let K be a simplicial complex. The family of simplicialmaps iL : L→ K for L ∈ π0(K) gives rise to isomorphisms

H∗(K)(H∗iL)

≈//

∏L∈π0(K)H

∗(L)

and⊕

L∈π0(K)H∗(L)P

H∗iL

≈// H∗(K) .


The homomorphisms H0f and H0f . We use the same notation 1 ∈ H0(K)and 1 ∈ H0(L) for the classes given by the unit cochains.

Lemma 5.7. Let f : K → L be a simplicial map. Then H0f(1) = 1.

Proof. The formula C0f(1) = 1 in C0(K) follows directly from Definition (5.2).

Corollary 5.8. Let f : K → L be a simplicial map with K and L connected.Then

H0f : Z2 = H0(L)→ H0(K) = Z2

andH0f : Z2 = H0(K)→ H0(L) = Z2

are the identity isomorphism.

Proof. By Proposition 4.3, the generator of H0(L) (or H0(K)) is the unitcocycle 1. By Lemma 5.7, this proves the cohomology statement. The homologystatement also follows from Proposition 4.3, since H0(K) and H0(L) are generatedby a cycle consisting of a single vertex.

More generally, one has H0(L) ≈ Zπ0(L)2 and H0(K) ≈ Z

π0(K)2 by Corollary 4.4.

Using this and Lemma 5.7, one gets the following corollary.

Corollary 5.9. Let f : K → L be a simplicial map. Then H0f : Zπ0(L)2 →

Zπ0(K)2 is given by H0f(λ) = λπ0f .

The degree of a map. Let f : K → L be a simplicial map between two finiteconnected n-dimensional pseudomanifolds. Define the degree deg(f) ∈ Z2 by

(5.10) deg(f) =

0 if Hnf = 0

1 otherwise.

By Proposition 4.6, Hn(K) ≈ Hn(L) ≈ Z2. Thus, deg(f) = 1 if and only if Hnfis the (only possible) isomorphism between Hn(K) and Hn(L). By Kroneckerduality, the homomorphism Hnf may be used instead of Hnf in the definitionof deg(f). Our degree is sometimes called the mod2-degree, since, for orientedpseudomanifolds, it is the mod 2-reduction of a degree defined in Z (see, e.g. [175,exercises of Ch. 4]).

Let f : K → L be a simplicial map between two finite n-dimensional pseudo-manifolds. For σ ∈ Sn(L), define

(5.11) d(f, σ) = ♯τ ∈ Sn(K) | f(τ) = σ ∈ N.

As an example, let K = L = P4, the polygon of Example 4.5 with 4 edges. Letf be defined by f(0) = 0, f(1) = 1, f(2) = 2, f(3) = 1. Then, d(f, 0, 1) =d(f, 1, 2) = 2, d(f, 2, 3) = d(f, 3, 0) = 0 and deg(f) = 0. This exampleillustrates the following proposition.

Proposition 5.12. Let f : K → L be a simplicial map between two finite n-dimensional pseudomanifolds which are connected. For any σ ∈ Sn(L), one has

deg(f) = d(f, σ) mod 2 .

Proof. By Proposition 4.6, Hn(L) = Z2 is generated by the cocycle formedby the singleton σ and Cnf(σ) represents the non-zero element of Hn(K) if andonly if ♯Cnf(σ) = d(f, σ) is odd.

5. THE HOMOMORPHISM INDUCED BY A SIMPLICIAL MAP 31

The interest of Proposition 5.12 is 2-fold: first, it tells us that deg(f) may becomputed using any σ ∈ Sm(L) and, second, it asserts that d(f, σ) is independentof σ. Proposition 5.12 is the mod 2 context of the identity between the degreeintroduced by Brouwer in 1910, [21, p.419], and its homological interpretation dueto Hopf in 1930, [96, § 2] For a history of the notion of the degree of a map, see[38, pp. 169–175].

Example 5.13. Let f : T 2 → K be the two-fold cover of the Klein bottle K bythe 2-torus T 2, given in Figure 5. In formulae: f(i) = i = f (i) for i = 1, . . . , 9.

f

1 2 3 1 2 3 1

2 3 1 2 3 1

4

5

6

7

8

9

5

4

6

7

8

9

4

5

1

a

α

1 2 3 1

4

5

1 2 3 1

5

4

6

7

8

9

a

α

Figure 5. Two-fold cover f : T 2 → K over the triangulation Kof the Klein bottle given in Figure 4.

The 1-dimensional (co)homology vector spaces of T 2 and K admit the followingbases:

(i) V = [a], [b] ⊂ H1(T 2), where a is drawn in Figure 5 and

b =2, 3, 3, 6, 6, 8, 8, 7, 7, 9, 9, 2

.

(ii) W = [α], [β] ⊂ H1(T2), where α is drawn in Figure 5 and

β =5, 7, 7, 9, 9, 4, 4, 7, 7, 9, 9, 5

.

(iii) V = [a], [b] ⊂ H1(K), where a and b are defined in Equations (4.21)and (4.22) (a drawn in Figure 5).

(iv) W = [α], [β] ⊂ H1(K), where α and β are defined in Equation (4.20)(α drawn in Figure 5).

The matrices for C∗f and C∗f in these bases are

C∗f =

(1 00 0

)and C∗f =

(1 00 0

).


Note that, under the isomorphism k : H1(−)≈−→ H1(−)♯, the bases V and V are

dual of W and W ; therefore, the matrix of C∗f is the transposed of that of C∗f .Now, T 2 and K are 2-dimensional pseudomanifolds and d(f, σ) = 2 for each

σ ∈ S2(K). By Proposition 5.12, deg(f) = 0 and both H∗f : H2(K) → H2(T 2)and H∗f : H2(T

2)→ H2(K) vanish.

Contiguous maps. Two simplicial maps f, g : K → L are called contiguousif f(σ) ∪ g(σ) ∈ S(L) for all σ ∈ S(K). We denote by τ(σ) the subcomplex of Lgenerated by the simplex f(σ)∪g(σ) ∈ S(L). For example, the inclusion K → CKof a simplicial complex K into its cone and the constant map of K onto the conevertex of CK are contiguous.

Proposition 5.14. Let f, g : K → L be two simplicial maps which are contigu-ous. Then H∗f = H∗g and H∗f = H∗g.

Proof. By Kronecker Duality, using Diagram (3.9), it is enough to prove thatH∗f = H∗g. By induction on m, we shall prove the truth of the following property:

Property H(m): there exists a linear map D : Cm(K)→ Cm+1(L) such that:

(i) ∂D(α) +D(∂α) = C∗f(α) + C∗g(α) for each α ∈ Cm(K).(ii) for each σ ∈ Sm(K), D(σ) ∈ Cm+1(τ(σ)) ⊂ Cm+1(L).

We first prove that Property H(m) for all m implies that H∗f = H∗g. Indeed,we would then have a linear map D : C∗(K)→ C∗+1(L) satisfying

(5.15) C∗f + C∗g = ∂D +D∂ .

Such a map D is called a chain homotopy from C∗f to C∗g. Let β ∈ Z∗(K). ByEquation (5.15), one has C∗f(β) +C∗g(β) = ∂D(β) which implies that H∗f([β]) +H∗g([β]) in H∗(L).

We now prove that H(0) holds true. We define D : C0(K) → C1(L) as theunique linear map such that, for v ∈ V (K):

D(v) =

f(v), g(v) = τ(v) if f(v) 6= g(v)

0 otherwise.

Formula (i) being true for any v ∈ S0(K), it is true for any α ∈ C0(K). Formula(ii) is obvious.

Suppose that H(m − 1) holds true for m ≥ 1. We want to prove that H(m)also holds true. Let σ ∈ Sm(K). Observe that D(∂σ) exists by H(m−1). Considerthe chain ζ ∈ Cm(L) defined by

ζ = C∗f(σ) + C∗g(σ) +D(∂σ)

Using H(m− 1), one has

∂ζ = ∂C∗f(σ) + ∂C∗g(σ) + ∂D(∂σ)

= C∗f(∂σ) + C∗g(∂σ) +D(∂∂σ) + C∗f(∂σ) + C∗g(∂σ)

= 0 .

On the other hand, ζ ∈ Cm(τ(σ)). As m ≥ 1, Hm(τ(σ)) = 0 by Corollary 4.14.There exists then η ∈ Cm+1(τ(σ)) such that ζ = ∂η. Choose such an η and setD(σ) = η. This defines D : Cm(K)→ Cm+1(L) which satisfies (i) and (ii), provingH(m).

6. EXACT SEQUENCES 33

Remark 5.16. The chain homotopy D in the proof of Proposition 5.14 is notexplicitly defined. This is because several of these exist and there is no canonicalway to choose one (see [152, p. 68]). The proof of Proposition 5.14 is an exampleof the technique of acyclic carriers which will be developed in § 9.

Remark 5.17. Let f, g : K → L be two simplicial maps which are contiguous.Then |f |, |g| : |K| → |L| are homotopic. Indeed, the formula F (µ, t) = (1−t)|f |(µ)+t|g|(µ) (t ∈ [0, 1]) makes sense and defines a homotopy from |f | to |g|.

6. Exact sequences

In this section, we develop techniques to obtain long (co)homology exact se-quences from short exact sequences of (co)chain complexes. The results are usedin several forthcoming sections. All vector spaces in this section are over a fixedarbitrary field F.

Let (C∗1 , δ1), (C∗2 , δ2) and (C∗, δ) be cochain complexes of vector spaces, givingrise to cohomology graded vector spaces H∗1 , H∗2 and H∗. We consider morphismsof cochain complexes J : C∗1 → C∗ and I : C∗ → C∗2 so that

(6.1) 0→ C∗1J−→ C∗

I−→ C∗2 → 0

is an exact sequence. We call (6.1) a short exact sequence of cochain complexes.Choose a GrV-morphism S : C∗2 → C∗ which is a section of I. The section Scannot be assumed in general to be a morphism of cochain complexes. The linearmap δS : Cm2 → Cm+1 satisfies

I δS(a) = δ2I S(a) = δ2(a) ,

thus δS(Zm2 ) ⊂ J(Cm+11 ). We can then define a linear map δ∗ : Zm2 → Cm+1

1 bythe equation

(6.2) J δ∗ = δS .

If a ∈ Zm2 , then J δ1(δ∗(a)) = δδ(S(a)) = 0. Therefore, δ∗(Zm2 ) ⊂ Zm+1

1 .

Moreover, if b ∈ Cm−12 and a = δ2(b), then

I δS(b) = δ2I S(b) = a ,

whence δS(b) = S(a) + J(c) for some c ∈ Cm1 . Therefore, δ∗(a) = δ1(c) which

shows that δ∗(B∗2 ) ⊂ B∗1 . Hence, δ∗ induces a linear map

δ∗ : H∗2 → H∗+11

which is called the cohomology connecting homomorphism for the short exact se-quence (6.1).

Lemma 6.3. The connecting homomorphism δ∗ : H∗2 → H∗+11 does not depend

on the linear section S.

Proof. Let S′ : Cm2 → Cm be another section of I, giving rise to δ′∗ : Zm2 →Zm+1

1 , via the equation J δ′∗ = δS′. Let a ∈ Zm2 . Then

S′(a) = S(a) + J(u)

for some u ∈ Cm1 . Therefore, the equations

J δ′∗(a) = δ(S(a)) + δ(J(u)) = δ(S(a)) + J(δ1(u))


hold in Cm+1. This implies that δ′∗(a) = δ∗(a)+ δ1(u) in Zm+11 , and then δ′∗(a) =

δ∗(a) in Hm+11 .

Proposition 6.4. The following long sequence

· · · → Hm1

H∗J−−−→ Hm H∗I−−−→ Hm2

δ∗−→ Hm+11

H∗J−−−→ · · ·is exact.

The exact sequence of Proposition 6.4 is called the cohomology exact sequenceassociated to the short exact of cochain complexes (6.1).

Proof. The proof involves 6 steps.

1. H∗I H∗J = 0: As H∗I H∗J = H∗(I J), this comes from that I J = 0.

2. δ∗H∗I = 0: Let b ∈ Zm. Then I(b + S(I(b))) = 0. Hence, b+ S(I(b)) = J(c)for some c ∈ Cm1 . Therefore,

J δ∗I(b) = δ(S(I(b)) = δ(b+ J(c)) = δ(b) + J δ1(c) = J δ1(c) ,

which proves that δ∗I(b) = δ1(c), and then δ∗H∗I = 0 in H∗1 .

3. H∗J δ∗ = 0: Let a ∈ Zm2 . Then, J δ∗(a) = δ(S(a)) ⊂ Bm+1, soH∗J δ∗([a]) =0 in Hm+1(K).

4. kerH∗J ⊂ Image δ∗: Let a ∈ Zm+11 representing [a] ∈ kerH∗J . This means

that J(a) = δ(b) for some b ∈ Cm. Then, I(b) ∈ Zm2 and S(I(b)) = b + J(c) forsome c ∈ Cm1 . Therefore,

δS I(b) = δ(b) + δ(J(c)) = J(a) + J(δ1(c)) .

As J is injective, this implies that δ∗(I(c)) = a+δ1(c), proving that δ∗([I(c)]) = [a].

5. kerH∗I ⊂ ImageH∗J : Let a ∈ Zm representing [a] ∈ kerH∗I. This meansthat I(a) = δ2(b) for some b ∈ Cm−1

2 . Let c = δ(S(b)) ∈ Cm. One has I(a+ c) = 0,so a+ c = J(e) for some e ∈ Cm1 . As δ(a+ c) = 0 and J is injective, the cochain eis in Zm1 . As c ∈ Bm, H∗J([e]) = [a] in Hm.

6. ker δ∗ ⊂ ImageH∗I: Let a ∈ Zm2 representing [a] ∈ ker δ∗. This means that

δ∗(a) = δ1(b) for some b ∈ Cm1 . In other words,

δ(S(a)) = J(δ1(b)) = δ(J(b)) .

Hence, c = J(b) + S(a) ∈ Zm and H∗I([c]) = [a].

We now prove the naturality of the connecting homomorphism in cohomology.We are helped by the following intuitive interpretation of δ∗: first, we consider C∗1a cochain subcomplex of C∗ via the injection J . Second, a cocycle a ∈ Zm2 may berepresented by a cochain in a ∈ Cm such that δ(a) ∈ C∗1 . Then, δ∗([a]) = [δ(a)].More precisely:

Lemma 6.5. Let

0→ C∗1J−→ C∗

I−→ C∗2 → 0

be a short exact sequence of cochain complexes. Then

(a) I−1(Zm2 ) = b ∈ Cm | δ(b) ∈ J(Cm+11 ).

(b) Let a ∈ Zm2 representing [a] ∈ Hm2 . Let b ∈ Cm with I(b) = a. Then

δ∗([a]) = [J−1(δ(b))] in Hm+11 .


Proof. Point (a) follows from the fact that I is surjective and from the equalityδ2I = I δ. For Point (b), choose a section S : Cm2 → Cm of I. By Lemma 6.3,δ∗([a]) = [J−1(δ(S(a))]. The equality I(b) = a implies that b = S(a) + J(c) forsome c ∈ Cm1 . Therefore,

[J−1(δ(b))] = [J−1(δS(a))] + [δ1(c)] = δ∗([a]) .

Let us consider a commutative diagram

(6.6)

0 // C∗1

F1

J // C∗

F

I // C∗2

F2

// 0

0 // C∗1J // C∗

I // C∗2 // 0

of morphisms of cochain complexes, where the horizontal sequences are exact. Thisgives rise to two connecting homomorphisms δ∗ : H∗2 → H∗+1

1 and δ∗ : H∗2 → H∗+11 .

Lemma 6.7 (Naturality of the cohomology exact sequence). The following di-agram

· · · // Hm1

H∗F1

H∗J // Hm

H∗F

H∗ I // Hm2

H∗F2

δ∗ // Hm+11

H∗F1

H∗J // · · ·

. . . // Hm1

H∗J // Hm H∗I // Hm2

δ∗ // Hm+11

H∗J // · · ·

is commutative.

Proof. The commutativity of two of the square diagrams follows from thefunctoriality of the cohomology: H∗F H∗J = H∗J H∗F1 since F J = J F1 andH∗F2H

∗I = H∗I H∗F since F2 I = I F . It remains to prove that H∗F1 δ∗ =

H∗δ∗F2.Let a ∈ Zm2 representing [a] ∈ Hm

2 . Let b ∈ Cm with I(b) = a. Then,I F (b) = F2(a). Using Lemma 6.5, one has

δ∗H∗F2([a]) = [J−1δF (b)]

= [J−1F δ(b)]

= [F1 J−1 δ(b)]

= H∗F1 δ∗([a]) .

We are now interested in the case where the cochain complexes (C∗i , δi) and(C∗, δ) are part of Kronecker pairs

P1 =((C∗1 , δ1), (C∗,1, ∂1), 〈 , 〉1

), P2 =

((C∗2 , δ2), (C∗,2, ∂2)〈 , 〉2

)

andP =

((C∗, δ), (C∗, ∂), 〈 , 〉

).

Let us consider two morphism of Kronecker pairs, (J, j) from P to P1 and (I, i)from P2 to P . We suppose that the two sequences

(6.8) 0→ C∗1J−→ C∗

I−→ C∗2 → 0


and

(6.9) 0→ C∗,2i−→ C∗

j−→ C∗,1 → 0

are exact sequences of (co)chain complexes. Note that, by Lemma 3.12, (6.8) isexact if and only if (6.9) is exact. Exact sequence (6.8) gives rise to the cohomologyconnecting homomorphism δ∗ : H∗2 → H∗+1

1 . We construct a homology connectinghomomorphism in the same way. Choose a linear section s : C∗,1 → C∗ of j, notrequired to be a morphism of chain complexes. As in the cohomology setting, onecan defines ∂∗ : Zm+1,1 → Zm,2 by the equation

(6.10) i ∂∗ = ∂s .

We check that ∂∗(Bm+1,1) ⊂ Bm,2. Hence ∂∗ induces a linear map

∂∗ : H∗+1,1 → H∗,2

called the homology connecting homomorphism for the short exact sequence (6.9).

Lemma 6.11. The connecting homomorphism ∂∗ : H∗+1,1 → H∗,2 does not de-pend on the linear section s.

Proof. The proof is analogous to that of Lemma 6.3 and is left as an exerciseto the reader.

Lemma 6.12. The connecting homomorphisms δ∗ : Hm2 → Hm+1

1 and ∂∗ : Hm+1,1 →Hm,1 satisfy the equation

〈δ∗(a), α〉1 = 〈a, ∂∗(α)〉2for all a ∈ Hm

2 , α ∈ Hm+1,1 and all m ∈ N. In other words, (δ∗, ∂∗) is a morphismof Kronecker pairs from (H∗1 , H∗,1, 〈 , 〉1) to (H∗2 , H∗,2, 〈 , 〉2).

Proof. Let a ∈ Zm2 represent a and α ∈ Zm+1,1 represent α. Choose linearsections S and s of I and j. Using Formulae (6.2) and (6.10), one has

〈δ∗(a), α〉1 = 〈δ∗(a), α〉1= 〈δ∗(a), j s(α)〉1= 〈J δ∗(a), s(α)〉= 〈S(a), ∂s(α)〉= 〈S(a), i ∂∗(α)〉= 〈I S(a), ∂∗(α)〉2= 〈a, ∂∗(α)〉2 = 〈a, ∂∗(α)〉2 .


· · · → Hm,2H∗i−−→ Hm

H∗j−−→ Hm,1∂∗−→ Hm−1,2

H∗i−−→ · · ·is exact.

The exact sequence of Proposition 6.13 is called the homology exact sequenceassociated to the short exact of chain complexes (6.9). It can be established directly,in an analogous way to that of Proposition 6.4. To make a change, we shall deduceProposition 6.13 from Proposition 6.4 by Kronecker duality.


Proof. By our hypotheses couples (I, i) and (J, j) are morphisms of Kroneckerpairs, and so is (δ∗, ∂∗) by Lemma 6.12. Using Diagram (3.9), we get a commutativediagram

· · · (Hm,2)♯oo (Hm)♯

(H∗i)♯

oo (Hm,1)♯

(H∗j)♯

oo H♯m−1,2

∂♯∗oo · · ·oo

· · · Hm1

oo

k≈

OO

HmH∗Ioo

k≈

OO

Hm1

H∗Joo

k≈

OO

Hm−12

δ∗oo

k≈

OO

· · ·oo

.

By Proposition 6.4, the bottom sequence of the above diagram is exact. Thus,the top sequence is exact. By Lemma 3.12, the sequence of Proposition 6.13 isexact.

Let us consider commutative diagrams

(6.14)

0 // C∗1

F1

J // C∗

F

I // C∗2

F2

// 0

0 // C∗1J // C∗

I // C∗2 // 0

and

(6.15)

0 C∗,1oo C∗joo C∗,2

ioo 0oo

0 C∗,1oo

f1

OO

C∗joo

f

OO

C∗,2ioo

f2

OO

0oo

such that the horizontal sequences are exact, Fi and F are morphisms of cochaincomplexes and fi and f are morphisms of cochain complexes.

Lemma 6.16 (Naturality of the homology exact sequence). Suppose that (Fi, fi)and (F, f) are morphisms of Kronecker pairs. Then, the diagram

· · · // Hm,2

H∗f2

H∗i // Hm

H∗f

H∗j // Hm,1

H∗f1

∂∗ // Hm−1,2

H∗f2

H∗i // · · ·

. . . // Hm,2H∗ i // Hm

H∗ j // Hm,1∂∗ // Hm−1,2

H∗ i // · · ·

is commutative.

Proof. By functoriality of the homology, the square diagrams not involving ∂∗commute. It remains to show that ∂∗H∗f1 = H∗f2∂∗. As H∗F1 δ

∗ = δ∗H∗F2

by Lemma 6.7, one has

〈a, ∂∗ H∗f1(α)〉2 = 〈H∗F1 δ∗(a), α〉1 = 〈δ∗ H∗F2(a), α〉1 = 〈a,H∗f1∂∗(α)〉2

for all a ∈ Hm−12 and α ∈ Hm,1. By Lemma 3.4, this implies that ∂∗H∗f1 =

H∗f2∂∗.


7. Relative (co)homology

A simplicial pair is a couple (K,L) where K is a simplicial complex and L is asubcomplex of K. The inclusion i : L → K is a simplicial map. Let a ∈ Cm(K).If, using Definition I.a of § 2, we consider a as a subset of Sm(K), then C∗i(a) =a ∩ Sm(L). If we see a as a map a : Sm(K)→ Z2, then C∗i(a) is the restriction ofa to Sm(L). We see that C∗i : C∗(K)→ C∗(L) is surjective. Define

Cm(K,L) = ker(Cm(K)

C∗i−−→ Cm(L))

and C∗(K,L) = ⊕m∈NCm(K,L). This definition implies that

• Cm(K,L) is the set of subsets of Sm(K)− Sm(L);• if K is a finite simplicial complex, Cm(K,L) is the the vector space with

basis Sm(K)− Sm(L).

As C∗i is a morphism of cochain complexes, the coboundary δ : C∗(K) →C∗(K) preserves C∗(K,L) and gives rise to a coboundary δ : C∗(K,L)→ C∗(K,L)so that (C∗(K,L), δ) is a cochain complex. The cocycles Z∗(K,L) and the cobound-aries B∗(K,L) are defined as usual, giving rise to the definition

Hm(K,L) = Zm(K,L)/Bm(K,L) .

The graded Z2-vector space H∗(K,L) = ⊕m∈NHm(K,L) is the simplicial relative

cohomology of the simplicial pair (K,L).When useful, the notations δK , δL and δK,L are used for the coboundaries of

the cochain complexes C∗(K), C∗(L) and C∗(K,L). We denote by j∗ the inclusionj∗ : C∗(K,L) → C∗(K), which is a morphism of cochain complexes, and use thesame notation j∗ for the induced linear map j∗ : H∗(K,L)→ H∗(K) on cohomol-ogy. We also use the notation i∗ for both C∗i and H∗i. We get thus a short exactsequence of cochain complexes

(7.1) 0→ C∗(K,L)j∗−→ C∗(K)

i∗−→ C∗(L)→ 0 .

If a ∈ Cm(L), any cochain a ∈ Cm(K) with i∗(a) = a is called a extension of a asa cochain in K. For instance, the 0-extension of a is defined by a = a ∈ Sm(L) ⊂Sm(K). Using § 6, Exact sequence (7.1) gives rise to a (simplicial cohomology)connecting homomorphism

δ∗ : H∗(L)→ H∗+1(K,L) .

It is induced by a linear map δ∗ : Zm(L)→ Zm+1(K,L) characterised by the equa-

tion j∗ δ∗ = δK S for some (or any) linear section S : Cm(L)→ Cm(K) of i∗, notrequired to be a morphism of cochain complex. For instance, one can take S(a) tobe the 0-extension of a. Using that C∗(K,L) is a chain subcomplex of C∗(K), thefollowing statement makes sense and constitutes a useful recipe for computing theconnecting homomorphism δ∗.

Lemma 7.2. Let a ∈ Zm(L) and let a ∈ Cm(K) be any extension of a as am-cochain of K. Then, δK(a) is a (m+ 1)-cocycle of (K,L) representing δ∗(a).

Proof. Choose a linear section S : Cm(L)→ Cm(K) such that S(a) = a. The

equation j∗ δ∗ = δK S proves the lemma.

We can now use Proposition 6.4 and get the following result.

7. RELATIVE (CO)HOMOLOGY 39


· · · → Hm(K,L)j∗−→ Hm(K)

i∗−→ Hm(L)δ∗−→ Hm+1(K,L)

j∗−→ · · ·is exact.

The exact sequence of Proposition 7.3 is called the simplicial cohomology exactsequence of the simplicial pair (K,L).

We now turn our interest to homology. The inclusion L → K induces aninclusion i∗ : C∗(L) → C∗(K) of chain complexes. We define Cm(K,L) as thequotient vector space

Cm(K,L) = coker(i∗ : Cm(L) → Cm(K)

).

As i∗ is a morphism of chain complexes, C∗(K,L) = ⊕m∈NCm(K,L) inherits aboundary operator ∂ = ∂K,L : C∗(K,L)→ C∗−1(K,L). The projection j∗ : C∗(K)→→ C∗(K,L) is a morphism of chain complexes and one gets a short exact sequenceof chain complexes

(7.4) 0→ C∗(L)i∗−→ C∗(K)

j∗−→ C∗(K,L)→ 0 .

The cycles and boundaries Z∗(K,L) and B∗(K,L) are defined as usual, giving riseto the definition

Hm(K,L) = Zm(K,L)/Bm(K,L) .

The graded Z2-vector space H∗(K,L) = ⊕m∈NHm(K,L) is the relative homologyof the simplicial pair (K,L). As before, the notations ∂K and ∂L may be used forthe boundary operators in C∗(K) and C∗(L) and i∗ and j∗ are also used for theinduced maps in homology.

Since the linear map i∗ : C∗(L) → C∗(K) is induced by the inclusion of basesS(L) → S(K), the quotient vector space C∗(K,L) may be considered as the vectorspace with basis S(K) − S(L). This point of view provides a tautological linearmap s : C∗(K,L) → C∗(K), which is a section of j∗ but not a morphism of chaincomplexes.

The Kronecker pairings for K and L are denoted by 〈 , 〉K and 〈 , 〉L, both atthe levels of (co)chains and of (co)homology. As 〈j∗(K,L), i∗(L)〉K = 0, we get abilinear map

Cm(K,L)× Cm(K,L)〈,〉K,L−−−−→ Z2 .

The formula

(7.5) 〈a, α〉K,L = 〈j∗(a), s(α)〉Kholds for all a ∈ Cm(K,L), α ∈ Cm(K,L) and all m ∈ N. Observe also that theformula

(7.6) 〈S(b), i∗(β)〉K = 〈b, β〉Lholds for all b ∈ Cm(L), β ∈ Cm(L) and all m ∈ N.

Lemma 7.7.(C∗(K,L), δK,L, C∗(K,L), ∂K,L, 〈 , 〉K,L

)is a Kronecker pair.

Proof. We first prove that 〈δK,L(a), α〉K,L = 〈a, ∂K,L(α)〉K,L for all a ∈Cm(K,L) and all α ∈ Cm+1(K,L) and all m ∈ N. Indeed, one has

〈δK,L(a), α〉K,L = 〈j∗δK,L(a), s(α)〉K(7.7)

= 〈δK j∗(a), s(α)〉K= 〈j∗(a), ∂K s(α)〉K


Observe that j∗∂K s(α) = ∂K,L(α) and therefore ∂K s(α) = s∂K,L(α) + i∗(c)for some c ∈ Cm(L). Hence, the chain of equalities in (7.7) may be continued

〈δK,L(a), α〉K,L = 〈j∗(a), ∂K s(α)〉K(7.7)

= 〈j∗(a), s∂K,L(α) + i∗(c)〉K= 〈j∗(a), s∂K,L(α)〉K + 〈j∗(a), i∗(c)〉K︸︷︷︸

0= 〈a, ∂K,L(α)〉K,L .

It remains to prove that the linear map k : C∗(K,L) → C∗(K,L)♯ given byk(a) = 〈a, 〉 is an isomorphism. As the inclusion i : L → K is a simplicial map, thecouple (C∗i, C∗i) is a morphism of Kronecker pairs by Lemma 5.3 and the resultfollows from Lemma 3.15.

Passing to homology then produces three Kronecker pairs with vanishing (co)boundaryoperators:

PL = (H∗(L), H∗(L), 〈, 〉L) , PK = (H∗(K), H∗(K), 〈, 〉K)

and

PK,L = (H∗(K,L), H∗(K,L), 〈, 〉K,L) .

Using § 6, short exact sequence (7.4) gives rise to the (simplicial homology)connecting homomorphism

∂∗ : H∗(K,L)→ H∗−1(L) .

It is induced by a linear map ∂ : Zm(K,L)→ Zm−1(L) characterised by the equa-tion

j∗ ∂∗ = ∂K s ,

using the section s of j∗ defined above (or any other one).

Lemma 7.8. The following couples are morphisms of Kronecker pairs:

(a) (i∗, i∗), from PL to PK .(b) (j∗, j∗), from PK to PK,L.(c) (δ∗, ∂∗), from PK,L to PL.

Proof. As the inclusion L → K is a simplicial map, Point (a) follows fromLemma 5.3. Point (c) is implied by Lemma 6.12. To prove Point (b), let a ∈Cm(K,L) and α ∈ Cm(K). Observe that s(j∗(α)) = α+ i∗(β) for some β ∈ Cm(L)and that 〈j∗(a), i∗(β)〉K = 0. Therefore:

〈a, j∗(α)〉K,L = 〈j∗(a), sj∗(α)〉K = 〈j∗(a), α〉K

Proposition 6.13 now gives the following result.


· · · → Hm(L)i∗−→ Hm(K)

j∗−→ Hm(K,L)∂∗−→ Hm−1(L)

i∗−→ · · ·is exact.

7. RELATIVE (CO)HOMOLOGY 41

The exact sequence of Proposition 7.9 is called the (simplicial) homology exactsequence of the simplicial pair (K,L).

We now study the naturality of the (co)homology exact sequences. Let (K,L)and (K ′, L′) be simplicial pairs. A simplicial map f of simplicial pairs from (K,L)to (K ′, L′) is a simplicial map fK : K → K ′ such that the restriction of f to Lis a simplicial map fL : L → L′. The morphism C∗fK : C∗(K ′) → C∗(K) thenrestricts to a morphism of cochain complexes C∗f : C∗(K ′, L′) → C∗(K,L) andthe morphism C∗fK : C∗(K) → C∗(K

′) descends to a morphism of chain com-plexes C∗f : C∗(K,L)→ C∗(K

′, L′). The couples (C∗fK , C∗fK) and (C∗fL, C∗fL)are morphisms of Kronecker pairs by Lemma 5.3. We claim that (C∗f, C∗f) is amorphism of Kronecker pair from (C∗(K,L), . . . ) to (C∗(K ′, L′), . . . ). Indeed, leta ∈ Cm(K ′, L′) and α ∈ Cm(K,L). One has

〈C∗f(a), α〉K,L = 〈j∗ C∗f(a), s(α)〉K= 〈C∗fK j′∗(a), s(α)〉K= 〈j′∗(a), C∗fK s(α)〉K′= 〈j′∗(a), C∗fK s(α)〉K′(7.9)

and

(7.10) 〈a, C∗f(α)〉K′,L′ = 〈j′∗(a), s′ C∗f(α)〉K′

The equation j′∗ s′C∗f(α) = j′∗ C∗fK s(α) = C∗f(α) implies that s′ C∗f(α) =

C∗fK s(α) + i′∗(β) for some β ∈ Cm(L′). As 〈j′∗(a), i′∗(β)〉K′ = 0, Equations (7.9)and (7.10) imply that 〈C∗f(a), α〉K,L = 〈a, C∗f(α)〉K′,L′ .

Lemma 6.7 and Lemma 6.16 then imply the following

Proposition 7.11. The cohomology and homology exact sequences are naturalwith respect to simplicial maps of simplicial pairs. In other words, given a simplicialmap of simplicial pairs f : (K,L)→ (K ′, L′), the following diagrams

· · · // Hm(K ′, L′)

H∗f

j′∗ // Hm(K ′)

H∗fK

i′∗ // Hm(L′)

H∗fL

δ′∗ // Hm+1(K ′, L′)

H∗f

j′∗ // · · ·

. . . // Hm(K,L)j∗ // Hm(K)

i∗ // Hm(L)δ∗ // Hm+1(K,L)

j∗ // · · ·

and

· · · // Hm(L)

H∗fL

i∗ // Hm(K)

H∗fK

j∗ // Hm(K,L)

H∗f

∂∗ // Hm−1(L)

H∗fL

i∗ // · · ·

. . . // Hm(L′)i′∗ // Hm(K ′)

j′∗ // Hm(K ′, L′)∂′∗ // Hm−1(L

′)i′∗ // · · ·

are commutative.

We finish this section by the exact sequences for a triple. A simplicial triple isa triplet (K,L,M) where K is a simplicial complex, L is a subcomplex of K andM is a subcomplex of L. A simplicial map f of simplicial triples, from (K,L,M)to (K ′, L′,M ′) is a simplicial map fK : K → K ′ such that the restrictions of fK toL and M are simplicial maps fL : L→ L′ and fM : M →M ′.


A simplicial triple T = (K,L,M) gives rise to pair inclusions

(L,M)i−→ (K,M)

j−→ (K,L)

and to a commutative diagram

(7.12)

0 // C∗(K,L)

C∗j

j∗K,L // C∗(K)OOid=

i∗K,L // C∗(L)

i∗L,M

// 0

0 // C∗(K,M)j∗K,M // C∗(K)

i∗K,M // C∗(M) // 0

where the horizontal lines are exact sequences of cochain complexes. A chasingdiagram argument shows that the morphism i∗K,L j

∗K,M , which sends C∗(K,M) to

C∗(L), has image C∗(L,M) and kernel the image of C∗j. This morphism coincideswith C∗i. We thus get a short exact sequence of cochain complexes

(7.13) 0→ C∗(K,L)C∗j−−→ C∗(K,M)

C∗i−−→ C∗(L,M)→ 0 .

The same arguments with the chain complexes gives a short exact sequence

(7.14) 0→ C∗(L,M)C∗i−−→ C∗(K,M)

C∗j−−→ C∗(K,L)→ 0 .

As above in this section, short exact sequences (7.13) and (7.14) produces con-necting homomorphisms δT : H∗(L,M) → H∗+1(K,L) and ∂T : H∗(K,L) →C∗−1(L,M). They satisfy 〈δT (a), α〉 = 〈a, ∂T (α)〉 as well as following proposition.

Proposition 7.15 ((Co)homology exact sequences of a simplicial triple). LetT = (K,L,M) be a simplicial triple. Then,

(a) the following sequences

· · · → Hm(K,L)H∗j−−−→ Hm(K,M)

H∗i−−→ Hm(L,M)δT−→ Hm+1(K,L)

H∗j−−−→ · · ·

and

· · · → Hm(L,M)H∗i−−→ Hm(K,M)

H∗j−−→ Hm(K,L)∂T−−→ Hm−1(L,M)

H∗i−−→ · · ·

are exact.(b) the exact sequences of Point (a) are natural for simplicial maps of simpli-

cial triples.

Remark 7.16. AsH∗(∅) = 0, we get a canonical GrV-isomorphismsH∗(K, ∅) ≈−→H∗(K), etc. Thus, the (co)homology exact sequences for the triple (K,L, ∅) giveback those of the pair (K,L)

(7.17) · · · → Hm(K,L)H∗j−−−→ Hm(K)

H∗i−−→ Hm(L)δ∗−→ Hm+1(K,L)

H∗j−−−→ · · ·

and

(7.18) · · · → Hm(L)H∗i−−→ Hm(K)

H∗j−−→ Hm(K,L)∂∗−→ Hm−1(L)

H∗i−−→ · · ·

where i : L → K and j : (K, ∅) → (K,L) denote the inclusions. This gives a moreprecise description of the morphisms j∗ and j∗ of Propositions 7.3 and 7.9.

8. MAYER-VIETORIS SEQUENCES 43

7.19. Historical note. The relative homology was introduced by S. Lefschetz in1927 in order to work out the Poincare duality for manifolds with boundary (see, e.g.[38, p. 58], [49, p. 47]). The use of exact sequences occured in several part of alge-braic topology after 1941 (see, e.g. [38, p. 86], [49, p. 47]). The (co)homology exactsequences play an essential role in the axiomatic approach of Eilenberg-Steenrod,[49].

8. Mayer-Vietoris sequences

Let K be a simplicial complex with two subcomplexes K1 and K2. We supposethat K = K1 ∪K2 (i.e. S(K) = S(K1) ∪ S(K2)). We call (K,K1,K2) a simplicialtriad. Then, K0 = K1 ∩ K2 is a subcomplex of K1, K2 and K, with S(K0) =S(K1) ∩ S(K2). The various inclusions are denoted as follows

(8.1)

K0i2

// i1 // K1j1

K2// j2 // K .

The notations i∗1, j∗1 , . . . , stand for both C∗i1, C

∗j1, etc, and H∗i1, H∗j1, etc. The

same holds for chains and homology: i1∗ for both C∗i1 andH∗i1, etc. Diagram (8.1)induces two diagrams

C∗(K)

j∗2

j∗1 // // C∗(K1)

i∗1C∗(K2)

i∗2 // // C∗(K0)

and

C∗(K0)

i2∗

// i1∗ // C∗(K1)j1∗

C∗(K2) // j2∗ // C∗(K) .

The cohomology diagram is Cartesian (pullback) and the homology diagram isco-Cartesian (pushout). Therefore, the following sequence

(8.2) 0→ C∗(K)(j∗1 ,j

∗2 )−−−−→ C∗(K1)⊕ C∗(K2)

i∗1+i∗2−−−→ C∗(K0)→ 0

is an exact sequence of cochain complexes and the following sequence

(8.3) 0→ C∗(K0)(i1∗,i2∗)−−−−−→ C∗(K1)⊕ C∗(K2)

j1∗+j2∗−−−−−→ C∗(K)→ 0

is an exact sequence of chain complexes.Consider the Kronecker pairs (C∗(Ki), C∗(Ki), 〈 , 〉i) for i = 0, 1, 2, and the

Kronecker pair (C∗(K), C∗(K), 〈 , 〉). A bilinear map

〈 , 〉⊕ :[C∗(K1)⊕ C∗(K2)

]×

[C∗(K1)⊕ C∗(K2)

]→ Z2

is defined by

〈(a1, a2), (α1, α2)〉⊕ = 〈a1, α1〉1 + 〈a2, α2〉2 .We check that (C∗(K1) ⊕ C∗(K2), C∗(K1) ⊕ C∗(K2), 〈 , 〉⊕) is a Kronecker pairand that the couples ((j∗1 , j

∗2 ), j∗1 + j∗2 ) and (i∗1 + i∗2, (i

∗1, i∗2)) are morphisms of Kro-

necker pairs. By § 6, there exist linear maps δMV : H∗(K0) → H∗+1(K) and∂MV : H∗(K) → H∗−1(K0) which, by Lemma 6.4 and 6.13, give the followingproposition.


Proposition 8.4 (Mayer-Vietoris sequences). The following long sequences

· · · → Hm(K)(j∗1 ,j

∗2 )−−−−→ Hm(K1)⊕Hm(K2)

i∗1+i∗2−−−→ Hm(K0)δMV−−−→ Hm+1(K)→ · · ·

and

· · · → Hm(K0)(i1∗,i2∗)−−−−−→ Hm(K1)⊕Hm(K2)

j1∗+j2∗−−−−−→ Hm(K)∂MV−−−→ Hm−1(K0)→ · · ·

are exact.

The homomorphisms δMV and ∂MV are called the Mayer-Vietoris connectinghomomorphisms in (co)homology. By Lemma 6.12, they satisfy 〈δMV (a), α〉 =〈a, ∂MV (α)〉0 for all a ∈ Hm(K0), all α ∈ Hm+1(k) and all m ∈ N. To define theconnecting homomorphisms, one must choose a linear section S of i∗1 + i∗2 and s ofj1∗ + j2∗. One can choose S(a) = (S1(a), 0), where S1 : C∗(K) → C∗(K1) is thetautological section of i∗1 given by the inclusion S(K0) → S(K1) (see § 7). A choiceof s is given, for σ ∈ S(K), by

s(σ) =

(σ, 0) if σ ∈ S(K1)

(0, 0) if σ /∈ S(K1) .

These choices produce linear maps δMV : Z∗(K0)→ Z∗+1(K) and ∂MV : Z∗(K)→Z∗−1(K0), representing δMV and ∂MV and defined by the equations

(j∗1 , j∗2 ) δMV = (δ1, δ2)S and (i1∗, i2∗) ∂MV = (∂1, ∂2)s .

(The apparent asymmetry of the choices has no effect by Lemma 6.3 and its ho-mology counterpart: exchanging 1 and 2 produces other sections, giving rise to thesame connecting homomorphisms.)

Finally, the Mayer-Vietoris sequences are natural for maps of simplicial triads.If T = (K,K1,K2) and T ′ = (K ′,K ′1,K

′2) are simplicial triads and if f : K → K ′

is a simplicial map such that f(Ki) ⊂ K ′i, then the Mayer Vietoris sequences ofT and T ′ are related by commutative diagrams, as in Proposition 7.11. This is adirect consequence of Lemma 6.7 and Lemma 6.16.

9. Appendix A: an acyclic carrier result

The powerful technique of acyclic carriers was introduced by Eilenberg andMcLane in 1953 [48], after earlier work by Lefschetz. Proposition 9.1 below is a veryparticular example of this technique, adapted to our needs. For a full developmentof acyclic carriers (see, e.g., [152, Ch. 1, § 13]).

Let (C∗, ∂) and (C∗, ∂) be two chain complexes and let ϕ : C∗ → C∗ be amorphism of chain complexes. We suppose that Cm is equipped with a basis Smfor each m and denote by S the union of all Sm. An acyclic carrier A∗ for ϕ withrespect to the basis S is a correspondence which associates to each s ∈ S a subchaincomplex A∗(s) of C∗ such that

(a) ϕ(s) ∈ A∗(s).(b) H0(A∗(s)) = Z2 and Hm(A∗(s)) = 0 for m > 0.(c) let s ∈ Sm and t ∈ Sm−1 such that t occurs in the expression of ∂ s in the

basis Sm−1. Then A∗(t) is a subchain complex of A∗(s) and the inclusionA∗(t) ⊂ A∗(s) induces an isomorphism on H0.

(d) if s ∈ S0 ⊂ C0 = Z0, then H0ϕ(s) 6= 0 in H0(A∗(s)).

10. APPENDIX B: ORDERED SIMPLICIAL (CO)HOMOLOGY 45

Proposition 9.1. Let ϕ and ϕ′ be two morphisms of chain complexes from(C∗, ∂) to (C∗, ∂). Suppose that ϕ and ϕ′ admit the same acyclic carrier A∗ withrespect to some basis S of C∗. Then H∗ϕ = H∗ϕ

′.

Proof. The proof is similar to that of Proposition 5.14. By induction on m,we shall prove the truth of the following property:

Property H(m): there exists a linear map D : Cm → Cm+1 such that:

(i) ∂D(α) +D(∂α) = ϕ(α) + ϕ′(α) for all α ∈ Cm.(ii) for each s ∈ Sm, D(s) ∈ Am+1(s).

Property H(m) for all m implies that H∗ϕ = H∗ϕ′. Indeed, we then have a

linear map D : C∗ → C∗+1 satisfying

(9.2) ϕ+ ϕ′ = ∂D +D∂ .

Let β ∈ Z∗. By Equation (9.2), one has ϕ(β) + ϕ(β) = ∂D(β) which implies thatH∗ϕ([β]) +H∗ϕ

′([β]) in H∗.Let us prove H(0). Let s ∈ S0. In H0(A∗(s)) = Z2, one has H0ϕ(s) 6= 0

and H0ϕ′(s) 6= 0. Therefore H∗ϕ(s) = H∗ϕ

′(s) in H0(A∗(s)). This implies thatϕ(s) + ϕ′(s) = ∂(ηs) for some ηs ∈ A1(s). We set D(s) = ηs. This procedure,for each s ∈ S0, provides a linear map D : C0 → C1, which, as ∂C0 = 0, satisfiesϕ(s) + ϕ′(s) = ∂D(α) +D(∂(α)).

We now prove that H(m− 1) implies H(m) for m ≥ 1. Let s ∈ Sm. The chainD(∂s) exists in Am(s) by H(m− 1). Let ζ ∈ Am(s) defined by

ζ = ϕ(s) + ϕ′(s) +D(∂s)

Using H(m − 1), one checks that ∂ζ = 0. Since Hm(A∗(s)) = 0, there existsν ∈ Am+1(s) such that ζ = ∂ν. Choose such an element ν and set D(σ) = ν. Thisdefines D : Cm → Cm+1 which satisfies (i) and (ii), proving H(m).

10. Appendix B: ordered simplicial (co)homology

This technical section may be skiped in a first reading. It shows that sim-plicial (co)homology may be defined using larger sets of (co)chains, based on or-dered simplexes. This will be used for comparisons between simplicial and singular(co)homology (see § 17) and to define the cup and cap products in Chapter 3.

Let K be a simplicial complex. Define

Sm(K) = (v0, . . . , vm) ∈ V (K)m+1 | v0, . . . , vm ∈ S(K) .Observe that dimv0, . . . , vm ≤ m and may be strictly smaller if there are repeti-

tions amongst the vi’s. An element of Sm(K) is an ordered m-simplex of K.The definitions of ordered (co)chains and (co)homology are the same those for

the simplicial case (see § 2), replacing the simplexes by the ordered simplexes. Wethus set

Definitions I (subset definitions):

(a) An ordered m-cochain is a subset of Sm(K).

(b) An ordered m-chain is a finite subset of Sm(K).

The set of ordered m-cochains of K is denoted by Cm(K) and that of ordered

m-chains by Cm(K). As in § 2, Definitions I are equivalent to

Definitions II (colouring definitions):


(a) An ordered m-cochain is a function a : Sm(K)→ Z2.

(b) An ordered m-chain is a function α : Sm(K)→ Z2 with finite support.

Definitions II endow Cm(K) and Cm(K) with a structure of a Z2-vector space.

The singletons provide a basis of Cm(K), in bijection with Sm(K). Thus, DefinitionII.b is equivalent to

Definition III: Cm(K) is the Z2-vector space with basis Sm(K):

Cm(X) =⊕

σ∈Sm(X)

Z2 σ .

We consider the graded Z2-vector spaces C∗(K) = ⊕m∈NCm(K) and C∗(K) =

⊕m∈NCm(K). The Kronecker pairing on ordered (co)chains

Cm(K)× Cm(K)〈 , 〉−−→ Z2

is defined, using the various above definitions, by the equivalent formulae

(10.1)


=∑


=∑

σ∈Sm(K) a(σ)α(σ) using Definitions II.a and II.b .

As in Lemma 2.2, we check that the map k : Cm(K)→ Cm(K)♯, given by k(a) =〈a, 〉, is an isomorphism.

The boundary operator ∂ : Cm(K) → Cm−1(K) is the Z2-linear map defined,

for (v0, . . . , vm) ∈ Sm(K) by

(10.2) ∂(v0, . . . , vm) =

m∑

i=0

(v0, . . . , vi, . . . , vm) ,

where (v0, . . . , vi, . . . , vm) ∈ Sm−1 is the m-tuple obtained by removing vi. The

coboundary operator δ : Cm(K)→ Cm+1(K) is defined by the equation

(10.3) 〈δa, α〉 = 〈a, ∂α〉 .With these definition, (C∗(K), ∂, C∗(K), δ, 〈 , 〉) is a Kronecker pair. We define

the vector spaces of ordered cycles Z∗(K), ordered boundaries B∗(K), ordered co-

cycles Z∗(K), ordered coboundaries B∗(K), ordered homology H∗(K) and ordered

cohomology H∗(K) as in § 3. By Proposition 3.7, the pairing on (co)chain descendsto a pairing

Hm(K)×Hm(K)〈 , 〉−−→ Z2

so that the map k : Hm → H♯m, given by k(a) = 〈a, 〉, is an isomorphism (Kronecker

duality).

Example 10.4. Let K = pt be a point. Then, Sm(pt) contains one element for

each integer m, namely the (m + 1)-tuple (pt, . . . , pt). Then, Cm(pt) = Z2 for allm ∈ N and the chain complex looks like

· · · ≈−→ C2k+1(pt)0−→ C2k(pt)

≈−→ C2k−1(pt)0−→ · · · ≈−→ C1(pt)

0−→ C0(pt)→ 0 .

Therefore,

H∗(pt) ≈ H∗(pt) ≈

0 if ∗ > 0Z2 if ∗ = 0 .

10. APPENDIX B: ORDERED SIMPLICIAL (CO)HOMOLOGY 47

One sees that, for a simplicial complex reduced to a point, the ordered (co)homologyand the simplicial (co)homology are isomorphic.

Example 10.5. The unit cochain 1 ∈ C0(K) is defined as 1 = S0(K). It is

a cocycle and defines a class 1 = H0(K). If K is non-empty and connected, then

H0(K) ≈ Z2 generated by 1. Then H0(K) ≈ Z2 by Kronecker duality; one has

Z0(K) = C0(K) and α ∈ Z0(K) represents the non-zero element of H0(K) if andonly if ♯α is odd. The proofs are the same as for Proposition 4.3.

Example 10.6. Let L be a simplicial complex and CL be the cone on L. Then

H∗(CL) ≈ H∗(CL) ≈

0 if ∗ > 0Z2 if ∗ = 0 .

The proof is the same as for Proposition 4.11, even simpler, since D : Cm(CL) →Cm+1(CL) is defined, for (v0, . . . , vm) ∈ Sm(CL) by the single line formulaD(v0, . . . , vm) = (∞, v0, . . . , vm).

Let f : L → K be a simplicial map. We define C∗f : C∗(L) → C∗(K) as thedegree 0 linear map such that

C∗f(v0, . . . , vm) = (f(v0), . . . , f(vm))

for all (v0, . . . , vm) ∈ S(L). The degree 0 linear map C∗f : C∗(K) → C∗(L) isdefined by

〈C∗f(a), α〉 = 〈a, C∗f(α)〉 .By Lemma 3.10, (C∗f, C∗f) is a morphism of Kronecker pairs.

We now construct a functorial isomorphism between the ordered and non-ordered (co)homologies, its existence being suggested by the previous examples.

Define ψ∗ : C∗(K)→ C∗(K) by

ψ∗((v0, . . . , vm)) =

v0, . . . , vm if vi 6= vj for all i 6= j

0 otherwise.

We check that ψ is a morphism of chain complexes. We define ψ∗ : C∗(K)→ C∗(K)by requiring that the equation 〈ψ∗(a), α〉 = 〈a, ψ∗(α)〉 holds for all a ∈ C∗(K) and

all α ∈ C∗(K). By Lemma 3.10, ψ∗ is a morphism of cochain complexes and (ψ∗, ψ∗)

is a morphism of Kronecker pairs between (C∗(K), C∗(K)) and (C∗(K), C∗(K)). It

thus defines a morphism of Kronecker pairs (H∗ψ,H∗ψ) between (H∗(K), H∗(K))

and (H∗(K), H∗(K)).To define a morphism of Kronecker pairs in the other direction, choose a sim-

plicial order ≤ on K (see 1.9). Define φ≤∗ : C∗(K) → C∗(K) as the unique linearmap such that

φ≤∗(v0, . . . , vm) = (v0, . . . , vm) ,

where v0 ≤ v1 ≤ · · · ≤ vm. We check that φ≤∗ is a morphism of chain complexes

and define φ≤∗ : C∗(K) → C∗(K) by requiring that the equation 〈φ≤∗(a), α〉 =

〈a, φ≤∗(α)〉 holds for all a ∈ C∗(K) and all α ∈ C∗(K). By Lemma 3.10, (φ≤∗, φ≤∗)

is a morphism of Kronecker pairs between (C∗(K), C∗(K)) and (C∗(K), C∗(K)). Itthen defines a morphism of Kronecker pairs (H∗φ≤, H

∗φ≤) between (H∗(K), H∗(K))

and (H∗(K), H∗(K)).

Proposition 10.7. H∗ψH∗φ≤ = idH∗(K) and H∗φ≤H∗ψ = idH∗(K).


Proof. As ψ∗φ≤∗ = idC∗(K), the first equality follows from Lemma 3.11.

For the second one, let (v0, . . . , vm) ∈ Sm(K). Let σ = v0, . . . , vm ∈ Sk(K) with

k ≤ m. Clearly, φ≤∗ψ∗(v0, . . . , vm) ∈ C∗(σ). By what was seen in Examples 10.5

and 10.6, the correspondence (v0, . . . , vm) 7→ C∗(v0, . . . , vm) is an acyclic carrier

A∗, with respect to the basis S∗(K), for both idC(K) and φ≤∗ψ∗. Therefore, the

equality H∗φ≤H∗ψ = idH∗(K) follows by Lemma 3.11 and Proposition 9.1.

Corollary 10.8. H∗ψH∗φ≤ = idH∗(K) and H∗φ≤H∗ψ = idH∗(K).

Proof. Corollary 10.8 is obtained from Proposition 10.7 by Kronecker duality.

Corollary 10.9. H∗ψ and H∗ψ are isomorphisms.

Corollary 10.10. H∗φ≤ and H∗φ≤ are isomorphisms which do not dependon the simplicial order ≤.

Proof. This follows from Proposition 10.7 and Corollary 10.8, since H∗ψ andH∗ψ do not depend on ≤.

We shall see in § 22 that H∗ψ and H∗φ≤ are isomorphisms of graded Z2-algebras. We now prove that they are also natural with respect to simplicial maps.Let f : L→ K be a simplicial map. Let C∗f : C∗(L)→ C∗(K) be the unique linearmap such that

C∗f((v0, . . . , vm)) = (f(v0), . . . , f(vm))

for each (v0, . . . , vm) ∈ Sm(K). Doing this for each min ∈ N produces a GrV-

morphism C∗f : C∗(L)→ C∗(K). The formula ∂ C∗f = C∗f ∂ is straightforward(much easier than that for non-ordered chains). Hence, we get a GrV-morphism

H∗f : H∗(L)→ H∗(K). A GrV-morphism C∗f : C∗(K)→ C∗(L) is defined by the

equation 〈C∗f(a), α〉 = 〈a, C∗f(α)〉 required to hold for all a ∈ Cm(L), α ∈ Cm(K)

and all m ∈ N. It is a cochain map and induces a GrV-morphism H∗f : H∗(K)→H∗(L), Kronecker dual to H∗f .

Proposition 10.11. Let f : L→ K be a simplicial map. Let ≤ be a simplicialorder on K and ≤′ be a simplicial order on L. Then the diagrams

H∗(L)

H∗ψ

H∗f // H∗(K)

H∗ψ

H∗(L)H∗f //

H∗φ≤′

KK

H∗(K)

H∗φ≤

KKand

H∗(K)

H∗φ≤

H∗f // H∗(L)

H∗φ≤′

H∗(K)

H∗f //

H∗ψ

OO

H∗(L)

H∗ψ

OO

are commutative.

Proof. By Kronecker duality, only the homology statement requires a proof.It is enough to prove that H∗f H∗ψ = H∗ψH∗f since the formula H∗f H∗φ≤′ =

H∗φ≤H∗f will follow by Corollary 10.8. Finaly, the formula C∗f C∗φ≤′ =C∗φ≤C∗f is straightforward.

The above isomorphism results also work in relative ordered (co)homology. Let(K,L) be a simplicial pair. Denote by i : L → K the simplicial inclusion. We definethe Z2-vector space of relative ordered (co)chain by

Cm(K,L) = ker(Cm(K)

C∗i−−→ Cm(L))

11. EXERCISES FOR CHAPTER 1 49

and

Cm(K,L) = coker(i∗ : Cm(L) → Cm(K)

).

These inherit (co)boundaries δ : C∗(K,L) → C∗(K,L) and ∂ = C∗(K,L) →C∗−1(K,L) which give rise to the definition of relative ordered (co)homology H∗(k, L)

and H∗(K,L). Connecting homomorphisms δ∗ : H∗(L)→ H∗+1(K,L) and

∂∗ : H∗(K,L)→ H∗−1(L) are defined as in § 7, giving rise to long exact sequences.

Our homomorphisms ψ∗ : C∗(K) → C∗(K) and φ≤∗ : C∗(K) → C∗(K) satisfy

ψ∗(C∗(L)) ⊂ C∗(L) and φ≤∗(C∗(L) ⊂ C∗(L), giving rise to homomorphisms on

relative (co)chains and relative (co)homology H∗ψ : H∗(K,L)→ H − ∗(K,L), etc.Proposition 10.7 and Corollary 10.8 and their proofs hold in relative (co)homology.Hence, as for Corollary 10.9 and 10.10, we get

Corollary 10.12. H∗ψ : H∗(K,L)→ H∗(K,L) and H∗ψ : H∗(K,L)→ H∗(K,L)are isomorphisms.

Corollary 10.13. H∗φ≤ : H∗(K,L) → H∗(K,L) and H∗φ≤ : H∗(K,L) →H∗(K,L) are isomorphisms which do not depend on the simplicial order ≤.

11. Exercises for Chapter 1

1.1. Let Fn be the full complex on the set 0, 1, . . . , n (see p. 24). What are the2-simplexes of the barycentric subdivision F ′2 of F2? How many n-simplexes doesF ′n contain?

1.2. Compute the Euler characteristic and the Poincare polynomial of the k-skeletonFkn of Fn.1.3. Let X be a metric space and let ε > 0. The Vietoris-Rips complex Xε of X isthe simplicial complex whose simplexes are the finite non-empty subset of X whosediameter is < ε (the diameter of A ⊂ X is the least upper bound of d(x, y) forx, y ∈ A). In particular, V (Xε) = X .

(a) Describe |Xε| for various ε when X is the set of vertices of a cube of edge 1

in R3. In particular, if√

2 < ε ≤√

3, show that |Xε| is homeomorphicto S3.

(b) Let X be the space n-th roots of unity, with the distance d(x, y) being theminimal length of an arc of the unit circle joining x to y. Suppose that4π/n < ε ≤ 6π/n.(i) If n = 6, show that |Xε| is homeomorphic to S2.(ii) If n ≥ 7 is odd, show that |Xε| is homeomorphic to a Mobius band.(iii) If n ≥ 7 is even, show that |Xε| is homeomorphic to S1 × [0, 1].

Note: the complex Xε was introduced by L Vietoris in 1927 [197]. After its re-introduction by E. Rips for studying hyperbolic groups, it has been popularizedunder the name of Rips complex. For some developments and applications, see[82, 126] and Wikipedia’s page “Vietoris-Rips complex”.

1.4. Let ℓ = (ℓ1, . . . , ℓn) ∈ Rn>0. A subset J of 1, . . . , n is called ℓ-short (orjust short) if

∑i∈J ℓi <

∑i/∈J ℓi. Show that short subsets are the simplexes of a

simplicial complex Sh(ℓ) with V (Sh(ℓ)) ⊂ J (used in § 62). Describe Sh(1, 1, 1, 1, 3),Sh(1, 1, 3, 3, 3) and Sh(1, 1, 1, 1, 1). Compute their Euler characteristics and theirPoincare polynomials.


1.5. Let K be the simplicial complex with V (K) = Z and S1(K) = r, r + 1 |r ∈ Z (|K| ≈ R). Then S1(K) is a 1-cocycle. Find all the cochains a ∈ C0(K)such that S1(K) = δ(a).

1.6. Find a simplicial pair (K,L) such that |K| is homeomorphic to S1 × I and|L| = Bd |K|. In the spirit of § 4.7, compute the simplicial cohomology of K andof (K,L) and find (co)cycles generating H∗(K), H∗(K,L), H∗(K) and H∗(K,L).Write completely the (co)homology exact sequence of (K,L).

1.7. Same exercise as before with |K| the Mobius band and |L| = Bd |K|.

1.8. Let f : K → L be a simplicial map between simplicial complexes. Supposethat L is connected and K is non-empty. Show that H0f is surjective.

1.9. Let m,n, q be positive integers. If m = nq, the quotient map Z → Z/nZdescends to a map Z/mZ → Z/nZ, giving rise to a simplicial map f : Pm → Pnbetween the simplicial polygons Pm and Pn (see Example 4.5). Compute H∗f .

1.10. Let M be a m-dimensional pseudomanifold. Let σ and σ′ be two distinctm-simplexes of M . Find a ∈ Cm−1(M) such that δ(a) = σ, σ′.

1.11. Let M be a finite non-empty n-dimensional pseudomanifold. Let 0 6= γ ∈Zm−1(M) which is a boundary. Prove that z is the boundary of exactly two nchains.

1.12. Let f : M → N be a simplicial map between finite n-dimensional pseudoman-ifolds. Show that the following two conditions are equivalent.

(a) Hnf 6= 0.(b) There exists σ ∈ S(N) such that ♯f−1(σ) is odd.

1.13. Let ±1 be the 0-dimensional simplicial complex with vertices −1 and 1.Let K be a simplicial complex. The simplicial suspension ΣK is the join K ∗ ±1.

(a) Let P4 be the polygon complex with 4-edges (see Example 4.5). Showthat P4 ∗K is isomorphic to the double suspension Σ(ΣK). [Hint: showthat the join operation is associative: (K ∗ L) ∗M ≈ K ∗ (L ∗M).]

(b) Prove that the suspension of a pseudomanifold is a pseudomanifold.(c) Prove that the correspondence K 7→ ΣK gives a functor from Simp to

itself.

1.14. Let A be a finite set. Show that FA is a pseudomanifold.

1.15. LetM be an n-dimensional pseudomanifold which is infinite. What isHn(M)?

1.16. Let (K,K1,K2) be a simplicial triad. Suppose that K1 and K2 are connectedand that K1 ∩K2 is not empty. Show that K is connected.

1.17. Let (K,K1,K2) be a simplicial triad and let K0 = K1 ∩K2.

(a) Prove that the homomorphism H∗(K1,K0) → H∗(K,K2) induced by theinclusion is an isomorphism (simplicial excision).

(b) Write the commutative diagram involving the homology exact sequencesof (K1,K0) and (K,K2). Using (a), construct out of this diagram theMayer-Vietoris sequence for the triad (K,K1,K2).


1.18. Let M1 and M2 be two finite n-dimensional pseudomanifolds. Let σi ∈ S(Mi)and let h : σ1 → σ2 be a bijection. The simplicial connected sum M = M1 ♯M2 isthe simplicial complex defined by

V (M) = V (M1) ∪V (M2)/v ∼ h(v) for v ∈ σ1

andS(M) =

(S(M1)− σ1

)∪

(S(M2)− σ2

).

Compute H∗(M) in terms of H∗(M1) and H∗(M2).

CHAPTER 2

Singular and cellular (co)homologies

12. Singular (co)homology

Singular (co)homology provides a functor associating to a topological space Xa graded Z2-vector space, whose isomorphism class depends only on the homotopytype of X . Such functors, from Top to categories of algebraic objects, constitutethe main subject of algebraic topology.

Invented by S. Eilenberg in 1944 [47] after earlier attempts by Lefschetz, sin-gular homology is formally akin to simplicial homology. However, in order to makecomputations for non-trivial examples, we need to establish some properties, such ashomotopy and excision, which require some work. When K is a simplicial complex,the simplicial homology of K and the singular cohomology of |K| are isomorphicin several ways, some of them being functorial (see § 17). Singular (co)homology isthe most powerful and relevant for spaces having the homotopy of a CW-complex,a notion introduced in § 15. For such spaces, singular (co)homology is isomorphicto other (co)homology theories (see § 18).

12.1. Definitions. The standard Euclidean m-simplex ∆m is defined by

∆m = (x0, . . . , xm) ∈ Rm+1 | xi ≥ 0 and∑

xi = 1 ,

endowed by the induced topology from that of Rn+1. In particular, ∆m = ∅ ifm < 0. Let X be a topological space. A singular m-simplex of X is a continuousmap σ : ∆m → X . The set of singular m-simplexes of X is denoted by Sm(X).

The definitions of singular (co)chains and (co)homology are copied from thosefor the simplicial case (see § 2), replacing the simplexes by the simplicial simplexes.We thus set

Definitions I (subset definitions):(a) An singular m-cochain of X is a subset of Sm(X).(b) An singular m-chain of X is a finite subset of Sm(X).

The set of singular m-cochains of X is denoted by Cm(X) and that of singularm-chains by Cm(X). As in § 2, Definitions I are equivalent to

Definitions II (colouring definitions):(a) An singular m-cochain is a function a : Sm(X)→ Z2.(b) An singular m-chain is a function α : Sm(X)→ Z2 with finite support.

Definitions II endow Cm(X) and Cm(X) with a structure of a Z2-vector space.The singletons provide a basis of Cm(X), in bijection with Sm(X). Thus, DefinitionII.b is equivalent to

53

54 2. SINGULAR AND CELLULAR (CO)HOMOLOGIES

Definition III: Cm(X) is the Z2-vector space with basis Sm(X):

Cm(X) =⊕

σ∈Sm(X)

Z2 σ .

We consider the graded vector spaces C∗(X) = ⊕m∈NCm(X) and C∗(X) =⊕m∈NC

m(X). By convention, Cm(X) = Cm(X) = 0 if m < 0 (so the index mcould be taken in Z in the previous formulae).

The Kronecker pairing on singular (co)chains

Cm(X)× Cm(X)〈 , 〉−−→ Z2

is defined, using the various above definitions, by the equivalent formulae

(12.1)


=∑


=∑

σ∈Sm(X) a(σ)α(σ) using Definitions II.a and II.b .

As in Lemma 2.2, we check that the map k : Cm(X) → Cm(X)♯, given by k(a) =〈a, 〉, is an isomorphism.

Let m, i ∈ N with 0 ≤ i ≤ m. Define the i-th face inclusion ǫi : ∆m−1 → ∆m

by

ǫi(x0, . . . , xm−1) = (x0, . . . , xi−1, 0, xi+1, . . . , xm−1) .

The boundary operator ∂ : Cm(X) → Cm−1(X) is the Z2-linear map defined, forσ ∈ Sm(X) by

(12.2) ∂(σ) =m∑

i=0

σǫi .

Lemma 12.3. ∂∂ = 0.

Proof. By linearity, it suffices to prove that ∂∂(σ) = 0 for σ ∈ Sm(X). Onehas

(12.4) ∂∂(σ) = ∂ (

m∑

i=0

σǫi) =∑

(i,j)∈A

σǫiǫj ,

where A = 0, . . . ,m × 0, . . . ,m − 1. The set B = (i, j) ∈ A | i ≤ j isin bijection with A − B, via the map (i, j) 7→ (j + 1, i). But if (i, j) ∈ B, thenǫiǫj = ǫj+1 ǫi, which implies that ∂∂ = 0.

The coboundary operator δ : Cm(X)→ Cm+1(X) is defined by the equation

(12.5) 〈δa, α〉 = 〈a, ∂α〉 .With these definition, ((C∗(X), ∂), (C∗(X), δ), 〈 , 〉) is a Kronecker pair. We

define the vector spaces of singular cycles Z∗(X), singular boundaries B∗(X), sin-gular cocycles Z∗(X), singular coboundaries B∗(X), singular homology H∗(X) andsingular cohomology H∗(X) as in § 3. By Proposition 3.7, the pairing on (co)chaindescends to a pairing

Hm(X)×Hm(X)〈 , 〉−−→ Z2

so that the map k : Hm → H♯m, given by k(a) = 〈a, 〉, is an isomorphism (Kronecker

duality in singular (co)homology). The Kronecker pairing extends to a bilinear map

12. SINGULAR (CO)HOMOLOGY 55

H∗(X) ×H∗(X)〈,〉−→ Z2 by setting 〈a, α〉 = 0 if a ∈ Hp(X) and α ∈ Hq(X) with

p 6= q.

Example 12.6. If X is the empty space, then Sm(X) = ∅ for all m and thusH∗(∅) = H∗(∅) = 0. Let X = pt be a point. Then, Sm(pt) contains one element foreach m ∈ N, namely the constant singular simplex ∆m → pt. Then, Cm(pt) = Z2

for all m ∈ N and the chain complex looks like

· · · ≈−→ C2k+1(pt)0−→ C2k(pt)

≈−→ C2k−1(pt)0−→ · · · ≈−→ C1(pt)

0−→ C0(pt)→ 0 .

Therefore,

(12.7) H∗(pt) ≈ H∗(pt) ≈

0 if ∗ > 0Z2 if ∗ = 0 .

Example 12.8. Let K be a simplicial complex. Choose a simplicial order “≤”for K. To an m-simplex σ = v0, . . . , vm ∈ Sm(K), with v0 ≤ · · · ≤ vm, weassociate the singular m-simplex R≤(σ) : ∆m → |K| defined by

(12.9) R≤(σ)(t0, . . . , tm) =

m∑

i=0

tivi .

The linear combination in (12.9) makes sense since v0, . . . , vm is a simplex of K.This defines a map R≤ : Sm(K)→ Sm(|K|) which extends to a linear map

R≤,∗ : C∗(K)→ C∗(|K|) .This map will be used several times in this chapter. The formula ∂R≤,∗ = R≤,∗∂is obvious, so R≤,∗ is a chain map from (C∗(K), ∂) to (C∗(|K|), ∂). We shallprove, in Theorem 17.5, that R≤,∗ induces an isomorphism between the simplicial(co)homology) of K and the singular (co)homology) of |K|.

Example 12.10. As the affine simplex ∆0 is a point, one can identify a singular0-simplex of X with its image, a point of X . This gives a bijection S0(X) ≈ X anda bijection between subsets of X and singular 0-cochains. For B ⊂ X and x ∈ X ,one has 〈B, x〉 = χB(x), where χB stands for the characteristic function for B. The1-cochain δB is the connecting cochain for B: if β ∈ S1(X), then

(12.11) 〈δ(B), β〉 = 〈B, ∂β〉 = 〈B, β(1, 0)〉+ 〈B, β(0, 1)〉 .In other words 〈δ(B), β〉 = 1 if and only if the (non-oriented) path β connects apoint in B to a point in X −B. Observe that δ(B) = δ(X −B).

Following Example 12.10, the unit cochain 1 ∈ C0(X) is defined by 1 =S0(X) ≈ X . By Equation (12.11) 〈δ1, β〉 = 0 for all β ∈ S1(X). This provesthat δ(1) = 0 by Lemma 2.2. Hence, 1 is a cocycle, whose cohomology class isagain denoted by 1 ∈ H0(X).

Proposition 12.12. Let X be a non-empty path-connected space. Then,

(i) H0(X) = Z2, generated by 1 which is the only non-vanishing singular0-cocycle.

(ii) H0(X) = Z2. Any 0-chain α is a cycle, which represents the non-zeroelement of H0(X) if and only if ♯α is odd.


Proof. The proof is analogous to that of Proposition 4.3. If X is non-emptythe unit cochain does not vanish and, as C−1(X) = 0, 1 6= 0 in H0(X).

Let a ∈ C0(X) with a 6= 0,1. Then there exists x, y ∈ X = S(X) witha(x) 6= a(y). Since X is path-connected, there exists σ ∈ S1(X) with σ(1, 0) = xand σ(0, 1) = y. As in Equation (12.11), this proves that 〈δ(a), σ〉 6= 0 so a is nota cocycle. This proves (i).

Now, H0(X) = Z2 since H0(X) ≈ H0(X)♯. Any α ∈ C0(X) is a cycle sinceC−1(X) = 0. It represents the non-zero homology class if and only if 〈1, α〉 = 1,that is if and only if ♯α is odd.

The reduced (singular) cohomology H∗(X) and homology H∗(X) of a topologicalspace X are the graded Z2-vector spaces defined by

(12.13)H∗(X) = coker

(H∗p : H∗(pt)→ H∗(X)

)

H∗(X) = ker(H∗p : H∗(X)→ H∗(pt)

)

where p : X → pt denotes the constant map to a point. In particular, H∗(pt) = 0 =

H∗(pt). One checks that the Kronecker pairing induces a bilinear map 〈 , 〉 : Hm(X)×Hm(X) → Z2 such that the correspondence a 7→ 〈a, 〉 gives an isomorphism

k : Hm(X)≈−→ Hm(X)♯.

The full strength of Definition (12.13) appears in other (co)homology theories,such as equivariant cohomology (see p. 223). For the singular cohomology, asH∗(pt) = Z21, one gets

Hm(X) =

H0(X)/Z21 if m = 0

Hm(X) if m 6= 0

and

Hm(X) =

ker

(H0(X)

〈1, 〉−−−→ Z2

)if m = 0

Hm(X) if m 6= 0 .

Thus, by Proposition 12.12, H0(X) = 0 = H0(X) if X is path-connected (see alsoCorollary 12.18).

Let f : Y → X be a continuous map between topological spaces. It induces amap Sf : S(Y )→ S(X) defined by Sf(σ) = f σ. The linear map C∗f : C∗(X)→C∗(Y ) is, using Definition II, defined by C∗f(a) = aS(f). As for C∗f : C∗(Y )→C∗(X), it is the linear map extending Sf , using Definition III. One checks that thecouple (C∗f, C∗f) is a morphism of Kronecker pair. It thus defines linear maps ofdegree zero H∗f : H∗(X) → H∗(Y ) and H∗f : H∗(Y ) → H∗(X). The functorialproperties are easy to prove: H∗ andH∗ are functors from the category Top of topo-logical spaces to the category GrV of graded vector spaces (see Proposition 12.33for a more general statement). Also, for any map f : Y → X , the following diagram

(12.14)

Y

p AAA

AAA

f // X

p~~

pt


is obviously commutative. This implies that the reduced cohomology H∗ and ho-mology H∗ are also functors from Top to GrV. The notations H∗f , H∗f , H∗fand H∗f are sometimes shortend in f∗ and f∗.

As in Lemma 5.7, we prove the following

Lemma 12.15. Let f : Y → X be a continuous map. Then H0f(1) = 1.

Lemma 12.15 implies the following result.

Lemma 12.16. Let (X,Y ) be a topological pair with X path-connected. Denoteby i : Y → X the inclusion. Then there are exact sequences

0→ H0(X)H∗i−−→ H0(Y )→ H0(Y )→ 0

and

0→ H0(Y )→ H0(Y )H∗i−−→ H0(X)→ 0 .

Let X be a topological space which is a disjoint union:

X =⋃

j∈JXj .

By this we mean that the above equality holds as sets and that each Xj is open(and therefore closed) in X . Denote the inclusion by ij : Xj → X .

Proposition 12.17. The family of inclusions ij : Xj → X for j ∈ J gives riseto isomorphisms

H∗(X)(H∗ij)

≈//

∏j∈J H

∗(Xj)

and⊕

j∈J H∗(Xj)P

H∗ij

≈// H∗(X) .

Proof. These isomorphisms come from the equality Sm(X) =⋃j∈J ij(Sm(Xj)).

Corollary 12.18. Let X be a topological space which is locally path-connected.Then, the family of inclusions iY : Y → X for Y ∈ π0(X) gives rise to isomorphisms

H∗(X)(H∗iY )

≈//

∏Y ∈π0(X)H

∗(Y )

and⊕

Y ∈π0(K)H∗(Y )P

H∗iY

≈// H∗(X) .

Proof. As X is locally path-connected, each Y ∈ π0(X) is open in X and X istopologically the disjoint union of its path-connected components. Corollary 12.18then follows from Proposition 12.17.

Corollary 12.19. Let X be a topological space which is locally path-connected.Then,

H0(X) = 0 ⇔ H0(X) = 0 ⇔ X is path-connected.

Also,H0(X) = H0(X)⊕ Z2 and H0(X) = H0(X)⊕ Z2

if X is not empty.


In the same spirit of reducing the computations of H∗(X) to those of smallersubspaces, another consequence of the definition of the singular (co)homology isthe following proposition.

Proposition 12.20. Let X be a topological space. Let K be the set of compactsubspaces of X, partially ordered by inclusion. Then, the natural homomorphisms

J∗ : lim−→

K∈K

H∗(K) −→ H∗(X)

and

J∗ : H∗(X) −→ lim←−

K∈K

H∗(K)

are isomorphisms.

Here, lim−→

denotes the direct limit (also called inductive limit or colimit) and lim←−

denotes the inverse limit (also called projective limit or just limit) in GrV.

Proof. Let A ∈ Hr(X), represented by α ∈ Zr(X). Then, α is a finite setof r-simplexes of X and K =

⋃σ∈α σ(∆r) is a compact subspace of X . One can

see α ∈ Zr(K), so J∗ is onto. Now, let K be a compact subspace of X andA ∈ Hr(K) mapped to 0 under Hr(K)→ Hr(X). Represent A by α ∈ Zr(K) andlet β ∈ Cr+1(X) with α = ∂(β). As before, there exists a compact subset L of Xcontaining K with β ∈ Cr+1(L), so A is mapped to 0 under Hr(K)→ Hr(L). Thisproves that J∗ is injective. Finally, the bijectivity of J∗ is deduced from that of J∗by Kronecker duality.

Remark 12.21. In Proposition 12.20 the morphism J∗ is an isomorphism forthe homology with any coefficients. The morphism J∗ is always surjective but,in general not injective (except for coefficients in a field, like Z2). Its kernel is

expressible using the derived functor lim←−

1 (see e.g. [80, Theorem 3F.8]). The same

considerations hold true for the following corollary.

Corollary 12.22. Let X be a topological space and let A be a family of sub-spaces of X, partially ordered by the inclusion. Suppose that each compact subspaceof X is contained in some A ∈ A. Then, the homomorphisms

j∗ : lim−→

A∈A

H∗(A) −→ H∗(X)

and

j∗ : H∗(X) −→ lim←−

A∈A

H∗(A)

are isomorphisms.

Proof. The hypothesis that each compact K ⊂ X is contained in some A ∈ Aimplies a factorisation of the homomorphism J∗ of Proposition 12.20:

lim−→

K∈K

H∗(K)

β

%%KKKKKKKK

J∗

≈// H∗(X)

lim−→

A∈A

H∗(A)

j∗

::uuuuuuuuu


The same hypothesis implies that β is onto, whence j∗ is an isomorphism. Theassertion for j∗ comes from Kronecker duality.

12.2. Relative singular (co)homology. A (topological) pair is a couple(X,Y ) where X is a topological space and Y is a subspace of X . The inclu-sion i : Y → X is a continuous map. Let a ∈ Cm(X). If, using Definition I.a,we consider a as a subset of Sm(X), then C∗i(a) = a ∩ Sm(Y ). If we see a as amap a : Sm(X) → Z2, then C∗i(a) is the restriction of a to Sm(Y ). We see thatC∗i : C∗(X)→ C∗(Y ) is surjective. Define

Cm(X,Y ) = ker(Cm(X)

C∗i−−→ Cm(Y ))

and C∗(X,Y ) = ⊕m∈NCm(X,Y ). As C∗i is a morphism of cochain complexes, the

coboundary δ : C∗(X)→ C∗(X) preserves C∗(X,Y ) and gives rise to a coboundaryδ : C∗(X,Y )→ C∗(X,Y ) so that (C∗(X,Y ), δ) is a cochain complex. The cocyclesZ∗(X,Y ) and the coboundaries B∗(X,Y ) are defined as usual, giving rise to thedefinition

Hm(X,Y ) = Zm(X,Y )/Bm(X,Y ) .

The graded Z2-vector space H∗(X,Y ) = ⊕m∈NHm(X,Y ) is the relative (singular)

cohomology of the pair (X,Y ). Observe that H∗(X, ∅) = H∗(X). We denote by j∗

the inclusion j∗ : C∗(X,Y ) → C∗(X), which is a morphism of cochain complexes,and use the same notation j∗ for the induced linear map j∗ : H∗(X,Y ) → H∗(X)on cohomology. We also use the notation i∗ for both C∗i and H∗i. We get thus ashort exact sequence of cochain complexes

(12.23) 0→ C∗(X,Y )j∗−→ C∗(X)

i∗−→ C∗(Y )→ 0 .

If a ∈ Cm(Y ), any cochain a ∈ Cm(X) with i∗(a) = a is called a extension ofa as a singular cochain in X . For instance, the 0-extension of a is defined bya = a ∈ Sm(Y ) ⊂ Sm(X).

With chains, the inclusion Y → X induces an inclusion i∗ : C∗(Y ) → C∗(X)of chain complexes. We define Cm(X,Y ) as the quotient vector space

Cm(X,Y ) = coker(i∗ : Cm(Y ) → Cm(X)

).

As i∗ is a morphism of chain complexes, C∗(X,Y ) = ⊕m∈NCm(X,Y ) inherits aboundary operator ∂ = ∂X,Y : C∗(X,Y )→ C∗−1(X,Y ). The projection j∗ : C∗(X)→→ C∗(X,Y ) is a morphism of chain complexes and one obtains a short exact se-quence of chain complexes

(12.24) 0→ C∗(Y )i∗−→ C∗(X)

j∗−→ C∗(X,Y )→ 0 .

The cycles and boundaries Z∗(X,Y ) and B∗(X,Y ) are defined as usual, giving riseto the definition

Hm(X,Y ) = Zm(X,Y )/Bm(X,Y ) .

The graded Z2-vector space H∗(X,Y ) = ⊕m∈NHm(X,Y ) is the relative (singular)homology of the pair (X,Y ). Observe that H∗(X, ∅) = H∗(X). The notations i∗and j∗ are also used for the induced maps in homology.

As in Sections 6 and 7 of Chapter 1, one gets a pairing 〈 , 〉 : Hm(X,Y ) ×Hm(X,Y )→ Z2 which makes (Hm(X,Y ), Hm(X,Y ), 〈 , 〉) a Kronecker pair. Also,the singular (co)homology connecting homomorphisms

δ∗ : H∗(Y )→ H∗+1(X,Y ) . and ∂∗ : H∗(X,Y )→ H∗−1(Y )


are defined and satisfy 〈δ∗(a), α〉 = 〈a, ∂∗(α)〉. The proof of the following lemma isthe same as that of Lemma 7.2.

Lemma 12.25. Let a ∈ Zm(Y ) and let a ∈ Cm(X) be any extension of a asa singular m-cochain of X. Then, δX(a) is a singular (m + 1)-cocycle of (X,Y )representing δ∗(a).

Remark 12.26. A class in A ∈ Hm(X,Y ) is represented by a relative singularcycle, i.e. a singular chain α ∈ Cm(X) such that ∂(α) is a singular chain (cycle)of Y . The homology class of ∂(α) in Hn−1(Y ) is ∂∗(A). This is the Kronecker dualstatement of Lemma 12.25.

As for the simplicial (co)homology (see § 7), the results of § 6 give the following(singular) (co)homology exact sequences of the pair (X,Y ).

Proposition 12.27 ((Co)homology exact sequences of a pair). Let (X,Y ) bea topological pair. Then, the following sequences

· · · → Hm(X,Y )j∗−→ Hm(X)

i∗−→ Hm(Y )δ∗−→ Hm+1(X,Y )

j∗−→ · · ·and

· · · → Hm(Y )i∗−→ Hm(X)

j∗−→ Hm(X,Y )∂∗−→ Hm−1(Y )

i∗−→ · · ·are exact.

These exact sequences are also available for reduced (co)homology. For this,the reduced (co)homology of a pair is defined as follows: when Y 6= ∅, then

H∗(X,Y ) = H∗(X,Y ) and H∗(X,Y ) = H∗(X,Y ); otherwise H∗(X, ∅) = H∗(X)

and H∗(X, ∅) = H∗(X).

Proposition 12.28 (Reduced (co)homology exact sequences of a pair). Theexact sequences of Proposition 12.27 hold with reduced (co)homology.

Proof. An argument is only required around m = 0. For the homology exactsequence, consider the following commutative diagram:

· · · // H1(X,Y )

=

∂∗ // H0(Y )

i∗ // H0(X)

j∗ // H0(X,Y )

=

// 0

· · · // H1(X,Y )∂∗ // H0(Y )

〈1, 〉

i∗ // H0(X)

〈1, 〉

j∗ // H0(X,Y ) // 0

Z2= // Z2

The commutativity of the bottom square is due to Lemma 12.15. As i∗∂∗ = 0,〈1, ∂(α)〉 = 0 for all α ∈ H1(X,Y ) and therefore ∂ : H1(X,Y ) → H1(Y ) exists.Since the sequence of the second line is exact, an easy chasing diagram argumentshows that the sequence of the first line is exact as well.

The reduced cohomology exact sequence can be established in an analogous wayor deduced from the homology one by Kronecker duality, using Lemma 3.12.

Remark 12.29. Let (X,Y ) be a topological pair with Y path-connected andnon-empty. By Proposition 12.28 and its proof, we get the isomorphisms

(12.30) j∗ : H0(X)≈−→ H0(X,Y ) and j∗ : H0(X,Y )

≈−→ H0(X) .


Also, if Y = x, we get the isomorphisms

(12.31) j∗ : H∗(X)≈−→ H∗(X,x) and j∗ : H∗(X,x)

≈−→ H∗(X) .

A direct proof of (12.31), say for cohomology, is given by the following diagram.

(12.32)

H∗(pt)

p∗

≈

$$JJJJ

JJJJ

H∗(X,x)j∗ //

≈

%%KKKKKKKKH∗(X)

i∗ //

H∗(x)

H∗(X)

where the line and the column are exact and p : X → pt is the constant map onto apoint. We see that the choice of x ∈ X produces a supplementary vector subspaceto p∗(H∗(pt)) in H∗(X).

We now study the naturality of the relative (co)homology and of the exactsequences. Let (X,Y ) and (X ′, Y ′) be topological pairs. A map f of (topological)pairs from (X,Y ) to (X ′, Y ′) is a continuous map f : X → X ′ such f(Y ) ⊂ Y ′.With these maps, topological pairs constitute a category Top2. The correspondenceX 7→ (X, ∅) makes Top a full subcategory of Top2.

Let f : (X,Y ) → (X ′, Y ′) be a map of topological pairs. The morphismC∗f : C∗(X ′)→ C∗(X) then restricts to a morphism of cochain complexes C∗f : C∗(X ′, Y ′)→C∗(X,Y ) and the morphism C∗f : C∗(X) → C∗(X

′) descends to a morphism ofchain complexes C∗f : C∗(X,Y )→ C∗(X

′, Y ′). As in § 7, we prove that (C∗f, C∗f)is a morphism of Kronecker pair. One then gets degree zero linear mapsH∗f : H∗(X ′, Y ′)→H∗(X,Y ) and H∗f : H∗(X,Y )→ H∗(X

′, Y ′) satisfying 〈H∗a, α〉 = 〈a,H∗α〉 for alla ∈ Hm(X ′, Y ′), α ∈ Hm(X,Y ) and all m ∈ N. Functorial properties are easy, sowe get the following lemma.

Proposition 12.33. The relative singular cohomology H∗( , ) is a contravariantfunctor from the category Top2 to the category GrV of graded Z2-vector spaces.The relative singular homology H∗( , ) is a covariant functor between these cate-gories. The same holds true for the reduced singular (co)homology.

As for Proposition 7.11, we can prove the following

Proposition 12.34. The (co)homology exact sequences are natural with respectto maps of topological pairs.

Here below, a special form of the cohomology exact sequence of a pair.

Proposition 12.35. Let A and B be topological spaces. Then the cohomologyexact sequence of the pair (A∪B,A) cuts into short exact sequences and there is acommutative diagram

(12.36)

0 // H∗(B) //

≈

H∗(A∪B) //

id

H∗(A) //

id

0

0 // H∗(A∪B,A) // H∗(A∪B) // H∗(A) // 0

.


Proof. If iA : A → A∪B and iB : B → A∪B denote the inclusions, Proposi-tion 12.17 provides a commutative diagram

H∗(A∪B)(i∗A,i

∗B)

≈//

i∗A %%LLLLLLLLH∗(A)×H∗(B)

proj1wwoooooooooo

H∗(A)

.

This proves that i∗a is surjective, which cuts the the cohomology exact sequence of(A∪B,A), giving the bottom line of (12.36). Also, ker i∗A is the image of H∗(B)under the monomorphism j : H∗(B) → H∗A∪B) given by j(u) = (i∗A, i

∗B)−1(0, u),

which we placed in the top line of (12.36).

As in simplicial (co)homology, the exact sequences of a pair generalize to that ofa triple. A (topological) triple is a triplet (X,Y, Z) where X is a topological spacesand Y , Z are subspaces of X with Z ⊂ Y . A map f of triples, from (X,Y, Z) to(X ′, Y ′, Z ′) is a continuous map f : X → X ′ such that f(Y ) ⊂ Y ′ and f(Z) ⊂ Z ′.

A triple T = (X,Y, Z) gives rise to pair inclusions

(Y, Z)i−→ (X,Z)

j−→ (X,Y )

and to a commutative diagram

(12.37)

0 // C∗(X,Y )

C∗j

j∗X,Y // C∗(X)

id=

i∗X,Y // C∗(Y )

i∗Y,Z

// 0

0 // C∗(X,Z)j∗X,Z // C∗(X)

i∗X,Z // C∗(Z) // 0

where the horizontal lines are exact sequences of cochain complexes As in (7.12),we get a short exact sequence of cochain complexes

(12.38) 0→ C∗(X,Y )C∗j−−→ C∗(X,Z)

C∗i−−→ C∗(Y, Z)→ 0 .

The same arguments with the chain complexes gives a short exact sequence

(12.39) 0→ C∗(Y, Z)C∗i−−→ C∗(X,Z)

C∗j−−→ C∗(X,Y )→ 0 .

As in § 7 of Chapter 1, short exact sequences (12.38) and (12.39) produces connect-ing homomorphisms δT : H∗(Y, Z)→ H∗+1(X,Y ) and ∂T : H∗(X,Y )→ C∗−1(Y, Z).They satisfy 〈δT (a), α〉 = 〈a, ∂T (α)〉 as well as following proposition.

Proposition 12.40 ((Co)homology exact sequences of a triple). Let T =(X,Y, Z) be a triple. Then,

(a) the following sequences

· · · → Hm(X,Y )H∗j−−−→ Hm(X,Z)

H∗i−−→ Hm(Y, Z)δT−→ Hm+1(X,Y )

H∗j−−−→ · · ·and

· · · → Hm(Y, Z)H∗i−−→ Hm(X,Z)

H∗j−−→ Hm(X,Y )∂T−−→ Hm−1(Y, Z)

H∗i−−→ · · ·are exact.

(b) the exact sequences of Point (a) are natural for maps of triples.


Remark 12.41. AsH∗(∅) = 0, we get a canonical GrV-isomorphismsH∗(X, ∅) ≈−→H∗(X), etc. Thus, the (co)homology exact sequences for the triple (X,Y, ∅) giveback those of the pair (X,Y )

(12.42) · · · → Hm(X,Y )H∗j−−−→ Hm(X)

H∗i−−→ Hm(Y )δ∗−→ Hm+1(X,Y )

H∗j−−−→ · · ·and

(12.43) · · · → Hm(Y )H∗i−−→ Hm(X)

H∗j−−→ Hm(X,Y )∂∗−→ Hm−1(Y )

H∗i−−→ · · ·where i : Y → X and j : (X, ∅)→ (X,Y ) denote the inclusions. This gives a moreprecise description of the morphisms j∗ and j∗ of Proposition 12.27.

We now draw a few consequences of Proposition 12.27. A topological pair(X,Y ) is of finite (co)-homology type if its singular homology (or, equivalently,cohomology) is of finite type. In this case, the Poincare series of (X,Y ) is that ofH∗(X,Y ):

Pt(X,Y ) =∑

i∈N

dimHi(X,Y ) ti =∑

i∈N

dimHi(X,Y ) ti ∈ N[[t]].

Corollary 12.44. Let (X,Y, Z) be a topological triple. Suppose that two ofthe pairs (X,Y ), (Y, Z) and (X,Z) are of finite cohomology type. Then, the thirdpair is of finite cohomology type and there is Qt ∈ N[[t]] such that the followingequality

(12.45) Pt(X,Y ) + Pt(Y, Z) = Pt(X,Z) + (1 + t)Qt ,

holds in N[[t]].

Proof. This follows from the cohomology exact sequence of T = (X,Y, Z) andelementary linear algebra. If δk

T: Hk(Y, Z) → Hk+1(X,Y ) denotes the connecting

homomorphism, one checks that (12.45) holds true for

Qt =∑

tk codim δkT.

Corollary 12.44 implies straightforwardly the following result.

Corollary 12.46. Let (X,Y, Z) be a topological triple. Suppose that dimH∗(Y, Z) <∞ and that dimH∗(X,Y ) <∞. Then dimH∗(X,Z) <∞ and

dimH∗(X,Z) ≤ dimH∗(X,Y ) + dimH∗(Y, Z) .

Corollary 12.22 has the following generalization with relative (co)-homology.

Proposition 12.47. Let (X,Y ) be a topological pair. Let A be family of sub-spaces of X, partially ordered by inclusion. Suppose that each compact subspace ofX is contained in some A ∈ A. Then, the natural homomorphisms

J∗ : lim−→

A∈A

H∗(A,A ∩ Y )≈−→ H∗(X,Y )

andJ∗ : H∗(X,Y )

≈−→ lim←−

A∈A

H∗(A,A ∩ Y )

are isomorphisms.


Proof. By Kronecker duality, only the bijectivity of J∗ must be proven. LetHr(Y ) = lim

−→A∈A

Hr(A∩ Y ), Hr(X) = lim−→

A∈A

Hr(A), and Hr(X,Y ) = lim−→

A∈A

Hr(A,A∩ Y ).

For each A ∈ A, one has the homology exact sequence of the pair (A,A ∩ Y ). Bynaturality of these exact sequences under inclusions, one gets the following diagram:

Hr(Y )

≈

// Hr(X)

≈

// Hr(X,Y )

J∗

∂ // Hr−1(Y )

≈

// Hr−1(X)

≈

Hr(Y ) // Hr(X) // Hr(X,Y )∂ // Hr−1(Y ) // Hr−1(X)

The top horizontal line is exact because the direct limit of exact sequences is exact.The bijectivity of the vertical arrows comes from Corollary 12.22. By the five-lemma, one deduces that J∗ is an isomorphism.

12.3. The homotopy property. Let f, g : (X,Y ) → (X ′, Y ′) be two mapsbetween topological pairs. Let I = [0, 1]. A homotopy between f and g is a map ofpairs F : (X × I, Y × I) → (X ′, Y ′) such that F (x, 0) = f(x) and F (x, 1) = g(x).If such a homotopy exists, we say that f and g are homotopic.

Proposition 12.48 (Homotopy property). Let f, g : (X,Y )→ (X ′, Y ′) be twomaps between topological pairs which are homotopic. Then H∗f = H∗g and H∗f =H∗g.

Proof. Note that H∗f = H∗g implies H∗f = H∗g by Kronecker duality, usingDiagram (3.9). We shall construct a Z2-linear map D : C∗(X) → C∗+1(X

′) suchthat

(12.49) C∗f + C∗g = ∂D +D∂ ,

i.e. D is a chain homotopy from C∗f to C∗g. The map D will satisfy D(C∗(Y )) ⊂C∗+1(Y

′) and will so induce a linear map D : C∗(X,Y ) → C∗+1(X′, Y ′) satisfy-

ing (12.49). As in the proof of Proposition 5.14, this will prove that H∗f = H∗g.That H∗f = H∗g is then deduced by Kronecker duality, using Diagram (3.9). LetF : (X × I, Y × I)→ (X ′, Y ′) be a homotopy from f to g.

By linearity, it is enough to define D on singular simplexes. Let σ : ∆m → X bea singular m-simplex of X . Consider the convex-cell complex P = ∆m×I. One hasV (P ) = V (∆m)×0, 1. Using the natural total order on V (∆m), we can define anaffine order on P by deciding that the elements of V (∆m) × 1 are greater than

those of V (∆m)×0. Lemma 1.11 thus provides a triangulation h≤ : |L≤(P )| ≈−→ P ,with V (L≤(P )) = V (P ). Set L = L≤(P ) and h = h≤. The order ≤ becomes asimplicial order on L, giving rise to a chain map R≤,∗ : C∗(L) → C∗(P ) from thesimplicial chains of L to the singular chains of P (see Example 12.8). ConsiderSm+1(L) as an (m+ 1)-simplicial cochain of L and define D(σ) to be the image ofSm+1(L) under the composed map(12.50)

Sm+1(L)R≤,∗−−−→ Sm+1(|L|) C∗h−−→ Sm+1(P )

C∗(σ×id)−−−−−−→ Sm+1(X × I) C∗F−−−→ Sm+1(X′) .

Observe that, if τ ∈ Sm(L) such that h(|τ |) hits the interior of P , then τ is theface of exactly two (m+1)-simplexes of L. Therefore, ∂(Sm+1(L)) = Sm(L(BdP )).But

BdP = ∆m × 0 ∪∆m × 1 ∪ Bd∆m × I .


As all the maps in (12.50) are chain maps, this permits us to prove that

(12.51) ∂D(σ) = C∗f(σ) + C∗g(σ) +D∂(σ)

As (12.51) holds true for all σ ∈ S(X), it implies (12.49).

Remark 12.52. In the proof of Proposition 12.48, the chain homotopy D is notunique. Some authors (e.g. [175, 41, 152]) just give an existence proof, based onan easy case of the acyclic carrier’s technique (like in our proof of Proposition 5.14).We used above an explicit triangulation of ∆m × I. The same triangulation occursthe proof of [80, p. 112], presented differently for the sake of sign’s control. Theidea of such triangulations of ∆m × I will be used again in the proof of the smallsimplex theorem 12.56.

A map of pairs f : (X,Y )→ (X ′, Y ′) is a homotopy equivalence if there exists amap of pairs g : (X ′, Y ′)→ (X,Y ) such that gf is homotopic to id(X,Y ) and f gis homotopic to id(X′,Y ′). The pairs (X,Y ) and (X ′, Y ′) are then called homotopyequivalent. Two spaces X and X ′ are homotopy equivalent if the pairs (X, ∅) and(X ′, ∅) are homotopy equivalent. Two homotopy equivalent spaces (or pairs) arealso said to have the same homotopy type.

By functoriality (Proposition 12.33), Proposition 12.48 implies that (co)homologyis an invariant of homotopy type:

Corollary 12.53 (Homotopy invariance of (co)homology). Let f : (X,Y ) →(X ′, Y ′) be a homotopy equivalence. Then H∗f : H∗(X,Y ) → H∗(X

′, Y ′) andH∗f : H∗(X ′, Y ′)→ H∗(X,Y ) are isomorphisms.

A (non-empty) topological space X is contractible if there exists a homotopyfrom idX to a constant map. For instance, the cone CX over a space X

(12.54) CX =(X × I

)/(X × 1

),

with the quotient topology, is contractible. A homotopy from idCX to a constantmap is given by F ((x, τ), t) = [x, t+ (1− t)τ ].

Corollary 12.55. The (co)homology of a contractible space is isomorphic tothat of a point:

H∗(X) ≈ H∗(X) ≈

0 if ∗ > 0

Z2 if ∗ = 0 .

Proof. Let x0 ∈ X such that there exists a homotopy from idX to the con-stant map onto x0 Then, the inclusion x0 → X is a homotopy equivalence andCorollary 12.55 follows from Corollary 12.53.

For a direct proof of Corollary 12.55, see Exercise 2.2.

12.4. Excision. Let X be a topological space. Let B be a family of subspacesof X . A map f : L → X is called B-small if f(L) is contained in an element ofB. Let SBm(X) be the set of singular m-simplexes of X which are B-small. Thevector spaces of (co)chains CmB (X) and CBm(X) are defined as in § 12, using B-small m-simplexes. We get, in the same way, a pairing 〈 , 〉 : CmB (X) × CBm(X) →Z2 identifying CmB (X) to CBm(X)♯. The boundary of a B-small simplex is a B-small chain, so (CB∗ (X), ∂) is a subcomplex of chains of (C∗(X), ∂), the inclu-sion being denoted by iB∗ : CmB (X) → Cm(X). Define δ : CmB (X) → Cm+1

B (X) by


〈δ(a), α〉 = 〈a, ∂(α)〉 and i∗BCm(X) → CmB (X) by 〈i∗B(a), α〉 = 〈a, iB∗ (α)〉. Then,

((C∗B(X), δ), (CB∗ (X), ∂), 〈 , 〉) is a Kronecker pair.The (co)homologies obtained by these definitions are denoted by H∗B(X) and

HB∗ (X). One uses the notations iB∗ : HBm(X)→ Hm(X) and i∗B : Hm(X)→ HmB (X)

for the induced linear maps.

Proposition 12.56 (Small simplexes theorem). Let X be a topological spacewith a family B of subspaces of X, whose interiors cover X. Then iB∗ : HB∗ (X) →H∗(X) and i∗B : H∗(X)→ H∗B(X) are isomorphisms.

The proof of Proposition 12.56 uses iterations of the subdivision operator,a chain map sd ∗ : C∗(X,Y ) → C∗(X,Y ) which replaces chains by chains with”smaller” simplexes. Intuitively, sd ∗ replaces a singular simplex σ : ∆m → X bythe sum of σ restricted to the barycentric subdivision of ∆m.

More precisely, consider the standard simplex ∆m as the geometric realizationof the full complex Fm over the set 0, 1, . . . ,m. The barycentric subdivision F ′mis endowed with its natural simplicial order ≤ of (1.15), p. 14. As explained inExample 12.8, we get a chain map

R∗ = R≤,∗ : C∗(F ′m)→ C∗(|F ′m|) = C∗(∆m) .

Let σ ∈ Sm(X). As a continuous map from ∆m to X , σ induces C∗σ : C∗(∆m)→

C∗(X). Define

sd ∗(σ) = C∗σ(Sm(F ′m)) .

This formula determines a unique linear map sd ∗ : C∗(X)→ C∗(X) which is clearlya chain map. If Y is a subspace of X , then sd ∗(C∗(Y )) ⊂ C∗(X), so we get a chainmap sd ∗ : C∗(X,Y )→ C∗(X,Y ), giving rise to a GrV-morphism

sd ∗ : H∗(X,Y )→ H∗(X,Y ) .

By Kronecker duality, we get a cochain map sd ∗ : C∗(X,Y ) → C∗(X,Y ) and aGrV-morphism sd ∗ : H∗(X,Y ) → H∗(X,Y ) satisfying 〈sd ∗(a), α〉 = 〈a, sd ∗(α)〉for all a ∈ C∗(X,Y ) and α ∈ C∗(X,Y ).

Observe that sd sends CB∗ (X,Y ) into CB∗ (X,Y ) and thusHB∗ (X,Y ) intoHB∗ (X,Y )and H∗B(X,Y ) into H∗B(X,Y )

Lemma 12.57. The subdivision operators sd ∗ : H∗(X,Y )→ H∗(X,Y ) andsd ∗ : H∗(X,Y ) → H∗(X,Y ) are equal to the identity. The same holds true forsd ∗ : H

B∗ (X,Y )→ HB∗ (X,Y ) and sd ∗ : H∗B(X,Y )→ H∗B(X,Y ).

Proof. We shall construct a Z2-linear map D : C∗(X)→ C∗+1(X) such that

(12.58) id + sd ∗ = ∂D +D∂ .

In other words, D is a chain homotopy from id to sd ∗ (see p. 32). The map Dwill satisfy D(C∗(Y )) ⊂ C∗+1(Y ) and will so induce a linear map D : C∗(X,Y )→C∗+1(X,Y ) satisfying (12.58). As in the proof of Proposition 5.14, this will provethat sd ∗ = id. That sd ∗ = id is then implied by Kronecker duality, using Dia-gram (3.9). Also, the map D will satisfy D(CB∗ (X)) ⊂ CB∗+1(X).

By linearity, it is enough to define D on singular simplexes. The proof issimilar to that of Proposition 12.48 (an idea of V. Puppe). Let σ : ∆m → X be asingular m-simplex of X . Consider the convex-cell complex P = ∆m × I, wherethe upper face ∆m × 1 is replaced by its barycentric subdivision |F ′m|. One hasV (P ) = V (∆m)×0 ∪V (F ′m)×1. We use the natural total order on V (∆m)×0


and the natural simplicial order on V (F ′m)×1 (see (1.15)). Deciding in additionthat the elements of V (F ′m)×1 are greater than those of V (∆m)×0 provides an

affine order≤ on P . Lemma 1.11 thus constructs a triangulation h≤ : |L≤(P )| ≈−→ P ,with V (L≤(P )) = V (P ). Set L = L≤(P ) and h = h≤. Seeing ≤ as a simplicialorder on L gives rise to a chain map R≤,∗ : C∗(L) → C∗(P ) from the simplicialchains of L to the singular chains of P (see Example 12.8). Consider Sm+1(L) asan (m + 1)-simplicial cochain of L and define D(σ) to be the image of Sm+1(L)under the composed map

(12.59) Sm+1(L)R≤,∗−−−→ Sm+1(|L|) C∗h−−→ Sm+1(P )

C∗(σ)−−−−→ Sm+1(X) ,

where σ : P → X is the map σ(x, t) = σ(x). Observe that the inclusion |F ′m| ⊂∆m × 1 is already the piecewise affine triangulation of ∆m × 1 determined bythe simplicial order on F ′m. Therefore, the construction of the proof of Lemma 1.11leaves ∆m × 1 unchanged. Formula (12.58) is then deduced as in the proof ofProposition 12.48. Finally, if σ is B-small, so is the map σ. HenceD(σ) ∈ CBm+1(X),which proves the lemma for the B-small (co)homology.

Proof of Proposition 12.56. By Kronecker duality, using Corollary 3.17,only the homology statement must be proved. Let sd k = sd · · · sd (k times).We shall need the following statement.

Claim: let α ∈ C∗(X). Then, there exists k(α) ∈ N such that sd k(α) ∈ CB∗ (X)for all k ≥ k(α).

Let us show that the claim implies Proposition 12.56. Let α ∈ Hm(X,Y )represented by α ∈ Cm(X) with ∂(α) ∈ Cm−1(Y ). The claim implies that, for k

big enough, sd k(α) ∈ CBm(X), and thus ∂(α) ∈ Cm−1(Y ). This implies that sd k(α)

is in the image of iB∗ . By Lemma 12.57, sd k(α) = α, so α is in the image of iB∗ , whichproves that iB∗ is surjective. For the injectivity, let β ∈ HBm(X,Y ) with iB∗ (β) = 0.Represent β by β ∈ CBm(X) with ∂(β) ∈ Cm−1(Y ). The hypothesis iB∗ (β) = 0 saysthat β = ∂(γ) + ω with γ ∈ Cm+1(X) and ω ∈ Cm(Y ). The claim tells us that, for

k big enough, sd k(γ) ∈ CBm+1(X) (and, so, sd k(ω) ∈ CBm(Y )). This implies that

sd k(β) = 0 in HBm(X,Y ). But sd k(β) ∈ CBm(X) and Lemma 12.57 tells us that

sd k coincides with the identity of HBm(X,Y ). Thus, β = 0 for all β ∈ ker iB∗ .It remains to prove the claim. Let ρ(m, k) be the maximal distance between

two points of a simplex of the k-th barycentric subdivision of ∆m. An elementaryargument of Euclidean geometry shows that

(12.60) ρ(m, k) ≤ ρ(m, 0)

(m

m+ 1

)k

(of course, ρ(m, 0) =√

2). For details, see e.g., [152, Proof of Thm 15.4] or [80,

p. 120]. By hypothesis, the family B = intB | B ∈ B is an open covering of X .

Consider the induced open covering σ−1B of ∆m. By (12.60), ρ(m, k) → 0 when

k → ∞. Using a Lebesgue number for the open covering σ−1B, this proves theclaim.

The main application of the small simplexes theorem is the invariance underexcision (see also § 12.6).


Proposition 12.61 (Excision property). Let (X,Y ) be a topological pair. LetU be a subspace of X with U ⊂ intY . Then, the linear maps induced by inclusions

i∗ : H∗(X,Y )≈−→ H∗(X − U, Y − U) and i∗ : H∗(X − U, Y − U)

≈−→ H∗(X,Y )

are isomorphisms.

Proof. By Corollary 3.17, i∗ is an isomorphism if and only if i∗ is an isomor-phism. We shall prove that i∗ is an isomorphism.

Let B = Y,X − U. One has a commutative diagram

0 // C∗(Y )

id=

// CB∗ (X)

iB∗

// CB∗ (X)/C∗(Y )

IB∗

// 0

0 // C∗(Y ) // C∗(X) // C∗(X)/C∗(Y ) // 0

where all arrows are induced by inclusions and the horizontal lines are short exactsequences of chain complexes. As in § 6, this gives a commutative diagram betweenthe corresponding long homology exact sequences

. . . // Hm(Y )

id=

// HBm(X)

iB∗

// Hm(CB∗ (X)/C∗(Y ))

IB∗

// Hm−1(Y )

id=

// . . .

. . . // Hm(Y ) // Hm(X) // Hm(X,Y ) // Hm−1(Y ) // . . .

As U ⊂ intY , the family B = Y,X−U satisfy the hypotheses of Proposition 12.56and iB∗ is an isomorphism. By the five-lemma, IB∗ is an isomorphism. Therefore, itsuffices to show that H∗(X − U, Y − U)→ H∗(C

B∗ (X)/C∗(Y )) is an isomorphism.

But it is easy to see that this is already the case at the chain level:

C∗(X − U, Y − U) = C∗(X − U)/C∗(Y − U)≈−→ CB∗ (X)/C∗(Y ) .

12.5. Well cofibrant pairs. Let (Z, Y ) be a topological pair and denote byi : Z → Y the inclusion. A (continuous) map r : Z → Y is called a retraction ifri = i. It is a retraction by deformation if ir is homotopic to the identity of Z.A retraction by deformation is thus a homotopy equivalence.

Note that Z retracts by deformation on Y if and only if there is a homotopyh : Z × I → Z which, for all (z, t) ∈ Z × I, satisfies h(z, 0) = z, h(z, 1) ∈ Y andh(y, t) = y when y ∈ Y . A topological pair (X,A) is called good if A is closed in Xand if there is a neighbourhood V of A which retracts by deformation onto A. Forinstance, (X, ∅) is a good pair (V = ∅ and h(x, t) = x).

Good pairs were introduced in [80] (with the additional condition that A isnon-empty). Earlier books rather rely on the notion of cofibration, developed inthe 1960’s essentially by D. Puppe and N. Steenrod (see [181] for references). Bothare useful in different circumstances, so we introduce below the mixed notion of awell cofibrant pair, specialy useful in the equivariant setting. We begin by cofibrantpairs, starting with the following lemma.

Lemma 12.62. For a topological pair (X,A), the following conditions are equiv-alent.

(1) There is a retraction from X × I onto X × 0 ∪A× I.


(2) Let f : X → Z and FA : A × I → Z be continuous maps such thatFA(a, 0) = f(a). Then, FA extends to a continuous map F : X × I → Zsuch that F (x, 0) = f(x) for all x ∈ X.

Proof. We give below the easier proof available when A is closed in X (for aproof without this hypothesis: see [37, (1.19)]). Let r : X× I → X×0∪A× I bea retraction. Given f and FA as in (2), define the map g : X ×0∪A× I → Z byg(x, 0) = f(x) and g(a, t) = FA(a, t). If A is closed, then g is continuous and themap F = gr satisfies the required condition. Hence, (1) implies (2). Conversely, iff and FA are the inclusions of X and A× I into Z = X×0∪A× I, the extensionF given by (2) is a continuous retraction from X × I onto Z.

A pair (X,A) with A closed in X which satisfies (1) or (2) of Lemma 12.62 iscalled cofibrant. According to the literature, the inclusion A → X is a cofibration,or satisfies the absolute homotopy extension property (AHEP) (see e.g. [42, 71]).See e.g. [36, Chapter 5] for other characterisations and properties of cofibrant pairs.

As a motivation of our concept of well cofibrant pair, we first give an example.

Example 12.63. Mapping cylinder neighbourhoods. Let (X,A) be a topologicalpair. A neighbourhood V of A is called a mapping cylinder neighbourhood if there isa continuous map ϕ : V → A (where V is the frontier of V ) and a homeomorphismψ : Mϕ → V where

Mϕ = [(V ×I) ∪A]/(x, 0) ∼ ϕ(x) | x ∈ V

is the mapping cylinder of ϕ. The homeomorphism ψ is required to satisfy ψ(x, 1) =

x and ψ(x, 0) = ϕ(x) for all x ∈ V . Here are examples of mapping cylinderneighbourhoods

• if X is a smooth manifold and A a smooth submanifold of codimension≥ 1, then a closed tubular neighbourhood of A [93, § 4.6] is a mappingcylinder neighbourhood.• if A is the boundary of a smooth manifold X , then a collar neighbourhood

of A [93, § 4.6] is a mapping cylinder neighbourhood.• a subcomplex of a CW-complex admits a mapping cylinder neighbour-

hood. The proof of this will be given in Lemma 15.3.

Given a mapping cylinder neighbourhood as above, a retraction F : X × I →X × 0 ∪A× I is defined by

F (x, t) =

(ϕ(v), t(1 − 2τ)) if x = ψ(v, τ) with τ ≤ 1/2.

ψ(v, 2τ − 1) if x = ψ(v, τ) with τ ≥ 1/2.

x if x ∈ X − intV .

Let u : X → I defined by

u(x) =

2τ if x = ψ(v, τ) with τ ≤ 1/2.

1 otherwise.

Let h : X × I → X defined by h = pX F , where pX : X × I → X is the projection.Then, u(h(x)) ≤ u(x) which implies that, for all T < 1, h restricts to a strongdeformation retraction form u−1([0, T ]) onto A = u−1(0). Hence, (X,A) is a goodand cofibrant pair.


A topological pair (X,A) is well cofibrant if there exists continuous mapsu : X → I and h : X × I → X such that

(1) A = u−1(0) (in particular, A is closed in X).(2) h(x, 0) = x for all x ∈ X .(3) h(a, t) = a for all (a, t) ∈ A× I.(4) h(x, 1) ∈ A for all x ∈ X such that u(x) < 1.(5) u(h(x, t)) ≤ u(x) for all (x, t) ∈ X × I.

We say that (u, h) is a presentation of (X,A) as a well cofibrant pair. Condi-tions (1)-(4) define a NDR-pair (neighbourhood deformation retract pair) in thesense of [181, 36, 137] (see also Remark 12.68 (b) below).

The pairs (X, ∅) and (X,X) are well cofibrant. One takes u(x) = 1 for (X, ∅),u(x) = 0 for (X,X) and h(x, t) = x for both pairs. Another basic example of wellcofibrant pairs is given by the following lemma.

Lemma 12.64. Suppose that A ⊂ X admits a mapping cylinder neighbourhoodin X. Then, (X,A) is well cofibrant.

Proof. The pair (u, h) in Example 12.63 is a presentation of (X,A) as a wellcofibrant pair.

Lemma 12.65. Let (X,A) and (Y,B) be two well cofibrant pairs. Then, the“product pair” (X × Y,A× Y ∪X ×B) is well cofibrant.

The following proof, coming from that of [181, Theorem 6.3], will be convenientfor the equivariant setting (see Lemma 40.23).

Proof. Let (u, h) and (v, j) be presentations of (X,A) and (Y,B) as wellcofibrant pairs. Define w : X×Y → I by w(x, y) = u(x)v(y). Define q : X×Y ×I →X × Y by

q(x, y, t) =

(x, y) if (x, y) ∈ A×B.(h(x, t), j(y, u(x)

v(y) t))

if v(y) ≥ u(x) and v(y) > 0.(h(x, v(y)u(x) t), j(y, t)

)if v(y) ≤ u(x) and u(x) > 0.

One checks that (w, q) is a presentation of (X×Y,A×Y ∪X×B) as a well cofibrantpair. Details for (1)-(4) are given in [181, p. 144] and (5) is obvious.

Lemma 12.66. Let (X,A) be a well cofibrant pair. Then, (X,A) is good andcofibrant.

Proof. Let (u, h) be a presentation of (X,A) as a well cofibrant pair. Asnoticed in Example 12.63, the condition u(h(x, t)) ≤ u(x) implies that, for allT < 1, h restricts to a strong deformation retraction form u−1([0, T ]) onto A. SinceA = u−1(0), it is closed. Hence, (X,A) is good. To see that (X,A) is cofibrant, let(Y,B) = (I, 0) presented as well cofibrant pair by (v, j) where v(y) = y/2 andj(y, t) = (1 − t)y. Let (w, q) be the presentation of

(X × Y,A× Y ∪X ×B) = (X × I,X × 0 ∪A× I)as a well cofibrant pair given in the proof of Lemma 12.65. As

w(x, y) = u(x)y/2 < 1 ,

the formula r(x, y) = q(x, y, 1) defines a retraction

(12.67) X × I r−→ X × 0 ∪A× I .


By Lemma 12.62, (X,A) is cofibrant.

Remarks 12.68. (a) The fact that the retraction r of (12.67) is a strong defor-mation retraction should not be a surprise. If r = (r1, r2) : X×I → X×0∪A×I ⊂X × I is any retraction, then the map R : X × I → X × I defined by

R(x, t, s) =(r1(x, (1− s)t), st+ (1− s)r2(x, t)

)

is a homotopy from idX×I to r [71, Lemma 16.28].

(b) The proof of Lemma 12.66 shows that a NDR-pair is cofibrant. The converseis also true (see [137, § 6.4]).

If (X,A) is a topological pair, we denote by X/A the quotient space where allpoints of A are identified in a single class. The projection π : (X,A)→ (X/A,A/A)is a map of pairs.

Lemma 12.69. Let (X,A) be a well cofibrant pair and let B ⊂ A. Then(X/B,A/B) is well cofibrant. In particular, the pair (X/A,A/A) is well cofibrant.

Proof. Let (u, h) be a presentation of (X,A) as a well cofibrant pair. By (1)and (3), u and h descend to continuous maps u : X/B → I and h : (X/B) × I →X/B, giving a presentation (u, h) of (X/B,A/B).

Lemma 12.70. Let (X,A) be a cofibrant pair such that A is contractible. Thenthe quotient map X → X/A is a homotopy equivalence.

Proof. If A is contractible, there is a continuous map FA : A × I → A ⊂ Xsuch that FA(a, 0) = a and F (A × 1) = a0. As (X,A) is cofibrant, there is acontinuous map F : X × I → X such that F (x, 0) = x and F (a, t) = FA(a, t) fora ∈ A. F|X×1 admits a factorisation

X≈ //

q)) ))TTTTTTTTTTTTTTTTTTT X × 1

$$ $$IIII

IIII

F|X×1 // X

X/A

g

==

Using F , gq is homotopic to idX . On the other hand, as F (A× I) ⊂ A, the mapqF descends to a continuous map F : X/A× I → X/A which is a homotopy fromidX/A to qg.

Proposition 12.71. Let (X,A) be a well cofibrant pair. Then, the homomor-phisms

π∗ : H∗(X/A,A/A) −→ H∗(X,A) and π∗ : H∗(X,A) −→ H∗(X/A,A/A)

are isomorphisms.

Proof. By Corollary 3.17, π∗ is an isomorphism if and only if π∗ is an isomor-phism. We shall prove that π∗ is an isomorphism (the proof is the same for both).There is nothing to prove if A = ∅, so we assume that A is not empty.

By Lemma 12.66, (X,A) is cofibrant. Let r be a retraction from X × I toX × 0 ∪ A × I. Let CA =

(A × I

)/(A × 1

)be the cone over A and let

X = X ∪A CA. As A is closed, r extends (by the identity on CA × I) to a


continuous retraction from X × I onto X ×0∪CA× I. Hence, the pair (X, CA)is cofibrant. But

H∗(X, CA)≈

excision// H∗(X − [A× 1], CA− [A× 1]) ≈

homotopy// H∗(X,A) .

On the other hand, X/A = X/CA. Set X = X/CA and C = CA/CA ≈ pt. The

quotient map q : (X, CA)→ (X, C) provides a morphism of exact sequences

Hk−1(X) //

q∗≈

Hk−1(C) //

q∗≈

Hk(X, C) //

q∗

Hk(X) //

q∗≈

Hk(C)

q∗≈

Hk−1(X) // Hk−1(CA) // Hk(X, CA) // Hk(X) // Hk(CA)

.

As C and CA are contractible, q∗ : H∗(C) → H∗(CA) is an isomorphism and so

is q∗ : H∗(X) → H∗(X) by Lemma 12.70. By the five lemma, q∗ : Hk(X, C) →Hk(X, CA) is an isomorphism, which proves Proposition 12.71

Remark 12.72. The proof of Proposition 12.71 uses only that the pair (X,A)is cofibrant. Another proof exists using that (X,A) is a good pair (see [80, Propo-sition 2.22] or Proposition 40.26). It is interesting to note that these relativelyshort proofs both use almost all the axioms of a cohomology theory (see § 20):functoriality, homotopy, excision and functorial exactness.

Corollary 12.73. Let (X,A) be a well cofibrant pair with A non-empty. Then,

π∗ : H∗(X/A)≈−→ H∗(X,A) and π∗ : H∗(X,A)

≈−→ H∗(X/A)

are isomorphisms.

Proof. If A 6= ∅, then A/A is a point. Therefore, by (12.31), H∗(X/A)≈−→

H∗(X/A,A/A) and H∗(X/A,A/A)≈−→ H∗(X/A). The results then follows form

Proposition 12.71

Corollary 12.74. Let (X,A) be a well cofibrant pair. Denote by i : A → Xthe inclusion and by j : X → X/A the quotient map. Then, there is a functorialexact sequence in reduced cohomology

· · · → Hk−1(X)H∗i−−→ Hk−1(A)

δ∗−→ Hk(X/A)H∗j−−−→ Hk(X)

H∗i−−→ Hk(A)→ · · ·The corresponding sequence exists in reduced homology.

Proof. The result is obvious if A is empty. Otherwise, this comes from the

exact sequence of Proposition 12.28 together with the isomorphism H∗(X/A)≈−→

H∗(X,A)≈−→ H∗(X,A) of Corollary 12.73.

One application of well cofibrant pairs is the suspension isomorphism. Let Xbe a topological space. The suspension ΣX of X is the quotient space

ΣX = CX/(X × 0

)

where CX is the cone over X (see (12.54)). The pairs (CX,X×0) and (ΣX,X× 12 )

are well cofibrant by Lemma 12.64, since the suspaces admits mapping cylinder


neighbourhoods. The (cohomology) suspension homomorphism is the degree-1 lin-

ear map Σ∗ : Hm(X)→ Hm+1(ΣX) given by the composition(12.75)

Σ∗ : Hm(X)δ∗−→ Hm+1(CX,X × 0) ≈ Hm+1(ΣX, [X × 0])

j∗−→ Hm+1(ΣX) ,

where the middle isomorphism comes from Proposition 12.71.

Proposition 12.76. For any topological space X, the suspension homomor-phism

Σ∗ : Hm(X)→ Hm+1(ΣX)

is an isomorphism for all m.

Because of Proposition 12.76, the homomorphism Σ∗ is called the suspensionisomorphism (in cohomology). Observe that Formula (12.75) can also be used to de-fine Σ∗ : Hm(X)→ Hm+1(ΣX). By Proposition 12.76, this unreduced suspensionhomomorphism is an isomorphism if m ≥ 1.

Proof. If X is empty, so is ΣX and Proposition 12.76 is trivial. Supposethen that X 6= ∅. As [X × 0] is a point, its reduced cohomology vanish and j∗ isan isomorphism in Formula (12.75), by the reduced cohomology exact sequence ofthe pair (ΣX, [X × 0]). As CX is contractible, its reduced cohomology also vanishand δ∗ is an isomorphism in Formula (12.75), by the reduced cohomology exactsequence of the pair (CX,X × 0). Finally, note that ΣX is path-connected, so

Hm(ΣX) = 0 for m ≤ 0, so Proposition 12.76 is also true for m ≤ 0.

Analogously, we define Σ∗ : Hm+1(ΣX)→ Hm(ΣX) by the composition

(12.77) Σ∗ : Hm+1(ΣX)j∗−→ Hm+1(ΣX, [X×0]) ≈ Hm+1(CX,X×0)

∂∗−→ Hm(X)

which satisfies 〈Σ∗(a), α〉 = 〈a,Σ∗(α)〉 for all a ∈ Hm(X) and all α ∈ Hm+1(X).By Proposition 12.76 (or directly), we deduce that Σ∗ is an isomorphism, calledthe suspension isomorphism (in homology).

Let (Xj , xj) (j ∈ J ) be a family of pointed spaces, i.e. xi ∈ Xi. Their bouquet(or wedge) X =

∨j∈J Xj is defined as the quotient space

X =∨

j∈J

Xj =⋃

j∈JXj

/⋃j∈Jxj .

By naming x ∈ X the equivalence class x =⋃j∈J xj, the couple (X,x) is a

pointed space. For each j ∈ J , one has a pointed inclusion ij : (Xj , xj) → (X,x).The bouquet plays the role of a sum in the category of pointed spaces and pointedmaps: if f j : (Xj , xj)→ (Y, y) are continuous pointed maps, then there is a uniquecontinuous pointed map f : (X,x)→ (Y, y) such that f ij = fj.

A well pointed space, is a pointed space (X,x) such that (X, x) is a wellcofibrant pair. Observe that this definition is stronger than that in other textbooks.

Lemma 12.78. If (Xj , xj) (j ∈ J ) are well pointed spaces, then (X,x) is a wellpointed space.

Proof. Let (uj , hj) be a presentation of (Xj , xj) as a well cofibrant pair. Then

(⋃j∈J u

j ,⋃j∈J h

j) is a presentation of (⋃j∈JX

j ,⋃j∈J xj) as a well cofibrant

pair. By Lemma 12.69, the quotient pair (X, x) is well cofibrant, so (X,x) is awell pointed space.


Proposition 12.79. Let (Xj , xj), with j ∈ J , be a family of well pointedspaces. Then, the family of inclusions ij : Xj → X =

∨j∈J Xj, for j ∈ J , gives

rise to isomorphisms on reduced (co)homology

H∗(X)(H∗ij)

≈// ∏

j∈J H∗(Xj)

and⊕

j∈J H∗(Xj)

P

H∗ij

≈// H∗(X) .

Proof. It is enough to establish that∑H∗ij is an isomorphism. The coho-

mology statement can be proved analogously or by Kronecker duality, using Dia-gram (3.9).

Write, as above, x =⋃j∈J xi ∈ X . The map of pairs (Xj , xi) → (X,x)

give rise, for each m ∈ N, to a commutative diagram between exact sequences

L

j∈J Hm+1(xj)

≈

//L

j∈J Hm+1(Xj)

≈

//L

j∈J Hm+1(Xj , xj)

L

H∗ij

∂∗ //

Hm+1( ˙Sj∈J xj) // Hm( ˙S

j∈J Xj) // Hm+1( ˙Sj∈J Xj , ˙S

j∈J xj)∂∗ //

∂∗ //⊕

j∈J Hm(xj)

≈

//⊕

j∈J Hm(Xj)

≈

∂∗ // Hm(⋃j∈J xj) // Hm(

⋃j∈JXj)

The isomorphisms for the vertical arrows are due to Proposition 12.17. By thefive-lemma,

⊕H∗ij is an isomorphism. As (Xj , xj) is well pointed, the pair

(⋃j∈JXj ,

⋃j∈J xj) is well cofibrant. By Proposition 12.71, the quotient map

q : (⋃j∈JXj,

⋃j∈J xj) → (X,x) induces an isomorphism on homology. One has

the following commutative diagram

⊕j∈J H∗(Xj)

≈

P

H∗ij // H∗(X)

≈

⊕j∈J H∗(Xj , xj)

L

H∗ij

≈// H∗(

⋃j∈JXj ,

⋃j∈J xj)

H∗q

≈// H∗(X, x) ,

where the vertical arrows are isomorphisms by Remark 12.29. Therefore,∑H∗ij

is an isomorphism.

The (co)homology of X =∨j∈J Xj may somehow be also controlled using the

projection πj : X → Xj defined by

(12.80) πj(z) =

z if z ∈ Xj

x otherwise,

where x =⋃j∈J xi ∈ X . As πj ij = idXj

, Proposition 12.79 implies the following


Proposition 12.81. Let (Xj , xj), with j ∈ J , be a family of well pointedspaces. Then, the composition

⊕j∈J H

∗(Xj)

P

H∗πj// H∗(X)(H∗ij)

≈// ∏

j∈J H∗(Xj)

is the inclusion of the direct sum into the product. Also, the composition

⊕j∈J H∗(Xj)

P

H∗ij

≈// H∗(X)

(H∗πj)// ∏j∈J H∗(Xj)

is the inclusion of the direct sum into the product.In particular, if J is finite, then

⊕j∈J H

∗(Xj)

P

H∗πj

≈// H∗(X) and H∗(X)

(H∗πj)

≈// ∏

j∈J H∗(Xj)

are isomorphisms.

12.6. Mayer-Vietoris sequences. Let X be a topological space. Let B =X1, X2 be a collection of two subspaces of X . Write X0 = X1 ∩X2 and

X0

i2

i1 // X1

j1

X2j2 // X

for the inclusions. We call (X,X1, X2, X0) a Mayer-Vietoris data. The followingsequence of cochain complexes

0→ C∗(X)(C∗j1,C

∗j2)−−−−−−−−→ C∗(X1)⊕ C∗(X2)C∗i1+C∗i2−−−−−−−→ CB∗ (X0)→ 0

is then exact, as well as the following sequence of chain complexes

0→ C∗(X0)(C∗i1,C∗i2)−−−−−−−→ C∗(X1)⊕ C∗(X2)

C∗j1+C∗j2−−−−−−−→ CB∗ (X)→ 0

By § 6, these short exact sequences give rise to connecting homomorphisms

δMV : H∗(X0)→ H∗+1B (X) and ∂MV : HB∗ (X)→ H∗−1(X0)

involved in long (co)homology exact sequences. If the interiors of X1 and X2

cover X , the theorem of small simplexes 12.56 implies that H∗B(X) ≈ H∗(X) andHB∗ (X) ≈ H∗(X). Therefore, we obtain the following proposition.

Proposition 12.82 (Mayer-Vietoris sequences I). Let (X,X1, X2, X0) be aMayer-Vietoris data. Suppose that X = intX1 ∪ intX2. Then, the following longsequences

→ Hm(X)(H∗j1,H

∗j2)−−−−−−−−→ Hm(X1)⊕Hm(X2)H∗i1+H∗i2−−−−−−−→ Hm(X0)

δMV−−−→ Hm+1(X)→and

→ Hm(X0)(H∗i1,H∗i2)−−−−−−−−→ Hm(X1)⊕Hm(X2)

H∗j1+H∗j2−−−−−−−→ Hm(X)∂MV−−−→ Hm−1(X0)→

are exact.

These Mayer-Vietoris sequences are natural for maps f : X → X ′ such thatf(Xi) ⊂ X ′i.

The hypotheses of Proposition 12.82 may not by directly satisfied in usualsituations. Here is a variant which is more useful in practice.


Proposition 12.83 (Mayer-Vietoris sequences II). Let (X,X1, X2, X0) be aMayer-Vietoris data, with Xi closed in X. Suppose that X = X1 ∪ X2 and that(Xi, X0) is a good pair for i = 1, 2. Then, the following long sequences

→ Hm(X)(H∗j1,H

∗j2)−−−−−−−−→ Hm(X1)⊕Hm(X2)H∗i1+H∗i2−−−−−−−→ Hm(X0)

δMV−−−→ Hm+1(X)→and

→ Hm(X0)(H∗i1,H∗i2)−−−−−−−−→ Hm(X1)⊕Hm(X2)

H∗j1+H∗j2−−−−−−−→ Hm(X)∂MV−−−→ Hm−1(X0)→

are exact.

Proof. Choose a neighbourhood Ui of X0 in Xi admitting a retraction bydeformation onto X0, called ρit : Ui → Ui, t ∈ I. Let X ′1 = X1 ∪ U2, X

′2 = X2 ∪ U1

and X ′0 = X ′1 ∩ X ′2 = X0 ∪ U1 ∪ U2. We claim that X = intX ′1 ∪ intX ′2. Indeed,as U2 is a neighbourhood of X0 in X2, there exists an open set V2 of X such thatX0 ⊂ V2 ∩X2 ⊂ U2. As X2 is closed, X1 −X2 = X −X2 is open in X . Therefore

X1 ⊂ (X1 −X2) ∪ (V2 ∩X2) = (X1 −X2) ∪ V2 ⊂ intX ′1 .

In the same way, X2 ⊂ intX ′2. Hence, X = intX ′1 ∪ intX ′2.As X = intX ′1 ∪ intX ′2, the Mayer-Vietoris sequences of Proposition 12.82

hold true with (X ′1, X′2, X

′0). But X ′1 retracts by deformation onto X1, using the

retraction ρ1t : X ′1 → X ′1 given by

ρ1t =

ρ2t (x) if x ∈ U2

x if x ∈ X1 .

In the same way, X ′2 retracts by deformation onto X2 and X ′0 retracts by deforma-tion onto X0. This proves Proposition 12.83.

For Mayer-Vietoris sequences with other hypotheses, see Exercise 2.11, fromwhich Proposition 12.83 may also be deduced.

13. Spheres, disks, degree

So far, we have not encountered any space whose (co)homology is not zero inpositive dimensions. The unit sphere Sn in Rn+1 will be the first example. Theshortest way to describe the (co)-homology of such simple spaces is by giving theirPoincare polynomials. The definitions are the same as for simplicial complexes.A topological pair (X,Y ) is of finite (co)-homology type if its singular homology(or, equivalently, cohomology) is of finite type. In this case, the Poincare series of(X,Y ) (or of X if Y is empty) is that of H∗(X,Y ):

Pt(X,Y ) =∑

i∈N

dimHi(X,Y ) ti =∑

i∈N

dimHi(X,Y ) ti ∈ N[[t]].

When the series is a polynomial, we speak of the Poincare polynomial of (X,Y ).

Proposition 13.1. The Poincare polynomial of the sphere Sn is

Pt(Sn) = 1 + tn .

Proof. The sphere S0 consists of two points, so the result for n = 0 followsfrom (12.7) and Corollary 12.18. We can then propagate the result by the suspen-

sion isomorphism Σ∗ : H∗(Sn)≈−→ H∗(Sn+1) (see Proposition 12.76), since Sn+1

is homeomorphic to ΣSn. The homology statement uses the homology suspension

isomorphism Σ∗ : H∗(Sn+1)

≈−→ H∗(Sn).

13. SPHERES, DISKS, DEGREE 77

As a consequence of Proposition 13.1, the sphere Sn is not contractible, thoughit is path-connected if n > 0. Also Sn and Sp are not homotopy equivalent if n 6= p.A useful corollary of Proposition 13.1 is the following

Corollary 13.2. Pt(Dn, Sn−1) = tn.

Proof. This follows from Proposition 13.1 and the (co)homology exact se-quence of the pair (Dn, Sn−1).

It will be useful to have explicit cycles for the generators of Hn(Dn, Sn−1) and

Hn(Sn). Let ∆n = ∆n−int∆n be the topological boundary of the standard simplex

∆n. The identity map in : ∆n → ∆n is a relative cycle of (∆n, ∆n), representing a

class [in] ∈ Hn(∆n, ∆n). The boundary ∂(in) belongs to Zm−1(∆n) and represents

∂∗([in]) in Hn−1(∆n).

Proposition 13.3. For all n ∈ N, the following two statements hold true:

An: [in] is the non-zero element of Hn(∆n, ∆n) = Z2.

Bn: [∂(in+1)] is the non-zero element of Hn(∆n+1) = Z2.

Proof. Statements An and Bn are proven together, by induction on n, asfollows:

(a) A0 and B0 are true.(b) An implies Bn.(c) Bn implies An+1.

As the affine simplex ∆0 is a point and ∆0 is empty, Statement A(0) follows from

the discussion in Example 12.6. To prove B(0), observe that ∆1 consists of twopoints p and q. Identifying a singular 0-simplex with a point, one has ∂(i1) = p+ q,

which represents a non-vanishing element of H0(∆1). But 〈1, p + q〉 = 0, which

shows that [∂(i1)] 6= 0 in H0(∆1).

Let us prove (b). Consider the inclusion ǫ : ∆n → ∆n+1 given by ǫ(t0, . . . , tn) =

(t0, . . . , tn, 0). Let Λn = adh (∆n+1 − ǫ(∆n)). Consider the homomorphisms:

Hn(∆n+1)

j∗

≈// Hn(∆n+1,Λn) Hn(∆

n, ∆n) .≈

H∗ǫoo

The arrow j∗ is bijective, as in (12.31), since Λm is contractible; the arrow H∗ǫ is

bijective by excision and homotopy. As in H∗(∆n+1,Λn) we neglect the singular

chains in Λn, one has

j∗([∂(in+1)]) = H∗ǫ ([in])

which proves (b).

To prove (c), we use that ∂∗ : Hn+1(∆n+1, ∆n+1)

≈−→ Hn(∆n) is an isomor-

phism, since ∆n+1 is contractible, and that ∂∗([in+1]) = [∂(in+1)].

For the sphere S1, Proposition 13.3 has the following corollary.

Corollary 13.4. Let σ : ∆1 → S1 given by σ(t, 1 − t) = e2iπt. Then σ ∈C1(S

1) is a singular 1-cycle of S1 and its homology class is the non-zero elementof H1(S

1) = Z2.


Proof. Since σ(1, 0) = σ(0, 1), the 1-cochain σ is a cycle. The map σ factorsas

∆1

p

##GGGGGG

G

σ // S1

∆1/∆1

s

≈

;;xxxxxxx

where s is a homeomorphism. Under the composed homomorphism

H1(∆1, ∆1)H∗p

≈// H1(∆1/∆1, [∆1])

≈ // H1(∆1)

H∗s

≈// H1(S

1) ,

the class [i1] goes to [σ]. By Proposition 13.3, [i1] is a generator of H1(S1), which

proves Corollary 13.4.

Let f : Sn → Sn be a continuous map. The linear map Hnf : Hn(Sn) →Hn(S

n) is a map between Z2 and itself. The degree deg(f) ∈ Z2 of f by

deg(f) =

0 if Hnf = 0

1 otherwise.

One can define the same degree using Hnf . For instance, the degree of a home-omorphism is 1 and the degree of a constant map is 0. Let f, g : Sn → Sn. ByProposition 12.48, deg(f) = deg(g) if f, g : Sn → Sn are homotopic. Also, usingthat Hn(gf) = HngHnf one gets

(13.5) deg(gf) = deg(g) · deg(f)

These simple remarks have the following surprisingly strong consequences. (For arefinement of Proposition 13.6 below using the integral degree (see S[152, Theo-rems 21.4 and 21.5].)

Proposition 13.6. Let f : Sn → Sn be a continuous map with deg f = 0.Then,

(a) f admits a fixed point.(b) there exists x ∈ Sn with f(x) = −x.

Proof. Suppose that there is no fixed point. Then f is homotopic to theantipodal map a(x) = −x: a homotopy is obtained by following the arc of greatcircle from f(x) to −x not containing x. Therefore deg f = deg a = 1 since a is ahomeomorphism. If f(x) 6= −x for all x, then deg f = 1 because f is homotopic tothe identity (following the arc of great circle from f(x) to x not containing −x).

We now give three recipes to compute the degree of a map from Sn to itself.A point u ∈ Sn is a topological regular value for f : Sn → Sn if there is a neigh-bourhood U of u such that U is “evenly covered” by f . By this, we mean thatf−1(U) is a disjoint union of Uj, indexed by a set J , such that, for each j ∈ J , therestriction of f to Uj is a homeomorphism from Uj to U . In particular, f−1(u) is adiscrete closed subset of Sn indexed by J , so J is finite since Sn is compact. Forinstance, a point u which is not in the range of f is a topological regular value of f(with J empty). For a topological regular value u of f , we define the local degreed(f, u) ∈ N of f at u by

d(f, u) = ♯f−1(u) .

13. SPHERES, DISKS, DEGREE 79

Proposition 13.7. Let f : Sn → Sn be a continuous map. For any topologicalregular value u of f , one has

deg(f) = d(f, u) mod 2 .

Example 13.8. The map S1 → S1 given by z 7→ zk has degree the residueclass of k mod 2.

Proof of Proposition 13.7. When n = 0, each of the two points of S0

is a regular value of f and the equality of Proposition 13.7 is easy to check byexamination of the various cases. We then suppose that n > 0.

If u is a topological regular value, there is a neighbourhood B of u which isevenly covered by f and which is homeomorphic to a closed disk Dn. Its preimageB = f−1(B) is a finite disjoint union of n-disks Bj , indexed by j ∈ J . DefineJ = J ∪0 and set B0 = B.

For j ∈ J , define Vj = Sn −Bj and set V = Sn − B. Consider the quotient

spaces Snj = Sn/Vj (j ∈ J ), which are homeomorphic to Sn. Thus, Sn/V ≈∨j∈J S

nj is homeomorphic to a bouquet of ♯J copies of Sn. Denote the quotient

maps by j : Sn → Snj and : Sn → Sn/V ≈ ∨

j∈J Snj .

If uj ∈ Uj is the point such that f(uj) = u (u0 = u), then Sn − uj is aneighbourhood of Vj which retracts by deformation onto Vj . Therefore, (Sn, Vj) isa good pair and, as Sn−uj is homeomorphic to Rn, the space Vj is contractible.Also, Vj admits a mapping cylinder neighbourhood in Sn, so the pair (Sn, Vj) iswell cofibrant by Lemma 12.64. By the reduced homology exact sequence of thepairs (Sn, Vj) and (Snj , [Vj ]) and Proposition 12.71, we get three isomorphisms inthe following commutative diagram

Hn(Sn)

≈

Hnj // Hn(Snj )

≈

Hn(Sn, Vj) ≈

// Hn(Snj , [Vj ]) ,

which shows that Hnj is an isomorphism.Let 0 6= α ∈ Hn(S

n), and, for j ∈ J , let 0 6= αj ∈ Hn(Snj ). The map f descends

to a continuous map f : Sn/V → Sn0 . Let us consider the following commutativediagram:

(13.9)

Hn(Sn)

Hnf

Hn // ⊕j∈J Hn(S

nj )

Hn f

Hn(Sn)

Hn0

≈// Hn(S

n0 )

The restriction of f to Snj is a homeomorphism, soHnf(αj) = α0. Let πk :∨j∈J S

nj →

Snk be the projection onto the kth component (see Equation (12.80)). Then j =

πj . By Proposition 12.81, this implies that Hn(α) = (αj). Then, Hnf Hn(α) =

d(f, u)α0. On the other hand, Hn0Hnf(α) = deg(f)α0. As Diagram (13.9) iscommutative, this proves Proposition 13.7.

The second recipe is the following lemma.


Lemma 13.10. Let f : Sn → Sn be a continuous map, with n > 0. LetB1, . . . , Bk be disjoint embedded closed n-disks of Sn with boundary Bi. Let Vbe the closure of Sn − ⋃

Bi. Suppose that f sends V onto a single point v ∈ Snand thus induces continuous maps fi : S

n ≈ Bi/Bi → Sn. Then

deg f =

k∑

i=1

deg fi .

Proof. Let Sni = Bi/Bi, homeomorphic to Sn. The map f factors in thefollowing way:

Sn

p

&&NNNNNNNNNN

f // Sn

Sn/V ≈ ∨ki=1 S

n

W

fi

88pppppppppp

Obviously, H∗p([Sn]) =

∑ki=1[S

ni ]. Hence,

deg f [Sn] =

k∑

i=1

H∗fi([Sni ]) =

( k∑

i=1

deg fi)[Sn] .

The third recipe concerns the self-maps of S1. Recall the elementary wayto prove that [S1, S1] ≈ Z. Let f : S1 → S1 be a (continuous) map. As t 7→exp(2iπt) is a local homeomorphism R → S1, there exists a map g : I → R suchthat f(exp(2iπt)) = exp(2iπg(t)). The integer

DEG (f) = g(1)− g(0) ∈ Z

depends only on the homotopy class of f . This defines a bijection

(13.11) DEG : [S1, S1]≈−→ Z .

For instance, for the map f(z) = zn, one can choose g(t) = nt. Thus, DEG (f) = nif and only if f is homotopic to z 7→ zn.

Proposition 13.12. For a map f : S1 → S1,

deg(f) = DEG (f) mod 2 .

Proof. If DEG (f) = n, then f is homotopic to z 7→ zn. This map satisfiesdeg(f) = n mod 2 by Proposition 13.7.

Remarks 13.13. (a) our degree is the reduction mod 2 of the integral degreeobtained using integral homology (see e.g. [80, § 2.2]). Proposition 13.7 would alsohold for the integral degree, provided one takes into account the orientations in thedefinition of the local degree.

(b) A continuous map f : Sn → Sn may not have any topological regular value.For example, S. Ferry constructed a map f : S3 → S3 with (integral) degree 2 sothat the preimage of every point is connected [60].

(c) Suppose that f = |g|, where g : K → L is a simplicial map, with |K| and|L| homeomorphic to Sn. Let τ ∈ Sn(L) and u be a point in the interior of |τ |.Then u is a regular value and d(f, u) = d(g, τ) (see Equation (5.11)).

14. CLASSICAL APPLICATIONS OF THE mod2-(CO)HOMOLOGY 81

(d) Let f : Sn → Sn be a smooth map. Any smooth regular value is a topolog-ical regular value. Then, deg(f) coincides with the degree mod 2 of f as presentedin e.g. [149, Section 4].

14. Classical applications of the mod 2-(co)homology

At our stage of development of homology, text books usually present a coupleof classical applications in topology. Several of them only require Z2-homology,other need the integral homology. We discuss this matter in this section.

Retractions and Brouwer’s fixed point Theorem. Let X be a topologicalspace with a subspace Y . A continuous retraction of X onto Y is a continuous mapr : X → Y extending the identity of Y . In other words, r i = idY , where i : Y → Xdenotes the inclusion. Therefore, H∗rH∗i = id and H∗iH∗r = id, which impliesthe following lemma.

Lemma 14.1. If there exists a continuous retraction from X onto Y , thenH∗i : H∗(Y ) → H∗(X) is injective and H∗i : H

∗(X) → H∗(Y ) is surjective. Thesame holds true for the reduced (co)homology.

As H∗(Dn) = 0 while Hn−1(S

n−1) = Z2, Lemma 14.1 has the following corol-lary.

Proposition 14.2. There is no continuous retraction of the n-disk Dn ontoits bondary Sn−1.

The most well known corollary of Proposition 14.2 is the fixed point theoremproved by Luitzen Egbertus Jan Brouwer around 1911 (see [38, Chapter 3]).

Corollary 14.3. A continuous map from the disk Dn to itself has at least onefixed point.

Proof. Suppose that f(x) 6= x for all x ∈ Dn. Then. a retraction r : Dn →Sn−1 is constructed using the following picture, contradicting Proposition 14.2.

f(x)

r(x)

x

Brouwer’s theorem says that, for a map f : Dn → Dn, the equation f(x) = xadmits a solution under the only hypothesis that f is continuous. Given the possiblewildness of a continuous map, this is a very deep theorem. It is impressive thatsuch a result is due to the fact that Hn(D

n) = 0 and Hn(Sn−1) = Z2.

Invariance of dimension. A n-dimensional topological manifold is a topolog-ical space such that each point has an open neighbourhood homeomorphic to Rn.The following result is known as the topological invariance of the dimension andgoes back to the work of Brouwer in 1911 (see [38, Chapter II]).

Theorem 14.4. Suppose that a non-empty m-dimensional topological manifoldis homeomorphic to an n-dimensional topological manifold. Then m = n.


Proof. Let M be a non-empty m-dimensional topological manifold and N bean n-dimensional topological manifold. Let h : M → N be a homeomorphism. Letx ∈ M . Then h restricts to a homeomorphism from M − x onto N − h(x).Hence, H∗h : H∗(M,M − x)→ H∗(N,N − h(x)) is an isomorphism. But thiscontradicts the fact that(14.5)

Hk(M,M −x) =

Z2 if k = 0,m

0 otherwiseand Hk(N,N−h(x)) =

Z2 if k = 0, n

0 otherwise.

Indeed, it enough to prove (14.5) in the case of M . As x has a neighbourhood in Mhomeomorphic to Rm, it has a neighbourhood B homeomorphic to closed m-ball.By the excision of M −B and homotopy, one has

H∗(M,M − x) ≈ H∗(B,B − x) ≈ H∗(B,BdB)

and (14.5) follows from Corollary 13.2.

Balls and spheres in spheres. The following results concerns the comple-ments of k-balls or k-spheres in Sn.

Proposition 14.6. Let h : Dk → Sn be an embedding. ThenH∗(S

n − h(Dk)) = 0.

Proof. We follow the classical proof (see e.g. [80, Proposition 2b.1]), whichgoes by induction on k. For k = 0, D0 is a point and Sn−h(D0) is then contractible.

For the induction step, suppose that Hi(Sn − h(Dk)) contains a non-zero element

α0. We use the homeomorphism Dk ≈ Dk−1 × I0 with I0 = [0, 1]. Then Sn −h(Dk) = A ∪B with A = Sn − (Dk−1 × [0, 1/2]) and B = Sn − (Dk−1 × [1/2, 1]).

Since, by induction hyphthesis, H∗(A ∩ B) = H∗(Sn − h(Dk−1 × 1/2) = 0,

the Mayer-Vietoris sequence implies that for I1 = [0, 1/2] or I1 = [1/2, 1], thehomomorphism Hi(S

n−h(Dk×I0)))→ Hi(Sn−h(Dk×I1))) sends α0 to 0 6= α1 ∈

Hi(Sn− (Dk−1× I1). Iterating this process produces a nested sequence Ij of closed

intervals converging to a point p ∈ I and a non-zero element αj ∈ lim→H∗(Xj)where Xj = Sn − (Dk−1 × Ij). Set X = Sn − (Dk−1 × p). As each compact

subspace of X is contained in some Xj , Corollary 12.22 implies that lim→H∗(Xj) is

isomorphic to H∗(X), contradicting the induction hypothesis.

Proposition 14.7. Let h : Sk → Sn be an embedding with k < n. ThenH∗(S

n − h(Sk)) ≈ H∗(Sn−k−1).

Proof. The proof is by induction on k. The sphere S0 consisting of twopoints, Sn − h(S0) is homotopy equivalent to Sn−1. For the induction step, writeSk as the union of two hemispheres D±. The Mayer-Vietoris sequence for Sn −h(D±) together with Proposition 14.6 gives the isomororphism H∗(S

n − h(Sk)) ≈H∗−1(S

n − h(Sk−1)).

The case k = n− 1 in Proposition 14.7 gives the following corollary.

Corollary 14.8 (Generalized Jordan Theorem). Let h : Sn−1 → Sn be anembedding. Then Sn − h(Sn−1) has two path-connected components.

Remarks 14.9. (a) Topological arguments show that, in Corollary 14.8, h(Sn−1)is the common frontier of each of the components of its complement (see e.g. [152,

15. CW-COMPLEXES 83

Theorem 6.3]). For a discussion about the possible homotopy types of these com-ponents, see e.g. [152, § 36] or [80, § 2B].

(b) A well known consequence of the generalized Jordan theorem is the invari-ance of domain: if U is an open set in Rn, then its image h(U) under an embeddingh : U → Rn is an open set in Rn. This can be deduced from Corollary 14.8 by apurely topological argument (see, e.g. [152, Theorem 36.5] or [80, Theorem 2B.3]).

Unavailabe applications. Some applications require integral homology andcannot be obtained using Z2-homology. The most well known are the following.

(a) The antipodal map in S2n is not homotopic to the identity and its con-sequence, the non-existence of non-zero vector fields on even-dimensionalspheres (see, e.g. [80, Theorem 2.28] or [152, Corollary 21.6])

(b) The determination of [S1, S1] and the fundamental theorem of algebra(using H1(S1; Z)) (see, e.g., [152, Exercice 2, § 21] or [80, Theorem 1.8])

15. CW-complexes

CW-complexes were introduced and developed by J.H.C. Whitehead in theyears 1940-50 [38, p. 221]. The spaces having the homotopy type of a CW-complex(CW-space) are closed under several natural construction (see [144]). They are thespaces for which many functors of algebraic topology, like singular (co)homology,are reasonably efficient.

Let Y be a topological space and let (Z,A) be a topological pair. Let ϕ : A→ Ybe a continuous map. Consider the space

Z ∪ϕ Y = Z ∪ Y/z = ϕ(z) | z ∈ A ,

endowed with the quotient topology. The space Y is naturally embedded intoZ ∪ϕ Y . We say that Z ∪ϕ Y is obtained from Y by attachment (or adjunction) ofZ, using the attaching map ϕ. When (Z,A) is homeomorphic to (Λ×Dn,Λ×Sn−1),where Λ is a set (considered as a discrete space), we say that Z is obtained from Yby attachment of n-cells, indexed by Λ. For λ ∈ Λ, the image of λ× intDn in Xis the open cell indexed by λ.

A CW-structure on the space X is a filtration

(15.1) ∅ = X−1 ⊂ X0 ⊂ X1 ⊂ · · · ⊂ X =⋃

n∈N

Xn ,

such that, for each n, the space Xn is homeomorphic to a space obtained fromXn−1 by attachment of n-cells, indexed by a set Λn = Λn(X). A space endowedwith a CW-structure is a CW-complex. We see Λn as the set of n-cells of X . Thespace Xn is called the n-skeleton of X . The topology of X is supposed to be theweak topology: a subspace A ⊂ X is open (or closed) if and only if A ∩Xn is open(or closed) for all k ∈ N.

IfX is a CW-complex, a subspace Y ⊂ X is a subcomplex ofX if Y n = Y ∩Xn isobtained from Y n−1 = Y ∩Xn−1 by attaching n-cells, indexed by Λn(Y ) ⊂ Λn(X),using the same attaching maps. For instance, the skeleta of X are subcomplexes ofX . A topological pair (X,Y ) formed by a CW-complex X and a subcomplex Y iscalled a CW-pair.

Let X be a CW-complex. With the above definition, the following propertieshold true:


(1) X is a Hausdorff space.(2) for each n and each λ ∈ Λk, there exists a continuous map ϕλ : (Dn, Sn−1)→

(Xn, Xn−1) ⊂ (X,Xn−1) such that its restriction to intDn is an embed-ding from intDn into X . Indeed, such a map, called a characteristic mapfor the n-cell λ, may be obtained by choosing a homeomorphism between(Xn, Xn−1) and (

[Λ×Dn

]∪ϕ Xn−1, Xn−1).

(3) a map f : X → Z to the topological space Z is continuous if and only ifits restriction to each skeleton is continuous. Also, f is continuous if andonly if f ϕλ is continuous for any characteristic map ϕλ and any cell λ.

(4) each subcomplex of X is a closed subset of X .(5) X0 is a discrete space.(6) A compact subset of a CW-complex meets only finitely many cells. In

consequence, a CW-complex is compact if and only if it is finite, i.e. itcontains a finite number of cells.

These properties are easy to prove (see, e.g. [80, pp. 519–523]).

Proposition 15.2. A CW-pair (X,A) is well cofibrant.

The literature contains many proofs that a CW-pair is good (see e.g. [80,Prop. A.5] or [62, Prop. 1.3.1]), or cofibrant (see e.g. [71, Prop. 14.13] or [36,Prop. 8.3.9]). The proof of Proposition 15.2 uses the following lemma.

Lemma 15.3. Let Z be a space obtained from a space Y by attaching a col-lection of n-cells. Then Y admits in Z a mapping cylinder neighbourhood (seeExample 12.63).

Proof. Let ϕ : Λ×Dn → Y be the attaching map. Let Cn = x ∈ Dn | |x| ≥1/2. Then, Z contains V = (Λ× Cn) ∪ϕ Y as a closed neighbourhood of Y . Thereader will check that V is homeomorphic to the mapping cylinder of ϕ.

Proof of Proposition 15.2. Let Xn = Xn∪A. By Lemmas 15.3 and 12.64,the pair (Xn, Xn−1) is well cofibrant for all n. Let (vn, gn) be a presentation of(Xn, Xn−1) as a well cofibrant pair. As X0 is the disjoint union of A with a discreteset, we may assume that v0(X0−A) = 1. Let Wn be the closure of (vn)−1([0, 1)).For n ≥ 1, by replacing vn by 2vn if necessary, we may assume that gn restricts toa strong deformation retraction of Wn onto Xn−1.

We now define a map u : X → I by constructing, inductively on n ∈ N, itsrestriction un = u|Xn . We set u0 = v0 and

un(x) =

max1, vn(x) + un−1(gn(x, 1)) if x ∈Wn

1 if x ∈ Xn − intWn .

We check that un is continuous. If x ∈ Xn, then uk(x) = un(x) for k ≥ n, thereforeu is well defined and continuous. The space V = u−1(([0, 1)) =

⋃n V

n is a closedneighbourhood of A in X , where V n = V ∩ Xn ⊂Wn.

Define hn : V n × I → V n by

hn(x, t) =

x if t ≤ 1/2n+1

gn(x, 2n+1t) if 1/2n+1 ≤ t ≤ 1/2n

gn(x, 1) if t ≥ 1/2n .

15. CW-COMPLEXES 85

Define hnt : V n → V n by hnt (x) = h(x, t). If x ∈ V n, define h(x, t) = h1t · · · hnt (x).

Note that hkt (x) = x for k > n so, if x ∈ V n ⊂ V m, then h1t · · · hnt (x) =

h1t · · · hmt (x). Therefore, h : V × I → V is well defined and continuous.

The pair (u, h) satisfies all the conditions for being a presentation of (X,A)as a well cofibrant pair, except that h is only defined on V × I instead of X × I.To fix that, choose a continuous map α : I → I such that α([0, 1/2]) = 1 andα vanish on a neighbourhood of 1. Let h : X × I → X and u : X → I defined byh(x, t) = h

(x, α(u(x) t

)and u(x) = 2u(x). One checks that (u, h) is a presentation

of (X,A) as a well cofibrant pair.

Here below are classical examples of CW-complexes.

Example 15.4. The sphere Sn has an obvious CW-structure with one 0-celland one n-cell (attached trivially).

Example 15.5. Observe that the sphere

Sn = x = (x0, x1, . . . , xn) ∈ Rn+1 | |x|2 = 1is obtained from Sn−1 by adjunction of two (n+1)-cellsDn+1

± , attached by the iden-

tity map of Sn−1. Indeed, the embeddings Dn± → Sn given by y = (y1, . . . , yn) 7→

(±√

1− |y|2, y1, . . . , yn) extend the inclusion Sn−1 → Sn and provide a homeo-morphism between Sn−1 ∪ Dn

± and Sn. Starting from S0 = ±1, we thus get a

CW-structure on Sn with two cells in each dimension and whose k-skeleton is Sk.Taking the inductive limit S∞ of those Sn gives a CW-complex known as the in-finite dimensional sphere. This is a contractible space (see e.g. [80, example 1.B.3p. 88]).

Example 15.6. The CW-structure on Sn of Example 15.5 is invariant underthe antipodal map. It then descends to a CW-structure on the projective spaceRPn = Sn/x ∼ −x, having one cell in each dimension. Its k-th skeleton is RP k

and the (k + 1)-cell is attached to RP k by the projection map Sk → RP k. Thisis called the standard CW-structure on RPn. Taking the inductive limit RP∞ ofthese CW-complexes gives a CW-complex known as the infinite dimensional (real)projective space. Analogous CW-decompositions for complex and quaternionic pro-jective spaces are given in § 35.

Example 15.7. If X and Y are CW-complexes, a CW-structure on X×Y may

be defined, with (X × Y )n =⋃p+q=nX

p ×Xq and Λn(X × Y ) =⋃p+q=nΛp(X)×

Λq(Y ) (see [62, Theorem 2.2.2]). The weak topology may have more open setsthan the product topology so the identity i : (X × Y )CW → (X × Y )prod is onlya continuous bijection. If X or Y is finite, or if both are countable, then i isa homeomorphism (see [62, p. 60]). These consideration are not important for ussince the two topologies have the same compact sets. Therefore, they have the samesingular simplexes, whence i induces an isomorphism on singular (co)homology.

We now establish a few lemmas useful for the cellular (co)homology. Let X bea CW-complex. Fix an integer n and chose, for each λ ∈ Λn, a characteristic mapsϕλ : (Dn, Sn−1)→ (Xn, Xn−1). These maps produce a global characteristic map

ϕn : (Λn ×Dn,Λn × Sn−1)→ (Xn, Xn−1) .

Lemma 15.8. Let X be a CW-complex and let n ∈ N. Let ϕn be a globalcharacteristic map for the n-cells. Then


(i) H∗ϕn : H∗(Λn ×Dn,Λn × Sn−1)

≈−→ Hk(Xn, Xn−1) is an isomorphism.

(ii) H∗ϕn : H∗(Xn, Xn−1)≈−→ Hk(Λn ×Dn,Λn × Sn−1) is an isomorphism.

Proof. By Kronecker duality, using Corollary 3.17, only statement (i) mustbe proved. The proof for n = 0 is easy and left to the reader, so we assume thatn ≥ 1.

By Lemma 15.3, (Xn, Xn−1) is a well cofibrant pair. As n ≥ 1, the space Xn−1

is not empty (unless X = ∅, a trivial case). The continuous map

ϕ : Λn ×Dn/Λn × Sn−1 → Xn/Xn−1

induced by ϕn is a homeomorphism, both spaces being homeomorphic to the bou-quet of copies of Sn indexed by Λn. In the commutative diagram

H∗(Λn ×Dn,Λn × Sn−1)

H∗ϕn

≈ // H∗(Λn ×Dn/Λn × Sn−1)

H∗ϕ≈

H∗(Xn, Xn−1)

≈ // H∗(Xn/Xn−1)

,

the horizontal maps are isomorphisms by Corollary 12.73. Therefore, H∗ϕn is an

isomorphism.

Corollary 15.9. Let X be a CW-complex and let n ∈ N. Then

(i)

Hk(Xn, Xn−1) ≈

⊕

Λn

Z2 if k = n.

0 if k 6= n.

(ii)

Hk(Xn, Xn−1) ≈

∏

Λn

Z2 if k = n.

0 if k 6= n.

Proof. Again, the easy case n = 0 is left to the reader. If n > 0, we use that,as noticed in the proof of Lemma 15.8, the map

Xn/Xn−1 →

∨

Λn

Sn

is a homeomorphism. Corollary 15.9 then follows from Proposition 12.79.

Lemma 15.10. Let X be a CW-complex and let n ∈ N. Then

(i) the homomorphism Hk(Xn)→ Hk(X) induced by the inclusion is an iso-

morphism for k < n and is surjective for k = n.(ii) the homomorphism Hk(X) → Hk(Xn) induced by the inclusion is an

isomorphism for k < n and is injective for k = n.

Proof. By Kronecker duality, using Corollary 3.17, only statement (i) mustbe proved. The homomorphisms induced by inclusions form a sequence

(15.11) Hk(Xk)→→ Hk(X

k+1)≈−→ Hk(X

k+2)≈−→ · · · → Hk(X) .

16. CELLULAR (CO)HOMOLOGY 87

The bijectivity or surjectivity of Hk(Xr)→ Hk(X

r+1) is deduced from the homol-ogy exact sequence of the pair (Xr+1, Xr) and Corollary 15.9. By Proposition 12.47,H∗(X) is the direct limit of H∗(K), for all compact sets K of X . Using that eachcompact set of X is contained in some skeleton, one checks that H∗(X) is the directlimit of H∗(X

k). By (15.11), this proves (i).

Lemma 15.12. Let X be a CW-complex and let n ∈ N. Then Hk(Xn) =Hk(X

n) = 0 if k > n.

Proof. The proof is by induction on n. The lemma is true for n = 0 since X0

is a discrete set. The induction step uses the exact sequence of the pair (Xn, Xn−1)together with Corollary 15.9.

Let (X,Y ) be a CW-pair. LetM = (r, s) ∈ N× N | r ≥ s endowed with thelexicographic order. The pairs (Xr, Y s) ((r, s) ∈ M), together with the inclusion

(Xr, Y s) → (Xr′ , Y s′

) when (r, s) ≤ (r′, s′), forms a direct system. The inclusionsjr,s : (Xr, Y s) → (X,Y ) induce a GrV-morphism

J∗ : lim−→

(r,s)∈M

H∗(Xr, Y s) −→ H∗(X,Y )

and a a GrA-morphism

J∗ : H∗(X,Y ) −→ lim←−

K∈K

H∗(Xr, Y s) .

To get a more general result, which will be useful, we can take the product with anarbitrary topological space Z.

Proposition 15.13. Let (X,Y ) be a CW-pair and M be as above. Let Z be atopological space. Then, the GrV-morphism

J∗ : lim−→

(r,s)∈M

H∗(Xr × Z, Y s × Z)

≈−→ H∗(X × Z, Y × Z)

and the GrA-morphism

J∗ : H∗(X × Z, Y × Z)≈−→ lim

←−K∈K

H∗(Xr × Z, Y s × Z) .

are isomorphisms.

Proof. By Kronecker duality, only the homology statement needs a proof.Let K be a compact subspace of X × Z. By Property (6) of p. 84, K is containedin Xr ×Z for some integer r. Hence, if Y is empty, Proposition 15.13 follows fromCorollary 12.22. When Y 6= ∅, we use the long exact sequences in homology andthe five lemma, as in the proof of Proposition 12.47.

16. Cellular (co)homology

Let X be a CW-complex. For m ∈ N, the m-cellular (co)chain vector spaces

Cm(X) and Cm(X) are defined as

Cm(X) = Hm(Xm, Xm−1) and Cm(X) = Hm(Xm, Xm−1) .

The cellular boundary operator ∂ : Cm(X)→ Cm−1(X) is defined by the composedhomomorphism

∂ : Hm(Xm, Xm−1)∂−→ Hm−1(X

m−1)→ Hm−1(Xm−1, Xm−2) .


The expression for ∂ ∂ contains the sequenceHm−1(Xm−1)→ Hm−1(X

m−1, Xm−2)∂−→

Hm−2(Xm−2) and then ∂ ∂ = 0

The cellular co-boundary operator δ : Cm(X) → Cm+1(X) is defined by thecomposed homomorphism

δ : Hm(Xm, Xm−1)→ Hm(Xm)δ−→ Hm+1(Xm+1, Xm) .

with again δ δ = 0. (Co)cycles Zm, Zm and (co)boundaries Bm, Bm are definedas usual, which gives to the definition

Hm(X) = Zm(X)/Bm(X) and Hm(X) = Zm(X)

/Bm(X) .

The graded Z2-vector space H∗(X) is the cellular homology of the CW-complex X

and the graded Z2-vector space H∗(X) is its cellular cohomology. The Kroneckerpairing

Hm(Xm, Xm−1)×Hm(Xm, Xm−1)〈 ,〉−−→ Z2

gives a pairing

Cm(X)× Cm(X)〈 ,〉−−→ Z2

which makes ((C∗(X), δ), (C∗(X), ∂), 〈 , 〉) a Kronecker pair.In the language of former sections, the cellular (co)chains admit the usual

equivalent definitions:

Definitions I (subset definitions):(a) A cellular m-cochain is a subset of Λm.(b) A cellular m-chain is a finite subset of Λm.

Definitions II (colouring definitions):(a) A cellular m-cochain is a function a : Λm → Z2.(b) A cellular m-chain is a function α : Λm → Z2 with finite support.

Definition II.b is equivalent to

Definition III: Cm(X) is the Z2-vector space with basis Λm:

Cm(X) =⊕

λ∈Λm(X)

Z2 λ .

The Kronecker pairing on (co)chains admits the usual equivalent formula

(16.1)〈a, α〉 = ♯(a ∩ α) (mod 2)

=∑

σ∈α a(σ) .

We now give a formula for the cellular boundary operator ∂ : Cm(X)→ Cm−1(X).

By Definition III, it is enough to define ∂(λ) for λ ∈ Λm. Choose an attaching map

ϕλ : Sm−1 → Xm−1 for the m-cell λ. When m = 1, the formula for ∂(λ) is easy:

(16.2) ∂(λ) =

0 if ♯ϕλ(S

0) = 1

ϕλ(S0) otherwise (using the subset definition) .

Let us now suppose that m > 1. For µ ∈ Λm−1, define ϕλ,µ : Sm−1 → Sm−1 as thecomposed map:

Sm−1 ϕλ−−→ Xm−1 →→ Xm−1/Xm−2 ≈∨

Λm−1

Sm−1 πµ−−→ Sm−1 ,


where πµ is the projection onto the µ-th component. Using the colouring definitionof cellular chains, we must give, for each µ ∈ Λm−1, the value ∂(λ)(µ) ∈ Z2.

Lemma 16.3. For m > 1, the cellular boundary operator ∂ : Cm(X)→ Cm−1(X)is the unique linear map satisfying

(16.4) ∂(λ)(µ) =∑

µ∈Λm−1

deg(ϕλ,µ)µ .

for each λ ∈ Λm.

Proof. The attaching map ϕλ : Sm−1 → Xm−1 extends to a characteristicmap ϕλ : Dm → Xm. Consider the commutative diagram:

Hm(Dm, Sm−1)

∂≈

H∗ϕλ// Hm(Xm, Xm−1)

∂

∂

))TTTTTTTTTTTT

Hm−1(Sm−1)

H∗ϕλ // Hm−1(Xm−1) // Hm−1(X

m−1/Xm−2)

H∗πµ

Hm−1(S

m−1)

Let α be the generator ofHm(Dm, Sm−1) = Z2 and let [Sm−1] be that ofHm−1(Sm−1).

Using Lemma 15.8 and its proof, one sees that

(a) λ ∈ Cm(X) corresponds to H∗ϕλ(a) ∈ Hm(Dm, Sm−1).(b) if γ ∈ Hm−1(X

m−1/Xm−2), then H∗πµ(γ) = γ(µ)·b (we use the colouringdefinition and see γ as a function from Λm to Z2).

As ∂(a) = b, one has

∂(λ)(µ) · b = H∗πµ ∂H∗ϕλ(a)

= H∗πµH∗ϕλ(b)

= deg(ϕλ,µ) · b ,which proves the lemma.

Formulae (16.2) and (16.4) for the cellular boundary operator take a specialunique form when X is a regular CW-complex, i.e. when each cell λ admits acharacteristic map ϕλ which is an embedding onto a subcomplex of X . A cell ofthis subcomplex is called a face of λ.

Lemma 16.5. Let X be a regular CW-complex. Let λ ∈ Λm and µ ∈ Λm−1.Then

∂(λ)(µ) =

1 if µ is a face of λ

0 otherwise.

Proof. When m = 1, this follows from (16.2), where the case ♯ϕλ(S0) = 1

does not happen since X is regular. When m > 1, we use Lemma 16.3 and computethe degree of ϕλ,µ by Proposition 13.7: since ϕλ is an embedding, any topologicalregular value of ϕλ,µ has exactly one element in its preimage.

We now prove the main result of this section.


Theorem 16.6. Let X be a CW-complex. Then, the cellular and the singular(co)homology of X are isomorphic:

H∗(X) ≈ H∗(X) and H∗(X) ≈ H∗(X) .

Proof. We consider the following commutative diagram:

(16.7)

Hm+1(Xm+1, Xm)

∂m+1

∂m+1

))RRRRRRRRRRR

0 // Hm(Xm+1)

≈

Hm(Xm)uu

jm

uulllllllllll

// //

77 77nnnnnnnnnn

0

Hm(X)

Hm(Xm, Xm−1)

∂m

∂m

((RRRRRRRRRRR

Hm−1(Xm−1)

vvjm−1

vvlllllllllll

Hm(Xm−1, Xm−2)

The properties of arrows (surjective, injective, bijective) come from Lemma 15.10,15.12 and Corollary 15.9. From Diagram (16.7), we get

Hm(X)≈←− Hm(Xm+1) ≈ Hm(Xm)/Im ∂m+1 ≈

jm // ker ∂m/Im ∂m+1 = Hm(X)

As the isomorphism H∗(X) ≈ H∗(X) does not come from a morphism of chain complex,we cannot invoke Kronecker duality to deduce the isomorphism in cohomology. Instead,we consider the Kronecker dual of Diagram (16.7)

(16.8)

Hm+1(Xm+1, Xm)

Hm(Xm)

δm+1iiRRRRRRRRRRR

Hm(X)

≈

oooo

Hm(Xm, Xm−1)

Jm

55 55lllllllllll

δm+1

OO

Hm(Xm+1)

gg

ggPPPPPPPPPP

Hm−1(Xm−1)

δm

iiRRRRRRRRRRR

0

OO

Hm(Xm−1, Xm−2)

Jm−166 66lllllllllll

δm

OO

which gives

Hm(X)≈−→ Hm(Xm+1)

≈−→ ker δm+1 ≈ J−1

m (ker δm+1)/Im δm = ker δm+1/Im δm = Hm(X) .


Here below are some application of the isomorphism between cellular and sin-gular (co)homology.

Corollary 16.9. Let X be a CW-complex with no m-dimensional cell. ThenHm(X) = Hm(X) = 0.

Proof. If Λm(X) = ∅, then Cm(X) = Cm(X) = 0, which implies Hm(X) =

Hm(X) = 0, and then Hm(X) = Hm(X) = 0 by Theorem 16.6.

Corollary 16.10. Let X be a CW-complex with only one m-dimensional cell.Then dimHm(X) = dimHm(X) ≤ 1.

Proof. One has Cm(X) ≈ Z2 which implies

dim Hm(X) ≤ dim Zm(X) ≤ dim Cm(X) = 1 .

Therfore, dimHm(X) ≤ 1 by Theorem 16.6. The result on cohomology is deducedby Kronecker duality.

A CW-complex is finite if it has a finite number of cells.

Corollary 16.11. Let X be a compact CW-complex. Then

dimH∗(X) = dimH∗(X) <∞ .

Proof. By the weak topology, a compact CW-complex is finite (see Remark (f)

p. 84). Hence, C∗(X) is a finite dimensional vector space, and so is Z∗(X) and

H∗(X). Proposition 16.11 then follows from Theorem 16.6 and Kronecker duality.

Let X be a finite CW-complex. Its Euler characteristic χ(X) is defined as

χ(X) =∑

m∈N

(−1)m ♯Λm(X) ∈ Z .

Proposition 16.12. Let X be a finite CW-complex. Then

χ(X) =∑

m∈N

(−1)m dimHm(X) =∑

m

(−1)m dimHm(X) .

Proof. If we use the cellular (co)homology, the proof of Proposition 16.12 isthe same as that of Proposition 4.15. The result then follows from Theorem 16.6.

A CW-complex X (or a CW-structure on X) is called perfect if the cellularboundary vanishes. For instance, ifX does not have cells in consecutive dimensions,then it is perfect. Also, the standard CW-structure on RPn (n ≤ ∞) is perfect (see

e.g. Proposition 35.1). If X is a perfect CW-complex, C∗(X) = H∗(X) and theidentification between the singular and cellular homologies, out of Diagram (16.7),is particularly simple:

(16.13) Hm(X)≈←− Hm(Xm)

≈−→ Hm(Xm, Xm−1)≈−→ Hm(X)

≈←− Cm(X) .

The natural functoriality of cellular (co)homology is for cellular maps. If Xand Y are CW-complexes, a continuous map f : Y → X is cellular if f(Y m) ⊂ Xm

for all m ∈ N. We thus get GrV-morphisms C∗f and C∗f making the followingdiagrams commute


Cm(Y )

=

C∗f // Cm(X)

=

Hm(Y m, Y m−1)

H∗f // Hm(Xm, Xm−1)

Cm(Y ) Cm(X)C∗foo

Hm(Y m, Y m−1)

=

OO

Hm(Xm, Xm−1)C∗foo

=

OO

They satisfy 〈C∗f(a), α〉 = 〈a, C∗f(α)〉 for all a ∈ C∗(X) and α ∈ C∗(X). It isuseful to have a formula for C∗f , using that

Cm(Y ) =⊕

λ∈Λm(Y )

Z2 λ and Cm(X) =⊕

µ∈Λm(X)

Z2 µ .

For λ ∈ Λm(Y ) and µ ∈ Λm(X), consider the map fλ,µ : Sm → Sm defined by thecomposition

fλ,µ : Smjλ−→

∨

λ∈Λm(Y )

Sm ≈ Y m/Y m−1 f−→ Xm/Xm−1 ≈∨

µ∈Λm(X)

Smπµ−−→ Sm ,

where jλ is the inclusion of the λ-component and πµ the projection onto the µ-component.

Lemma 16.14. For m ≥ 1, C∗f : Cm(Y ) → Cm(X) is the unique linear mapsuch that

C∗f(λ) =∑

µ∈Λm(X)

deg(fλ,µ)µ .

Proof. The map f : (Y m, Y m−1)→ (Xm, Xm−1) induces a map f :∨λ∈Λm(Y ) S

m →∨µ∈Λm(X) S

m making the following diagram commute

Cm(Y )

C∗f

≈ // Hm(Y m, Y m−1)

H∗f

≈ // Hm(∨λ∈Λm(Y ) S

m)

H∗f

Cm(X)≈ // Hm(Y m, Y m−1)

≈ // Hm(∨µ∈Λm(X) S

m)

As in the proof of Lemma 16.3, one checks that, under the top horizontal isomor-phisms, λ ∈ Cm(Y ) corresponds toH∗jλ([S

m−1]). Also, if γ ∈ Hm−1(Xm−1/Xm−2),

then H∗πµ(γ) = γ(µ) · [Sm−1] (seeing γ as a function from Λm(X) to Z2 by thecolouring definition). Hence,

C∗f(λ)(µ) = H∗πµ H∗f H∗jλ([Sm−1]) = H∗fλ,µ([S

m−1]) = deg(fλ,µ) [Sm−1] ,

which proves the lemma.

16.15. Homology-cell complexes. The results of this section and the previousone are also valid for complexes where cells are replaced by homology cells. A wellcofibrant pair (B, B) is a homology n-cell if H∗(B) = 0 and H∗(B) ≈ H∗(S

n−1).This GrV-isomorphism is “abstract”, i.e. not assumed to be given by any con-tinuous map. It follows that H∗(B, B) ≈ H∗(D

n, Sn−1). We also say that B is a

homology n-cell with boundary B. If λ ∈ Λ is indexing a family of homology n-cell(B(λ), B(λ)) and if ϕλ : B(λ) → Y is a family of continuous maps, we say thatthe quotient space

X = Y ∪ϕ(∪λ∈ΛB(λ)

)

17. ISOMORPHISMS BETWEEN SIMPLICIAL AND SINGULAR (CO)HOMOLOGY 93

is obtained from Y by attachement of homology n-cells (they may be different forvarious λ’s). We identify Λ with the set of homology n-cells.

A homology-cell complex is defined as in p. 83 with atachements of n-cells re-placed by atachements of a set Λn(X) of homology n-cells. The cellular (co)homology

H∗(X) and H∗(X) are defined accordingly and Theorem 16.6 holds true, with thesame proof. Homology-cell structures are used in the proof of Poincare duality (see§ 31).

17. Isomorphisms between simplicial and singular (co)homology

LetK be a simplicial complex. In this section, we prove three theorems showingthat the simplicial (co)homology of K and the singular (co)homology of |K| areisomorphic.

Theorem 17.1. Let K be a simplicial complex. Then

H∗(K) ≈ H∗(|K|) and H∗(K) ≈ H∗(|K|)Proof. The geometric realization |K| of K is naturally endowed with a struc-

ture of a regular CW-complex, with |K|m = |Km|, Λm(|K|) = Sm(K), with acanonical characteristic map for the m-cell σ ∈ Sm(K) given by the inclusion of |σ|into |K|. Thus, Cm(|K|) = Cm(K) and, using Lemma 16.5, the following diagram

Cm(|K|)

∂

= // Cm(K)

∂

Cm−1(|K|)= // Cm−1(K)

is commutative. Therefore, H∗(|K|) = H∗(K) and, by Theorem 16.6, the singularhomologyH∗(|K|) and the simplicial homologyH∗(K) are isomorphic. The equality

H∗(|K|) = H∗(K) is deduced from H∗(|K|) = H∗(K) by Kronecker duality and,using by Theorem 16.6 again, the singular cohomology H∗(|K|) and the simplicialcohomology H∗(K) are also isomorphic.

We now go to the second isomorphism theorem, which uses the ordered sim-plicial (co)homology of § 10. To an ordered m-simplex (v0, . . . , vm) ∈ Sm(K), weassociate the singular m-simplex R(v0, . . . , vm) : ∆m → |K| defined by

(17.2) R(v0, . . . , vm)(t0, . . . , tm) =

m∑

i=0

tivi .

The linear combination in (17.2) makes sense since v0, . . . , vm is a simplex of K.

This defines a map R : Sm(K)→ Sm(|K|) which extends to a linear map

R∗ : C∗(K)→ C∗(|K|) .The formula ∂R = R ∂ is obvious, so R is a morphism of chain complexes

from (C∗(K), ∂) to (C∗(|K|), ∂). Define the linear map R∗ : C∗(|K|) → C∗(K)by 〈R∗(a), α〉 = 〈a,R∗(α)〉. By Lemma 3.10, (R∗, R∗) is a morphism of Kroneckerpair. We also denote by R∗ and R∗ the induced linear maps on (co)homology:

R∗ : H∗(K)→ H∗(|K|) and R∗ : H∗(|K|)→ H∗(K) .


If f : L→ K be a simplicial map, the formulae

R∗C∗f = C∗|f |R∗ and C∗f R∗ = R∗C∗|f |

are easy to check. They induce the formulae

(17.3) R∗H∗f = H∗|f |R∗ and H∗f R∗ = H∗C∗|f |

on (co)homology. In particular, if f is the inclusion of a subcomplex L of K, theabove considerations permit us to construct degree zero linear maps

R∗ : H∗(K,L)→ H∗(|K|, |L|) and R∗ : H∗(|K|, |L|)→ H∗(K,L)

so that (R∗, R∗) is a morphism of Kronecker pair. Finally, if f : (K,L)→ (K ′, L′)is a simplicial map of simplicial pairs, then Formulae (17.3) hold true in relative(co)homology.

Theorem 17.4. Let (K,L) be a simplicial pair. Then the linear maps

R∗ : H∗(K,L)≈−→ H∗(|K|, |L|) and R∗ : H∗(|K|, |L|) ≈−→ H∗(K,L)

are isomorphisms. They are functorial for simplicial maps of simplicial pairs

Proof. The functoriality has already been established. By Kronecker duality,it is enough to prove that R∗ is an isomorphism. The proof goes through a coupleof particular cases.

Case 1: (K,L) = (FA, FA), where FA is the full complex on the finite set of m+1elements A = v0, . . . , vm (see p. 24), which is isomorphic to an m-simplex. Then

(|FA|, |FA|) ≈ (Dm, Sm−1). By Corollary 4.14 and Proposition 13.2,

Hk(FA, FA) = Hk(FA, FA) = Hk(|FA|, |FA|) = 0

if k 6= m and

Hm(FA, FA) ≈ Hm(FA, FA) ≈ Hm(|FA|, |FA|) ≈ Z2 .

Thus, it is enough to prove that R∗ : Hm(FA, FA)→ Hm(|FA|, |FA|) is not triv-

ial. The vector space Hm(FA, FA) is generated by the ordered simplex σ =

(v0, . . . , vm). Let r = R(σ) : (∆m, ∆m) → |FA|, |FA|). One has [r] = H∗r([im])

where im is the identity map of (∆m, ∆m). But [im] 6= 0 in Hm(∆m, ∆m) by Propo-

sition 13.3 and r is a homeomorphism of pairs. Thus R∗(σ) 6= 0 in Hm(|FA|, |FA|).Case 2: (K,L) = (Km,Km−1) with m ≥ 1. The non-vanishing homology groupsare

Hm(Km,Km−1) ≈ Hm(Km,Km−1) ≈ Hm(|Km|, |Km−1|) ≈⊕

Sm(K)

Z2

For each σ ∈ Sm(K) choose an ordered simplex σ = (v0, . . . , vm) with v0, . . . , vm =

σ. Then Hm(Km,Km−1) ≈ Hm(Km,Km−1) is the Z2-vector space with basisσ | σ ∈ Sm(K). Denote by σ the subcomplexes of K generated by σ and by ˙σthe subcomplex of the proper faces of σ. The map rσ : (|σ|, | ˙σ|)→ (|Km|, |Km−1|)is a characteristic map for the m-cells of |K| corresponding to σ. The union rm ofthe rσ is then a global characteristic map for the m-cells of |K|. Let us consider

17. ISOMORPHISMS BETWEEN SIMPLICIAL AND SINGULAR (CO)HOMOLOGY 95

the following commutative diagram

⊕σ∈Sm(K) Hm(σ, ˙σ)

≈

R∗

≈//⊕

σ∈Sm(K)Hm(|σ|, | ˙σ|)

≈ H∗rm

Hm(Km,Km−1)R∗ // Hm(|Km|, |Km−1|)

The bijectivity of the left vertical arrow was seen above. That of the right verticalarrow is Lemma 15.8. The bijectivity of the top horizontal arrow is guaranteed bythe Case 1. Hence, R∗ : Hm(Km,Km−1)→ Hm(|Km|, |Km−1|) is an isomorphism.

Case 2: (K,L) = (Km, ∅). This is proven by induction on m, the case m = 0being obvious. By the naturality of R∗, one has the commutative diagram of exactsequences:

H∗+1(Km, Km−1)

R∗ ≈

∂∗ // H∗(Km−1)

R∗ ≈

// H∗(Km)

R∗

// H∗(Km, Km−1)

R∗ ≈

∂∗ // H∗−1(Km−1)

R∗ ≈

H∗+1(Km, Km−1)

∂∗ // H∗(Km−1) // H∗(Km) // H∗(Km, Km−1)∂∗ // H∗−1(Km−1)

(one has to check that the diagrams with ∂∗ are commutative). The bijectivity ofthe vertical arrows come by induction hypothesis and by case 2. By the five-lemma,R∗ : H∗(Km)→ H∗(|Km|) is an isomorphism.

General case. We first prove that R∗ : Hm(K) → Hm(|K|) is an isomorphism forall m. By the naturality of R∗, one has the following commutative diagram:

Hm(Km+1)

≈

R∗

≈// Hm(|Km+1|)

≈

Hm(K)

R∗ // Hm(|K|) .

The bijectivity of the left vertical arrow is obvious. That of the right verticalarrow is Lemma 15.10. The bijectivity of the top vertical arrow was established inCase 3. Therefore, the bottom horizontal arrow is bijective. Finally, the generalcase (K,L) is deduced from the absolute cases using, as in Case 3, the homologyexact sequences of the pair (K,L) and the five-lemma.

Four our third isomorphism theorem, choose a simplicial order ≤ on K. Definea map R≤ : S(K) → S(|K|) by R≤(σ) = R(σ) where, if σ = v0, . . . , vm, thenσ = (v0, . . . , vm) with v0 ≤ · · · ≤ vm. As above, we check that R≤ induces linearmaps R≤,∗ : H∗(K,L)→ H∗(|K|, |L|) and R∗≤ : H∗(|K|, |L|)→ H∗(K,L) of degreezero.

Theorem 17.5. Let (K,L) be a simplicial pair. For any simplicial order ≤ onK, the linear maps

R≤,∗ : H∗(K,L)≈−→ H∗(|K|, |L|) and R∗≤ : H∗(|K|, |L|) ≈−→ H∗(K,L)

are isomorphisms. Moreover, these isomorphisms do not depend on the simplicialorder ≤.


Proof. By Kronecker duality, only the homology statement requires a proof.By our definitions, one has the following commutative diagram

H∗(K,L)R≤,∗ //

H∗φ≤

≈ &&MMMMMMM

MMH∗(|K|, |L|)

H∗(K,L)

R∗

≈

77ppppppppp

The bijectivity of the arrows come from Corollary 10.13 and Theorem 17.5. There-fore, R≤,∗ is an isomorphism. As H∗φ≤ in independent of ≤ by Corollary 10.13, sois R≤,∗.

18. CW-approximations

It is sometimes useful to know that any space has the (co)homology of a CW-complex or of a simplicial complex (see e.g. p. 113, 136 and 269 in this book).We give below classical functorial results about that. Relationships with similarconstructions in the literature are discussed in Remark 18.12 at the end of thesection.

We shall need a standard notion of category theory: naturals transformations.Let a and b be two (covariant) functors from a category C to a category C′. Anatural transformation associates to each object X in C a morphism φX : a(X)→b(X) in C′ such that the following diagram

(18.1)

a(X)a(f) //

ΦX

a(Y )

ΦY

b(X)

b(f) // b(Y )

is commutative for every morphism f : X → Y in C.We first consider the category of CW-spaces and cellular maps. It is denoted

by CW and, as usual, by CW2 for pairs of CW-complexes. We denote by j be theinclusion morphism from CW2 to Top2.

Theorem 18.2. There is a covariant functor cw : (X,Y )→ (XCW, Y CW) fromTop2 → CW2 and a natural transformation φ = φ(X,Y ) : (XCW, Y CW) → (X,Y )from jcw to the indentity functor of Top2, such that H∗φ and H∗φ are isomor-phisms.

The construction in the proof below is sometimes called in the literature thethick geometric realization of the singular complex of X .

Proof. We start with some preliminaries. If ∆m is the standard m-simplexand I ⊂ 0, 1, . . . ,m, we set

∆mI = (t0, . . . , tm) ∈ ∆m | ti = 0 if i /∈ I

which is a simplex of dimension ♯I. This gives rise to an obvious inclusion mapǫI : ∆♯I−1 → ∆m.

Let X be a topological space. The space XCW is defined as the quotient space

(18.3) XCW =⋃

m≥0

(Sm(X)×∆m

)/∼

18. CW-APPROXIMATIONS 97

where ∼ is the equivalence relation (σ, ǫI(u)) ∼ (σǫI , u) for all σ ∈ Sm(X), I ⊂0, . . . ,m and u ∈ ∆♯I−1. Then XCW is a CW-complex whose k-skeleton is

(XCW)k =⋃

0≥m≤k

(Sm(X)×∆m

)/∼ .

In particular, (XCW)0 is just the space X endowed with the discrete topology. Thek-cells are indexed by Sk(X). The characteristic map for the k-cell correspondingto σ ∈ Sk(X) is the restriction to σ × ∆k of the quotient map from the disjointunion in (18.3) onto (XCW)k.

A continuous f : X1 → X2 determines a cellular map fCW : XCW1 → XCW

2

induced by fCW(σ, u) = (f σ, u). Note that, if Y is a subspace of X , then Y CW

is a subcomplex of XCW. We thus check that cw is a covariant functor (X,Y )→(XCW , Y CW ) from Top2 to CW2.

For σ ∈ Sm(X), one has a continuous map φσ : σ × ∆m → X defined byφσ(σ, u) = σ(u). The disjoint union of those φσ descends to a continuous mapφ : XCW → X , or φ : (XCW, Y CW)→ (X,Y ). One has

φfCW(σ, u) = φ(f σ, u) = f σ(u) = f φ(σ, u)

which amounts, using (18.1), to φ being a natural transformation from jcw to theindentity functor of Top2.

It remains to prove that H∗φ is a GrV-isomorphism (that H∗φ is a GrA-isomorphism will follow by Kronecker duality). We start with the absolute caseY = ∅. We shall construct a diagram

(18.4)

H∗(XCW)

H∗φ // H∗(X)

αyyrrrrrrrr

H∗(XCW)

β

ffMMMMMMMMM

such that H∗φβα = id and α and β are isomorphisms. The bijection S(X)=−→

cells of XCW extends to a linear map α : C∗(X) → C∗(XCW) which satisfies

αδ = δα and thus induces the isomorphism α : H∗(X)≈−→ H∗(X

CW). For β, oneassociates to the k-cell of XCW indexed by σ the map

∆k ≈−→ σ ×∆k char.map−−−−−−→ XCW

which is an element of Sk(XCW). This extends to a linear map β : Ck(XCW) →

Ck(XCW). Again, we check that β δ = δβ. We thus get the linear map β : H∗(X

CW)→H∗(X

CW). The equation H∗φβα = id is straightforward.It remains to prove that β is an isomorphism. To simplify the notation,

write X = XCW. Note that β induces linear maps βk : H∗(Xk) → H∗(X

k)

and βk+1,k : H∗(Xk+1, Xk) → H∗(X

k+1, Xk). Obviously, H∗(X) = limk H∗(Xk).

By Corollary 12.22, one also has that H∗(X) = limkH∗(Xk). Therefore, it suf-

fices to show that βk is an isomorphism for all k. This is done by induction onk. It is obviously true for k = 0, since X0 is a discrete space. For the induc-tion step, suppose that βk : Hi(X

k) → Hi(Xk) is an isomorphism for all i ∈ N.

Then, βk+1 : Hi(Xk+1)→ Hi(X

k+1) is an isomorphism for all i, except perhaps for


i = k, k + 1 where we must consider the following commutative diagram(18.5)

0 // Hk+1(Xk+1)

βk+1

// Hk+1(Xk+1, Xk) //

βk+1,k

Hk(Xk) //

βk≈

Hk(Xk+1) //

βk+1

0

0 // Hk+1(Xk+1) // Hk+1(X

k+1, Xk) // Hk(Xk) // Hk(Xk+1) // 0

where the horizontal lines are the cellular and singular homology exact sequencesof the pair (Xk+1, Xk). By the five lemma, it thus suffices to prove that βk+1,k isan isomorphism. One has the following commutative diagram

(18.6)

Hk+1(Xk+1, Xk)

βk+1,k

oo ≈⊕

σ∈Sk+1(X)

Hk+1(σ × (∆k+1,Bd∆k+1))

⊕βσ

Hk+1(Xk+1, Xk) oo ≈

⊕

σ∈Sk+1(X)

Hk+1(σ × (∆k+1,Bd∆k+1))

where βσ sends the (k+1)-cell σ×(∆k+1 (generator of Hk+1(σ×(∆k+1,Bd∆k+1)) =Z2) to the tautological singular simplex ∆k+1 → σ × ∆k+1. The latter is thegenerator of Hk+1(σ × (∆k+1,Bd∆k+1)) = Z2 (see Proposition 13.3). Hence,βk+1,k is an isomorphism.

We have proven that H∗φ : H∗(XCW) → H∗(X) is a GrV-isomorphism for

all topological space X . Using the homology exact sequences and the five lemma,this implies that H∗φ : H∗(X

CW, Y CW)→ H∗(X,Y ) is a GrV-isomorphism for alltopological pairs (X,Y ).

A slightly more sophisticated construction for the functor of Theorem 18.7gives the following result. Let RCW be the category of regular CW-complexesand cellular maps and let j be the inclusion morphism from RCW2 to Top2.

Theorem 18.7. There is a covariant functor rcw : (X,Y )→ (XRCW, Y RCW)from Top2 → RCW2 and a natural transformation φ = φ(X,Y ) : (XRCW, Y RCW)→(X,Y ) from jrcw to the indentity functor of Top2, such that H∗φ and H∗φ areisomorphisms.

The proof of Theorem 18.7 requires some preliminaries.Let FN be the full simplicial complex with vertex set the integers N. If X is a

topological space, the set of N-singular simplexes of X is defined by

NS(X) = (s, τ) | s ∈ S(FN) and τ : |s| → X is a continuous map .where s is the simplicial complex formed by s and all its faces (see p. 10). LetNSn(X) be the subset of NS(X) formed by those pairs (s, τ), where s is of dimensionn and let NCn(X) be the Z2-vector space with basis NSn(X). Using the facets of s,we define a boundary operator ∂ : NCn(X)→ NCn−1(X) making NC∗(X) a chaincomplex. The homology of this chain complex is the N-singular homology of X ,denoted by NH∗(X). The relative homology NH∗(X,Y ) is defined as in § 12.2.

18. CW-APPROXIMATIONS 99

The order on N provides a simplicial order on FN. Thus, if s ∈ Sn(FN),

there is a canonical homeomorphism hs : |s| ≈−→ ∆n (see (12.9)). We define mapsµ : Sn(X)→ NSn(X) and ν : NSn(X)→ Sn(X) by:

• µ(σ) = (s0, σhs0(n)), where s0(n) = 0, 1, . . . , n and

• ν(s, τ) = τ h−1s .

The linear extensions C∗µ : Cn(X) → NCn(X) and C∗ν : NCn(X) → Cn(X) com-mute with the boundary operators and are thus morphisms of chain complexes.The constructions extend to pairs and we get GrV-morphisms H∗µ : Hn(X,Y )→NHn(X,Y ) and H∗ν : NHn(X,Y )→ Hn(X,Y ).

Lemma 18.8. H∗µ : Hn(X,Y )→ NHn(X,Y ) and H∗ν : NHn(X,Y )→ Hn(X,Y )are isomorphisms, inverse of each other.

Proof. Clearly, ν µ = id on S(X), thus H∗ν H∗µ = id on H∗(X,Y ). Tosee that H∗µH∗ν = id on NH∗(X,Y ), we first restrict ourselves to the absolutecase Y = ∅. We shall prove that C∗µC∗ν and the identity of NC∗(X) admit acommon acyclic carrier A∗ with respect to the basis NS(X). The condition thatH∗µH∗ν = id on NH∗(X) then follows from Proposition 9.1.

For (s, τ) ∈ NSn(X) and k ∈ N, define Bk(s, τ) by

Bk(s, τ) = (t, τ |p|) | t ∈ Sk(FN) and p : t→ s is a simplicial map .Let Ak(s, τ) be the Z2-vector space with basis Bk(s, τ). Using the restriction of pto the facets of t, one defines a boundary operator ∂ : Ak(s, τ)→ Ak−1(s, τ) makingA∗(s, τ) a subchain complex of NC∗(X).

For (s, τ) ∈ NS(X), one has (sτ) ∈ A∗(s, τ) (p = ids and

µν(s, τ) = (s0, τ h−1s hs0(n)) ∈ A∗(s, τ) ,

since h−1s hs0(n) = |p| for p : s → s0 the unique simplicial isomorphism preserv-

ing the order. The conditions for the correspondence (s, τ) 7→ A∗(s, τ) being anacyclic carrier (see § 9) are easy to check once we know that H0(A∗(s)) = Z2 andHm(A∗(s)) = 0 for m > 0 which we prove below.

Consider the simplicial complex K(s) with vertex set V (K(s)) = N×V (s) andwhose k-simplexes are the sets (n0, s0), . . . , (nk, sk) with n0 < · · · < nk. We checkthat the correspondence sending (t, τ |p|) to the graph of p : V (t)→ V (s) inducesan isomorphism of chain complex between A∗(s, τ) and the simplicial chain ofK(s).We have thus to prove that H∗(K(s)) ≈ H∗(pt). But K(s) is the union of Kn(s),where Kn(s) is the union of all simplexes of K(s) with vertices in 0, . . . , n×V (S).The inclusion V (Kn(s)) → V (Kn+1(s)) together with the map k 7→ (n + 1, k)

provides a bijection V (Kn(s)) ∪ V (s)≈−→ V (Kn+1(s)) and a simplicial isomorphism

Kn(s) ∗ s0 ≈−→ Kn+1(s) ,

where s0 is the 0-skeleton of s. Hence, the inclusion Kn(s) → Kn+1(s) factorsthrough the cones onKn(s) contained in the join Kn(s)∗s0. Therefore,H∗(K(s)) ≈lim→H∗(Kn(s)) ≈ H∗(pt).We have thus proved Lemma 18.8 in the case Y = ∅. Using the homology exact

sequences, this proves thatH∗µ : Hn(X,Y )→ NHn(X,Y ) andH∗ν : NHn(X,Y )→Hn(X,Y ) are both isomorphisms. But we have already noted that H∗ν H∗µ = idon H∗(X,Y ). Therefore, H∗µH∗ν = id on NH∗(X,Y ).

We are now ready for the proof of Theorem 18.7.


Proof of Theorem 18.7. If t ⊂ s are simplexes ofFN, we denote by it,s : t→s the simplicial map given by the inclusion. The space XRCW is defined as the quo-tient space

(18.9) XRCW =⋃

(s,τ)∈NS(X)

((s, τ)) × |s|

)/∼

where ∼ is the equivalence relation ((s, τ), |it,s|(u)) ∼ (t, τ |it,s|), u) for all (s, τ) ∈NS(X), all subsimplex t of s and all u ∈ |t|. As in the proof of Theorem 18.2,XRCW

is a naturally a CW-complex. The characteristic map for the k-cell corresponding to(s, τ) ∈ NSk(X) is the restriction to (s, τ))× |s| of the quotient map in (18.9). Inparticular, (XRCW)0 is the set N×X endowed with the discrete topology. Becauseof the role of N in the indexing of the cells, one checks that XRCW is a regular CW-complex. For (s, τ) ∈ NSm(X), one has a continuous map φ(s,τ) : (s, τ)×|s| → Xdefined by φ(s,τ)((s, τ), u) = τ(u). The disjoint union of those evaluation maps

descends to a continuous map φ : XRCW → X . if Y is a subspace of X , then Y RCW

is a subcomplex of XRCW. The functoriality of the correspondence (X,Y ) 7→(XRCW, Y RCW), as well as that φ is a natural transformation from jrcw to theindentity functor of Top2, are established as in the proof of Theorem 18.2.

We now prove that NH∗φ : NH∗(XRCW) → NH∗(X) is a GrV-isomorphism,following the pattern of the proof of Theorem 18.2. Similarly to (18.4), we constructthe diagram

(18.10)

NH∗(XRCW)NH∗φ // NH∗(X)

Nαxxqqqqqqqqq

H∗(XRCW)

Nβ

ggOOOOOOOOOO

such that NH∗φNβNα = id and Nα and Nβ are isomorphisms. As in (18.4),

the bijection NS(X)=−→ cells of XRCW gives the isomorphism Nα. The grv-

morphism β comes from associating to the k-cell of XRCW indexed by (s, τ) themap

|s| ≈−→ (s, τ) × |s| char.map−−−−−−→ XRCW

which is an element of NSk(XRCW). The equation NH∗φNβNα = id is straight-forward.

The proof that Nβ is an isomorphism is quite similar to to that (for β) inthe proof of Theorem 18.2. Indeed, using Lemma 18.8, NH∗( ) is a homologytheory and thus diagrams like in (18.5) and (18.6) do exist; the isomorphismH∗µ : NH∗(∆k−1,Bd∆k+1)fl≈H∗(∆k−1,Bd∆k+1) is also explicit enough and per-mits to proceed as in the proof of Theorem 18.2. Details are left to the reader.

The GrV-isomorphism H∗µ of Lemma 18.8 is natural; one has thus a commu-tative diagram

NH∗(XRCW )

µ≈

NH∗φ

≈// NH∗(X)

µ≈

H∗(XRCW )

H∗φ // H∗(X)

19. EILENBERG-MCLANE SPACES 101

which shows that H∗φ is an isomorphism. The relative case is obtained as at theend of the proof of Theorem 18.2 and that H∗φ is a GrA-isomorphism comes fromKronecker duality.

Theorem 18.11. There is a covariant functor symp: (X,Y )→ (KX ,KY ) fromTop2 → Simp2 and a natural transformation φ = φ(X,Y ) : (|KX |, |KY |) → (X,Y )from | symp| to the indentity functor of Top2 such that H∗φ and H∗φ are isomor-phisms.

Proof. Let X be a topological space. By the proof of Theorem 18.7, the regu-lar CW-complexXRCW comes equipped with characteristic embeddings ϕ(s,τ) : (s, τ))×|s| → XRCW ((s, τ)) ∈ NS(X)), satisfying the following condition: if t is a face of swith simplicial inclusion it,s : t→ s, then ϕ(t,τ |it,s|) = ϕ(s,τ) |it,s|. The only miss-

ing thing to make XRCW a simplicial complex is that several simplexes may havethe same boundary. But this can be avoided by taking the barycentric subdivisionof each cell, with the characteristic embedding ϕ′(s,τ) : (s, τ)) × |(s)′| → XRCW .

We thus get a functorial triangulation of XRCW .

Remarks 18.12. The following facts about the above constructions should benoted.

(a) The proof of Theorem 18.2 goes back to Giever [65]; for a more recenttreatment, see [71, p. 146]. Theorem 18.7 may be obtained from Theo-rem 18.7 by subdivision techniques in semi-simplicial complexes (see [71,Theorem 16.41]). Our proof of Theorem 18.7 is different.

(b) The construction XCW in the proof of Theorem 18.2 is sometimes calledin the literature the thick geometric realization of the singular complex

of X . A quotient XCW of XCW was introduced by J. Milnor [143], inwhich the degenerate simplexes are collapsed. Thus, XCW has one k-cellfor each non-degenerate singular k-simplex of X . Under mild conditions,the Milnor functor behaves well with products (see [143, § 2]).

(c) The maps φ of Theorem 18.2, 18.7 and 18.11 are actually weak homotopyequivalences (see e.g. [71, Corollary 16.43]). Such maps are called CW-approximations [80] or resolutions [71]. In particular, if X is itself aCW-complex, these maps are homotopy equivalences by the Whiteheadtheorem [80, Theorem 4.5] (but they are not homeomorphisms). Somehowsimpler (but not functorial) proofs that a spaces has the weak homotopytype of a CW-complex may be found in e.g. [80, Proposition 4.13] or [71,Proposition 16.4].

(d) By its construction in the proof of Theorem 18.7, XRCW is a regular ∆-set is the sense of [80, p. 533–34]. In this appendix of [80], the readermay find enlightening considerations related to our constructions in thissection.

19. Eilenberg-McLane spaces

The Eilenberg-McLane spaces are used to make the cohomology H∗(−) a rep-resentable functor. With Z2 as coefficients, they admit an ad hoc presentation givenbelow, which only uses the material developed in this book. The equivalence withthe usual definition using the homotopy groups is proven at the end of the section.

A CW-complex K is an Eilenberg-McLane space in degree m if


(i) Hm(K) = Z2; we denote by ι the generator of Hm(K).(ii) for any CW complex X , the correspondence f 7→ H∗f(ι) gives a bijection

φ : [X,K]≈−→ Hm(X) ,

where [X,K] denotes the set of homotopy classes of continuous maps fromX to K.

If f : X → K is a map, the class H∗f(ι) is said to be represented by f . Property(ii) says that the functorH∗(−) would be representable by K in the sense of categorytheory [131].

The notation K(Z2,m) is usual for a CW-complex which is an Eilenberg-McLane space in degree m. We shall also use the notation Km. The unambiguityof these notations is guaranteed by the following existence and uniqueness result.

Proposition 19.1. (a) For any integer m, there exists an Eilenberg-McLanespace in degree m.

(b) Let Km and K′m be two Eilenberg-McLane spaces in degree m. Then,there exists a homotopy equivalence g : K′m → Km whose homotopy classis unique.

Example 19.2. By Corollary 12.18, we see that the point is an Eilenberg-McLane space in degree 0.

Proof. We start by the uniqueness statement (b). Let K and K′ be twoEilenberg-McLane spaces in degree m. Then, there is a bijection

Z2 = Hm(K′) ≈ [K′,K]

under which the constant maps corresponds to 0. Let g : K′ → K be a continuousmap representing the non-vanishing class (unique up to homotopy). In the sameway, let h : K → K′ represent the non-vanishing class of Z2 = Hm(K) ≈ [K,K′].Then, gh represent the non-vanishing class of Z2 = Hm(K) ≈ [K,K] and hgrepresent the non-vanishing class of Z2 = Hm(K′) ≈ [K′,K′]. As idK and idK′ dothe same, we deduce that gh is homotopic to idK and hg is homotopic to idK′ .Therefore, h and g are homotopy equivalences.

We now construct an Eilenberg-McLane space K in degree m ≥ 1 (K0 = pt,as noticed in Example 19.2). Its m-skeleton Km is the sphere Sm, with one 0-cellv and one m-cell called ε. Then, for each map ϕ : Sm → Km of degree 0, an(m+1)-cell is attached to to Km via ϕ, thus getting Km+1. Finally, for k ≥ m+ 2,Kk is constructed by induction by attaching to Kk−1 a k-cell for each continuousmap f : Sk−1 → Kk−1.

As the (m+ 1)-cells of K are attached to Km by maps of degree 0, the cellular

boundary ∂ : Cn+1(K)→ Cn(K) vanishes by Lemma 16.3. Therefore, Hm(K) = Z2

by Theorem 16.6 and Hm(K) = Z2 by Kronecker duality. The singleton ε, seenas a cellular m-cycle of K, is called ι ∈ Zm(K). Seen as an m-cocycle, we denote it

by ι ∈ Zm(K) (it represents ι ∈ Hm(K)).Let us prove the surjectivity of φ : [X,K] → Hm(X). Let a ∈ Hm(X), repre-

sented by a cellular cocycle a ∈ Cm(X) ⊂ Cm(Xm). We shall construct a mapf : X → K such that H∗f(ι) = a. Let j : Dm → K be a characteristic map for theunique m-cell of K. The map f sends Xm−1 to the point v = K0. Its restrictionto an m-cell e of X is equal to j if e ∈ a and the constant map onto v otherwise.

19. EILENBERG-MCLANE SPACES 103

This gives a map f : Xm → Km which, by construction and Lemma 16.14, satisfies

(19.3) C∗f(α) = 〈a, α〉 ιfor all α ∈ Cm(X). Hence

〈C∗f(ι), α〉 = 〈ι, C∗f(α)〉 = 〈ι, 〈a, α〉ι〉 = 〈a, α〉for all α ∈ Cm(X). By Lemma 3.4, we deduce that C∗f(ι) = a.

To extend f to Xm+1, let λ ∈ Λm+1(X) with attaching map ϕλ : Sm → Xm.As a is a cocycle, one has 〈a, ∂λ〉 = 〈δ(a), λ〉 = 0. Using Lemmas 16.3 and 16.14together with Equation (19.3), we get that fλ = f ϕλ : Sm → Km ≈ Sm is amap of degree 0. By construction of K, a (m + 1)-cell e is attached to Km via fλ,so f may be extended to λ, using a characteristic map for e extending fλ. Thisproduces a cellular map fm+1 : xm+1 → Km+1. Finally, suppose, by inductionon k ≥ m + 1, that fm+1 extends to fk : Xk → Kk. Let λ ∈ Λk+1(X) withattaching map ϕλ : Sk → Xk. Set gλ = fk ϕλ. By construction of K, there existsegλ∈ Λk+1(K) with attaching map gλ. Thus fk may be extended to the cell λ, using

a characteristic map for eλ extending gλ. Doing this for each λ ∈ Λk+1(X) producesthe desired extension fk+1 : Xk+1 → Kk+1. the surjectivity of φ : [X,K]→ Hm(X)is thus established.

For the injectivity of φ, let f0, f1 : X → K such that H∗f0(ι) = H∗f1(ι).Since any map between CW-complexes is homotopic to a cellular map (see e.g. [62,Theorem 2.4.11]), we may assume that f0 and f1 are cellular. We must constructa homotopy F : X × I → K between f0 and f1, which will be done cell by cell. Asf0(X

m−1) = f1(Xm−1) = v, The maps f0 and f1 descend to cellular maps from

X/Xm−1 to K. Hence, we can assume that Xm−1 = X0 is a single point w, withf0(w) = f1(w) = v. The homotopy F is defined to be constant on w: F (w, t) = v.

Let λ ∈ Λm(X) with characteristic map ϕλ : Sm → X .

As Xm−1 is a point, the homology class H∗f0(ι) = H∗f1(ι) is represented by a

single cellular cocycle a ∈ Cm(X) (Bm(X) = 0). Let λ ∈ Λm(X) with characteristicmap ϕλ : Sm → X . By Lemma 16.14, one has, for j = 0, 1:

(19.4) 〈a, λ〉 = 〈C∗fj(ι), λ〉 = 〈ι, C∗fj(λ)〉 = 〈ι, deg(fj ϕλ)ι〉 = deg(fj ϕλ) .

Let Σm be the boundary of Dm × I, homeomorphic to Sm. A map Fλ : Σm → Kmis defined by

(19.5) Fλ(x, t) =

f0(ϕλ(x)) if t = 0

f1(ϕλ(x)) if t = 1

v if x ∈ Sm−1 .

Using (19.4) together with Lemma 13.10 (with B1 = Dm×0 and B2 = Dm×1),we deduce that degFλ = 0. Then, there is an (m+ 1)-cell of K is attached to Kmwith Fλ. This implies that F extends to Fλ : Dm× I → Km+1 which is a homotopyfrom f0 to f1 over Xm union the cell λ. Doing this for each λ ∈ Λm(X) producesa homotopy Fm : Xm × I → Km+1 between f0 and f1. We can thus assume, byinduction on k ≥ m, that a homotopy F k : Xk × I → Kk+1 between f0 and f1has been constructed. We must extend it to F k+1 : Xk+1 × I → Kk+2, which canbe done individually over each cell λ ∈ Λk+1(X). We define Fλ : Σk+1 → Kk+1 asin (19.5). As k + 1 > m, a (k + 2)-cell of K is attached to Kk+1 with Fλ, whichpermits us, as above, to extend the homotopy F k over the cell λ.

The proof of Proposition 19.1 is now complete.


The above construction of an Eilenberg-McLane space uses a lot of cells sowe may expect that the (co)-homology of Kn is complicated. It was computed byJ.P. Serre [171, § 2], whose theorem will be given in Theorem 49.11. In degree 1however, we have the following simple example of an Eilenberg-McLane space.

Proposition 19.6. The projective space RP∞ is an Eilenberg-McLane spacein degree 1 ( RP∞ ≈ K(Z2, 1)).

Proof. We use the standard CW-structure on K = RP∞ of Example 15.6,with one cell in each dimension and so that Kk = RP k. Let pk : S1 → S1 given bypk(z) = zk. The following properties hold true:

(i) the 2-cell of K is attached to K1 ≈ S1 by the map p2 which, by Proposi-tion 13.7 is of degree 0.

(ii) each map g : S1 → K1 of degree 0 is null-homotopic (i.e. homotopic to aconstant map) in K2. Indeed, it is classical that any map from S1 to S1 ≈S1 is homotopic to pk for some integer k (see e.g. [133, Theorem 5.1] or[80, Theorem 1.7]). By Proposition 13.7, deg pk = 0 if and only if k = 2r.Point (i) implies that g = p2 is null homotopic and so is p2r = p2pr.

(iii) for k ≥ 2, each map g : Sk → Kk is null-homotopic into Kk+1. Indeed,the lifting property of covering spaces tells us that g admits a lifting

Sk

p

Sk

g<<y

yy

yg // RP k

and the (k + 1)-cell of K is attached via the covering map p.

By Point (i), H1(K) = Z2. Points (ii) and (iii) imply that the argument of theproof of Proposition 19.1 may be used to prove that φ : [X,RP 1] → H1(X) is abijection. Hence, K = RP∞ is an Eilenberg-McLane space in degree 1.

Corollary 19.7. Let f : RPn → RP k be a continuous map, with n < k ≤ ∞.Then f is either homotopic to a constant map or to the inclusion RPn → RP k.

Proof. The lemma is true for k = ∞ by Proposition 19.6. Therefore, thereis a homotopy from the composition of f with the inclusion RP k → RP∞ toeither a constant map or the inclusion. Making this homotopy cellular (see [203,Corollary 4.7, p. 78]) produce a homotopy whose range is in RPn+1.

We finish this section with the relationship between our definition of Eilenberg-McLane spaces and the usual one involving the homotopy groups. Recall that thei-th homotopy group πi(X,x) of a pointed space (X,x) is defined by πi(X,x) =[Si, X ]•, for some fixed base point in Sn. Below, the base points are omitted fromthe notation.

Proposition 19.8. A CW-complex X is an Eilenberg-McLane space Km if andonly if πi(X) = 0 if i 6= m and πm(X) = Z2.

Proof. We first prove that Km satisfies the conditions. By Propositions 19.1and 19.6, the space K1 is homotopy equivalent to RP 1. The statement then followsusing the 2-fold covering S∞ → RP∞ and the fact that S∞ is contractible (see [80,example 1.B.3 p. 88]). By Proposition 19.1 and its proof, the space Km admits a

20. GENERALIZED COHOMOLOGY THEORIES 105

CW -structure whose (m− 1)-skeleton is a point. Thus, when m > 1, Km is simplyconnected and [Si,Km]• ≈ [Si,Km] (see [80, proposition 4A.2]). The cohomology ofSi, computed in Proposition 19.1, implies that the set [Si,Km] ≈ Hm(Si) containsone element if i 6= m and two elements if i = m.

Conversely, if X is a CW-complex satisfying πi(X) = 0 if i 6= m and πm(X) =Z2, we must prove that X is homotopy equivalent to Km. This requires techniquesnot developed in this book. When m = 1, there exists a map f : X → RP∞ ≈ K1

inducing an isomorphism on the fundamental group (see (24.2)). The map f theninduces an isomorphism on all the homotopy groups, what is called a weak homotopyequivalence. By the Whitehead theorem [80, Theorem 4.5], a weak homotopy equiv-alence between connected CW complexes is a homotopy equivalence. When m > 1,let α : Sm → X representing the non-zero element of πm(X). By the Hurewicztheorem [80, Theorem 4.32], the integral homology Hm(X ; Z) = Z2 and, from theuniversal coefficient theorem [80, Theorem 3B.5], it follows that Hm(X) = Z2 andH∗α : Hm(Sm)→ Hm(X) is an isomorphism. By Kronecker duality, Hm(X) = Z2

and H∗α : Hm(X) → Hm(Sm) is an isomorphism. Let g : X → Km representingthe non-zero element of Hm(X). As H∗α : Hm(X)→ Hm(Sm) is an isomorphism,the map g induces an isomorphism from πm(X) to πm(Km) (we have proved abovethat πm(Km) = Z2). Hence, g is a weak homotopy equivalence and therefore ahomotopy equivalence by the Whitehead theorem.

20. Generalized cohomology theories

The axiomatic viewpoint for (co)homology was initiated by Eilenberg andSteenrod in the late 1940’s [50, 49] and had a great impact on the general un-derstanding of the theory. We give below a version in the spirit of [80, Sections 2.3and Chapter 3]. Our application will be the Kunneth theorem 27.15.

A cohomology theory is a contravariant functor h∗ from the category Top2 oftopological pairs to the category GrV of graded Z2-vector spaces, together witha natural connecting homomorphism δ∗ : h∗(A)→ h∗+1(X,A) (the notation h∗(A)stands for h∗(A, ∅)). In addition, the following axioms must be satisfied.

(1) Homotopy axiom: if f, g : (X,A) → (X ′, A′) are homotopic, then h∗f =h∗g.

(2) Exactness axiom: for each topological pair (X,A) there is a long exactsequence

· · · δ∗−−→ hm(X,A)→ hm(X)→ hm(A)δ∗−→ hm+1(X,A)→ · · ·

where the unlabelled arrows are induced by inclusions. This exact se-quence is functorial, i.e. if f : (X ′, A′)→ (X,A) is a map of pair, there isa commutative diagram

· · · // h∗(X)

h∗f

// h∗(A)

h∗f

δ∗ // h∗+1(X, A)

h∗f

// h∗+1(X)

h∗f

// · · ·

· · · // h∗(X ′) // h∗(A′)δ∗ // h∗+1(X ′, A′) // h∗+1(X ′) // · · ·

(3) Excision axiom: let (X,A) be a topological pair, with U be a subspace ofX satisfying U ⊂ intA. Then, the GrV-morphism induced by inclusionsi∗ : h∗(X,A)→ h∗(X − U,A− U) is an isomorphism.


(4) Disjoint union axiom: for a disjoint union (X,A) =⋃j∈J (Xj , Aj) the

homomorphism

h∗(X,A)→∏

j∈J

h∗(Xj , Aj)

induced by the family of inclusions (Xj , Aj) → (X,A) is an isomorphism.

Examples 20.1. The singular cohomology H∗ is a generalized cohomologytheory. Axioms (1)–(3) are fulfilled, as seen in § 12.2–12.4. The disjoint unionaxiom corresponds to Proposition 12.17 for a pair (X, ∅); it may be extended toarbitrary topological pairs, using the exactness axiom and the five lemma.

K-theory and cobordism are other examples of generalized cohomology theo-ries.

Let h∗ and k∗ be two cohomology theories. A natural transformation µ from h∗

to k∗ is a natural transformation of functors commuting with the connecting homo-morphisms. In particular, for each topological pair (X,A), one has a commutativediagram of exact sequences:

(20.2)

· · · // h∗(X)

µ

// h∗(A)

µ

δ∗ // h∗+1(X, A)

µ

// h∗+1(X)

µ

// · · ·

· · · // k∗(X) // k∗(A)δ∗ // k∗+1(X, A) // k∗+1(X) // · · ·

The aim of this section is to prove the following theorem.

Proposition 20.3. Let h∗ and k∗ be two cohomology theories and let µ be

a natural transformation from h∗ to k∗. Suppose that µ : h∗(pt)≈−→ k∗(pt) is an

isomorphism. Then µ : h∗(X,A)≈−→ k∗(X,A) is an isomorphism for all CW-pairs

(X,A) where X is finite dimensional.

The hypothesis that X is finite dimensional is not necessary in Proposition 20.3(see [80, Proposition 3.19]), but it simplifies the proof considerably. Proposi-tion 20.3 is enough for the applications in this book (see § 27).

Proof. We essentially recopy the proof of [80, Proposition 3.19]. By Dia-gram (20.2) and the five-lemma, it suffices to show that µ is an isomorphism whenA = ∅. The proof goes by induction on the dimension of X . When X is 0-dimensional, the result holds by hypothesis and by the axiom for disjoint unions.Diagram (20.2) for (X,A) = (Xm, Xm−1) and the five-lemma reduce the induc-tion step to showing that µ is an isomorphism for the pair (Xm, Xm−1). Letϕm : Λm × (Dm, Sm−1)→ (X ,Xm−1) be a global characteristic maps for all the mcells of X. Like in the proof of Lemma 15.8, the axioms (essentially excision) implythat h∗ϕm and k∗ϕm are isomorphisms so, by naturality, it suffices to show thatµ is an isomorphism for Λm × (Dm, Sm−1). The axiom for disjoint unions gives afurther reduction to the case of the pair (Dm, Sm−1). Finally, this case follows byapplying the five-lemma to Diagram (20.2), since Dm is contractible and hence iscovered by the 0-dimensional case, and Sn−1 is (n− 1)-dimensional.


2.1. Give the list of the (maximal) simplexes of the triangulation of ∆m × I usedin the proof of Proposition 12.48. Draw them for m = 1, 2. Same question for thetriangulation used in the proof of Lemma 12.57.


2.2. Let X be a topological space.

(a) Show that X is contractible if and only if there is a correspondence f 7→ fassociating to a continuous map f : A → X an extension f : CA → X ,where CA is the cone over A. This correspondence is natural in the

following sense: if g : B → A is a continuous map, then f g = f Cg.(b) Using (a), find a direct proof of that the (co)homology of a contractible

space is isomorphic to that of a point. [Hint: use that ∆n+1 ≈ C∆n.]

2.3. Let X be a 2-sphere or a 2-torus. Let A be a subset of X containing n points.Compute H∗(X −A) and H∗(X,A).

2.4. Find topological pairs (X,Y ) and (X ′, Y ′) such that H∗(X,Y ) 6≈ H∗(X′, Y ′)

while X is homeomorphic to X ′ and Y is homeomorphic to Y ′.

2.5. Let X be a topological space. Let A be a subspace of X which is open andclosed. Show that (X,A) is well cofibrant.

2.6. Show that there is no retraction from the Mobius band onto its boundary.

2.7. Show that the Klein bottle K is made out of two copies of the Mobius bandglued along their common boundaries. Compute H∗(K), using the Mayer-Vietorisexact sequence for this decomposition.

2.8. Let f : Sn → Sn be a continuous map such that no antipodal pair of pointsgoes to an antipodal pair of points. Show that the degree of f is 0.

2.9. Let (X,Y, Z) be a topological triple. Find a commutative diagram linking theexact sequences of the pairs (X,Y ), (X,Z), (Y, Z) and that of the triple (X,Y, Z).

2.10. Let (X,X1, X2, X0) be a Mayer-Vietoris data with X = X1 ∪ X2. Supposethat X is a CW-complex and that Xi are subcomplexes. Find a short proof of theexistence of the Mayer-Vietoris for the cellular (co)homology. [Hint: analogous tothe simplicial case.]

2.11. Let (X,X1, X2, X0) be a Mayer-Vietoris data with X = X1 ∪ X2. Sup-pose that the homomorphism H∗(X1, X0) → H∗(X,X2) induced by the inclusionis an isomorphism. Deduce the Mayer-Vietoris (co)homology exact sequences for(X,X1, X2, X0).

2.12. Using a tubular neighbourhood and the Mayer-Vietoris sequence, computethe homology of the complement of a (smooth) knot in S3.

2.13. Let X be a countable CW-complex. Show that H∗(X) is countable. Is ittrue for H∗(X)?

2.14. Let X be a non-empty CW-complex. Show that X is contractible if and onlyif for any CW-pair (P,Q), any continuous map f : Q → X admits a continuous

extension f : P → X .

2.15. For n ∈ N≥1, consider the circle Cn := z ∈ C | |z − 1/n| = 1/n. TheHawaiian earring is the subspace B of C constituted by the union of Cn for n ≥ 1.

(a) Show that H1(B) surjects onto∏∞n=1 Z2.

(b) Show that [B,RP∞] is countable.(c) Deduce from (a) and (b) that B has not the homotopy type of a CW-

complex.


2.16. Let Rq, q ∈ N≥1 be a sequence of Z2-vector spaces. Find a path-connectedspace X such that Hq(X) ≈ Rq.2.17. Let X be a 2-dimensional CW-complex with a single 0-cell, m 1-cells and n2-cells. Show that m = n if and only if b1(X) = b2(X).

2.18. Find perfect CW-decompositions for the 2-torus and the Klein bottle.

2.19. Let P = 〈A|R〉 be a presentation of a group G with a set A of generatorsand a set B of relators. The presentation complex XP is the 2-dimensional complexobtained from a bouquet CA of circles indexed by A by attaching, for each relatorr ∈ R, a 2-cell according to r ∈ π1(CA). Hence, π1(X

1P ) ≈ G. Compute H∗(XP )

in the following cases.

• P = 〈a, b, c | abc−1b−1, bca−1c−1〉 and P = 〈x, y |xyxy−1x−1y−1〉 (two pre-sentations of the trefoil knot group).• P = 〈a, b, c | a5, b3, (ab)2〉 (a presentation of the alternate group A5).

2.20. Let K = K(Z2, n) be an Eilenberg-McLane space. Let f : K → K be acontinuous map. Show that f is either homotopic to the identity or to a constantmap.

2.21. LetKn = K(Z2, n) be an Eilenberg-McLane space. LetX be a non-contractible

CW-complex. Suppose that there are continuous maps Xj−→ Kn r−→ X such that

rj is homotopic to the identity. Show that X and Kn have the same homotopytype.

CHAPTER 3

Products

So far, the reader may not have been impressed by essential differences betweenhomology and cohomology: the latter is dual to the former via the Kroneckerpairing, so they are even isomorphic for spaces of finite homology type. However,cohomology is a definitely more powerful invariant than homology, thanks to itscup product, making H∗(−) a graded Z2-algebra. Thus, the homotopy types of twospaces with isomorphic homology may sometimes be distinguished by the algebra-structure of their cohomology. Simple examples are provided by RP 2 versus S1∨S2,or by the 2-torus versus the Klein bottle.

In this chapter, we present the cup product for simplicial and singular coho-mology, out of which the cap and cross products are derived, with already manyapplications (more will come in other chapters).

Cohomology and its cup product occured in 1935 (40 years after homology) inthe independent works of Kolmogoroff and Alexander, soon revisited and improvedby Chech and by Whitney [27, 205]. These people were all present in the inter-national topology conference held in Moscow, September 1935. Vivid recollectionsof this memorable meeting were later written by Hopf and by Whitney [100, 207].For surveys of the interesting history of cohomology and products, see [38, ChapterIV] and [134].

22. The cup product

22.1. The cup product in simplicial cohomology. Let K be a simplicialcomplex. Choose a simplicial order ≤ on K. Let a ∈ Cp(K) and b ∈ Cq(K). UsingPoint (c) of Lemma 3.4, we define a cochaina ≤ b ∈ Cp+q(K) by the formula

〈a ≤ b, σ〉 = 〈a, v0, . . . , vp〉〈b, vp, . . . , vp+q〉 ,required to be valid for all σ = v0, . . . , vp+q ∈ Sp+q(K), with v0 < v1 < · · · <vp+q. This defines a map

Cp(K)× Cq(K)≤−−→ Cp+q(K) .

We can see ≤ as a composition law on C∗(K):

C∗(K)× C∗(K)≤−−→ C∗(K) .

Lemma 22.1. (C∗(K),+,≤) is a (non-commutative) graded Z2-algebra.

Proof. The associativity and distributivities are obvious. The neutral elementfor ≤ is the unit cochain 1 ∈ C0(K).

Lemma 22.2. δ(a ≤ b) = δa ≤ b+ a ≤ δb.

109

110 3. PRODUCTS

Proof. Set a ∈ Cp(K), b ∈ Cq(K) and σ = v0, . . . , vp+q+1 with v0 < v1 <· · · < vp+q+1.

One has

〈δa ≤ b, σ〉 = 〈δa, v0, . . . , vp+1〉〈b, vp+1, . . . , vp+q+1〉= 〈a, ∂v0, . . . , vp+1〉〈b, vp+1, . . . , vp+q+1〉

=

p+1∑

i=0

〈a, v0, . . . , vi, . . . , vp+1〉〈b, vp+1, . . . , vp+q+1〉 .(22.2)

In the same way,

(22.3) 〈a ≤ δb, σ〉 =p+q+1∑

i=p+1

〈a, v0, . . . , vp+1〉〈b, vp+1, . . . , vi, . . . , vp+q+1〉 .

The last term in the sum of (22.2) is equal to the first term in the sum of (22.3).Hence, these terms cancel if we add up the two sums and the remaining terms give〈a ≤ b, ∂(σ)〉 = 〈δ(a ≤ b), σ〉.

Lemma 22.2 implies that Z∗(K) ≤ Z∗(K) ⊂ Z∗(K), B∗(K) ≤ Z

∗(K) ⊂B∗(K) and Z∗(K) ≤ B

∗(K) ⊂ B∗(K). Therefore, ≤ induces a map Hp(K) ×Hq(K)

−→ Hp+q(K), seen as a composition law on H∗(K):

H∗(K)×H∗(K)−→ H∗(K)

called the cup product on simplicial cohomology. The notation and the name cupproduct (the latter due to the former) were first used by Whitney [205]. It followsfrom Lemma 22.1 that (H∗(K),+,) is a graded Z2-algebra. Dropping the index“≤” is justified by the following proposition.

Proposition 22.4. The cup product on H∗(K) does not depend on the sim-plicial order “≤”.

Proof. The procedure to define the cup product may be done with the ordered

cochains. For a ∈ Cp(K) and b ∈ Cq(K), we define a b ∈ Cp+q(K) by theformula

〈a b, σ〉 = 〈a, (v0, . . . , vp)〉〈b, (vp, . . . , vp+q)〉 ,required to be valid for all (v0, . . . , vp+q) ∈ Sp+q(K). This defines a graded Z2-

algebra structure on C∗(K). The formula δ(a b) = δa b+ a δb is proven as

for Lemma 22.2, whence a graded algebra structure on H∗(K). These definitionsimply that the isomorphism

H∗φ≤ : (H∗(K),+,)≈−→ (H∗(K),+,≤)

of § 10 is an isomorphism of graded algebras. As, by Corollary 10.10, H∗φ≤ isindependent of the simplicial order “≤”, so is the cup product on H∗(K).

Corollary 22.5 (Commutativity of the cup product). The cup product insimplicial cohomology is commutative, i.e. a b = b a for all a, b ∈ H∗(K).

Proof. Let a, b ∈ Z∗(K) representing a, b. Let “≤” be a simplicial order onK. In Z∗(K), one has

a ≤ b = b ≥ a ,

where “≥” is the opposite order of “≤”. By Proposition 22.4, this proves Corol-lary 22.5.

22. THE CUP PRODUCT 111

Let GrA be the category whose objects are commutative graded Z2-algebrasand whose morphisms are algebra maps. Corollary 22.5 says that H∗(K) is anobject of GrA. There is an obvious forgetful functor from GrA to GrV.

Proposition 22.6 (Functoriality of the cup product). Let f : L → K be asimplicial map. Then H∗f : H∗(K) → H∗(L) is multiplicative: H∗f(a b) =H∗f(a) H∗f(b) for all a, b ∈ H∗(K).

Proof. The proof of Proposition 22.4 shows that H∗(K) is an object of GrA.

Using Corollary 10.8, it also shows that the isomorphism H∗ψ : H∗(K) → H∗(K)

is a GrA-isomorphism. Let a ∈ Cp(K) and b ∈ Cq(K). Then, for all σ =

(v0, . . . , vp+q) ∈ Sp+q(K), one has

〈C∗f(a b), σ〉 = 〈a b, (f(v0), . . . , f(vp+q))〉= 〈a, (f(v0), . . . , f(vp))〉〈b, (f(vp), . . . , f(vp+q))〉= 〈C∗f(a), (v0, . . . , vp)〉〈C∗f(b), (vp, . . . , vp+q)〉= 〈C∗f(a) C∗f(b), σ〉 .

By Lemma 3.4, this implies that C∗f(a b) = C∗f(a) C∗f(b). We deduce that

H∗f : H∗(K)→ H∗(L) is multiplicative. Using Proposition 10.11, this implies thatH∗f is multiplicative.

Proposition 22.6 has the following corollary.

Corollary 22.7. The simplicial cohomology is a contravariant functor fromSimp to GrA.

The cup product may also be defined in relative simplicial cohomology. Let L1

and L2 be two subcomplexes of K. For any simplicial order “≤” on K, one has

C∗(K,L1) ≤ C∗(K,L2) ⊂ C∗(K,L1 ∪ L2) .

Hence, we get a map

H∗(K,L1)×H∗(K,L2)−→ H∗(K,L1 ∪ L2)

which is bilinear and commutative.In particular, we get relative cup products

H∗(K,L)×H∗(K)−→ H∗(K,L) and H∗(K)×H∗(K,L)

−→ H∗(K,L)

which are related as described by the following two lemmas.

Lemma 22.8. Let (K,L) be a simplicial pair. Denote by j : (K, ∅) → (K,L)the inclusion. Let a ∈ Hp(K,L) and b ∈ Hq(K,L). Then, the equality

H∗j(a) b = a b = a H∗j(b)

holds in Hp+q(K,L).

Proof. Denote also by a ∈ Zp(K,L) and b ∈ Zq(K,L) cocycles representingthe cohomology classes a and b. Choose a simplicial order on K and let σ =v0, . . . , vp+q ∈ Sp+q(K) − Sp+q(L) with v0 ≤ · · · ≤ vp+q. Let σ1 = v0, . . . , vpand σ2 = vp, . . . , vp+q. One has

(22.9) 〈C∗j(a) b, σ〉 = 〈C∗j(a), σ1〉〈b, σ2〉 = 〈a, C∗j(σ1)〉〈b, σ2〉

112 3. PRODUCTS

and

(22.10) 〈a b, σ〉 = 〈a, σ1〉〈b, σ2〉 .If σ1 ∈ Sp(L), then C∗j(σ1) = 0 and the right hand sides of (22.9) and (22.10)both vanish. If σ1 /∈ Sp(L), then C∗j(σ1) = σ1 and the right hand sides of (22.9)and (22.10) are equal. As Cp+q(K,L) is the vector space with basis Sp+q(K) −Sp+q(L), this proves that H∗j(a) b = a b. The other equation is provensimilarly.

The proof of the following lemma, quite similar to that of Lemma 22.8, is leftto the reader (Exercise 3.1).

Lemma 22.11. Let (K,L) be a simplicial pair. Denote by j : (K, ∅) → (K,L)the inclusion. Let a ∈ Hp(K) and b ∈ Hq(K,L). Then, the equality

H∗j(a b) = a H∗j(b)

holds in Hp+q(K).

There is also a relationship between the relative cup product and the connectinghomomorphism of simplicial a pair.

Lemma 22.12. Let (K,L) be a simplicial pair. Denote by i : L→ K the inclu-sion. Let a ∈ Hp(K) and b ∈ Hq(L). Then, the equality

δ∗(b H∗i(a)) = δ∗b a

holds true in Hp+q+1(K,L).

Proof. Denote also by a ∈ Zp(L) and b ∈ Zq(L) the cocycles represent-ing the cohomology classes a and b. Let b ∈ Cq(K) be an extension of thecochain b. The cochain b a ∈ Cp+q(K) is then an extension of b C∗i(a).By Lemma 7.2, δK(b a) ∈ Zp+q+1(K,L) represents δ∗(b H∗i(a)), whereδK : C∗(K) → C∗+1(K) is the coboundary homomorphism for K. As a is a co-cycle, δK(b a) = δK(b) a. By Lemma 7.2 again, δK(b) a represents thecohomology class δ∗b a. This proves the lemma.

22.2. The cup product in singular cohomology. Let X be a topologicalspace and let σ : ∆m → X be an element of Sm(X). For 0 ≤ p, q ≤ m, we definepσ ∈ Sp(X) and σq ∈ Sq(X) bypσ(t0, . . . , tp) = σ(t0, . . . , tp, 0 . . . , 0) and σq(t0, . . . , tq) = σ(0, . . . , 0, t0, . . . , tq) .

The singular simplexes pσ and σq are called the front and back faces of σ. Leta ∈ Cp(X) and b ∈ Cq(X). Using Point (c) of Lemma 3.4, we define a cochaina b ∈ Cp+q(X) by the formula

(22.13) 〈a b, σ〉 = 〈a, pσ〉〈b, σq〉 ,required to be valid for all σ ∈ Sp+q(X). This defines a bilinear map

(22.14) Cp(X)× Cq(X)−→ Cp+q(X) .

The formula of Lemma 22.2 holds true, with the same proof. Hence, we get a cupproduct in singular cohomology: Hp(X) × Hq(X)

−→ Hp+q(X), giving rise to acomposition law

H∗(X)×H∗(X)−→ H∗(X) .

Proposition 22.15. (H∗(X),+,) is a commutative graded Z2-algebras.

22. THE CUP PRODUCT 113

Proof. The associativity and distributivities are easily deduced from the def-initions, like for the cup product in simplicial cohomology. If X is empty, thenH∗(X) = 0 and there is nothing to prove. Otherwise, the neutral element for is the class of the unit cochain 1 ∈ H0(X). Proving the commutativity directly israther difficult. We shall use that the singular cohomology of X is that of a sim-plicial complex (see Theorem 18.11), together with Proposition 22.16 below, whoseproof is straightforward.

Proposition 22.16. Let K be a simplicial complex. For any simplicial order≤ on K, the isomorphism

R∗≤ : H∗(|K|) ≈−→ H∗(K)

of Theorem 17.5 is an isomorphism of graded algebras.

Proposition 22.17. The singular cohomology is a contravariant functor fromTop to GrA.

Proof. By Proposition 22.15, we already know that H∗(X) is an object ofGrA. We also know, by Proposition 12.33, that H∗() is a contravariant functorfrom Top to GrV. It remains to prove the multiplicativity of H∗f : H∗(X) →H∗(Y ) for a continuous map f : Y → X . If σ ∈ Sp+q(X), then f pσ = p(f σ) andf σq = (f σ)q. Thus, the proof that C∗f(a b) = C∗f(a) C∗f(b) is the sameas for Proposition 22.6.

To get relative cup products as in simplicial cohomology, some hypothesis re-lated to the techniques of small simplexes (§ 12.4) is required. Let Y1 and Y2 besubspaces of a topolgical space X . Let Y = Y1 ∪ Y2 and B = Y1, Y2. We say that(Y1, Y2) is an excisive couple if H∗(Y )→ H∗B(Y ) is an isomorphism.

Lemma 22.18. A couple (Y1, Y2) of subspaces of X is excisive if and only if theinclusion (Y1, Y1 ∩ Y2) → (Y1 ∪ Y2, Y2) induces an isomorphism in (co)homology.

Proof. Let Y = Y1 ∪ Y2 and B = Y1, Y2 as above. There is a morphism

0 // C∗(Y, Y2) //

C∗(Y ) //

C∗(Y2) //

0

0 // C∗(Y1, Y1 ∩ Y2) // C∗B(Y ) // C∗(Y2) // 0

of short exact sequences of singular cochain complex. It induces a morphism of theassociated long exact sequences on cohomology which, by the five-lemma, impliesthe result.

Lemma 22.19. Let (Y1, Y2) be an excisive couple of a topological space X. Then,(22.14) defines a relative cup product

(22.20) H∗(X,Y1)×H∗(X,Y2)−→ H∗(X,Y1 ∪ Y2)

which is bilinear. The analogues of Lemmas 22.8–22.12 hold true.

Proof. Let Y = Y1 ∪ Y2 and B = Y1, Y2. Equation (22.13) gives a bilinearmap

C∗(X,Y1)× C∗(X,Y2)−→ C∗(X,Y B)

114 3. PRODUCTS

where C∗(X,Y B) = ker(C∗(X)→ C∗B(Y )). There is a commutative diagram

Hk(X)

≈

// Hk(Y )

// Hk+1(X,Y )

// Hk+1(X)

≈

// Hk+1(Y )

Hk(X) // HkB(Y ) // Hk+1(X,Y B) // Hk+1(X) // Hk+1

B (Y )

where the lines are exact. By the five-lemma If H∗(Y )→ H∗B(Y ) is an isomorphism,so isH∗(X,Y )→ H∗(X,Y B), whch gives (22.20). The properties of the relative cupproduct listed at the end of Lemma 22.19 are proved as in the simplicial case.

Remark 22.21. The couple (Y1, Y2) is excisive in X if and only if it is excisivein Y1 ∪ Y2. Thus, by Proposition 12.56, (Y1, Y2) is excisive when Y1 and Y2 areboth open. Also, (Y1, Y2) is excisive when one of the subspaces Yi is containedin the other, for instance if one is empty or if Y1 = Y2. In some situations, thehypothesis can be fulfilled by enlarging Yi to Y ′i without changing the homotopytype, and then (22.20) makes sense. As in Proposition 12.83 and its proof, this isthe case if X is a CW-complex and Yi are subcomplexes. Note that, if (Y1, Y2) isexcisive, then the Mayer-Vietoris sequence for (Y1 ∪ Y2, Y1, Y2, Y1 ∩ Y2) holds true,by Lemma 22.18 and Exercise 2.11.

23. Examples

Disjoint unions. Let X be a topological space which is a disjoint union:

X =⋃

j∈JXj .

By Proposition 12.17, the family of inclusions ij : Xj → X induce an isomorphismin GrV

H∗(X)(H∗ij)

≈//

∏j∈J H

∗(Xj) .

By Proposition 22.17, H∗ij is a homomorphism of algebras for each j ∈ J . Hence,the above map (H∗ij) is an isomorphism of graded algebras.

Bouquets. Let (Xj , xj), with j ∈ J , be a family of well pointed spaces whichare path-connected. By Proposition 12.79, the family of inclusions ij : Xj → X =∨j∈J Xj, for j ∈ J , gives rise to isomorphisms on reduced cohomology

H∗(X)(H∗ij)

≈// ∏

j∈J H∗(Xj) .

The reduced and unreduced cohomologies share the same positive parts: H>0( ) =

H>0( ). As each space Xj is path-connected, so is the their bouquet X . Thus

H>0(X) = H∗(X) and we get a GrV-morphism

H>0(X)(H∗ij)

≈// ∏

j∈J H>0(Xj) .

Being induced by continuous maps, (H∗ij) is multiplicative. AsX is path-connected,this produces the GrA-isomorphism

(23.1) H∗(X)≈−→ Z2 1⊕

∏

j∈J

H>0(Xj) .

23. EXAMPLES 115

When J is finite, one can also use the projections πj : X → Xj defined in (12.80).By Proposition 12.81, they produce a GrV-isomorphism

⊕j∈J H

>0(Xj)P

H∗πj

≈// H>0(X) (J finite) .

Being induced by continuous maps,∑H∗πj is multiplicative. As X is path-

connected, this produces the GrA-isomorphism

(23.2) Z2 1⊕⊕

j∈J

H>0(Xj)≈−→ H∗(X) (J finite) .

Connected sum(s) of closed topological manifolds. A closed n-dimensionaltopological manifold is a compact space such that each point has an open neigh-bourhood homeomorphic to Rn.

Let M1 and M2 be two closed n-dimensional topological manifolds. We supposethat M1 and M2 are connected. Let Bj ⊂ Mj be two embedded compact n-ballswith boundary Sj. We suppose that each ballBj is nicely embedded in a bigger ball;this implies that (Mj , Bj) and (Mj, Sj) are good pairs. Given a homeomorphism

h : B1≈−→ B2, form the closed topological manifold

M = M1♯hM2 = (M1 − intB1) ∪h (M2 − intB2) .

The manifold M is called a connected sum of M1 and M2. Though connectedtopological manifolds are homogeneous for nicely embedded balls (see e.g. [93,Theorem 6.7], the homeomorphism type of M may depend on h: for example, if h isobtained from h by precomposition with a homeomorphism of B1 which reverses theorientation, then M1♯hM2 has not, in general, the same homotopy type as M1♯hM2

(for instance, if M1 = M2 = CP 2, the two cases are distinguished by their integralintersection form). In most applications in the literature, the connected sum isdefined for oriented manifolds and one requires that h reverses the orientation;this makes the oriented homeomorphism type of M1♯M2 well defined. However, byProposition 23.3 below, the mod 2-cohomology algebra of M1♯hM2 does not dependon h, up to algebra isomorphism.

If each Mj admit a triangulation |Kj| ≈ Mj, then Kj is a connected n-dimensional pseudomanifold (see Corollary 31.7). The connected sum may be donein the world of pseudomanifolds, using n-simplexes for the balls Bj . By Proposi-tion 4.6, Hn(Mj) = Z2, generated by the fundamental class [Mi]. The statementHn(Mj) = Z2 also holds for closed connected topological manifolds (see e.g. [80,Theorem 3.26]). We denote by [Mj ]

♯ the generator of Hn(Mi) = Z2.

Proposition 23.3. Under the above hypotheses, the cohomology ring H∗(M1♯hM2)is isomorphic to the quotient of Z2 1⊕H>0(M1)⊕H>0(M2) by the ideal generatedby [M1]

♯ + [M2]♯:

H∗(M1♯hM2) ≈ Z2 1⊕H>0(M1)⊕H>0(M2)/

([M1]♯ + [M2]

♯) .

In particular, under this isomorphism, the classes [M1]♯ and [M2]

♯ both correspondto the fundamental class [M ]♯ of M .

Proof. Form the space M = M1∪hM2 and let B ⊂ M be the common image

ofB1 and B2 in M , with boundary S. As (Mj , Bj) are good pairs, Proposition 12.83

116 3. PRODUCTS

provides a Mayer-Vietoris sequence for (M,M1,M2, B). As B has the cohomologyof a point, one gets a multiplicative GrV-isomorphism:

α : H>0(M)≈−→ H>0(M1)⊕H>0(M2) .

As Mj is connected, so is M and α extend to a GrA-isomorphism

α : H∗(M)≈−→ Z2 1⊕H>0(M1)⊕H>0(M2) .

Let M = M1♯hM2 ⊂ M . The pair (M,M) is obviously a good pair, whence, by

excision and homotopy, the isomorphism H∗(M,M) ≈ H∗(B,S). The non-zero

part of H∗(B,S) is Hn(B,S) = Z2. Therefore, the homomorphism β∗ : Hk(M)→Hk(M) induced by the inclusion is an isomorphism, except possibly for k = n− 1

or n. In these degrees, the cohomology exact sequence of (M,M) looks like

Hn−1(M)β∗ // Hn−1(M)

δ∗ // Hn(M, M) //

≈

Hn(M)

≈

β∗ // Hn(M)

≈

// 0

Z2 Z2 ⊕ Z2 Z2

Therefore, β∗ : Hk(M)→ Hk(M) is an isomorphism for k ≤ n−1 and the GrA ho-

momorphism β∗ : H∗(M)→ H∗(M) is onto. The kernel of β∗ : Hn(M)→ Hn(M)is of dimension 1 and, by symmetry (M1∪hM2 = M2∪h−1M1), it must be generatedby [M1]

♯ + [M2]♯.

Remark 23.4. If we work simplicially with pseudomanifolds, the fact that

ker(β∗ : Hn(M) → Hn(M)) contains [M1]♯ + [M2]

♯ may be seen directly. Indeedthe n-cocycle consisting of the n-simplex Bj represents [Mj ]

♯ by Proposition 4.6.

Hence, the n-cocycle B represents [M1]♯ + [M2]

♯ in Hn(M) and and is in kerβ∗.

Cohomology algebras of surfaces. We start with the triangulation M ofRP 2 of p. 26, drawn in Figure 2 p. 26. We use the simplicial order given by thenumeration 0, . . . , 5 of the vertices. The computation of H∗(M) is given in (4.18)and the generator of H1(M) = Z2 is given by the cocycle a given in (4.19):

a = α =1, 2, 2, 3, 3, 4, 4, 5, 5, 1

⊂ S1(RP

2) .

We see that, in C2(M),

a a =1, 2, 3, 2, 3, 4, 3, 4, 5

.

Containing an odd number of 2-simplexes, Proposition 4.6 implies that a a isthe generator [M ]♯ of H2(M). Therefore, one gets a GrA-isomorphism

H∗(RP 2) ≈ Z2[a]/(a3)

from H∗(RP 2) to the quotient of the polynomial ring Z2[a] by the ideal generatedby a3. Using (23.1), this shows that RP 2 and S1∨S2 have not the same homotopytype though they have the same Betti numbers.

Our next example is the torus T 2. We use the triangulation given in Figure 3on p. 27 which shows two 1-cocycles a, b ∈ C1(T 2) whose cohomology classes, again

23. EXAMPLES 117

denoted by a and b, form a basis of H1(T 2) ≈ Z2 ⊕ Z2. One checks that thefollowing equations hold in C2(T 2):

a a =4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9

b b =2, 3, 6, 3, 6, 8

a b =6, 7, 8

b a =7, 8, 9

.

In H2(T 2) = Z2, generated by [T 2]♯, Proposition 4.6 implies that

a a = b b = 0 and a b = b a = [T 2]♯ .

Observe that a b 6= b a in C2(T 2), the equality only holding true in cohomol-ogy. We get a GrA-isomorphism

H∗(T 2) ≈ Z2[a, b]/(a2, b2) .

Our third example is the Klein bottle K, using the triangulation given in Fig-ure 4 on p. 28: analogously to the case of the torus, Figure 4 shows two 1-cocyclesa, b ∈ C1(K) whose cohomology classes, again denoted by a and b, form a basis ofH1(K) ≈ Z2 ⊕ Z2. The following equations hold in C2(K):

a a =4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 4, 5, 9

b b =2, 3, 6, 3, 6, 8

a b =6, 7, 8

b a =7, 8, 9

.

In H2(K) = Z2, generated by [K]♯, Proposition 4.6 implies

a a = [K]♯ , b b = 0 and a b = b a = [K]♯ .

Though H∗(T 2) and H∗(K) are GrV-isomorphic, we see that they are notGrA-isomorphic. Indeed, for a space X , consider the cup-square map

H∗(X)2

−−→ H∗(X)

given by 2 (x) = x x. Note that this map is linear, since the ground field isZ2. Our above computations show that 2= 0 for X = T 2 but not for X = K.It does not vanish either for X = RP 2, as seen above. Now, it is classical thata connected closed surface X is a connected sum of tori if X is orientable and aconnected sum of projective spaces otherwise. Hence, Proposition 23.3 implies thatthe orientability of a connected surface may be seen on its cohomology algebra:

Proposition 23.5. Let M be a closed connected surface. Then M is orientableif and only if its cup-square map H1(M)→ H2(M) vanishes.

Remark 23.6. As a consequence of Wu’s formula, we shall see in Corol-lary 58.11 that Proposition 23.5 generalizes in the following way: a closed connectedn-dimensional manifoldM is orientable if and only if the linear map sq1 : Hn−1(M)→Hn(M) vanishes.

Finally, we see that closed surfaces are distinguished by their cohomology al-gebra.

118 3. PRODUCTS

Proposition 23.7. Two closed surfaces are diffeomorphic if and only if theircohomology algebra are GrA-isomorphic.

Proof. By Proposition 23.5, the cohomology algebra determines whether aclosed surfaceM is orientable or not. If M is orientable, then M is a connected sumof m tori and, by Proposition 23.3H1(M) ≈ Z2m

2 . If M is not orientable, then M isa connected sum of m projective planes and, by Proposition 23.3H1(M) ≈ Zm2 .

24. Two-fold coverings

24.1. H1, fundamental group and 2-fold coverings. Let (Y, y) and (Y ′, y′)be two pointed spaces. Let [Y, Y ′]• be the set of homotopy classes of pointed mapsfrom Y to Y ′ (the homotopies also preserving the base point). Let F : [Y, Y ′]• →[Y, Y ′] be the obvious forgetful map.

Let (X,x) be a pointed topological space. We first define a map e : H1(X)→map(π1(X,x),Z2). Let a ∈ H1(X). If c : S1 → X is a pointed map representing[c] ∈ [S1, X ]• = π1(X,x), we set e(a)([c]) = H∗c(a) ∈ H1(S1) = Z2. As H∗c =H∗c′ if c is homotopic to c′, the map is well defined. Observe that map(π1(X,x),Z2)is naturally a Z2-vector space, containing hom(π1(X,x),Z2) as a linear subspace.

Lemma 24.1. Let X be a connected CW-complex, pointed by x ∈ X0. Then themap e is an isomorphism

e : H1(X)≈−→ hom(π1(X,x),Z2) .

Proof. We first prove that the image of e lies in hom(π1(X,x),Z2). Themultiplication in π1(X,x) = [S1, X ]• may be expressed using the comultiplicationµ : S1 →→ S1/S0 ≈ S1 ∨ S1. Then [c][c′] = [(c ∨ c′)µ]. Using that H1(S1 ∨ S1) ≈H1(S1)×H1(S1) (see Proposition 12.79), one has

e(a)([c][c′]) = H∗µ(e(a)([c]), e(a)([c′])

)= e(a)([c]) + e(a)([c′])

for all a ∈ H1(X). This proves that e([c][c′]) = e([c]) + e([c′]). The equalitye(a+ b)([c]) = e(a)([c]) + e(b)([c]) is obvious, so e is a homomorphism.

Let us consider RP∞ with its standard CW-structure of Example 15.6, withone cell in each dimension, pointed by its 0-cell a. Van Kampen’s Theorem impliesthat π1(RP∞, a) = Z2. The fundamental group functor gives rise to a map

(24.2) [X,RP∞]•≈−→ hom(π1(X,x),Z2)

which is a bijection. Indeed, the bijectivity is established in the same way as, inProposition 19.6, the fact that φ : [X,RP∞]→ H1(X) is a bijection. The forgetfulmap F : [X,RP∞]• → [X,RP∞] and the homomorphism e fit in the commutativediagram

(24.3)

[X,RP∞]•F //

≈

[X,RP∞]

≈

hom(π1(X,x),Z2) H1(X)eoo

The map F is surjective: if f : X → RP∞, any path γ from f(x) to a extendsto homotopy from f to a pointed map. This follows from the fact that (X, a)is cofibrant (see Proposition 15.2). Hence, the commutativity of Diagram (24.3)implies that e (and F ) are bijective.

24. TWO-FOLD COVERINGS 119

We now turn our attention to 2-fold coverings. The reader is assumed somefamiliarity with the theory of covering spaces, as presented in many textbooks (seee.g. [175, Chapter 2] or [80, section 1.3]).

Let X be a connected CW-complex, pointed by x ∈ X0. Two covering pro-jections pi : Xi → X are equivalent if there exists a homeomorphism h : X1 → X2

such that p2h = p1. Denote by Cov2(X) the set of equivalence classes of 2-foldcoverings of X .

Let p : X → X be a 2-fold covering. Choose x ∈ p−1(x). Then p∗(π1(X, x))is a subgroup of index ≤ 2 of π1(X,x). Let Grp2(π1(X,x)) be the set of such

subgroups. A subgroup of index ≤ 2 being normal, the subgroup p∗(π1(X, x)) doesnot depend on the choice of x ∈ p−1(x). We thus get a map

Cov2(X)≈−→ Grp2(π1(X,x))

which is a bijection (see, e.g, [80, Theorem 1.38]). For example, the trivial 2-foldcovering ±1×X → X corresponds to the whole group π1(X,x) which is of index1 ≤ 2. An element H ∈ Grp2(π1(X,x)) is the kernel of a unique homomorphismπ1(X,x)→→ π1(X,x)/H → Z2. This gives a bijection

Grp2(π1(X,x))≈−→ hom(π1(X,x),Z2) .

If f : X → RP∞ is a continuous map, one can form the pullback diagram

(24.4)

Xf //

p

S∞

p∞

X

f // RP∞ .

In detail, X = (u, z) ∈ X×S∞ | f(u) = p∞(z), with p(u, z) = u and f(u, z) = z.We say that the covering projection p is induced from p∞ by the map f and writep = f∗p∞. Observe that p correspond to the subgroup kerπ1f . Thus, homotopicmaps induce equivalent coverings and we get a map ind: [X,RP∞] → Cov2(X).These various maps, together with those of (24.3) sit in the commutative diagram

(24.5)

[X,RP∞]ind //

φ

≈ &&LLLL

LLLL

LCov2(X)

≈

H1(X)

e

≈ ''PPPPPPPPPPGrp2(π1(X,x))

≈

[X,RP∞]•

≈F

OO

≈ // hom(π1(X,x),Z2)

The commutativity of Diagram (24.5) implies the following proposition.

Proposition 24.6. ind: [X,RP∞]→ Cov2(X) is a bijection.

Let p : X → X be a 2-fold covering. A continuous map f : X → RP∞ suchthat p is equivalent to f∗p∞ is called a characteristic map for the covering p.Proposition 24.6 implies the following corollary.

Corollary 24.7. Let X be a connected CW-complex. Then, any 2-fold cov-ering admits a characteristic map. Two such characteristic maps are homotopic.

120 3. PRODUCTS

Let p : X → X be a 2-fold covering. The correspondence which, over eachx ∈ X , exchanges the two points of p−1(x) defines a homeomorphism τ : X → X,which is an involution (i.e. τ τ = id) without fixed point. Also, τ is a decktransformation, i.e. pτ = p. We call τ the deck involution of p. For the coveringp∞ : S∞ → RP∞, one has τ(z) = −z.

Lemma 24.8. A continuous map f : X → RP∞ is a characteristic map for the2-fold covering p : X → X if and only if there exists a commutative diagram

Xf //

p

S∞

p∞

Xf // RP∞ ,

where f is a continuous map such that f τ(v) = −f(v).

Proof. Let X → X be the covering induced by f (see (24.4)). If f is a

characteristic map for p, there is a homeomorphism g : X≈−→ X such that pg = p.

Therefore, g satisfies gτ = τ g. As f τ(v) = −f(v), the map f = f g satisfies

the requirements of Lemma 24.8. Conversely, given f , let g : X → X given byg(v) = (p(v), f (v)). The map g satisfies pg = p and gτ = τ g. Hence, g issurjective and is a covering projection. Since both p and p are 2-folds coverings, gis a homeomorphism and p is equivalent to p.

Example 24.9. The inclusion i : RPn → RP∞ is covered by the τ -equivariantmap i : Sn → S∞. By Lemma 24.8, the map i is characteristic map for the coveringSn → RPn. In particular, the identity of RP∞ is a characteristic map for thecovering S∞ → RP∞.

24.2. The characteristic class. Diagram (24.5) together with Proposition 24.6provides a bijection

(24.10) w : Cov2(X)≈−→ H1(X) .

This associates to a 2-fold covering p : X → X its characteristic class w(p) ∈H1(X). For instance, the characteristic class w(p∞) for the covering p∞ : S∞ →RP∞ is the non-zero element ι ∈ H1(RP∞) = Z2. Indeed, as H1(RP∞) = Z2, theset Cov2(RP∞) has two elements and the trivial covering corresponds to 0. As S∞

is connected, p∞ is not the trivial covering, hence w(p∞) = ι. The following lemmais obvious.

Lemma 24.11. Let p : X → X be a 2-fold covering over a CW-complex. Then

(1) if f : X → RP∞ is a characteristic map for the covering p, thenw(p) = H∗f(w(p∞)) = H∗f(ι).

(2) if g : Y → X is a continuous map, then w(g∗p) = H∗g(w(p)).(3) p is the trivial covering if and only if w(p) = 0.

Let us give a geometric descriptions of the characteristic class w(p). Chose

a set-theoretic section b : X → X of p and let B = b(X) ⊂ X. We consider B

as a singular 0-cochain of X. Using the coboundary δ : C0(X) → C1(X), we get

δB ∈ C1(X) which is the connecting 1-cochain for B: for σ ∈ S1(X), 〈δ(B), σ〉 = 1if and only if the (non-oriented) path σ connects a point in B to a point in X −B


(see Example 12.10). Observe that δ(B) = δ(X − B) = δ(τ(B)), where τ is thedeck involution of p. Hence

〈δ(B), τ σ〉 = 〈δ(B), C∗τ(σ)〉 = 〈C∗τ δ(B), σ〉= 〈δC∗τ(B), σ〉 = 〈δ(τ(B)), σ〉= 〈δ(B), σ〉 .

Thus, 〈δ(B), σ〉 depends only on pσ ∈ S1(X). This permits us to define a singular1-cochain wb(p) ∈ C1(X) by the formula

〈wb(p), σ〉 = 〈δ(B), σ〉where σ ∈ S1(X) is any lifting of σ ∈ S1(X).

Proposition 24.12. wb(p) is a 1-cocycle representing w(p) ∈ H1(X).

Proof. Let wb = wb(p). Let σ2 ∈ S2(X). If σ2 ∈ S2(X) is a lifting of σ2, then

the 1-simplexes in ∂(σ2) are liftings of those in ∂(σ2). Therefore

(24.13) 〈δ(wb), σ2〉 = 〈wb, ∂(σ2)〉 = 〈δ(B), ∂(σ2)〉 = 0 ,

which proves that wb is a cocycle.We next prove that the cohomology class wb ∈ H1(X) of wb does not depend

on the set-theoretic section b. Let b′ : X → X ′ another such section, giving B′ =b′(X) ∈ C0(X). Define r ∈ C0(X) by

〈r, x〉 = 〈B, x〉+ 〈B′, x〉 ,where x is a chosen element in p−1(x). If ˜x is another choice, one has 〈B, ˜x〉 =

〈B, x〉+ 1 and 〈B′, ˜x〉 = 〈B′, x〉+ 1 in Z2, so r is well defined. Let σ ∈ S1(X) with

end points u and v. Let σ ∈ S1(X) be a lifting of σ with end points u and v. Then

〈wb + wb′ , σ〉 = 〈δ(B) + δ(B′), σ〉= 〈B, u〉+ 〈B′, u〉+ 〈B, v〉+ 〈B′, v〉= 〈δ(r), σ〉 .

This proves that wb′ = wb + δ(r) and thus [wb′ ] = [wb]. Denote by w(p) ∈ H1(X)the cohomology class [wb].

We can now prove that w(p) = w(p′) if p′ : X ′ → X is a 2-fold covering equiv-

alent to p. Indeed, if h : X≈−→ X ′ is a homeomorphism such that p′ h = p, then,

wb(p) = wh b(p′), which implies that w(p) = w(p′).

Choosing a characteristic map f : X → RP∞ for p, we now have w(p) = w(p),

where p : X → X is the induced covering of Diagram (24.4). Choose a set-theoreticsection b0 : RP∞ → S∞ of p∞ and set B0 = p∞(RP∞) and w0 = [wb0(p∞)] ∈H1(RP∞). This gives rise to a set-theoretic section b of p by the formula b(x) =

(x, b0f(x)). It satisfies B = b(X) = f−1(B0), where f : X → S∞ is the map

covering f , as in (24.4). Let σ ∈ S1(X) with a lifting σ ∈ S1(X). Then, f σ is alifting of f σ in S1(S

∞) and we have

〈C ∗ f(wb0), σ〉 = 〈wb0 , f σ〉 = 〈δ0(B0), f σ〉= 〈δ0(B0), C∗f(σ)〉 = 〈C∗f δ0(B0), σ〉= 〈δC∗f(B0), σ〉 = 〈δ(f−1(B0)), σ〉 = 〈δ(B), σ〉= 〈wb(p), σ〉 .

122 3. PRODUCTS

Hence,

(24.14) w(p) = C∗f(w0) .

Together with Lemma 24.11, Equation (24.14) reduces the proof of Proposition 24.12,to showing that w0 = w(p∞). As 0 6= w(p∞) ∈ H1(RP∞) = Z2, it is enough toprove that w0 6= 0. By Equation (24.14) again, it is enough to find a covering

q : Y → Y over some CW-complex Y for which w(q) 6= 0.We take for q the double covering q : S1 → S1. Let σ : ∆1 → S1 given by

σ(t, 1 − t) = e2iπt. As σ(0, 1) 6= σ(1, 0), one has 〈w(q), σ〉 6= 0. But σ is a 1-cocycle representing the generator of H1(S1) = Z2 (see Corollary 13.4). Hence,w(q) 6= 0.

Remark 24.15. Let p : X → X be a two-fold covering over a CW-complex X .To describe the characteristic class w(p) in the cellular cohomology of X , choose a

section b : X0 → X and see B = b(X0) ⊂ X0 as a cellular 0-cochain of X (for thecellular decomposition induced from that of X). Let ϕ : Λ1(X)×I → X be a global

characteristic maps for the 1-cells of X . Let ϕ : Λ1(X) × I → X be the lifting of

ϕ for which ϕ(λ, 0) ∈ B. Consider the cellular 1-cochain wB ∈ C1(X) defined, forλ ∈ Λ1(X), by

〈wB , λ〉

1 if ϕ(λ, 1) /∈ B0 otherwise.

Note that 〈wB , λ〉 = 〈δ(B), λ〉 where λ is any one-cell of X above λ, which, as

in (24.13), proves that wB is a cellular cocycle. We claim that [wB] ∈ H1(X)

corresponds to w(p) ∈ H1(X), under the identification of H1(X) and H1(X) asthe same subgroup of H1(X1) (see (16.8)). We can thus suppose that X = X1.We can also suppose that X is connected. If T is a maximal tree of X1, thenthe quotient map X1 → X1/T is a homotopy equivalence by Proposition 15.2 andLemma 12.70. The covering p is then induced from one over X1/T , so we canassume that X1 is a bouquet of circles indexed by Λ1. For each one cell λ, acharacteristic map ϕλ : D1 → X1 gives a singular 1-simplex of X1 (identifying D1

with ∆1). If ϕλ is a lifting of ϕλ, one has

(24.16) 〈wB , λ〉 = 〈wb, ϕλ〉 =

1 if ϕλ is a loop

0 otherwise,

where b : X → X is a set theoritic section of p extending b. As [ϕλ] | λ ∈ Λ1 is

a basis for H1(X1), Equation (24.16) implies that [wB ] ∈ H1(X1) corresponds tow(p) ∈ H1(X).

24.3. The transfer exact sequence of a 2-fold covering. Let p : X → Xbe a 2-fold covering projection with deck involution τ . To each singular simplexσ : ∆m → X , one can associate the set of the two liftings of σ into X. This definesa map from Sm(X) to Cm(X), extending to a linear map tr∗ : Cm(X) → Cm(X).The map tr∗ is clearly a chain map. By § 3, this gives risen to two GrV-morphisms

tr∗ : Hm(X)→ Hm(X) and tr∗ : Hm(X)→ Hm(X)

satisfying 〈tr∗(a), α〉 = 〈a, tr∗(α)〉. The linear maps tr∗ and tr∗ are called thetransfer maps for the covering p.


The transfer homomorphism in cohomology and the characteristic class w(p) ∈H1(X) are related by the following exact sequence.

Proposition 24.17 (Transfer exact sequence). The following sequence

· · · → Hm(X)H∗p−−−→ Hm(X)

tr∗−−→ Hm(X)w(p)−−−−−−→ Hm+1(X)

H∗p−−−→ · · ·is exact. It is functorial with respect to induced coverings.

Proof. The following sequence

0→ C∗(X)tr∗−−→ C∗(X)

C∗p−−→ C∗(X)→ 0 .

is clearly an exact sequence of chain complexes and it is functorial with respect toinduced coverings. By Kronecker duality, it gives a short exact sequence of cochaincomplexes

(24.18) 0→ C∗(X)C∗p−−→ C∗(X)

tr∗−−→ C∗(X)→ 0 .

By Proposition 6.4, this gives rise to a connecting homomorphism d∗ : H∗(X) →H∗+1(X) and a functorial long exact sequence

· · · → Hm(X)H∗p−−−→ Hm(X)

tr∗−−→ Hm(X)d∗−→ Hm+1(X)

H∗p−−−→ · · · .It just remains to identify d∗ with w(p) −.

To construct the connecting homomorphism d∗ we need a GrV-section of tr∗ inSequence (24.18). Choose a set-theoretic section b : X → X of p. If σ : ∆m → X is

a singular 1-simplex of X , define b×(σ) : ∆m → X to be the unique lifting of σ with

b×(σ)(1, 0, . . . , 0) ∈ b(X). This defines a map b× : S(X) → S(X). If a ∈ Cm(X),

we consider a as a subset of Sm(X) and so its direct image b×(a) ⊂ Sm(X) is an

m-cochain of X . This determines a GrV-morphism b× : C∗(X)→ C∗(X) which isa section of tr∗. By Equation (6.2), the connecting homomorphism d∗ is determinedby the equation

(24.19) 〈C∗pd∗(a), β〉 = 〈δb×(a), β〉 ,for all a ∈ Cm(X) and β ∈ Sm(X), where δ : C∗(X)→ C∗+1(X) is the coboundary.The equality 〈C∗pd∗(a), β〉 = 〈d∗(a), C∗p(β)〉 together with (24.19) shows that

〈δb×(a), β〉 depends only on C∗p(β) = pβ = β. Therefore, by taking τ β insteadof β if necessary, we may assume that β /∈ b×(Sm(X)). Then, the faces βǫi of βare not in b×(Sm(X)), except possibly for i = 0 and

(24.20) 〈δb×(a), β〉 = 〈b×(a), ∂(β)〉 = 〈b×(a), βǫ0〉 .The number 〈b×(a), βǫ0〉 equals 1 if and only if β(0, 1, 0, . . . , 0) ∈ b(X) and βǫ0 ∈a. In other words, if and only if the front and back faces of β satisfy 1β ∈ wb(p) (seeprevious subsection) and βm−1 ∈ a. Hence, Equations (24.19) and (24.20) implythat

〈d∗(a), β〉 = 〈b×(a), βǫ0〉 = 〈wb(p) a, β〉for all a ∈ Cm(X) and β ∈ Sm(X). This proves that d∗(−) = wb(p) − in C∗(X).By Proposition 24.12, this implies that d∗(−) = w(p) − in H∗(X).

An important application of the transfer exact sequence is the determinationof the cohomology ring of RPn.

124 3. PRODUCTS

24.4. The cohomology ring of RPn. Let Z2[a] be the polynomial ring overa formal variable a in degree 1. This is an object of GrA, as well as its trun-cation Z2[a]/(a

n+1), the quotient of Z2[a] by the ideal generated by an+1. ByProposition 19.6, H1(RP∞) = Z2, generated by the class ι. Therefore, there is aGrA-morphism Z2[a]→ H∗(RP∞) sending ak to ιk, where the latter denotes thecup product of k copies of ι. The composition Z2[a] → H∗(RP∞) → H∗(RPn)factors by a GrA-morphism Z2[a]/(a

n+1)→ H∗(RPn).

Proposition 24.21. The above GrA-morphisms

Z2[a]→ H∗(RP∞) and Z2[a]/(an+1)→ H∗(RPn)

are GrA-isomorphisms. In particular, the GrA-morphism H∗(RP∞)→ H(RPn),induced by the inclusion, is surjective.

Proof. As S∞ is contractible [80, example 1.B.3 p. 88], the transfer ex-act sequence of the covering p∞ : S∞ → RP∞ shows that the cup product with

w(p∞) ∈ H1(RP∞) gives an isomorphism H∗(RP∞)≈−→ H∗+1(RP∞). In particu-

lar, w(p∞) is the generator of H1(RP∞). This proves the statement for RP∞.For RPn we use the covering p : Sn → RPn. The transfer exact sequence

proves at once that the cup product with w(p) ∈ H1(RPn) gives an isomorphism

Hm(RPn)≈−→ Hm+1(RPn) for 0 ≤ m < n− 1. As RPn has a CW-structure with

one k-cell for 0 ≤ k ≤ n, the end of the transfer exact sequence of the coveringp : Sn → RPn involves the following Z2-vector spaces

0→ Hn−1(RPn)︸︷︷︸dim=1

w(p)−−−−−−→ Hn(RPn)︸︷︷︸dim≤1

H∗p−−−→ Hn(Sn)︸︷︷︸dim=1

tr∗−−→ Hn(RPn)︸︷︷︸dim≤1

→ 0

Thus, the cup product with w(p) is also an isomorphismHn−1(RPn)≈−→ Hn(RPn).

This proves the proposition for RPn.

25. Nilpotency, Lusternik-Schnirelmann categories and topologicalcomplexity

Let X be a topological space. A subspace U of X is categorical if the in-clusion U → X is homotopic to a constant map. The Lusternik-Schnirelmanncategory cat (X) is the minimal cardinality of an open covering of X with categori-cal subspaces. Some authors (see, e.g. [33]) adopt a different normalization for theLusternik-Schnirelmann category, equal to one less than the definition above.

For a survey paper about the Lusternik-Schnirelmann category (see [106]).Amongst its properties, cat (X) is an invariant of the homotopy type of X . Forexample, cat (X) = 1 if and only if X is contractible and cat (Sn) = 2. Moregenerally, one has the following result (see [106, Proposition 1.2] for a more generalstatement and a different proof).

Proposition 25.1. Let K be a connected simplicial complex of dimension n.Then cat (|K|) ≤ n+ 1.

Proof. We use the barycentric subdivision K ′ of K (see p. 13 for the nota-tions). For σ ∈ S(K), define

U(σ) = OstK′(σ) ,

25. NILPOTENCY, LUSTERNIK-SCHNIRELMANN CATEGORIES, ETC 125

where OstK′ denotes the open star in K ′. Then, U(σ) is an open neighborhoods ofσ in |K ′| = |K| and U(σ) retracts by deformation onto σ. Let U = U0, . . . , Un,where

(25.2) Ui =⋃

σ∈Si(K)

U(σ) .

One checks that U is an open covering of |K|. Also, U(σ) and U(σ′) are disjointif σ, σ′ ∈ Si(K) with σ 6= σ′. As K is connected, |K| is path-connected. Being adisjoint union of categorical sets, Ui is thus categorical.

Let X be a topological space and B be a vector subspace of H∗(X). Thenilpotency class nilB of B is the minimal integer m such that

B · · · B︸︷︷︸m

= 0 .

If no such integer exists, we set nilB =∞.

Proposition 25.3. Let X be a topological space. Then nilH>0(X) ≤ cat (X).

Proof. Let U1, . . . , Um ⊂ X be open subspaces of X which are categorical.By the homotopy property, the homomorphism H>0(X) → H>0(Ui) induced bythe inclusion vanishes. Hence, the exact sequence of the pair (X,Ui) implies thatthe restriction homomorphism H>0(X,Ui) → H>0(X) is surjective. Then, in thefollowing diagram

m∏

i=1

H>0(X,Ui)

// //m∏

i=1

H>0(X)

H>0(X,U1 ∪ · · · ∪ Um) // H∗(X)

,

which is commutative by the functoriality of the cup product, the upper horizontalarrow is surjective. But, if X = U1 ∪· · · ∪ Um, the lower left vector space vanishes.This proves that nilH>0(X) ≤ m.

A classical consequence is the vanishing of the cup products in a suspension.

Corollary 25.4. Let Y be a topological space. Then, all cup products inH>0(ΣY ) vanish.

Proof. As ΣY is the union of two cones, cat (ΣY ) ≤ 2, which proves thecorollary.

The Lusternik-Schnirelmann category admits several generalizations, for in-stance the category of a map (see, e.g. [106, § 7]). Here, we introduce the categorycat (X,A) of a topological pair (X,A). A subspace U of a topological space X isA-categorical if the inclusion U → X is homotopic to a map with value in A. Thencat (X,A) is defined to be the minimal cardinality of an open covering of X withA-categorical subspaces. For instance, X is path-connected, then

(25.5) cat (X,A) ≤ cat (X) = cat (X, pt)

Lemma 25.6. cat (X,A) is an invariant of the homotopy type of the pair (X,A).

126 3. PRODUCTS

Proof. Let f : (X,A)→ (X ′, A′) be a homotopy equivalence of pair. It sufficesto prove that, if U ′ is a subset of X ′ which is A′-categorical, then U = f−1(U ′)is A-categorical in X . This will imply that cat (X,A) ≤ cat (X ′, A′). Homotopyequivalence being an equivalence relation, we also get cat (X ′, A′) ≤ cat (X,A).

Let g : (X ′, A′) → (X,A) be a homotopy inverse of f . Let βt : U′ → X ′ be a

homotopy with β0(v) = v and β1(U′) ⊂ A′. Then, the map αt : U → X defined by

αt(u) = gβtf(u) is a homotopy satisfying α0(u) = gf(u) and α1(U) ⊂ A. Asgf is homotopic to id(X,A), this proves that U is A-categorical.

Proposition 25.3 generalizes in the following statement.

Proposition 25.7. Let (X,A) be a topological pair with X path-connected.Then

nilB ≤ cat (X,A) ,

where B = kerH∗(X)→ H∗(A) .Proof. As X is path-connected, B ⊂ H>0(X). Let U1, . . . , Um ⊂ X be

open subspaces of X which are A-categorical Then, the homomorphism H∗(X)→H∗(Ui) factors:

H∗(X)

$$JJJJ

JJJJ

// H∗(Ui)

H∗(A)

99.

Therefore, if a ∈ B, then a is in the image of H>0(X,Ui). The proof of Proposi-tion 25.7 is then the same as that of Proposition 25.3.

This category for pairs is related to the topological complexity, a notion ofmathematical robotics introduced by M. Farber [52, 53]. Let Y be a topologicalspace and PY be the space of continuous paths γ : I → Y , endowed with thecompact-open topology. Let π : PY → Y × Y be the origin-end map: π(γ) =(γ(0), γ(1)). A motion planning algorithm is a section of π. It is not possible to finda continuous motion planning algorithm unless Y is contractible [52, Theorem 1].The topological complexity TC (Y ) is the minimal cardinality of an open covering Uof Y × Y such that π : PY → Y × Y admits a continuous section over each U ∈ U .Let ∆Y be the diagonal subset of Y ×Y . The following proposition is the contentsof [53, Corollary 18.2].

Proposition 25.8. TC (Y ) = cat (Y × Y,∆Y ).

In consequence, TC (Y ) is an invariant of the homotopy type of Y .

Proof. Let U ⊂ Y × Y . Suppose that a continuous section s : U → PYof π exists. Then σ(y, y′, t) = (s(y, y′)(t), y′) satisfies σ(y, y′, 0) = (y, y′) andσ(y, y′, 1) = (y′, y′) ∈ ∆Y , showing that U is ∆Y -categorical. Conversely, ifc(t) = (c1(t), c2(t)) ∈ Y × Y is a path from (y, y′) to (u, u) ∈ ∆Y , then the pathc1 c−12 joins y to y′. This process being continuous in (y, y′), it provides a sections

of π over ∆Y -categorical subsets of Y × Y .

Proposition 25.8 together with (25.5) implies that TC (Y ) ≤ cat (Y × Y ).The inequality cat (Y ) ≤ TC (Y ) also holds true [53, Lemma 9.2], but is not aconsequence of Proposition 25.8.

26. THE CAP PRODUCT 127

If Y is path-connected, Propositions 25.8 and 25.7 give the inequality

(25.9) TC (Y ) ≥ nil ker(H∗(Y × Y )→ H∗(∆Y )

).

We shall see with the Kunneth theorem that there is a commutative diagram(see Remark 27.5):

(25.10)

H∗(Y × Y )

×≈

// H∗(∆Y )

≈

H∗(Y )⊗H∗(Y ) // H∗(Y )

.

(to use the Kunneth theorem, we need that Y is of finite cohomology type). Underthe cross product, the image of B in the ring H∗(Y ) ⊗ H∗(Y ) is the ideal of thedivisors of zero for the cup product. The inequality (25.9) thus corresponds to [52,Theorem 7]

For results concerning the topological complexity of the projective space (seethe end of § 36.2).

26. The cap product

Let K be a simplicial complex. Choose a simplicial order ≤ on K. We definethe cap product

Cp(K)× Cn(K)≤−−→ Cn−p(K)

to be the unique bilinear map such that

(26.1) a ≤ v0, . . . , vn = 〈a, v0, . . . , vp〉 vp, . . . , vnfor all a ∈ Cp(K) and all v0, . . . , vn ∈ Sn(K), with v0 < v1 < · · · < vn (thismakes sense if n ≥ p; otherwise, the cap product just vanishes). If a ∈ Cp(K), b ∈Cn−p(K) and γ ∈ Cn(K) the following formula follows directly from the definitions

(26.2) 〈a ≤ b, γ〉 = 〈b, a ≤ γ〉 .

Lemma 26.3. If a ∈ Cp(K) and γ ∈ Cn(K), then

∂(a ≤ γ) = δ(a) ≤ γ + a ≤ ∂(γ) .

Proof. Let q = n− p and b ∈ Cq(K). Denote ≤ and ≤ by just and .Using (26.2), one has

(26.4) 〈δ(a b), γ〉 = 〈(a b, ∂(γ)〉 = 〈b, a ∂(γ)〉 .In the other hand

〈δ(a b), γ〉 = 〈δ(a) b, γ〉+ 〈a δ(b), γ〉(26.4)

= 〈b, δ(a) γ〉+ 〈δ(b), a γ〉= 〈b, δ(a) γ〉+ 〈b, ∂(a γ)〉 .

Equations (26.4) and (26.4) imply that

〈b, ∂(a γ)〉 = 〈b, δ(a) γ + a ∂(γ)〉 .for all b ∈ Cq(K). By Part (b) of Lemma 3.4, this implies Lemma 26.3.

128 3. PRODUCTS

Lemma 26.3 implies that Z∗(K) ≤ Z∗(K) ⊂ Z∗(K), B∗(K) ≤ Z∗(K) ⊂B∗(K) and Z∗(K) ≤ B∗(K) ⊂ B∗(K). Therefore, ≤ induces a map Hp(K) ×Hn(K)

−→ Hn−p(K), or

H∗(K)×H∗(K)−→ H∗(K)

called the cap product (on simplicial cohomology). As in the case of the cup productwe drop the index “≤” from the notation because of the following proposition.

Proposition 26.5. The cap product on H∗(K)×H∗(K)−→ H∗(K) does not

depend on the simplicial order “≤”.

Proof. Let ≤ and ≤′ be two simplicial orders on K. Let a ∈ Hp(K) andγ ∈ Hn(K). For any b ∈ Hn−p(K), Formula (26.2) and Proposition 22.4 implythat

〈b, a ≤ γ〉 = 〈a ≤ b, γ〉= 〈a ≤′ b, γ〉= 〈b, a ≤′ γ〉 .

By Part (b) of Lemma 3.4, this implies that a ≤ γ = a ≤′ γ in Hn−p(K).

Proposition 26.6. The cap product H∗(K) × H∗(K)−→ H∗(K) endows

H∗(K) with a structure of H∗(K)-module.

Proof. By definition, is bilinear and the equality 1 γ = γ is obvious. Itremains to prove that

(26.7) (a b) γ = a (b γ)

for all a ∈ Hp(K), b ∈ Hq(K) and γ ∈ Hn(K). As the cup product is associativeand commutative (Corollary 22.5), one has, for any c ∈ Hn−p−q(K),

〈c, (a b) γ〉 = 〈(a b) c, γ〉= 〈(b a) c, γ〉= 〈(b (a c), γ〉= 〈(a c, b γ〉= 〈c, a (b γ)〉 .

By Part (b) of Lemma 3.4, this proves Equation (26.7).

Proposition 26.8 (Functoriality of the cap product). Let f : L → K be asimplicial map. Then, the formula

a H∗f(γ) = H∗f(H∗f(a) γ)

holds in H∗(K) for all a ∈ H∗(K) and all γ ∈ H∗(L).

Proof. Suppose that a ∈ Hp(K) and γ ∈ Hn(L). Using the functoriality ofthe cup product established in Proposition 22.6, one has, for any b ∈ Hn−p(K),

〈b, a H∗f(γ)〉 = 〈a b,H∗f(γ)〉= 〈H∗f(a b), γ〉= 〈H∗f(a) H∗f(b), γ〉= 〈H∗f(b), H∗f(a) γ〉= 〈b,H∗f(H∗f(a) γ)〉 .

26. THE CAP PRODUCT 129

By Part (b) of Lemma 3.4, this proves Proposition 26.8.

There are several version of the cap product in relative simplicial (co)homology.Let (K,L) be a simplicial pair. Choose a simplicial order ≤ on K. We note firstthat C∗(K) ≤ C∗(L) ⊂ C∗(L), whence a cap product

(26.9) Hp(K)×Hn(K,L)−→ Hn−p(K,L) .

One may also compose

Hp(K,L)×Hn(K)j∗×id−−−−→ Hp(K)×Hn(K)

−→ Hn−p(K)

to obtain a cap product

(26.10) Hp(K,L)×Hn(K)−→ Hn−p(K) .

The latter cap product may be post-composed with Hn−p(K) → Hn−p(K,L) andget a cap product

(26.11) Hp(K,L)×Hn(K)−→ Hn−p(K,L) .

As the restriction of Cp(K)× Cn(K)−→ Cn−p(K) to Cp(K,L)× Cn(L) vanishes,

we obtain a cap product

(26.12) Hp(K,L)×Hn(K,L)−→ Hn−p(K) .

As for Formula (26.7), the following equation

(26.13) (a b) γ = a (b γ)

holds true in Hn−p−q(K) for all a ∈ Hp(K,L), b ∈ Hq(K,L) and γ ∈ Hn(K,L).The cap products (26.10) and (26.12) are used in (26.13).

More generally, suppose that L is the union of two subcomplexes L = L1 ∪L2.Then, the restriction of Cp(K) × Cn(K)

−→ Cn−p(K) to Cp(K,L1) × Cn(L) hasimage contained in Cn−p(L2). This gives a cap product

(26.14) Hp(K,L1)×Hn(K,L)−→ Hn−p(K,L2) .

The functoriality holds for a simplicial map f : (K ′, L′)→ (K,L) satisfying f(L′i) ⊂Li for i = 1, 2: the formula

(26.15) a H∗f(γ) = H∗f(H∗f(a) γ)

holds in H∗(K,L2) for all a ∈ H∗(K,L1) and all γ ∈ H∗(K ′, L′). The proof is thesame as for Proposition 26.8.

The next two lemmas express the compatibility between these relative capproducts, the absolute one and the connecting homomorphisms for a simplicialpair (K,L).

Lemma 26.16. Let (K,L) be a simplicial pair. Denote by i : L → K andj : (K, ∅) → (K,L) the inclusions. Let x ∈ Hn(K,L). Then, for all integer p, thefollowing diagram

Hp(K,L)

x

H∗j // Hp(K)

x

H∗i // Hp(L)

∂∗x

δ∗ // Hp+1(K,L)

x

Hn−p(K)H∗j // Hn−p(K,L)

∂∗ // Hn−p−1(L)H∗i // Hn−p−1(K)

is commutative.

130 3. PRODUCTS

Proof. For the left hand square diagram, let a ∈ Hp(K,L) and b ∈ Hn−p(K,L).One has

〈b,H∗j(a) x〉 = 〈b H∗j(a), x〉and

〈b,H∗j(a x)〉 = 〈H∗j(b), a x〉 = 〈H∗j(b) a, x〉 .Hence, the left hand square diagram commutes if and only if b H∗j(a) =H∗j(b) a, which was established in Lemma 22.8.

For the middle square diagram, let a ∈ Hp(K) and b ∈ Hn−p−1(L). One has

〈b,H∗i(a) ∂∗x〉 = 〈b H∗i(a), ∂∗x〉 = 〈δ∗(b H∗i(a)), x〉 .On the other hand:

〈b, ∂∗(a x)〉 = 〈δ∗(b), a x〉 = 〈δ∗(b) a, x〉 .The commutativity of the middle square diagram is thus equivalent to the formulaδ∗(b H∗i(a)) = δ∗(b) a holding true in Hn−p(K,L) for all a ∈ Hp(K) andb ∈ Hn−p−1(L). This formula was proven in Lemma 22.12. In the same way, wesee that the commutativity of the right hand square diagram is a consequence ofLemma 22.12 (intertwining the role of a and b).

Lemma 26.17. Let (K,L) be a simplicial pair. Denote by j : (K, ∅) → (K,L)the pair inclusion. Then, the equation

H∗j(a α) = a H∗j(α) .

holds true in Hn−p(K,L) for all a ∈ Hp(K) and all α ∈ Hn(K).

Proof. It is then enough to prove that 〈b,H∗j(a α)〉 = 〈b, a H∗j(α)〉 forall b ∈ Hn−p(K,L). But,

〈b,H∗j(a α)〉 = 〈H∗j(b), a α〉= 〈H∗j(b) a,α〉= 〈H∗j(b a), α〉 by Lemma 22.11

= 〈b a,H∗j(α)〉= 〈b, a H∗j(α)〉 .

The cap product is also defined in the singular (co)homology of a space X . Onthe (co)chain level, it is the unique bilinear map

Cp(X)× Cn(X)−→ Cn−p(X)

such that

a σ = 〈a, pσ〉σqfor all a ∈ Cp(X) and all σ ∈ Sn(X), where the back and front faces pσ and σq

are defined as in p 112. If a ∈ Cp(X), b ∈ Cn−p(X) and γ ∈ Cn(X) the followingformula follows directly from the definition

(26.18) 〈a b, γ〉 = 〈b, a γ〉 .Therefore, as for the simplicial cap product, properties follows from those of thecup product. The formula ∂(a γ) = δ(a) γ + a ∂(γ) is proved as for

27. THE CROSS PRODUCT AND THE KUNNETH THEOREM 131

Lemma 26.3 and we get an induced bilinear map Hp(X) ×Hn(X)−→ Hn−p(X),

or

H∗(X)×H∗(X)−→ H∗(X)

called the cap product in singular (co)homology. This cap product endows H∗(X)with a structure of H∗(X)-module, as in Proposition 26.6 and is functorial forcontinuous maps f : Y → X , as for Proposition 26.8.

For a topological pair (X,Y ), the three relative versions of the cap products:

(26.19) Hp(X)×Hn(X,Y )−→ Hn−p(X,Y ) ,

(26.20) Hp(X,Y )×Hn(X)−→ Hn−p(X,Y )

and

(26.21) Hp(X,Y )×Hn(X,Y )−→ Hn−p(X)

hold true, as for (26.9)–(26.12). When Y = Y1∪Y2, a relative cap product analogousto (26.14)

(26.22) Hp(X,Y1)×Hn(X,Y )−→ Hn−p(X,Y2) .

is available under some conditions, for instance if (Y, Yi) is a good pair for i = 1, 2,so one can use the small simplexes technique, as for the Mayer-Vietoris sequencein Proposition 12.83. The functoriality formula (26.15) as well as the analogues ofLemmas 26.16 and 26.17 hold true.

Finally, the simplicial and singular cap products are intertwined by the isomor-phisms

R≤,∗ : H∗(K)≈−→ H∗(|K|) and R∗≤ : H∗(|K|) ≈−→ H∗(K)

of Theorem 17.5. For any simplicial order ≤, the following equation

(26.23) a R≤,∗(γ) = R≤,∗(R∗≤(a) γ)

holds in H∗(|K|) for all a ∈ H∗(|K|) and all γ ∈ H∗(K): The proof of (26.23) isstraightforward for γ a simplex of K.

27. The cross product and the Kunneth theorem

Let X and Y be topological spaces. Results computing H∗(X ×Y ) in terms ofH∗(X) and H∗(Y ) are known as Kunneth theorems (or Kunneth formulas). Thisgeneric name comes from the thesis of Hermann Kunneth in 1923 (see [38, pp. 55–56]). To give an example, when X and Y are discrete spaces, the cohomology ringsare concentrated in dimension 0 and

(27.1) H0(X) = ZX2 , H0(Y ) = ZY2 , H0(X × Y ) = Z2X×Y .

The cross product of maps

ZX2 × ZY2×−→ Z2

X×Y

defined by (f × g)(x, y) = f(x)g(y) is bilinear. The associated linear map

(27.2) ZX2 ⊗ ZY2×−→ Z2

X×Y

is also called the cross product. The map (27.2) is clearly injective. It is notsurjective if both X and Y are infinite; for instance, if X = Y is infinite, it is easyto see that the characteristic function of the diagonal in X×X is not in the image of×. On the other hand, suppose that X or Y is finite (say Y ). Let F : X×Y → Z2.

132 3. PRODUCTS

For y ∈ Y , define Fy : X → Z2 by Fy(x) = F (x, y) and let χy be the characteristicfunction of y. Then

F =∑

y∈Y

Fy × χy .

Thus, if Y is finite, the cross product of (27.2) is an isomorphism.Such finiteness conditions will occur in the statements of this section, under

the form that Y should be of finite cohomology type (see definition p. 76).Observe that, under the identification of (27.1), the cross product H0(X) ×

H0(Y )→ H0(X × Y ) satisfies the formula

f × g = π∗Xf π∗Y g ,

where πX and πY are the projections of X × Y onto X and Y .More generally, let X and Y be two topological spaces. Using the usual tensor

product ⊗ of vector spaces over Z2, we define the tensor product of the Z2-algebras(H∗(X),+,) and (H∗(Y ),+,) as the Z2-algebra (H∗(X)⊗H∗(Y ),+, •) definedby

[H∗(X)⊗H∗(Y )]m =⊕

i+j=m

Hi(X)⊗Hj(Y ) ,

with the product

(27.3) (a1 ⊗ b1) • (a2 ⊗ b2) = (a1 a2)⊗ (b1 b2) .

The projections πX : X × Y → X et πY : X × Y → Y give GrA-morphismsπ∗X : H∗(X)→ H∗(X×Y ) et π∗Y : H∗(Y )→ H∗(X×Y ). This permits us to definea bilinear map

H∗(X)×H∗(Y )×−→ H∗(X × Y )

by

(27.4) a× b = ×(a, b) = π∗X(a) π∗Y (b)

called the cross product. By the universal property of the tensor product (analogousto that of vector spaces), this gives a GrV-morphism

H∗(X)⊗H∗(Y )×−→ H∗(X × Y ) ,

also called the cross product.

Remark 27.5. Let ∆ : X → X ×X be the diagonal map ∆(x) = (x, x). Thecomposition

(27.6) H∗(X)×H∗(X)×−→ H∗(X ×X)

∆∗−−→ H∗(X)

is equal to the cup product (see also Diagram (25.10), p. 127). This relation,due to Lefschetz (see [179, pp. 38–41] for historical considerations), was quiteinfluential: in some books (e.g. [175, 133]), the cross product is introduced firstusing homological algebra (the Eilenberg-Zilber theorem) and the cup product isdefined via Formula (27.6). Our opposite approach follows the viewpoint of [72, 80].

Under some hypotheses, the cross product may be defined in relative coho-mology. Let (X,A) and (Y,B) be topological pairs. The projections πX and πYgive homomorphisms π∗X : H∗(X,A) → H∗(X × Y,A × Y ) and π∗Y : H∗(Y,B) →


H∗(X × Y,B ×X). Suppose that A or B is empty, or one of the pairs (X,A) or(Y,B) is a good pair. Then formula (27.4) defines a relative cross product

(27.7) H∗(X,A)⊗H∗(Y,B)×−→ H∗(X × Y,A× Y ∪X ×B) .

Indeed, we must just check that the relative cup product

H∗(X × Y,A× Y )⊗H∗(X × Y,B ×X)−→ H∗(X × Y,A× Y ∪X ×B) .

is defined. By Lemma 22.19, it is enough to show that (A×Y,X×B) is excisive inX×Y . This is obvious if A or B is empty. Otherwise, suppose that one of the pair,say (Y,B), is a good pair. Let V be a neighbourhood of B in Y which retracts by

deformation onto B. Let Z = A× Y ∪X ×B. Then, A× (Y − V ) ⊂ intZ(A× Y ).By excision of A× (Y − V ) and homotopy, we get isomorphisms

H∗(Z,A× Y )≈−→ H∗(A× V ∪X ×B,A× V )

≈−→ H∗(X ×B,A×B) .

By Lemma 22.18, this implies that (A× Y,X ×B) is excisive.We first establish the functoriality of the cross product. In Lemmas 27.8

and 27.9 below, we assume that the conditions for the relative cross product tobe defined are satisfied.

Lemma 27.8. Let f : (X ′, A′) → (X,A) and g : (Y ′, B′) → (Y,B) be maps ofpairs. Then, for all a ∈ H∗(X,A) and b ∈ H∗(Y,B) the following formula holds:

H∗(f × g)(a× b) = H∗f(a)×H∗g(b) .Proof. As πX (f × g) = f πX′ and πY (f × g) = gπY ′ , one has

H∗(f × g)(a× b) = H∗(f × g)(H∗πX(a) H∗πY (b)

)

= H∗(f × g)(H∗πX(a)) H∗(f × g)(H∗πY (b)))= H∗πX′ H

∗f(a) H∗πY ′ H∗g(b)

= H∗f(a)×H∗g(b) .

Formula (27.3) provides a product “•” on H∗(X,A)⊗H∗(Y,B).

Lemma 27.9. The cross product H∗(X,A)⊗H∗(Y,B)×−→ H∗(X×Y,A×Y ∪X×

B) is multiplicative. In particular, the cross product H∗(X)⊗H∗(Y )×−→ H∗(X×Y )

is a GrA-morphism.

Proof.

×((a1 ⊗ b1) • (a2 ⊗ b2)

)= (a1 a2)× (b1 b2)

= π∗X(a1 a2) π∗Y (b1 b2)

= π∗X(a1) π∗X(a2) π∗Y (b1) π∗Y (b2)

= π∗X(a1) π∗Y (b1) π∗X(a2) π∗Y (b2)

= (a1 × b1) (a2 × b2)= ×(a1 ⊗ b1) ×(a2 ⊗ b2)

Remark 27.10. In the proof of Lemma 27.9, we have established that

(a1 a2)× (b1 b2) = (a1 × b1) (a2 × b2)for all ai ∈ H∗(X,A) and bj ∈ H∗(Y,B).

134 3. PRODUCTS

Observe that the Kronecker pairing

[H∗(X)⊗H∗(Y )]× [H∗(X)⊗H∗(Y )]〈 , 〉−−→ Z2

given by

(27.11) 〈a⊗ b, α⊗ β〉 = 〈a, α〉〈b, β〉

is a bilinear map (By convention, 〈a, α〉 = 0 if a ∈ Hp(−) and α ∈ Hq(−) withp 6= q.)

Lemma 27.12. Let X and Y be topological spaces with Y of finite cohomologytype. Then, for all n ∈ N, the linear map

⊕

p+q=n

Hp(X)⊗Hq(Y )k−→ [

⊕

p+q=n

Hp(X)⊗Hq(Y )]♯

given by k(a⊗ b) = 〈a⊗ b,−〉 is an isomorphism.

Proof. It suffices to prove that k : Hp(X)⊗Hq(Y )→ [Hp(X)⊗Hq(Y )]♯ is anisomorphism for all integers p, q. As Hr(−) ≈ Hr(−)♯ via the Kronecker pairing,this amounts to prove that, for vector spaces V and W , the homomorphism

k : V ♯ ⊗W ♯ → [V ⊗W ]♯ ,

given by k(r×s)(v⊗w) = r(v)s(w), is an isomorphism whenW is finite dimensional.This classical fact (true over any base field) is easily proven by induction on dimW(see, e.g., [41, Chapter VI, Proposition 10.18] for a proof in a more general setting).

Lemma 27.12 permits us to define a Kronecker dual

× : H∗(X × Y )→ H∗(X)⊗H∗(Y )

to the cross product, by requiring that the formula

(27.13) 〈a× b, γ〉 = 〈a⊗ b,×(γ)〉

holds true for all a ∈ H∗(X), b ∈ H∗(Y ) and γ ∈ H∗(X × Y ). We call theGrV-morphism × the homology cross product.

Lemma 27.14. Let X and Y be topological spaces, with Y of finite cohomologytype. Let M be a basis for H∗(Y ) and let M∗ = m∗ ∈ H∗(Y ) | m ∈ M be thedual basis for the Kronecker pairing. Then, for all γ ∈ H∗(X × Y ), one has

×(γ) =∑

m∈M

H∗πX

(H∗πY (m∗) γ

)⊗m.


Proof. Denote by ×(γ) the right member of the above equation. Let a ∈H∗(X), b ∈ H∗(Y ) and γ ∈ H∗(X × Y ). Then,

〈a⊗ b, ×(γ)〉 = 〈a⊗ b,∑

m∈M

H∗πX

(H∗πY (m∗) γ

)⊗m〉

=∑

m∈M

〈a,H∗πX

(H∗πY (m∗) γ

)〉〈b,m〉

=∑

m∈M

〈H∗πX(a), H∗πY (m∗) γ〉〈b,m〉

= 〈H∗πX(a),

[ ∑

m∈M

〈b,m〉H∗πY (m∗)

] γ〉

= 〈H∗πX(a), H∗πY

[ ∑

m∈M

〈b,m〉(m∗)] γ〉

= 〈H∗πX(a), H∗πY (b) γ〉 = 〈H∗πX(a) H∗πY (b), γ〉= 〈a× b, γ〉 ,

which proves that × = ×.

Theorem 27.15. [Kunneth Theorem] Let X and Y be topological spaces. Sup-pose that Y is of finite cohomology type. Then, the cross product

× : (H∗(X)⊗H∗(Y ),+, •)≈−→ (H∗(X × Y ),+,)

is a GrA-isomorphism and the homology cross product

× : H∗(X × Y )→ H∗(X)⊗H∗(Y )

is a GrV-isomorphism.

The finiteness condition on one of the space (here Y ) is necessary, as seen inthe beginning of the section. It is used in the proof through the following lemma.

Lemma 27.16. Let V be a family of vector spaces over a field F. Let W be afinite dimensional F-vector space. Then the linear map

Φ:( ∏

V ∈V

V)⊗W −→

∏

V ∈V

(V ⊗W

)

given byΦ((v) ⊗ w) = (v ⊗ w)

is an isomorphism.

Proof. The proof is by induction on n = dimW . The case n = 1 followsfrom the canonical isomorphism T ⊗ F ≈ T for any vector space T . The inductionstep uses that, in the category of F-vector spaces, tensor and Cartesian productscommute with direct sums.

Proof of the Kunneth theorem. By Lemma 27.9, we know that the crossproduct is a GrA-morphism. It is then enough to prove that it is a GrV-isomorphism.Assuming that Y is of finite cohomology type, the proof goes as follows.

(1) We prove that the cross product is a GrV-isomorphism when X is a finitedimensional CW-complex.

136 3. PRODUCTS

(2) By Kronecker duality, Point (1) implies that the homology cross productis a GrV-isomorphism when X is a finite dimensional CW-complex. Anycompact subspace of X × Y is contained in Xn × Y for some n ∈ N.Therefore, H∗(X×Y ) is the direct limit of H∗(X

n×Y ) by Corollary 12.22.Also, H∗(X) is the direct limit of H∗(X

n). The homology cross productbeing natural by Lemma 27.8 and Kronecker duality, we deduce that × isa GrV-isomorphism when X is any CW-complex. By Kronecker duality,the cross product is a GrV-isomorphism for any CW-complex X .

(3) If X is any space, there is a map fX : X → X , where X is a CW-complexand f is a weak homotopy equivalence, i.e. the induced map on the

homotopy groups π∗f : π∗(X, u)≈−→ π∗(X, f(u)) is an isomorphism for

all u ∈ X (see [80, p.352] or Remark 18.12). As π∗(A × B, (a, b))≈−→

π∗(A, a) × π∗(B, b), the map fX × id : X × Y → X × Y is also a weakhomotopy equivalence. But, weak homotopy equivalences induce isomor-phisms on singular (co)homology (see [80, Prop. 4.21]). The followingdiagram

H∗(X)⊗H∗(Y )

H∗fX⊗ id ≈

× // H∗(X × Y )

H∗(fX×id)≈

H∗(X)⊗H∗(Y )×

≈// H∗(X × Y )

is commutative by Lemma 27.8. This proves that the cross product insingular cohomology is a GrA-isomorphism for any space X . The corre-sponding diagram for the homology cross product, or Kronecker duality,proves that the homology cross product is a GrV-isomorphism

It thus remains to prove Point (1). We follow the idea of from [80, p. 218].Let us fix the topological space Y . To a topological pair (X,A), we associate twograded Z2-vector spaces:

h∗(X,A) = H∗(X,A)⊗H∗(Y ) and k∗(X,A) = H∗(X × Y,A× Y ) .

We shall prove the following lemma:

Lemma 27.17. Let Y be a topological space of finite cohomology type. Then

(a) k∗ and h∗ are two generalized cohomology theories in the sense of § 20,with h∗(pt) ≈ k∗(pt) ≈ H∗(Y ).

(b) The cross product provides a natural transformation from h∗ to k∗, re-

stricting to an isomorphism h∗(pt)≈−→ k∗(pt).

Using Proposition 20.3, Point (1) follows from Lemma 27.17

Proof of Lemma 27.17. If f : (X,A) → (X ′, A′) is a continuous map ofpairs, we define h∗f = H∗f ⊗ idH∗(Y ) and k∗f = H∗(f × idY ). This makes h∗

and k∗ functors from Top2 to GrV. The connecting homomorphism δ∗h : h∗(A)→h∗+1(X,A) and δ∗k : k∗(A)→ k∗+1(X,A) are defined by

δ∗h = δ∗ ⊗ idH∗(Y ) and δ∗k = δ∗ : H∗(A× Y )→ H∗+1(X × Y,A× Y ) ,

using the homomorphism δ∗ of singular cohomology; δ∗h and δ∗k are then functorialfor continuous maps.


We now check that Axioms (1)–(3) of p. 105 hold both for h∗ and k∗. Thehomotopy and excision axioms are clear. For a topological pair (X,A), the longexact sequence for k∗ is that in singular cohomology for the pair (X × Y,A × Y ).The exact sequence for h∗ is obtained by tensoring with H∗(Y ) the exact sequenceof (X,A) for H∗. We use that a direct sum of exact sequences is exact and that,over a field, tensoring with a vector space preserves exactness. The disjoint unionaxiom holds trivially for k∗. For h∗, we use that Hm(Y ) if of finite dimension for allm and Lemma 27.16. Thus, both h∗ and k∗ are generalized cohomology theories.

We now check Point (b). Let f : (X ′, A′) → (X,A) be a continuous map ofpairs. We must prove that the following diagram

(27.18)

h∗(X,A)

h∗f

× // k∗(X,A)

k∗f

h∗(X ′, A′)× // k∗(X ′, A′)

is commutative. This amounts to show that

(27.19) H∗f(a)× y = H∗(f × idY )(a× y)for all a ∈ H∗(X,A) and y ∈ H∗(Y ). This follows from Lemma 27.8.

For the second part of Point (b), we must show the commutativity of thediagram

(27.20)

h∗(A)δ∗h //

×

h∗+1(X,A)

×

k∗(A)δ∗k // k∗+1(X,A) .

This is equivalent to the commutativity of the diagram

(27.21)

Hp(A) ×Hq(Y )δ∗×id //

×

Hp+1(X,A)×Hq(Y )

×

Hp+q(A× Y )δ∗× // Hp+q+1(X × Y,A× Y ) ,

for all p, q ∈ N. Here, we have introduced more precise notations, distinguishing theconnecting homomorphisms in singular cohomology δ∗ : H∗(A)→ H∗+1(X,A) andδ∗× : H∗(A × Y ) → H∗+1(X × Y,A × Y ). We shall also distinguish the homomor-phisms π∗X : H∗(X) → H∗(X × Y ) and π∗X : H∗(A) → H∗(A × Y ) induced by theprojections onto A and X , as well as the homomorphisms π∗Y : H∗(Y )→ H∗(X×Y )and π∗Y : H∗(Y ) → H∗(A × Y ) induced by the projections onto Y . Analogousnotations are used for cochains. The commutativity of Diagram (27.21) is thusequivalent to the formula(27.22)

π∗X δ∗(a) π∗Y (y) = δ∗×

(π∗X(a) π∗Y (y)

)for all a ∈ Hp(A) , y ∈ Hq(Y ) .

Let a ∈ Zp(A) and y ∈ Zq(Y ) represent a and y. Let a ∈ Cp(X) be an extension ofa as a p-cochain ofX . By the recipe of Lemma 12.25, δ(a) is a cocycle of Cp+1(X,A)

138 3. PRODUCTS

representing δ∗(a). Thus, the left hand member of (27.22) is represented by thecocycle

(27.23) π∗X δ(a) π∗Y (y) .

To compute the right hand member, we need an extension of π∗X(a) π∗Y (y) asa cochain of X × Y . But, as cochains of X × Y , π∗X(a) is an extension of π∗X(a)and the cocycle πY (y) is an extension of π∗Y (y). Therefore, π∗X(a) πY (y) is anextension of π∗X(a) π∗Y (y). By Lemma 12.25, the right hand member of (27.22)is then represented by the cocycle δ×

(π∗X(a) πY (y)

). As δ×(πY (y)) = 0, one has

(27.24) δ×(π∗X(a) πY (y)

)= δ×π

∗X(a) πY (y) = π∗X δ(a) π∗Y (y)

Comparing (27.23) and (27.24) proves Formula (27.22) and then the commutativityof Diagram (27.20).

Under some hypotheses, there are relative versions of the Kunneth theorem,generalizing Theorem 27.15.

Theorem 27.25 (Relative Kunneth theorem). Let (X,A) be a topological pair.Let (Y,B) be a good pair such that Y and B are of finite cohomology type. Then,the cross product

(27.26) × : H∗(X,A)⊗H∗(Y,B)≈−→ H∗(X × Y,A× Y ∪X ×B)

is a GrA-isomorphism.

The classical proof of the Kunneth theorem (see e.g. [175]) gives the moregeneral statement that (27.26) is an isomorphism if (A × Y,X × A) is excisive inX × Y and (Y,B) is of finite cohomology type. If (X,A) and (Y,B) are CW-pairs,the condition that (Y,B) is of finite cohomology type is also sufficient (see [80,Theorem 3.21]; see also Corollary 28.41 below).

Proof. As (Y,B) is a good pair, the relative cross product (27.7) is defined.Let Z = A× Y ∪X ×B and let p ∈ N. Let us consider the following commutativediagram.

Hp(X, A)⊗Hq−1(Y )×

≈//

Hp+q−1(X×Y, A×Y )

Hp(X, A)⊗Hq−1(B)

×

≈//

Hp+q−1(X×B, A×B) oo J∗

≈Hp+q−1(Z, A×Y )

Hp(X, A)⊗Hq(Y, B)

× //

Hp+q(X×Y, Z)

Hp(X, A)⊗Hq(Y )

×

≈//

Hp+q(X×Y, A×Y )

Hp(X, A)⊗Hq(B)

×

≈// Hp+q(X×B, A×B) oo J∗

≈Hp+q(Z, A×Y )

The left column is the cohomology exact sequence for (Y,B) tensored by Hp(X,A).It is still exact since we work in the category of Z2-vector spaces. The right column isthe cohomology exact sequence for the triple (X×Y, Z,A×Y ). The homomorphism

28. SOME APPLICATIONS OF THE KUNNETH THEOREM 139

J∗, induced by inclusion, is an isomorphism: if V is a neighbourhood of B in Ywhich retracts by deformation onto B, J∗ is the composition

H∗(A×Y ∪X×B,A×Y )≈−→ H∗(A×V ∪X×B,A×V )

≈−→ H∗(X×B,A×B)

The left arrow is an isomorphism by excision of A × (Y − V ) and the right oneby the homotopy property. As Y and B are of finite cohomology type, the crossproducts involving the absolute cohomology H∗(Y ) or H∗(B) are isomorphisms, asestablished during the proof of Theorem 27.15. By the five-lemma, this proves thatthe middle cross product is an isomorphism.

28. Some applications of the Kunneth theorem

28.1. Poincare series and Euler characteristic of a product. One appli-cation of the Kunneth theorem is the multiplicativity of Poincare series and Eulercharacteristic.

Proposition 28.1. Let X and Y be spaces of finite cohomology type. Then,X × Y is of finite cohomology type and

(28.2) Pt(X × Y ) = Pt(X) ·Pt(Y ) .

If X and Y are finite complexes, then

(28.3) χ(X × Y ) = χ(X) · χ(Y ) .

Proof. Let ai = dimHi(X), bi = dimHi(Y ). The Kunneth theorem impliesthat dimHn(X × Y ) =

∑i+j=n aibj which proves (28.2). Equation (28.3) follows,

since χ is the evaluation of Pt at t = −1. Note that (28.3) also follows moreelementarily from the cellular decomposition of X × Y (see Example 15.7).

28.2. Slices. Let y0 ∈ Y . The slice inclusion sX : X → X × Y at y0 is thecontinuous map defined by sX(x) = (x, y0). The slice inclusion sY : Y → X × Y atx0 ∈ X is defined accordingly.

Using the bijection Y ≈ S0(Y ), we see y0 ∈ Y as a 0-homology class [y0] ∈H0(Y ). Hence, for b ∈ H0(Y ) the number 〈b, y0〉 ∈ Z2 is defined.

Lemma 28.4. Let sX : X → X × Y be the slice inclusion at y0 ∈ Y . Leta ∈ Hm(X) and b ∈ Hn(Y ). Then,

H∗sX(a× b) =

〈b, y0〉 a if n = 0

0 otherwise.

Proof. One has πX sX = idX , while πY sX is the constant map c onto y0.Thus, H∗c (b) = 0 if n 6= 0. When n = 0, H∗c (b) = 〈b, y0〉1. Thus,

H∗sX(a× b) = H∗sX(π∗X(a) π∗

Y(b)) = a H∗c (b) = 〈b, y0〉 a .

Here below, two corollaries of Lemma 28.4 which enable us to detect cohomol-ogy classes via the slice homomorphisms.

Corollary 28.5. Let X and Y be path-connected topological spaces such thatHk(X) = 0 for k < n. Then, the following equation

a = 1×H∗sY (a) +H∗sX(a)× 1

is satisfied for all a ∈ Hn(X × Y ).

140 3. PRODUCTS

Proof. By the hypotheses and the Kunneth theorem, the cross product pro-vides an isomorphism

× : H0(X)⊗Hn(Y )⊕Hn(X)⊗H0(Y )≈−→ Hn(X × Y ) .

and H0(X) ≈ Z2 ≈ H0(Y ). This implies that a = 1×v+u×1 for some unique u ∈Hn(X) and v ∈ Hn(Y ). By Lemma 28.4, one has H∗sY (a) = v and H∗sX(a) = u,which proves the corollary.

The case n = 1 in Corollary 28.5 gives the following statement.

Corollary 28.6. Let X and Y be path-connected spaces. Then, the followingequation

a = 1×H∗sY (a) +H∗sX(a)× 1

is satisfied for all a ∈ H1(X × Y ).

28.3. The cohomology ring of a product of spheres. We first note theassociativity of the cross product.

Lemma 28.7. Let X, Y and Z be three topological spaces. In H∗(X × Y ×Z),the cross product is associative: (x× y)× z = x× (y× z) for all x ∈ H∗(X), y ∈ Yand z ∈ Z.

Proof. We have to consider the various projections π12 : X×Y ×Z → X×Y ,π23 : X × Y × Z → Y × Z, π1 : X × Y × Z → X , etc. Also, π12

1 : X × Y → X , etc.

They satisfy πijj πij = πj . Using the associativity and the functoriality of the cupproduct, we get

(x× y)× z = π∗12(π121∗(x) π12

2∗(y)) π∗3(z) = π∗1(x) π∗2(y) π∗3(z) .

In the same way, x× (y × z) = π∗1(x) π∗2(y) π∗3(z).

The cohomology of the sphere Sd being concentrated in dimension 0 and d, onehas a GrA-isomorphism

(28.8) Z2[x]/(x2)≈−→ H∗(Sd) (x of degree d) ,

sending x to the generator [Sd]♯ ∈ Hd(Sd). Here, Z2[x]/(x2) denotes the quotient

of the polynomial ring Z2[x], where x is a formal variable (here of degree d), bythe ideal generated by x2. The following proposition then follows directly from theKunneth theorem.

Proposition 28.9. Let X be a topological space. The GrA-homomorphism

H∗(X)[x]/(x2) −→ H∗(X × Sd) (x of degree d) ,

induced by a 7→ a× 1, for a ∈ H∗(X), and x 7→ 1× [Sd]♯, is a GrA-isomorphism.

Using Proposition 28.9 together with Lemma 28.7, we get the following propo-sition.

Proposition 28.10. For i = 1 . . . ,m, let xi be a formal variable of degree di.Then, the GrA-homomorphism

Z2[x1, . . . , xm]/(x2

1, . . . , x2m) −→ H∗(Sd1 × · · · × Sdm)

induced byxi 7→ 1× · · · × 1× [Sdi ]♯ × 1× · · · × 1

is a GrA-isomorphism.


28.4. Smash products and joins. Let (X,x) and (Y, y) be two pointedspaces. The base points provide an inclusion

X ∨ Y ≈ X × y ∪ x × Y → X × Y .The smash product X ∧ Y of X and Y is the quotient space

X ∧ Y = X × Y/X ∨ Y .

It is pointed by x ∧ y, the image of X ∨ Y in X ∧ Y .Recall that (X,x) is well pointed if the pair (X, x) is well cofibrant. The

following lemma is the first place where the full strength of this definition is used.

Lemma 28.11. If (X,x) and (Y, y) are well pointed, so is (X ∧ Y, x ∧ y).Proof. By Proposition 12.65, the pair (X × Y,X ∨ Y ) is well cofibrant and,

by Lemma 12.69, so is (X ∧ Y, x ∧ y).

By Proposition 12.71 and (12.31), one has the isomorphisms

(28.12) H∗(X × Y,X ∨ Y ) ≈ H∗(X ∧ Y, x ∧ y) ≈ H∗(X ∧ Y ) .

Proposition 28.13. Let (X,x) and (Y, y) be well pointed spaces. Then, thehomomorphisms induced by the inclusion i : X ∨ Y → X × Y and the projectionp : X × Y → X ∧ Y give rise to the following short exact sequence

0→ H∗(X ∧ Y )H∗p−−−→ H∗(X × Y )

H∗i−−→ H∗(X ∨ Y )→ 0 .

Proof. Using the isomorphism (28.12) and the exact sequence of Corollary 12.74,

it is enough to prove that H∗i is onto. Consider the following commutative diagram

H∗(X)id //

H∗π∗X

&&LLLL

LLLLL

H∗(X)

H∗(X × Y )H∗i // H∗(X ∨ Y )

H∗j∗Y &&LLLLLLLL

H∗jX88rrrrrrrr

H∗(Y )id //

H∗π∗Y

88rrrrrrrr

H∗(Y )

where πX , πY are the projections and jX , jY the inclusions. We note that πY jX andπX jY are homotopic to constant maps. By Proposition 12.79, the homomorphism

H∗(X∨Y )(H∗jX ,H

∗j∗Y )−−−−−−−−−→ H∗(X)×H∗(Y ) is an isomorphism. Hence H∗i is onto.

Remark 28.14. Using the relationship between the exact sequence of the pair(X × Y,X ∨ Y ) and that of Corollary 12.74, Proposition 28.13 implies that thehomorphism H∗i : H∗(X × Y )→ H∗(X ∨ Y ) is surjective, whence the short exactsequence

(28.15) 0→ H∗(X × Y,X ∨ Y ) −→ H∗(X × Y )H∗i−−→ H∗(X ∨ Y )→ 0 .

As (X, x) and (Y, y) are good pairs, the relative cross product

H∗(X, x)⊗H∗(Y, y) ×−→ H∗(X × Y,X ∨ Y )

142 3. PRODUCTS

is defined by (27.7). Using the isomorphisms of (12.31), one constructs the followingcommutative diagram

(28.16)

H∗(X, x)⊗H∗(Y, y)

×

≈ // H∗(X)⊗ H∗(Y )

×

H∗(X × Y,X ∨ Y )≈ // H∗(X ∧ Y )

which defines the reduced cross product ×. The relative Kunneth theorem 27.25gives the following reduced Kunneth theorem.

Proposition 28.17. Let (X,x) and (Y, y) be well pointed spaces, with Y offinite cohomology type. Then, the reduced cross product

× : H∗(X)⊗ H∗(Y )≈−→ H∗(X ∧ Y )

is a multiplicative GrV-isomorphism.

For a pointed space (Z, z), Diagram (12.32) provides an injective homomor-

phism H∗(Z) → H∗(Z). Using this together with Proposition 28.13 (or Re-mark 28.14), the Kunneth theorem and its reduced form are summed up by thefollowing diagram

(28.18)

H∗(X)⊗ H∗(Y )

×≈

// // H∗(X)⊗H∗(Y )

×≈

H∗(X ∧ Y ) // // H∗(X × Y )

.

Example 28.19. Proposition 28.17 says that Hk(Sp∧Sq) = 0 for k 6= p+q andHp+q(Sp ∧ Sq) = Z2. Actually, Sp ∧ Sq is homeomorphic to Sp+q by the followinghomeomorphisms. Let Dr be the compact unit disk of dimension r with boundary∂Dr = Sr−1. Then Dr/∂Dr is homeomorphic to Sr and

Sp ∧ Sq ≈−→(Dp/∂Dp ×Dq/∂Dq

)/[∂Dp]×Dq ∪Dp × [∂Dq]

≈−→ Dp ×Dq

/∂Dp ×Dq ∪Dp × ∂Dq

≈−→ Dp ×Dq/∂(Dp ×Dq) ≈ Sp+q .

Let (X,x) be a well pointed space. The smash product X ∧ S1 is called thereduced suspension of X , which has the same homotopy type than the suspensionΣX . Indeed, let ∂I = 0, 1. The map

F : ΣX = (X × I)/(X × ∂I)→ X ∧ S1

given by F (x, t) = [(x, e2iπt] descends to a homeomorphism

F : ΣX = ΣX/(x × I) ≈−→ X ∧ S1 .

This homeomorphism preserves the base points, if we choose those to be [x] ∈ ΣXand 1 ∈ S1. The pair (I, ∂I) is well cofibrant by Lemma 12.64. By Lemma 12.65, sois the product pair (X×I,X×∂I∪x×I). By Lemma 12.69, the pair (ΣX, x×I)


is well cofibrant. As x×I is contractible, the projection F : ΣX →→ ΣX ≈ X∧S1

is a homotopy equivalence by Lemma 12.70.

Let b be the generator of H1(S1) = Z2. By Propositions 28.17 and 12.76 andthe above, the three arrows in the following proposition are isomorphisms.

Lemma 28.20. The following diagram

Hn(X)

−×b

≈wwoooooooooΣ∗

≈ &&MMMMMMMMM

Hn+1(X ∧ S1)H∗F

≈// Hn+1(ΣX)

is commutative.

Proof. As all these isomorphisms are functorial, it is enough to prove thelemma for X = Kn. This is possible since the cellular decomposition of Kn given inProposition 19.1 has 0-skeleton K0

n = x0, so (Kn, x0) is a well pointed space byProposition 15.2. In this particular case, the statement is obvious since the threegroups are isomorphic to Z2.

The smash product gives a geometric interpretation of the cup product. Leta ∈ Hm(X) and b ∈ Hm(X), given by maps fa : X → Km and fm : X → Knto Eilenberg-McLane spaces. By Proposition 28.17, Hm+n(Km ∧ Kn) = Z2, withgenerator corresponding to g : Km ∧ Kn → Km+n.

Proposition 28.21. The composed map

X(fa,fb)−−−−→ Km ×Kn → Km ∧ Kn g−→ Km+n

represents the class a b ∈ Hm+n(X).

Proof. By Proposition 28.17, the generator of Hm+n(Km ∧ Kn) = Z2 isthe reduced cross product ım×ın. By Diagram (28.18), it is send to ım × ın inHm+n(Km×Kn). Now, the composed map of Proposition 28.21 coincides with thecomposition

X∆−→ X ×X fa×fb−−−−→ Km ×Kn → Km ∧ Kn g−→ Km+n .

Proposition 28.21 then follows from Remark 27.5.

If we consider the composed map f : X(fa,fb)−−−−→ Km × Kn → Km ∧ Kn, Propo-

sition 28.21 gives the following corollary.

Corollary 28.22. The following diagram

Hm(Km)⊗Hn(Kn)

f∗a⊗f∗b

×

≈// Hm+n(Km ∧ Kn)

f∗

Hm(X)⊗Hn(X)

// Hm+n(X)

is commutative.

144 3. PRODUCTS

Let X and Y be two topological spaces. Their join X ∗ Y is the quotientof X × Y × I by the equivalence relation (x, y, 0) ∼ (x, y′, 0) for y, y′ ∈ Y and(x, y, 1) ∼ (x′, y, 1) for x, x′ ∈ X . This topological join is related to the simplicialjoin in the following way: if K and L are locally finite simplicial complexes, then|K ∗ L| is homeomorphic to |K| ∗ |L| (see [152, Lemma 62.2]).

The two open subspacesX×Y ×[0, 1) andX×Y ×(0, 1] ofX×Y ×I define opensubspaces UX and UY of X ∗Y . The space UX retracts by deformation onto X andUY retracts by deformation onto Y . Moreover, UX ∩ UY retracts by deformationonto X × Y × 1

2. The following diagram is homotopy commutative,

UX ∩ UY incl // UX

X × Y πX //

≃

OO

X

≃

OO

as well as the corresponding diagram for Y . Consider the homomorphismm

Hk(X)⊕Hk(Y )π∗X+π∗Y−−−−−→ Hk(X × Y ) .

If k > 0, then (π∗X

+π∗Y)(a, b) = a×1+1×b and, by the Kunneth theorem, π∗

X+π∗

Y

is injective. As X ∗ Y is path-connected, the Mayer-Vietoris sequence for the data(X ∗Y, UX , UY , UX ∩UY ) splits and gives, for all integers k ≥ 0, the exact sequence

(28.23) 0→ Hk(X)⊕ Hk(Y )π∗X+π∗Y−−−−−→ Hk(X × Y ) −→ Hk+1(X ∗ Y )→ 0 .

Example 28.24. The join Sp ∗ Sq is homeomorphic to Sp+q+1. Considering

Sp+q+1 ⊂ Rp+1 ×Rq+1, a homeomorphism Sp+q+1 ≈−→ Sp ∗ Sq is given by (x, y) 7→[(x, y, |x|]. The reader can check (28.23) on this example, including the case p =q = 0.

Observe that UX and UY are contractible in X ∗ Y . Hence, the Lusternik-Schnirelmann category of X ∗Y is equal to 2. By Proposition 25.3, the cup productin H>0(X ∗ Y ) vanishes.

When the Kunneth theorem is valid, one sees that the cohomology ring ofX ∗ Y is isomorphic to that of Σ(X ∧ Y ). Actually, under some hypotheses, thesetwo spaces have the same homotopy type (see [80, Ex. 24, p. 20]).

28.5. The theorem of Leray-Hirsch. An important generalization of aproduct space is a locally trivial fiber bundle. A map p : E → B is a locally trivialfiber bundle with fiber F (in short: a bundle) if there exists an open covering Uof B and, for each U ∈ U , a homeomorphism ψU : U × F ≈−→ p−1(U) such thatpψ(x, v) = x for all (x, v) ∈ U × F . The space E is the total space and B is thebase space of the bundle. If A is a subspace of B, we set EA = p−1(A), getting abundle p : EA → A. If b ∈ B, we set Eb = Eb and by ib : Eb → E the inclusion.A fiber inclusion is an embedding i : F → E which is a homeomorphism onto somefiber Eb. As elswhere in the literature, we shall often speak about a (locally trivial)

bundle Fi−→ E

p−→ B, meaning a locally trivial bundle p : E → B with fiber Ftogether with a chosen fiber inclusion i.

If p : E → B is a bundle, then the homomorphism p∗ = H∗p : H∗(B)→ H∗(E)provides a structure of graded H∗(B)-module on H∗(E).


A cohomology extension of the fiber is a GrV-morphism θ : H∗(F ) → H∗(E)such that, for each b ∈ B, the composite map

H∗(F )θ−→ H∗(E)

H∗ib−−−→ H∗(Eb)

is a GrV-isomorphism. We do not require that θ is multiplicative. In the presen-

tation of a bundle by a sequence Fi−→ E

p−→ B, a cohomology extension θ of thefiber exists if and only if H∗i is surjective.

A cohomology extension θ of the fiber provides a morphism of graded H∗(B)-modules

H∗(B)⊗H∗(F )θ−→ H∗(E)

given by θ(a⊗ b) = p∗(a) θ(b).As in (27.13), we define the Kronecker dual

θ : H∗(E)→ H∗(B)⊗H∗(F ) .

to θ, by requiring that the formula

(28.25) 〈θ(b⊗ u), γ〉 = 〈b ⊗ u, θ(γ)〉holds true for all b ∈ H∗(B), u ∈ H∗(F ) and γ ∈ H∗(E).

Suppose that F is of finite cohomology type. LetM be a basis for H∗(F ) andletM∗ = m∗ ∈ H∗(F ) | m ∈M be the dual basis for the Kronecker pairing. Asin Lemma 27.14, the formula

θ(γ) =∑

m∈M

H∗p(θ(m∗) γ

)⊗m.

is satisfied for all γ ∈ H∗(E)

Theorem 28.26 (Leray-Hirsch). Let Ep−→ B be a locally trivial fiber bundle

with fiber F . Suppose that F is of finite cohomology type. Let θ : H∗(F )→ H∗(E)

be a cohomology extension of the fiber. Then, θ is an isomorphism of graded H∗(B)-modules and θ is a GrV-isomorphism.

Proof. By Kronecker duality, only the cohomology statement must be proven.Let A ⊂ B and let h∗(A) = H∗(A) ⊗H∗(F ). The composition

θA : H∗(F )θ−→ H∗(E) −→ H∗(EA)

is a cohomology extension of the fiber for the bundle p : EA → A, giving rise to

θA : h∗(A) → H∗(EA). We want to prove that θB is an isomorphism. Consideringthe following commutative diagram

h∗(B)θB //

≈

H∗(E)

≈

( ∏A∈π0(B)H

∗(A))⊗H∗(F )

Φ ≈∏

A∈π0(B) h∗(A)

Q

θA //∏A∈π0(B)H

∗(EA)

146 3. PRODUCTS

where Φ is the linear map of Lemma 27.16, which is an isomorphism since Hk(F )is finite dimensional for all k, permits us to reduce to the case where the base ispath-connected.

We now suppose that B is path-connected and that the bundle E → B istrivial, i.e. there exists a homeomorphism ϕ : B ×F → E such that pϕ = πB, theprojection to the factor B. Since F is of finite cohomology type, one may use H∗ϕtogether with the Kunneth formula to identify H∗(E) with H∗(B)⊗H∗(F ).

Fix an integer n and consider the vector subspace Pnk ofH∗(B)⊗H∗(F ) definedby

Pnk =⊕

p,q

Hp(B)⊗Hq(F ) | p+ q = n and q ≤ k.

These subspaces provide a filtration

(28.27) 0 = Pn−1 ⊂ Pn0 ⊂ · · · ⊂ Pnn = Hn(E) .

We denote by ψ : H∗(F )→ H∗(F ) the GrV-morphism defined by ψ = H∗iθ.The fiber inclusion i is a slice over one point. Let u ∈ Hq(F ). As B is path-connected, Lemma 28.4 implies that

(28.28) θ(u) = 1⊗ ψ(u) mod P qq−1 .

Hence, if a ∈ Hp(B) with p+ q = n, one has

(28.29)θ(a⊗ u) = (a⊗ 1) θ(u)

= (a⊗ 1) (1⊗ ψ(u)) mod Pnn−1

= a⊗ ψ(u) mod Pnn−1

In particular, θ preserves the filtration (28.27). It thus induces homomorphismsθ : Pnk /P

nk−1 → Pnk /P

nk−1. Moreover, one has a natural identification Pnk /P

nk−1 ≈

Hn−k(B) ⊗Hk(F ) under which θ(a⊗ u) = a⊗ ψ(u). This enables us to prove by

induction on k that θ : Pnk → Pnk is an isomorphism, using the five-lemma in thediagram

0 // Pnk−1//

θ≈

Pnk //

θ

Hn−k(B)⊗Hk(F ) //

id⊗ψ≈

0

0 // Pnk−1// Pnk // Hn−k(B)⊗Hk(F ) // 0

Indeed, the left vertical arrow is an isomorphism by induction hypothesis. SincePn−1 = 0, the induction starts with k = 0, using (28.29). Therefore, the Leray-Hirsch theorem is true for a trivial bundle.

Let Bi, i = 1, 2, be two open sets of B with B = B1 ∪B2. Let B0 = B1 ∩ B2

and Ei = p−1(Bi). The Mayer-Vietoris cohomology sequence for (B,B1, B2, B0)may be tensored with Hk(F ) and remains exact, since we are dealing with vectorspaces. The sum of these sequences provides the exact sequence of the top line ofthe commutative diagram

hk−1(B1)⊕ hk−1(B2)

θ1⊕θ2

// hk−1(B0)

θ0

// hk(B)

θ

//

Hk−1(E1)⊕Hk−1(E2) // Hk−1(E0) // Hk(E) //


// hk(B1)⊕ hk(B2) //

θ1⊕θ2

hk(B0)

θ0

// Hk(E1)⊕Hk(E2) // hk(E0)

The bottom line is the Mayer-Vietoris sequence for the data (B,B1, B2, B0). By

the five-lemma, this shows that, if θi are isomorphisms for i = 0, 1, 2, then θ is anisomorphism.

What has been done so far implies that the θ is an isomorphism for a bundle of

finite type, i.e. admitting a finite covering U such that EUp−→ U is trivial for U ∈ U .

By Kronecker duality, θ is an isomorphism in this case. As in Point (2) of theproof of the Kunneth theorem (p. 136), θ is the direct limit of θA for A ⊂ B suchthat EA → A is of finite type. Therefore, θ is an isomorphism and, by Kronecker

duality, θ is an isomorphism for any bundle.

The Leray-Hirsch theorem also has the following version, in which the finitetype hypothesis is on the base rather than on the fiber. The proof, involving theSerre spectral sequence, may be found in [138, Theorem.10].

Theorem 28.30 (Leray-Hirsch II). Let Ep−→ B be a locally trivial fiber bun-

dle with fiber F . Suppose that B is path-connected and of finite cohomology type.

Let θ : H∗(F ) → H∗(E) be a cohomology extension of the fiber. Then, θ is anisomorphism of graded H∗(B)-modules and θ is a GrV-isomorphism.

Here below a few corollaries of the above Leray-Hirsch theorems.

Corollary 28.31. Let Fi−→ E

p−→ B be a locally trivial fiber bundle whose baseB is path-connected and whose fiber F (or base B) is of finite cohomology type.Suppose that H∗i : H∗(E)→ H∗(F ) is surjective. Then

(1) H∗p : H∗(B)→ H∗(E) is injective.(2) kerH∗i is the ideal generated by the elements of positive degree in the

image of H∗p.

Proof. Let θ : Hk(F ) → Hk(E) be a cohomology extension of the fiber such

that H∗iθ is the identity of H(F ). As H∗p(b) = θ(b⊗1), the homomorphism H∗pis injective. On the other hand,

H∗i θ(b⊗ a) = H∗i(H∗p(b) θ(a))= H∗iH∗p(b) H∗i(θ(a))= H∗(pi)(b) a .

As pi is a constant map, one has H∗(pi)(1) = 1 and H∗(pi)(b) = 0 if b haspositive degree. This proves (2).

The considerations in the above proof about the bases of H∗(B) and H∗(F )implies the following result.

Corollary 28.32. Let Fi−→ E

p−→ B be a locally trivial fiber bundle whosewhose base B is path-connected. Suppose that H∗i : H∗(E)→ H∗(F ) is surjective.

148 3. PRODUCTS

Suppose that F and B are of finite cohomology type. Then, E is of finite cohomologytype and the Poincare series of F , E and B satisfy(28.33)

Pt(E) = Pt(F )Pt(B) .

Actually, Equation (28.33) is equivalent toH∗i being surjective (see [14, Propo-sition 2.1]). Here below another kind of corollaries of the Leray-Hirsch theorem.

Corollary 28.34. Let p : E → B be a locally trivial fiber bundle with fiber F .Suppose that H∗(F ) = 0. Then, H∗p : H∗(B) → H∗(E) is a GrA-isomorphism.

Remarks 28.35. (1) By the Leray-Hirsch theorem, the existence of a coho-mology extension of the fiber implies that p∗ : H∗(B) → H∗(E) is injective. Theconverse is not true, even if the map p has a section (see e.g. [70]).

(2) In the Leray-Hirsch theorem the isomorphism θ is not a morphism of al-gebras, unless θ is multiplicative. It is possible that there exists a cohomologyextension of the fiber but that none of them is multiplicative (see Examples 28.72or Example 39.30).

(3) The proof of the Leray-Hirsch theorem shows the following partial result.Let θ : Hk(F ) → Hk(E) be a linear map defined for all k ≤ n. Suppose that,for each b ∈ B, the composition H∗ibθ : Hk(F ) → Hk(Eb) is an isomorphismfor k ≤ n. Then, with the notation of the proof of Theorem 28.26, the linear map

θ : hk(B)→ Hk(E) is an isomorphism for k ≤ n. For instance, we get the followingproposition.

Proposition 28.36. Let p : E → B be a locally trivial fiber bundle with fiber F .Suppose that Hk(F ) = 0 for all k ≤ m. Then, H∗p : Hk(B) → Hk(E) is anisomorphism for k ≤ m.

The Leray-Hirsch theorem admits a version for bundle pairs. A bundle pairwith fiber (F, F ′) is a topological pair (E,E′) and a map p : (E,E′)→ (B,B) suchthat there exists an open covering U of B and, for each U ∈ U , a homeomorphism

ψU : U × (F, F ′)≈−→ (p−1(U), p−1(U) ∩ E′) such that pψ(x, v) = x for all (x, v) ∈

U × F . In consequence, p : E → B is a bundle with fiber F and the restriction ofp to E′ is a bundle with fiber F ′. A cohomology extension of the fiber is a GrV-morphism θrel : H

∗(F, F ′)→ H∗(E,E′) such that, for each b ∈ B, the composite

H∗(F, F ′)θrel−−→ H∗(E,E′)

H∗ib−−−→ H∗(Eb, E′b)

is a GrV-isomorphism. A cohomology extension θ of the fiber provides a morphismof graded H∗(B)-modules

H∗(B)⊗H∗(F, F ′) θrel−−→ H∗(E,E′)

given by θrel(a⊗ b) = p∗(a) θrel(b).As in (27.13), we define the Kronecker dual

θrel : H∗(E,E′)→ H∗(B)⊗H∗(F, F ′) .

to θrel, by requiring that the formula

(28.37) 〈θrel(b⊗ u), γ〉 = 〈b ⊗ u, θrel(γ)〉holds true for all b ∈ H∗(B), u ∈ H∗(F, F ′) and γ ∈ H∗(E,E′).


Suppose that (F, F ′) is of finite cohomology type. Let M be a basis forH∗(F, F

′) and let M∗ = m∗ ∈ H∗(F, F ′) | m ∈ M be the dual basis for theKronecker pairing. As in Lemma 27.14, the formula

(28.38) θrel(γ) =∑

m∈M

H∗p(θrel(m

∗) γ)⊗m.

is satisfied for all γ ∈ H∗(E,E′)

Theorem 28.39 (Leray-Hirsch relative). Let p : (E,E′) → (B,B) be a bundlepair with fiber (F, F ′). Suppose that (F, F ′) is a well cofibrant pair and is of finitecohomology type. Let θrel : H

∗(F, F ′) → H∗(E,E′) be a cohomology extension of

the fiber. Suppose that (E,E′) is a well cofibrant pair. Then, θrel is an isomorphismof graded H∗(B)-modules and θrel is a GrV-isomorphism.

The hypothesis that (E,E′) is well cofibrant may be removed but this wouldnecessitate some preliminary work. Besides, this hypothesis is easily fullfiled in ourapplications.

Proof. By Kronecker duality, only the cohomology statement must be proven.We first reduce to the case where F ′ is a point. Let E = E/∼ where ∼ is theequivalence relation

x ∼ y ⇐⇒ p(x) = p(y) and x, y ∈ E′

and let E′ = E′/∼. Then the map p descends to a map p : (E, E′)→ (B,B) whichis a bundle pair with fiber (F/F ′, y0), where y0 is the point given by F ′ in F/F ′. In

particular, p : E′ → B is a homeomorphism. Consider the following commutativediagram:

(28.40)

H∗(B)⊗H∗(F/F ′, y0)

θrel

≈ // H∗(B)⊗H∗(F, F ′)

θrel

H∗(E, E′)

≈ // H∗(E,E′)

We shall show that the horizontal homomorphisms, induced by the quotient maps,are isomorphisms. Therefore, the right vertical arrow is bijective if and only if theleft one is.

The top horizontal homomorphism of Diagram (28.40) is an isomorphism byProposition 12.71 since (F, F ′) is a well cofibrant pair. To see that the bottomhorizontal map is also bijective, consider the commutative diagram

H∗(E/E′, [E′])≈ //

≈

H∗(E/E′, [E′])

≈

H∗(E, E′) // H∗(E,E′)

The top horizontal map is an isomorphism because the quotient spaces E/E′ and

E/E′ are equal. As (E,E′) is well cofibrant, the right hand vertical map is bijective

by Proposition 12.71. Also, Lemma 12.69 implies that (E, E′) is well cofibrant and

150 3. PRODUCTS

thus, the left hand vertical map is an isomorphism by Proposition 12.71. Now, thefollowing diagram

H∗(F/F ′, [F ′])

≈

θrel // H∗(E, E′)

≈

// H∗(Eb, E′b)

≈

H∗(F, F ′)θrel //

≈

33H∗(E,E′) // H∗(Eb, E′b)

shows that the bundle pair p inherits a cohomology extension of the fiber θrel.We are then reduced to the case F ′ = y0 being a single point. Consider the

following commutative diagram:

H∗(B)⊗H∗(F, y0)

θrel

// // H∗(B)⊗H∗(F )

θ≈

// // H∗(B)⊗H∗(y0)

θ≈

H∗(E,E′) // // H∗(E) // // H∗(E′)

The top line is the exact sequence of the pair (F, y0) tensored by H∗(B). It is exactsince we are dealing with vector spaces and its splits since y0 is a retract of F ′.The bottom exact sequence of the pair (E,E′) also splits since p : E → B ≈ E′

is a retraction of E onto E′. We shall check below the existence of a cohomology

extension of the fiber θ : H∗(F ) → H∗(E), whence the middle vertical map θ.

The two maps θ are bijective by the absolute Leray-Hirsch theorem 28.26. By thefive-lemma, θrel is then also an isomorphism.

The existence of a cohomology extension of the fiber θ : H∗(F )→ H∗(E) comesfrom θrel : H

∗(F, y0) → H∗(E,E′) when ∗ > 0, since Hk(F, y0) ≈ Hk(F ) andHk(Eb, E

′b) ≈ Hk(Eb) for k > 0. When k = 0, we consider the following diagram

H0(F, y0)

//θrel

//

≈,,

H0(E,E′)

H∗j

// // H0(Eb, E′b)

H0(F )

θ // H0(E)

// // H0(Eb)

H0(y0)

θ′ //

≈

22

r∗

XX

H0(E′) //

p∗

XX

H0(E′b)

,

where j : (E, ∅) → (E,E′) denotes the inclusion. The retraction r : F → y0produces a section r∗ : H0(y0)→ H0(F ) of the homomorphism induced by the in-clusion; this section provides an isomorphism H0(F ) ≈ H0(F, y0) ⊕ H0(y0). Asp : E′ → B is a homeomorphism, one gets a section p∗ : H0(E′) → H0(E) ofthe homomorphism induced by the inclusion. The homomorphism θ : H0(F ) ≈H0(F, y0)⊕H0(y0)→ H0(E) given by θ(a, b) = H∗j θrel(a) + p∗θ′(b) completesthe definition of the cohomology extension of the fiber θ in degree 0.

Corollary 28.41 (Relative Kunneth theorem II). Let X be a topological spaceand (Y,C) be a well cofibrant pair which is of finite cohomology type. Then, the


cross product

× : H∗(X)⊗H∗(Y,C)≈−→ H∗(X × Y,X × C)

is a GrV-isomorphism.

Proof. We see the projection π1 : (X × Y,X × C) → (X,X) as a trivialbundle pair with fiber (Y,C). Then θ = H∗π2 : H∗(Y,C) → H∗(X × Y,X × C)is a cohomology extension of the fiber. As (Y,C) is a well cofibrant pair, so is(X × Y,X × C). The cohomological result then follows from the relative Leray-Hirsch theorem 28.39.

28.6. The Thom isomorphism. We start by some preliminary results.

Lemma 28.42. Let p : (E, E)→ (B,B) be a bundle pair whose fiber (F , F ) is a

well cofibrant pair. Suppose that Hk(F , F ) = 0 for k < r and that Hr(F , F ) = Z2.

Then, Hk(E, E) = 0 for k < r and there is a unique isomorphism Φ∗ : Hr(E, E)≈−→

H0(B) such that, for each b ∈ B, the following diagram

(28.43)

Hr(Eb, Eb)

=

// // Hr(E, E)

Φ∗≈

H0(b) // // H0(B)

is commutative, where the horizontal homomorphisms are induced by the inclusions.

In Diagram (28.43), the left vertical isomorphism is abstract but well defined,

since both Hr(Eb, Eb) ≈ Hr(F , F ) and H0(b) are equal to Z2.

Proof. When B is path-connected, Lemma 28.42 implies that Hr(E, E) = Z2.Therefore, the homomorphism Φ∗, if it exists, is unique.

If the bundle pair is trivial, the lemma follows from the relative Kunneth the-orem 28.41. Suppose that B = B1 ∪B2, where B1 and B2 are two open sets withB1∩B2 = B0. Suppose that the the conclusion of the lemma is satisfied for (Ei, Ei)

for i = 0, 1, 2. Then, the Mayer-Vietoris sequence for the data (Ei, Ei) implies that

Hk(E, E) = 0 for k < r and gives the diagram

Hr(E0, E0) //

Φ∗≈

Hr(E1, E1)⊕Hr(E2, E2) //

Φ∗≈

Hr(E, E)

Φ∗≈

// 0

H0(B0) // H0(B1)⊕H0(B2) // H0(B) // 0

.

Diagram (28.43) for each b ∈ B0 implies that the left square is commutative.Therefore, the middle vertical isomorphism descends to a unique homomorphismΦ∗ : Hr(E, E) → H0(B) making the right square commutative, which an isomor-phism by the five-lemma. It remains to prove the commutativity of Diagram (28.43)

152 3. PRODUCTS

for Φ∗. Let b ∈ B. Without loss of generality, we may suppose that b ∈ B1. Con-sider the following diagram

Hr(Eb, Eb) //

=

Hr(E1, E1) //

Φ∗≈

Hr(E, E)

Φ∗≈

H0(b) // H0(B1) // H0(B)

.

Both square commuting, this gives the commutativity of Diagram (28.43) for Φ∗.

We have so far proven the lemma when the bundle pair (E, E) → (B,B)is of finite type. Let A be the set of subspaces A of B such that the bundle pair(EA, EA)→ (A,A) is of finite type. Each compact of B is contained in some A ∈ Aand each compact of E is contained in EA for some A ∈ A. By Proposition 12.47,this provides isomorphisms

(28.44) lim−→

A∈A

Hr(EA, EA) ≈ Hr(E, E) and lim−→

A∈A

H0(A) ≈ H0(B) .

Now, if A,A′ ∈ A with A ⊂ A′, Diagram (28.43) for each b ∈ A implies that thefollowing diagram

(28.45)

Hr(EA, EA)

Φ∗≈

// Hr(EA′ , EA′)

Φ∗≈

H0(A) // H0(A′)

is commutative. We therefore get isomorphisms

lim−→

A∈A

Hr(EA, EA)≈−→ lim

−→A∈A

H0(A)

which, together with the isomorphisms of (28.44), produce the required isomor-

phism Φ∗ : Hr(E, E)≈−→ H0(B).

By Kronecker duality, Lemma 28.42 gives the following lemma.

Lemma 28.46. Let p : (E, E)→ (B,B) be a bundle pair whose fiber (F , F ) is a

well cofibrant pair. Suppose that Hk(F , F ) = 0 for k < r and that Hr(F , F ) = Z2.

Then, Hk(E, E) = 0 for k < r and there is a unique isomorphism Φ∗ : H0(B)≈−→

Hr(E, E) such that, for each b ∈ B, the following diagram

(28.47)

H0(B)

Φ∗≈

// // H0(b)=

Hr(E, E) // // Hr(Eb, Eb)

is commutative, where the horizontal homomorphisms are induced by the inclusions.

Let p : (E, E)→ (B,B) a bundle pair satisfying the hypotheses of Lemma 28.46.

The class U = Φ∗(1) ∈ Hr(E, E) is called the Thom class of the bundle pair p. If B

is path-connected, the Thom class is just the non-zero element of Hr(E, E) = Z2,whence the following characterisation of the Thom class.


Lemma 28.48. The Thom class of p is the unique class in Hr(E, E) which

restricts to the generator of Hr(Eb, Eb) for all b ∈ B.

Let Σ be a topological space having the same homology (mod 2) as the sphereSk−1. For example, Σ = Sk−1 or a lens space with odd fundamental group. Letp : E → B be a bundle with fiber Σ. Let E be the mapping cylinder of p:

E = (E × I) ∪B/(x, 1) ∼ p(x) .

Let CΣ be the cone over Σ. Then p extends to a bundle pair p : (E, E) → (B,B)with fiber (CΣ,Σ), called the mapping cylinder bundle pair of p. As (CΣ,Σ) is a wellcofibrant pair (by Lemma 12.64) and Hk(CΣ,Σ) = 0 for k 6= r and Hr(CΣ,Σ) =

Z2, the Thom class U ∈ Hr(E, E) is defined.

Theorem 28.49 (The Thom isomorphism theorem). Let p : E → B be a bundle

with fiber Σ, where Σ has the homology of the sphere Sr−1. Let p : (E, E)→ (B,B)

be its mapping cylinder bundle pair. Let U ∈ Hr(E, E) be the Thom class. Then,the homomorphisms

Φ∗ : Hk(B)→ Hk+r(E, E) and Φ∗ : Hk(E, E)→ Hk−r(B)

given by

Φ∗(a) = H∗p(a) U and Φ∗(γ) = H∗p(U γ)

are isomorphism for all k ∈ Z.

Observe that Lemma 28.46 gives the result for k ≤ 0.

Proof. As Hj(CΣ,Σ) = 0 for j 6= r, the homomorphism θrel : H∗(CΣ,Σ) →

H∗(E, E) sending the generator of Hr(CΣ,Σ) = Z2 onto the Thom class U is a

cohomology extension of the fiber. Also, (E, E) and the fiber (CΣ,Σ) are wellcofibrant by Lemma 12.64. The relative Leray-Hirsch theorem 28.39 then provides

a GrV-isomorphism θrel : H∗(B) ⊗H∗(CΣ,Σ)

≈−→ H∗(E, E). Let Φ∗ be the com-posite isomorphism

Φ∗ : Hk(B) ≈ Hk(B)⊗Hr(CΣ,Σ)θrel−−→ Hk+r(E, E)

satisfy, by definition of θrel, the formula Φ∗(a) = H∗p(a) U . This proves thecohomology statement.

For the isomorphism Φ∗, let 0 6= m ∈ Hr(CΣ,Σ). Then m and U areKronecker dual bases for (co)homology of (CΣ,Σ) in degree r. By Theorem 28.39and Formula (28.38) implies that the composite isomorphism

Φ∗ : Hk+r(E, E)θrel−−→ Hk(B)⊗Hr(CΣ,Σ) ≈ Hk(B)

satisfies Φ∗(γ) = H∗p(U γ).

Let q : E → B be a bundle with fiber F and let f : A → B be a continuousmap. The induced bundle f∗q : f∗E → A is defined by

f∗E = (a, y) ∈ A× E | f(a) = q(y) , f∗q(a, y) = a ,

where f∗E is topologized as a subspace of A × E. Then f∗q is a bundle over Awith fiber F . The projection onto E gives a map f : f∗E → E and a commutative

154 3. PRODUCTS

diagram

f∗E

f∗q

f // E

q

A

f // B

.

Let p : E → B be a bundle with fiber Σ, where Σ has the homology of thesphere Sr−1. Let p : (E, E) → (B,B) be its mapping cylinder bundle pair. Let

f : A → B be a map. Then (f∗E, f∗E) → (A;A) is the mapping cylinder bundlepair of the induced bundle f∗E. The following lemma states the functoriality ofthe Thom class.

Lemma 28.50. If U ∈ Hr(E, E) is the Thom class of p, then H∗f(U) ∈Hr(f∗E, f∗E) is the Thom class of f∗p.

Proof. For a ∈ A, let consider the commutative diagram

Hr(E, E)

H∗ f // Hr(f∗E, f∗E)

Hr(Ef(a), Ef(a))H∗ f

≈// Hr((f∗E)a, (f

∗E)a)

.

Both cohomology groups downstairs are equal to Z2. The left vertical arrow sendsthe Thom class U to the non-zero element. Therefore, H∗f(U) goes, by the rightvertical arrow, to the non-zero element. As this is true for all a ∈ A, we deducefrom Lemma 28.48 that H∗f(U) is the Thom class of f∗p.

Let p : E → B be a bundle with fiber Σ, where Σ has the homology of thesphere Sr−1. Let p : (E, E)→ (B,B) be its mapping cylinder bundle pair. As CΣ

is contractible, Proposition 28.36 implies that H∗p : H∗(B) → H∗(E) is a GrA-isomorphism. Therefore, there is a unique class e ∈ Hr(B) such that

(28.51) H∗j(U) = H∗p(e) ,

where U ∈ Hr(E, E) is the Thom class and j : (E, ∅)→ (E, E) is the pair inclusion.The class e = e(p) is called the Euler class of the bundle p. If Φ∗ : Hr(B) →H2r(E, E) is the Thom isomorphism, one has the following formula

(28.52) Φ∗(e) = U U .

Indeed:

Φ∗(e) = H∗p(e) U = H∗j(U) U = U U ,

the last equality coming from Lemma 22.8. The Euler class is functorial by thefollowing lemma.

Lemma 28.53. Let p : E → B be a bundle with fiber Σ, where Σ has the homol-ogy of the sphere Sr−1. Let f : A → B be a map. If e ∈ Hr(B) is the Euler classof p, then H∗f(e) ∈ Hr(A) is the Euler class of f∗p.


Proof. This follows from the definition of the Euler class, Lemma 28.50 andthe commutativity of the following diagram.

Hr(E, E)

H∗ f

// H(E)

H∗ f

(H∗p)−1

≈// Hr(B)

H∗ f

Hr(E, E) // Hr(E)(H∗f∗p)−1

≈// Hr(A)

.

The Euler class occurs in the Gysin exact sequence.

Proposition 28.54 (Gysin exact sequence). Let p : E → B be a bundle withfiber Σ, where Σ has the homology of the sphere Sr−1. Let e ∈ Hr(B) be its Eulerclass. Then, there is a long exact sequence

· · · → Hk−1(B)H∗p−−−→ Hk−1(E)→ Hk−r(B)

−e−−−→ Hk(B)H∗p−−−→ Hk(E)→ · · ·

which is functorial with respect to induced bundles.

Proof. Let p : (E, E) → (B,B) be the mapping cylinder pair of p. One uses

the cohomology exact sequence of the pair (E, E) and the following diagram

· · · // Hk−1(E) // Hk(E, E)H∗j // Hk(E) // · · ·

Hk−r(B)

Φ∗ ≈

OO

−e // Hk(B)

≈

OO

where j : (E, E) → (E, ∅) denotes the inclusion and Φ∗ is the Thom isomorphism.The diagram is commutative since, for a ∈ Hk−r(B),

H∗j Φ∗(a) = H∗j(H∗p(a) U

)

= H∗p(a) H∗j(U)= H∗p(a) H∗p(e)= H∗p(a e) .

(The second equality is the singular analogue of Lemma 22.11). The functorialityof the Gysin exact sequence comes from Lemma 28.50 and 28.53.

Corollary 28.55. Let p : E → B be a bundle with fiber Σ, where Σ has thehomology of the sphere Sr−1. If p admits a continuous section, then the Euler classof p vanishes.

Proof. In the following segment of the Gysin sequence:

H0(B)−e−−−→ Hr(B)

H∗p−−−→ Hr(E) ,

the class 1 ∈ H0(B) is sent to the Euler class e. If p admits a section, then H∗p isinjective, which implies that e = 0.

Remark 28.56. The vanishing of the Euler class of p : E → B does not implythat p admits a section. As an example, let p : SO(3) → S2 the map sending a

156 3. PRODUCTS

matrix to its first column vector. Then p is an S1-bundle, equivalent to the unittangent bundle of S2. The Gysin sequence gives

H0(S2)−e−−−→ H2(S2)

H∗p−−−→ H2(SO(3))→ 0

As SO(3) is homeomorphic to RP 3, H2(SO(3)) = Z2 by Proposition 24.21 and allthe cohomology groups in the above sequence are equal to Z2. Hence, e = 0. Butit is classical that S2 admits no nowhere zero vector field [80, Theorem 2.28].

Proposition 28.57. Let p : E → B be a bundle with fiber Σ, where Σ has thehomology of the sphere Sr−1. Let e ∈ Hr(B) be its Euler class. Then, the followingassertions are equivalent.

(1) e = 0.(2) The restriction homomorphism H∗(E)→ H∗(Σ) is surjective.(3) Hr−1(E) ≈ Hr−1(B) ⊕ Z2.

Proof. Let p : (E, E)→ (B,B) be the mapping cylinder pair of p. IdentifyingΣ as the fiber over some point of B, we get a commutative diagram

0 // Hr−1(E) // Hr−1(E) //

Hr(E, E) //

≈

Hr(E)

Hr−1(Σ)≈ // Hr(CΣ,Σ)

where the top line is the cohomology exact sequence of (E, E). But Hr(E, E) ≈Z2 generated by the Thom class which, under the homomorphism Hr(E, E) →Hr(E) ≈ Hr(B), goes to the Euler class. This proves the proposition.

Let us consider the particular case of the Gysin sequence for an S0-bundle.Such a bundle is simply a 2-fold covering ξ = (p : X → X). The Gysin sequencemay thus be compared to the transfer exact sequence of Proposition 24.17.

Proposition 28.58. Let ξ = (p : X → X) be a 2-fold covering (an S0-bundle).Then, the Gysin and the transfer exact sequences of ξ coincide, i.e. the followingdiagram

· · · // Hk(X)OOid

H∗p // Hk(X)OOid

tr∗ // Hk(X)w(ξ)−//

OOid

Hk+1(X)OOid

// · · ·

· · · // Hk(X)H∗p // Hk(X) // Hk(X)

e(p)−// Hk+1(X) // · · ·

is commutative. In particular, the Euler class e(ξ) ∈ H1(X) and the characteristicclass w(p) ∈ H1(X) are equal.

Proof. By Corollary 24.7, ξ is induced from ξ∞ = (p∞ : S∞ → RP∞) by acharacteristic map f : X → RP∞. Both the Gysin and transfer exact sequencesbeing functorial with respect to induced bundles, it suffices to prove the propositionfor ξ∞. This is trivial since the vector spaces occurring in the diagram are eitherequal to 0 or Z2.


The Thom isomorphism is classically used for vector bundles. Recall that a(real) vector bundle ξ of rank r is a map p : E → B together with a R-vector spacestructure on Eb = p−1(b) for each b ∈ B, satisfying the following local trivialitycondition: there is an open covering U of B and for each U ∈ U , a homeomorphism

ψU : U × Rr≈−→ p−1(U) such that, for all (b, v) ∈ U × Rr, pψ(b, v) = b and

ψU : b × Rr → Eb is a R-linear isomorphism. In consequence, p is a bundlewith base B = B(ξ), total space E = E(ξ) and fiber Rr. The map σ0 : B → Esending b ∈ B to the zero element of Eb is called the zero section of ξ (it satisfiespσ0(b) = b).

An Euclidean vector bundle is a vector bundle ξ together with a continuousmap v 7→ |v| ∈ R≥0 defined on E(ξ) whose restriction to each fiber is quadraticand positive definite. Such a map is called an Euclidean structure (or Riemannianmetric) on ξ. It is of course the same as defining a positive definite inner product oneach fiber which varies continuously. Vector bundles with paracompact basis admitan Euclidean structure, [103, Chapter 3, Theorems 9.5 and 5.5]. If ξ = (p : E → B)is an Euclidean vector bundle, the restriction of p to

S(E) = v ∈ E | |v| = 1 and D(E) = v ∈ E | |v| ≤ 1

gives the associated unit sphere and disk bundles. These bundles do not dependon the choice of the Euclidean structure on ξ. Indeed, using the map (v, t) 7→ tvfrom S(E) × I → D(V ) together with the zero section, the reader will easily con-

struct a homeomorphism (S(E), S(E))≈−→ (D(E), S(E)) over the identity of B,

where (S(E), S(E)) → (B,B) is the mapping cylinder bundle pair of S(E) → B.Thus, the Thom class U ∈ Hr(D(E), S(E)) exists by Lemma 28.46 and, by Theo-

rem 28.49, gives rise to the Thom isomorphisms Φ∗ : Hk(B)≈−→ Hk+r(D(E), S(E))

and Φ∗ : Hk+r(D(E), S(E))≈−→ Hk(B). Let E0 = E − σ0(B) and D(E)0 =

D(E) ∩ E0. By excision and homotopy, one has

H∗(E,E0)≈−→ H∗(D(E), D(E)0)

≈−→ H∗(D(E), S(E)) .

Hence the Thom class may be seen as an element U(ξ) ∈ Hr(E,E0) and one hasthe following theorem.

Theorem 28.59 (The Thom isomorphism theorem for vector bundles). Letξ = (p : E → B) be a vector bundle of rank r with B paracompact. Let U(ξ) ∈Hr(E,E0) be the Thom class. Then, the homomorphisms

Φ∗ : Hk(B)→ Hk+r(E,E0) and Φ∗ : Hk(E,E0)→ Hk−r(B)

given by

Φ∗(a) = H∗p(a) U(ξ) and Φ∗(γ) = H∗p(U(ξ) γ)

are isomorphism for all k ∈ Z.

Let ξ = (p : E → B) be a vector bundle of rank r. The map E × I → E givenby 7→ tv is a retraction by deformation of E onto the zero section of ξ. Hence,H∗p : H∗(B) → H∗(E) is a GrA-isomorphism. Therefore, there is a unique classe(ξ) ∈ Hr(B) such that H∗j(U(ξ)) = H∗p(e(ξ)), where j : (E, ∅) → (E,E0). Theclass e(ξ) is called the Euler class of ξ. Lemma 28.53 and Corollary 28.55 implythe following two lemmas.

158 3. PRODUCTS

Lemma 28.60. Let ξ = (p : E → B) be a vector bundle of rank r, with Bparacompact. Let f : A → B be a continuous map. Then the equality e(f∗ξ) =H∗f(e(ξ)) holds in Hr(A).

Lemma 28.61. Let ξ = (p : E → B) be a vector bundle of rank r, with Bparacompact. If ξ admits a nowhere zero section, then e(ξ) = 0.

Let ξi = (pi : Ei → Bi) (i = 1, 2) be two vector bundles of rank ri, with Biparacompact. The product bundle ξ1× ξ2 is the vector bundle of rank r1 + r2 givenby p1 × p2 : E1 × E2 → B1 × B2. If B1 = B2 = B, the Whitney sum ξ1 ⊕ ξ2 isthe the vector bundle of rank r1 + r2 over B given by ξ1 ⊕ ξ2 = ∆∗(ξ1 × ξ2) where∆: B → B × B is the inclusion of the diagonal, ∆(x) = (x, x). The behaviour ofthe Euler class under these constructions is as follows.

Proposition 28.62. .

(1) e(ξ1 × ξ2) = e(ξ1)× e(ξ2).(2) e(ξ1 ⊕ ξ2) = e(ξ1) e(ξ2).

Proof. Using Euclidean structures on ξi the Thom class U(ξi) may be seen asan element of Hri(D(Ei), S(Ei)). Let E = E1 ×E2, B = B1×B2 and r = r1 + r2.

Let ji : (D(Ei), ∅) → (D(Ei), S(Ei)) and j : (D(E), ∅) → (D(E), S(E)) denotethe inclusions. There are homeomorphisms of pairs making the following diagramcommutative

(D(E), ∅)≈

j // (D(E), S(E))

≈

(D(E1), ∅)× (D(E2), ∅)j1×j2 // (D(E1), S(E1))× (D(E2), S(E2))

.

In the same way, if b = (b1, b2) ∈ B, there is a homeomorphism of pairs

(28.63)(D(E)b, S(E)b

)≈

(D(E1)b1 , S(E1)b1

)×

(D(E2)b2 , S(E2)b2

).

By the relative Kunneth theorem 27.25, the generator of Hr(D(E)b, S(E)b) =Z2 is the cross product of the generators ofHri(D(Ei)bi

, S(Ei)bi). Using Lemma 28.48,

we deduce that

(28.64) U(ξ1 × ξ2) = U(ξ1)× U(ξ2) .

Using Lemma 27.8, one has

H∗(p1 × p2)(e(ξ)) = H∗j(U(ξ))= H∗j(U(ξ1)× U(ξ2))= H∗j1(U(ξ1))×H∗j2(U(ξ2))= H∗p1(e(ξ1))×H∗p2(e(ξ2))= H∗(p1 × p2)(e(ξ1)× e(ξ2)) .

As H∗(p1 × p2) is an isomorphism, this proves (1). Point (2) is deduced from (1)using the definition of ξ1 ⊕ ξ2 and Remark 27.5:

e(ξ1 ⊕ ξ2) = H∗∆(e(ξ1 × ξ2)) = H∗∆(e(ξ1)× e(ξ2)) = e(ξ1) e(ξ2) .

The Thom class of a product bundle was computed in (28.64). For the Whitneysum, we use the projections πi : E(ξ1 ⊕ ξ2)→ E(ξi).


Proposition 28.65. Let ξ1 and ξ2 be two vector bundles over a paracompactbasis. Let U(ξi) ∈ Hri(D(Ei), S(Ei)) be the Thom classes (for an Euclidean struc-ture). Then

U(ξ1 ⊕ ξ2) = H∗π1(U(ξ1)) H∗π2(U(ξ2)) .

Proof. Restricted to the fiber over b ∈ B, the right hand side of the formulagives the cross product of the generators of Hri(D(Ei)b, S(Ei)b). The latter is thegenerator of Hr(D(E)b, S(E)b). The proposition thus follows from Lemma 28.48.

28.7. Bundles over spheres. In this section, we study bundles ξ = (p : E →Sm) over the sphere Sm with fiber F . If A ⊂ Sm we set EA = p−1(A). Consider thecellular decomposition of Sm with one 0-cell b and one m-cell with characteristicmap ϕ : Dm → Sm sending Sm−1 onto b. We denote by φ : Sm−1 → b thisconstant map. Identify F with Eb, getting thus an inclusion i : F → E. As Dm iscontractible, any bundle over Dm is trivial [177, Corollary 11.6]. Therefore, thereexists a trivialization ϕ∗E ≈ Dm × F of the induced bundle ϕ∗ξ. The map (ϕ, φ)

are covered by a bundle maps ϕ : Dm ×F → E and φ : Sm−1 ×F → F . The latter

satisfies, for each x ∈ Sm−1, that φ : x × F ≈−→ F is a homeomorphism. Observethat

(28.66) E = (Dm × F ) ∪ϕ F .Let x0 ∈ Sm−1 be the base point corresponding to 1 ∈ S0 ⊂ .Sm−1. By changingthe trivialization of ϕ∗ξ if necessary, we shall assume that φ : x0 × F → F is theprojection onto F . The map ϕ is called the bundle characteristic map and the mapφ is called the bundle gluing map of the bundle ξ.

Lemma 28.67. The bundle characteristic map ϕ : Dm × F → E induces anisomorphism

H∗ϕ : H∗(E,F )≈−→ H∗(Dm × F, Sm−1 × F ) .

Proof. Consider the decomposition Dm = B ∪ C, where B is the disk withcenter 0 and radius 1/2 and C the adherence of Dm −B; let S = B ∩C. As ϕ(C)is a disk around b, the bundle ξ is trivial above ϕ(C): Eϕ(C) ≈ ϕ(C) × F . Asϕ : B×F → Eϕ(B) is a homeomorphism, the lemma follows from the commutativediagram

H∗(E,F )H∗ϕ // H∗(Dm × F, Sm−1 × F )

H∗(E,Eϕ(C))

≈ excision

≈ excision

OO

H∗(Dm × F,C × F )

≈ excision

OO

≈ excision

H∗(Eϕ(B), Eϕ(S))H∗ϕ

≈// H∗(B × F, S × F )

Proposition 28.68. Let p : E → Sm be a bundle with fiber F . There is a longexact sequence

· · · → Hk−1(E)H∗i−−→ Hk−1(F )

Θ−→ Hk−m(F )J−→ Hk(E)

H∗i−−→ Hk(F )→ · · · .

160 3. PRODUCTS

The exact sequence of Proposition 28.68 is called the Wang exact sequence.

Proof. We start with the exact sequence of the pair (E,F )

(28.69) · · · → Hk−1(E)H∗i−−→ Hk−1(F )

δ∗−→ Hk(E,F )H∗j−−−→ Hk(E)→ · · ·

where j : (E, ∅) → (E,F ) denotes the pair inclusion. The following commutativediagram defines the homomorphism Θ and J .

(28.70)

Hk−1(F )

Θ

''

δ∗ // Hk(E,F )H∗j //

H∗ϕ≈

Hk(E)

Hk(Dm × F, Sm−1 × F )

Hk−m(F )

e×−≈

OO J

DD

Here, H∗ϕ is an isomorphism by Lemma 28.67, e ∈ Hm(Dm, Sm−1) = Z2 is thegenerator and the map e × − is an isomorphism by the relative Kunneth theo-rem 28.41.

We now give some formulae satisfied by the homomorphism Θ: Hk−1(F ) →Hk−m(F ). We start with the case m = 1 which deserves a special treatment. Thebundle gluing map φ : S0×F → F satisfy φ(1, x) = x and φ(−1, x) = h(x) for somehomeomorphism h : F → F . The decomposition of (28.66) amounts to say that Eis the mapping torus Mh of h:

E = Mh =([−1, 1]× F

)/(1, x) ∼ (−1, h(x)) .

The bundle projection p : Mh → S1 is given by p(t, x) = exp(2iπt). The corre-spondence x → [(x, 0)] gives an inclusion j : F → Mh. Let e ∈ H1(S1) = Z2 bethe generator. Proposition 28.68 may be rephrased and made more explicit in thefollowing way.

Proposition 28.71 (Mapping torus exact sequence). Let h : F → F be a home-omorphism. Then, there is a long exact sequence

· · ·Hk−1(Mh)H∗i−−→ Hk−1(F )

Θ−→ Hk−1(F )J−→ Hk(Mh)

H∗i−−→ Hk(F )→ · · · ,with Θ = id +H∗h.

Proof. We use the exact sequence (28.69) with E = Mh and Diagram (28.70).It remains to identify Θ with id +H∗h. Let i± : ±1 × F → S0 × F denote theinclusions. Let α : Hk−1(F ) → Hk−1(S0 × F ) be the homomorphism such thatH∗i+α(a) = a and H∗i−α(a) = H∗h(a). Consider the following diagram.

Hk−1(F )

δ∗

α // Hk−1(S0 × F )

δ∗

i∗

≈// Hk−1(1 × F )⊕Hk−1(−1 × F )

+

Hk(Mh, F )H∗ϕ

≈// Hk(D1 × F, S0 × F ) Hk−1(F )

e×−

≈oo

where i∗ = (H∗i+, H∗i−). Let Ψ± : Hk−1(F ) → Hk−1(F ) be the composed ho-

momorphisms through the upper right or lower left corners. Then ψ+ = id +H∗h


and ψ− = Θ. The left square of the diagram being commutative by construction ofMh, it then suffices to prove the commutativity of the right square, that is δ∗ = ψ,where ψ(a) = e×

(H∗i+(a) +H∗i−(a)

). The homomorphisms δ∗ and ψ are both

functorial. As, by § 19, a class a ∈ Hk−1(F ) is represented by a map F → Kk−1,it suffices to prove that δ∗ = ψ for F = Kk−1. Observe that δ∗ and ψ are bothsurjective and have the same kernel, the image of Hk−1(D1×F )→ Hk−1(S0×F ).As Hk−1(Kk−1) = Z2, this proves that δ∗ = ψ when F = Kk−1.

Example 28.72. Let h : S1 → S1 be the complex conjugation. Then, Mh is

homeomorphic to the Klein bottle K and we get a bundle S1 → Kp−→ S1. The

homomorphism Θ of Proposition 28.71 satisfies Θ = id+H∗h = 0. By the mappingtorus exact sequence, we deduce that H∗(K)→ H∗(S1) is surjective (this can alsobe obtained using a triangulation like on p. 28 and computations like on p. 117).A cohomology extension of the fiber σ : H∗(S1)→ H∗(K) produces, by the Leray-

Hirsch theorem 28.26, a GrV-isomorphism σ : H∗(S1) ⊗H∗(S1)≈−→ H∗(K). But

σ is not a morphism of algebra. Indeed, the square map x 7→ x•x vanishes inH∗(S1)⊗H∗(S1) while the cup-square map x 7→ x x does not vanish in H∗(K)(see, Proposition 23.5).

When m > 1, some information about the homomorphism Θ: Hk−1(F ) →Hk−m(F ) may be obtained via the composition

Hk−1(F )Θ // Hk−m(F ) // e×− // Hk−1(Sm−1 × F ) ,

where e ∈ Hm−1(Sm−1) = Z2 is the generator. The map e× − is injective by theKunneth theorem.

Proposition 28.73. Suppose that m > 1. Then

e×Θ(a) = H∗φ(a)−H∗p2(a) ,

where φ, p2 : Sm−1 × F → F are the bundle gluing map and the projection onto F .

Proof. As F is a retract of Sm−1 × F , the cohomology exact sequence of thepair (Sm−1 × F, F ) splits into short exact sequences and, by the Kunneth theoremand Lemma 28.4, there is a commutative diagram(28.74)

0 // Hk−m(F )

≈

e×−

**TTTTTTTTTTTTTT// Hk−m(F )⊕Hk−1(F )

α≈

// Hk−1(F )

id≈

// 0

0 // Hk−1(Sm−1 × F, F ) // Hk−1(Sm−1 × F )H∗i // Hk−1(F ) // 0

where i : F → Sm−1 × F is the slice inclusion at the base point x0 ∈ §m−1 andα(a, b) = e× a+ 1× b. Recall that we assume the restriction of φ to x0 × F tocoincide with the projection p2. Therefore, the composition

Hk−1(F )H∗φ−H∗p2−−−−−−−→ Hk−1(Sm−1 × F )→ Hk−1(F )

162 3. PRODUCTS

vanishes. Using Diagram (28.74), we get a factorisation

Hk−1(F )H∗φ−H∗p2 //

Θ′

&&MMMMMMMMMHk−1(Sm−1 × F )

Hk−m(F )

e×−

66mmmmmmmmmm

which we introduce in the following diagram

(28.75)

Hk−1(F )

Θ′

&&

δ∗ //

H∗φ+H∗p2

Hk(E,F )

H∗ϕ≈

Hk−1(Sm−1 × F )δ∗ // Hk(Dm × F, Sm−1 × F )

Hk−m(F )= //

e×−

OO

Hk−m(F )

e×−≈

OO.

We claim that the two square of Diagram (28.75) are commutative. By Dia-gram (28.70), this will imply that Θ′ = Θ and will prove the lemma.

As φ is the restriction of ϕ, the naturality of the connecting homomorphism δ∗

implies that δ∗ H∗φ = H∗ϕδ∗. Since p2 extends to Dn × F , the homomorphismH∗p2 : Hk−1(F ) → Hk−1(Sm−1 × F ) factors through Hk−1(Dm × F ) and thusδ∗H∗p2 = 0. Hence, the top square is commutative. For the bottom one, leta ∈ Hk−1(F ). By § 19, a = H∗f(ι) for some map f from F into the Eilenberg-McLane space Kk−1. The bottom square being functorial for the map f , it sufficesto prove its commutativity for F = Kk−1. As the source and range vector spaceare both then isomorphic to Z2, the commutativity holds trivially.

As an exercise, the reader may adapt the proof of Proposition 28.73 to the casem = 0, thus getting an alternative proof of Proposition 28.71. The main point isto replace e (which has no meaning in H0(S0) by the class of −1.

The family of homomorphisms Θ: Hk−1(F ) → Hk−m(F ) forms an endomor-phism of H∗(F ) of degree m− 1 (it sends Hq(F ) to Hq−m+1(F )).

Proposition 28.76. As an an endomorphism of H∗(F ), Θ satisfies

Θ(a b) = Θ(a) b + a Θ(b) .

Proof. Proposition 28.73 may be rephrased as

H∗φ(a) = H∗p2(a) + e×Θ(a) = 1× a+ e×Θ(a) .

Therefore, if a ∈ Hp(F ) and b ∈ Hq(F ),

H∗φ(a b) = 1× (a b) + e×Θ(a b)

and, using Lemma 27.9,

H∗φ(a) H∗φ(b) =[1× a+ e×Θ(a)

]

[1× b+ e×Θ(b)

]

= 1× (a b) + e× (Θ(a) b) + e× (a Θ(b))= 1× (a b) + e×

[Θ(a) b+ a Θ(b)

].

As H∗φ(ab) = H∗φ(a)H∗φ(b) and e×− is injective, this proves the proposi-tion.


Remark 28.77. The material of this section was inspired by [203, § 1, Chap-ter VII]. As in this this reference, the following fact can also be proved:

(1) The Wang exact sequence holds for Serre fibrations. It also has a gener-alization to bundles over a suspension.

(2) A Wang exact sequence for homology exists.

Further properties of the Wang sequences are given in [203, § 2, Chapter VII].

28.8. The face space of a simplicial complex. Let K be simplicial com-plex. Fix an integer d > 0. For each v ∈ V (K), consider a copy Sdv of the sphereSd. It is pointed by e1 = (1, 0, . . . , 0) ∈ Sdv . For σ ∈ S(K), consider the space

Fd(σ) = (zv) | zv = e1 if v /∈ σ ⊂∏

v∈V (K)

Sdv ,

which is homeomorphic to∏v∈σ S

dv . The face space ofK is the subset of

∏v∈V (K) S

dv

defined by

Fd(K) =⋃

σ∈S(K)

Fd(σ) ⊂∏

v∈V (K)

Sdv .

Remark 28.78. Let K be a flag simplicial complex (i.e. if K contains a graphL isomorphic to the 1-skeleton of an r-simplex, then L is contained in an r-simplexof K). Then the complex F1(K) is the Salvetti complex of the right-angled Coxetergroup determined by the 1-skeleton of K (see [28]).

The interest of the face space appears in the following proposition, based on aalgebraic theorem of J. Gubeladze.

Proposition 28.79. Let K and K ′ be two finite simplicial complexes. Let dbe a positive integer. Then, K is isomorphic to K ′ if and only if H∗(Fd(K)) andH∗(Fd(K

′)) are GrA-isomorphic.

To explain the proof of Proposition 28.79, we compute the cohomology algebraof Fd(K) for a finite simplicial complex K. Let us number the vertices of K:V (K) = 1, . . . ,m. Consider the polynomial ring Z2[x1, . . . , xm] with formalvariables x1, . . . , xm which are of degree d. If J ⊂ 1, . . . ,m, we denote by xJ ∈Z2[x1, . . . , xm] the monomial

∏j∈J xj . Let I(K) be the ideal of Z2[x1, . . . , xm]

generated by the squares x2i of the variables and the monomials xJ for J /∈ S(K)

(non-face monomials). The quotient algebra

Λd(K) = Z2[x1, . . . , xm]/I(K)

is called the face exterior algebra (because u2 = 0 for all u ∈ Λd(K); however,because the ground field is Z2, Λd(K) is commutative).

Lemma 28.80. The ring H∗(Fd(K)) is isomorphic to Λd(K).

Proof. (Compare [57, Proposition 4.3].) Let ∆K = F(V (K)) be the fullcomplex over the set V (K) = 1, . . . ,m. The simplicial inclusion K ⊂ ∆K inducesan inclusion

j : Fd(K) → Fd(∆K) =∏

v∈V (K)

Sdv .

For σ ⊂ 1, . . . ,m, the fundamental class [Fd(σ)] ∈ H(dimσ+1)d(Fd(σ)) deter-mines a class [σ] ∈ H(dimσ+1)d(Fd(∆K)) (by convention, [∅] is the generator of

164 3. PRODUCTS

H0(Fd(∆K))). If σ ∈ S(K), [σ] is the image underH∗j of a class inH(dimσ+1)d(Fd(K)),also called [σ]. Let

A = [σ] ∈ H∗(Fd(K)) | σ ∈ S(K) ∪ ∅ ⊂ H∗(Fd(K))

and

B = [σ] ∈ H∗(Fd(∆K)) | σ ⊂ 1, . . . ,m ⊂ H∗(Fd(∆K)) .

By the Kunneth theorem and Corollary 12.22, H∗(Fd(K)) is generated by A andB is a basis of H∗(Fd(∆K)). It follows that A is a basis of H∗(Fd(K)) and thatH∗j is injective. By Kronecker duality, H∗j is surjective and the Kronecker-dualbasis B♯ of B is sent onto Kronecker-dual basis A♯ of A by

(28.81) H∗j([σ]♯) =

[σ]♯ if σ ∈ S(K)

0 otherwise.

By the Kunneth theorem again,

(28.82) H∗(Fd(∆K)) ≈ Z2[x1, . . . , xm]/(x2

1, . . . , x2m)

and, if σ ⊂ 1, . . . ,m, then [σ]∗ = xσ. By (28.81), kerH∗j is the Z2-vector space inH∗(Fd(∆K)) with basis xσ | σ /∈ S(K). Using (28.82), we check that, under theepimorphism Z2[x1, . . . , xm]→→ H∗(Fd(∆K)), kerH∗j is the image of I(K).

The proof of Lemma 28.80 provides the following corollary.

Corollary 28.83. The Poincare polynomial of the algebra Λd(K) is

Pt(Λd(K)) = 1 +∑

σ∈S(K)

t(dimσ+1)d .

The proof of Proposition 28.79 follows from Lemma 28.80 and the followingtheorem of J. Gubeladze. For a proof, see [74, Theorem 3.1].

Theorem 28.84 (J. Gubeladze). Let K and K ′ be two finite simplicial com-plexes. Suppose that Λd(K) = Z2[x1, . . . , xm]/I(K) and Λd(K

′) = Z2[y1, . . . , ym′ ]/I(K ′)are isomorphic as graded algebras. Then m = m′ and there is a bijection

φ : x1, , . . . , xm ≈−→ y1, . . . , ym′such that φ(I(K)) = I(K ′).

28.9. Continuous multiplications on K(Z2,m). A continuous multiplica-tion µ : X ×X → X on a space X is homotopy commutative if the maps (x, y) 7→µ(x, y) and (x, y) 7→ µ(y, x) are homotopic. A element u ∈ X is a homotopy unitfor µ if the maps x 7→ µ(u, x) and x 7→ µ(x, u) are homotopic to the identity of X .Note that, if u0 ∈ X is a homotopy unit for µ and if X is path-connected, then anyu ∈ X is also a homotopy unit.

Let K ≈ K(Z2,m) be an Eilenberg-McLane space in degree m, with its class0 6= ι ∈ Hm(K). Recall from § 19, the map φ : [X,K] −→ Hm(X) given by φ(f) =H∗f(ι) is a bijection. In particular, if K and K′ are two Eilenberg-McLane spacesin degree m, there is a a homotopy equivalence g : K′ → K whose homotopy classis unique.

Proposition 28.85. Let K be an Eilenberg-McLane space in degree m.

(1) There exists a continuous multiplication on K admitting a homotopy unitand which is homotopy commutative.


(2) Any two continuous multiplications on K admitting a homotopy unit arehomotopic.

(3) Let (K, µ) and K′, µ′) be two Eilenberg-McLane spaces in degree m withcontinuous multiplications admitting homotopy units, Let g : K′m → Kma (unique up to homotopy) homotopy equivalence. Then, the followingdiagram

K′ ×K′ µ′ //

g×g

K′

g

K ×K µ // K

commutes up to homotopy.

Proof. Consider the class

(28.86) p = ι× 1 + 1× ι ∈ Hm(K ×K) .

Since [K × K,K] is in bijection with Hm(K × K), one has p = H∗µ(ι) for somecontinuous map µ : K × K → K, which we see as a continuous multiplication. Theinvolution τ exchanging the coordinates on K × K satisfies H∗τ(p) = p and thenH∗(µτ) = H∗µ. Hence, µτ is homotopic to µ, which says that µ is homotopycommutative.

Choose u ∈ K and let i1, i2 : K → K × K be the slice inclusions i1(x) = (x, u)and i2(x) = (u, x). By Lemma 28.4, i∗j H

∗µ(ι) = ι for j = 1, 2. Hence, µij is

homotopic to the identity, which proves that u is a homotopy unit. Point (1) isthus established.

For Point (2), let µ is continuous multiplication on K admitting a homotopyunit u. Let i1, i2 : K → K×K be the slice inclusions i1(x) = (x, u) and i2(x) = (u, x).As u is a homotopy unit, hij is homotopic to the identity for j = 1, 2, and thusH∗ij H

∗µ(a) = a for all a ∈ H∗(X). by Lemma 28.4, this implies that

(28.87) H∗µ(a) = a× 1 + 1× a+∑

y × y′ ,where the degrees of y and y′ are both positive. By the Kunneth theorem, the crossproduct gives an isomorphism isomorphism Hm(K)⊗H0(K)⊕H0(K)⊗Hm(K) ≈Hm(K). Therefore, H∗µ(ι) = p, which says that the homotopy class of µ is welldetermined.

For Point (3), let h : K → K ′ be a homotopy inverse for g. Then, the formulaµ′′(x, y) = hµ(g(x), g(y)) is a continuous multiplication of K′ with a homotopyunit. By (2), µ′′ is homotopic to µ′, which proves (3).

Examples 28.88. The following classical multiplications occur in Eilenberg-McLane spaces Km ≈ K(Z2,m) (or more generally on K(G,m) for an abeliangroup G).

• The loop space ΩKm+1 is an Eilenberg-McLane space in degree m [80,pp. 407 and ff.]. One can use the loop multiplication.• Using semi-simplicial techniques, J. Milnor has shown that there exists an

Eilenberg-McLane space Km which is an abelian topological group [143,§ 3].

The following property of the multiplication µ of Proposition 28.85 will beuseful in § 47.

166 3. PRODUCTS

Lemma 28.89. Let K be an Eilenberg-McLane space in degree m. Let a ∈Hk(K) for n ≤ k < 2n. Then

(28.90) H∗µ(a) = a× 1 + 1× a .Proof. This comes from (28.87) since

(28.91) Hk(K)⊗H0(K) ⊕ H0(K) ⊗Hk(K)≈−→ Hk(K ×K)

for n ≤ k < 2n by the Kunneth theorem.

Remark 28.92. Together with the cup product, the mapH∗µmakesH∗(K(Z2,m))a Hopf algebra (see [80, Section 3.C]). In this setup, an element a ∈ Hk(K(Z2,m))satisfying (28.90) is called primitive.

Let X be a CW-complex. the multiplication µ on K = K(Z2;m) induces acomposition law

[X,K]× [X,K]⋆−→ [X,K]

given by f ⋆ g (x) = µ(f(x), g(x)). It admits the following interpretation.

Proposition 28.93. Let X be a CW-complex. Then, the bijection φ : Hm(X)≈−→

[X,K] satisfies

φ(a) ⋆ φ(b) = φ(a+ b) .

for all a, b,∈ Hm(X).

Proof. Let f, g : X → K represent φ(a) and φ(b). Then φ(a) ⋆ φ(b) is repre-sented by the composition

X(f,g)−−−→ K×K h−→ K .

The two projections π1, π2 : K × K → K satisfy π1 (f, g) = f and π2(f, g) = g.Using that H∗µ(ι) = p (see the proof of Proposition 28.85), one has

φ(a) ⋆ φ(b) = H∗(f, g)H∗f(ι)

= H∗(f, g)(ι× 1 + 1× ι)= H∗(f, g)(H∗π1(ι) +H∗π2(ι))

= H∗f(ι) +H∗g(ι) = φ(a) + φ(b) .


3.1. Write the proof of Lemma 22.11.

3.2. As H∗(S1 ∨ S1) has 4 elements, the bouquet of two circle has 4 inequivalent2-fold coverings by the bijection (24.10). For each of them, describe the total spaceand the transfer exact sequence.

3.3. Same exercise as the previous one, replacing S1 ∨ S1 by the Klein bottle.Compare with the discussion on p. 31.

3.4. Write the transfer exact sequence for a trivial 2-fold covering.

3.5. Let p : X → X be finite covering an odd number of sheets. Prove that H∗p isinjective.


3.6. Let M and N be closed surfaces, with M orientable and N non-orientable.Prove that there is no continuous map f : M → N which is of degree one.

3.7. Show that there are no continuous map of degree one between the torus andthe Klein bottle, in either direction. Same things for S1 × S2 and RP 3.

3.8. Let M be a closed topological manifold of dimension n. Let h : Dn → M bean embedding of the closed disk Dn into M . Form the manifold M as the quotientof M − inth(Dn) by the identification h(x) ∼ h(−x) for x ∈ BdDn. Compute the

ring H∗(M). [Hint: express M as a connected sum.]

3.9. Show that the cohomology algebras of (S1×S1) ♯RP 2 and of RP 2 ♯RP 2 ♯RP 2

are GrA-isomorphic. (Actually, these two spaces are homeomorphic, see [133,Lemma 7.1]).

3.10. Using the triangulation of the Klein bottle given in Figure 4, compute all thesimplicial cap products.

3.11. Show that the join and the smash product of two homology spheres is ahomology sphere.

3.12. Compute the cohomology ring H∗(X) for (a) X = RP∞ × · · · × RP∞ (ntimes); (b) X = CP 2 ∧CP 3; (c) X = CP 2 ∗CP 3.

3.13. Write the Mayer-Vietoris cohomology sequence for the decomposition

S1 × Sn = [(S1 − 1))× Sn]× [(S1 − −1))× Sn] .and describe its various homomorphisms. If a ∈ H1(S1) and b ∈ Hn(Sn) are thegenerators, describe how the elements a× 1, 1× b and a× b behave with respect tothe homomorphisms of the Mayer-Vietoris sequence.

3.14. Show that the product of two perfect CW-complexes is a perfect CW-complex.

3.15. What is the Lusternik-Schnirelmann category of RP 2 × RP 3?

3.16. Prove the relevant functoriality property for the homology cross product.

3.17. Cap product in the (co)homology of X × Y . Let X and Y be topologicalspaces, with Y being of finite cohomology type. Let a ∈ H∗(X), b ∈ H∗(Y ),α ∈ H∗(X) and β ∈ H∗(Y ). Prove that the formula

(29.1) ×((a× b) ×−1(α⊗ β)

)= (a α)⊗ (b β)

holds in H∗(X)⊗H∗(Y ), using the (co)homology cross products × and × of § 27.

3.18. Slices in homology. Let X and Y be topological spaces, with Y being offinite cohomology type. Let y0 ∈ Y and let sX : X → X × Y be the slice inclusionof X at y0. Let α ∈ H∗(X). Prove that

H∗sX(α) = ×(α⊗ y0

),

where y0 is seen as a 0-homology class of Y , using the bijection Y ≈ S0(Y ).

3.19. Let K be a finite simplicial complex and let Fd(K) its face complex for aninteger d > 0. What is the relationship between the Euler characteristic of Fd(K)and that of K?

CHAPTER 4

Poincare Duality

30. Algebraic topology and manifolds

Manifolds studied by algebraic topology tools occur in several categories: smooth,piecewise linear, topological, homology manifolds, etc. Here below a few wordsabout this matter.

Henri Poincare’s paper analysis situs [158], published 1895, is considered as thehistorical start of algebraic topology (for the “prehistory” of the field, see [160]).The aim of Poincare was to use tools of algebraic topology in order to distinguishsmooth manifolds up to diffeomorphism (which he called “homeomorphism”). So,differential and algebraic topology were born together. The importance of study-ing smooth manifolds up to diffeomorphism was reaffirmed all along the twentiethcentury by many great mathematicians (Thom, Smale, Novikov, Atiyah, etc). It isbased on the deep role played by global properties of smooth manifolds in analysis,differential geometry, dynamical systems and physics.

After the failure of defining homology using submanifolds (see [38, § I.3]),Poincare initiated a new approach [159], in which smooth manifolds are equippedwith a triangulation. This permitted him to define what will later become simplicialhomology. The existence and essential uniqueness of smooth triangulations wereof course a problem, solved only in 1940 by J.H.C. Whitehead [204, Theorems 7and 8]. Also, besides some developments in the twenties (Veblen, Morse), thereal foundations of differential topology rose only after 1935 with the works ofH. Whitney. As a result, homology was for three decades seen as combinatorial innature and smooth manifolds were not considered as a right object to study. Inthe prominent book written in 1934 by H. Seifert and W. Threfall [170], smoothmanifolds are not even mentioned, but replaced by a simplicial counterpart, i.e.combinatorial or piecewise linear (PL) manifolds (see definition in § 31 below).Techniques analogous to those for smooth manifolds were later developed in thePL-framework (see [102]). Polyhedral homology manifolds were later introduced(see § 31), whose importance may grow with the development of computationalhomology. For even more general objects, like ANR homology manifolds, see e.g.[202].

Topological manifolds have also long attracted the attention of topologists,mostly to know whether they carry smooth or piecewise linear structures (see,e.g. [7, p. 235], [129, p. 183]). Their status remained however mysterious till the1960s. R. Kirby and L.C. Siebenmann produced examples in all dimension ≥ 5of topological manifolds without PL-structures and developed much techniques todeal with these topological manifolds [114]. The field of topological versus smoothmanifolds developed very much in dimension four, after 1980, with the work ofM. Freedman and S. Donaldson.

169

170 4. POINCARE DUALITY

Poincare duality is one of the most remarkable properties of closed manifolds.In its strong form, it gives, for a compact n-manifoldM , that Hk(M) andHn−k(M)are isomorphic under the cap product with the fundamental class [M ]. This resultcan be obtained in two contexts:

• by working with homology manifolds, using simplicial topology and dualcells. Taking its origin in the early work of Poincare, this was achievedaround 1930 in the work by L. Pontryagin, L. Vietoris and S. Lefschetz(see [38, § II.4.C]). In the next sections, we follow this approach, akinto the presentation of [152, Chapter 8] This proves Poincare duality fortriangulable topological manifold, whence for smooth manifolds. Observethat smooth manifolds techniques (Morse theory or handle presentations)give an isomorphism from Hk(M) to Hn−k(M) but not the identificationof this isomorphism with a cap product (see e.g. [118, §VII.6]).• by working with topological manifolds, using Cech cohomology techniques

(see, e.g. [175, § 6.2] or [80, § 3.3]). This is not done in this book.

31. Poincare Duality in polyhedral homology manifolds

A polyhedral homology n-manifold is a simplicial complex such that, for eachσ ∈ Sk(M), the link Lk(σ) of σ in M is a simplicial complex of dimension n−k−1which has the homology of the sphere Sn−k−1. (Recall that our homology is mod2by default; thus, in a broader context, these objects may more acurately be calledpolyhedral Z2-homology manifolds).

Remark 31.1. (1) Let X be topological space satisfying the followinglocal property: for any x ∈ X ,

Hj(X,X − x) =

Z2 j = n

0 j 6= n .

Such a space is called a homology n-manifold. For instance, an n-dimensionaltopological manifold is a homology n-manifold by (14.5). The followingresult is proven in e.g. [152, Theorem 63.2]: if K is a simplicial complexsuch that |K| is a homology n-manifold, then K is a polyhedral homologyn-manifold.

(2) Special kind of polyhedral homology n-manifold are PL-manifolds. Asimplicial complex M is a PL-manifold, or a combinatorial manifold if, foreach σ ∈ Sk(M), the link Lk(σ) of σ in M has a subdivision isomorphicto a subdivision of the boundary of the (n − k)-simplex. PL-manifoldswere the combinatorial objects replacing smooth manifolds for algebraictopologists around 1930.

(3) A smooth manifold M admits a so-called C1-triangulation, making M aPL-manifold. Two C1-triangulations have isomorphic subdivisions. Thiswas proven by J.H.C. Whitehead in [204, Theorems 7 and 8].

(4) By a result of R. Edwards (see [125]), any PL-manifold of dimension ≥ 5admits non-PL triangulations (which are then polyhedral homology man-ifolds by (1) above). It is an open problem whether a closed topologicalmanifold of dimension ≥ 5 admits a (possibly non-PL) triangulation. Thisis wrong in dimenison 4 (see [165, § 5]).

31. POINCARE DUALITY IN POLYHEDRAL HOMOLOGY MANIFOLDS 171

(5) There are polyhedral homological manifolds M such that |M | is not atopological manifold. For instance, the suspension of a homology n-manifold N which has the (mod 2) homology of Sn (a homology sphere)is an (n + 1)-dimensional homology manifold. But there are many PL-homology sphere (even for integral homology) with non-trivial fundamen-tal group, [113]. More examples are given, for instance, by lens spaceswith odd fundamental groups.

Here are two first consequences of the definition of a polyhedral homology n-manifold.

Lemma 31.2. Let M be a polyhedral homology n-manifold. Then

(1) any simplex of M is contained in some n-simplex of M .(2) any (n− 1)-simplex of M is a face of exactly two n-simplexes of M .

Proof. If v is a vertex of M , then Lk(v) is n − 1 dimensional, so M is n-dimensional. Let σ ∈ Sk(M). If Lk(σ) = ∅, σ must be an n-simplex by the above.If Lk(σ) is not empty, it must contain a (n−k−1)-simplex τ . Then, σ is containedin the join σ ∗ τ which is an n-simplex. This proves (1).

If σ ∈ Sn−1(M) then Lk(σ) is a 0-dimensional complex having the homologyof S0. Hence, Lk(σ) consists of 2 points, which proves (2).

Let M be a finite polyhedral homology n-manifold. It follows from Point (2)of Lemma 31.2 that the n-chain Sn(M) is a cycle and represent a homology class[M ] ∈ Hn(M) called the fundamental class of M .

Theorem 31.3 (Poincare Duality). Let M be a finite polyhedral homology n-manifold. Then, for any integer k, the linear map

−[M ] : Hk(M) −→ Hn−k(M)

is an isomorphism.

The proof of this Poincare duality theorem will start after Proposition 31.8below. We first give some corollaries of Theorem 31.3. By Kronecker duality, weget

Corollary 31.4 (Poincare Duality, weak form). Let M be a finite polyhedralhomology n-manifold. Then, for any integer k,

dimHk(M) = dimHn−k(M) .

Thus, in the computation of the Euler characteristic of M , the Betti numbersessentially come in pairs, which gives the following corollary.

Corollary 31.5. Let M be a finite polyhedral homology n-manifold. Then,the Euler characteristic χ(M) satisfies the following:

(1) if n is odd, then χ(M) = 0.(2) if n = 2m, then χ(M) ≡ dimHm(M) (mod 2).

Expressed in terms of Poincare polynomial, Corollary 31.4 has the followingform.


Corollary 31.6. Let M be a finite polyhedral homology n-manifold. Then,

Pt(M) = tn P1/t(M).

Another easy consequence of Poincare duality is the following.

Corollary 31.7. A finite polyhedral homology n-manifold which is connectedis an n-dimensional pseudomanifold.

Proof. Let M be a finite polyhedral homology n-manifold. We may supposethat M is non-empty, otherwise there is nothing to prove. By Lemma 31.2, Msatisfies Conditions (a) and (b) of the definition of an n-dimensional pseudoman-ifold. If M is connected, then H0(M) = Z2. By Poincare duality, this impliesthat Hn(M) = Z2. Using Proposition 4.9, we deduce that M is an n-dimensionalpseudomanifold.

By Corollary 31.7, a continuous map between connected finite polyhedral man-ifolds of the same dimension has a degree (see (5.10)).

Proposition 31.8. Let f : M ′ → M be a continuous map of degree one be-tween connected finite n-dimensional polyhedral manifolds. Then H∗f : H∗(M

′)→H∗(M) is surjective.

Proof. The hypotheses imply that H∗f([M ′]) = [M ]. By Proposition 26.8,this implies that the following diagram

Hk(M ′) oo H∗f

−[M ′]

Hk(M)

−[M ]

Hn−k(M′)

H∗f // Hn−k(M)

is commutative for all integer k ≥ 0. This provides a section for H∗f .

The remaining of this section is devoted to the proof of Theorem 31.3. LetM ′ be the barycentric subdivision of M , with the notations introduced in p. 14.The simplicial complex M ′ is endowed with its natural simplicial order ≤ definedin (1.15). For σ ∈ S(M), define D(σ) ⊂ S(M ′) by

D(σ) = t ∈ S(M ′) | σ = min t .The simplicial subcomplex D(σ) of M ′ generated by D(σ) is called the dual cell

of σ. Observe that dimD(σ) = n − dim(σ). The simplicial subcomplex D(σ) =Lk(σ,D(σ)) is called the boundary of D(σ). Its dimension is one less than that of

D(σ). We are interested in the topological spaces E(σ) = |D(σ)| and E(σ) = |D(σ)|.Lemma 31.9. Let σ ∈ Sk(M), where M is a polyhedral homology n-manifold.

Then,

(a) the space E(σ) is a homology (n− k)-cell with boundary E(σ);

(b) D(σ) is an (n− k − 1)-dimensional pseudomanifold.

Proof. The space E(σ) is compact. Observe that D(σ) is the cone over D(σ),

with cone vertex σ. Hence, E(σ) is the topological cone over E(σ). Therefore,

(E(σ), E(σ)) is a good pair and H∗(E(σ)) = 0. It then suffices to prove that


b

b

b

b

b

b

b

1

2

3

4

5

6

7

16

15615715

D(15)

D(7)

s(156)

Figure 1. Dual cells and the map s : S(M)→ S(M ′) of (31.20).In the simplexe’s notation, brackets and commas have been om-mited: 156 = 1, 5, 6, etc.

H∗(E(σ)) ≈ H∗(Sn−k−1). We shall see below that D(σ) and Lk(σ,M)′ are iso-

morphic simplicial complexes. As |Lk(σ,M)′| = |Lk(σ,M)| and M is a polyhedralhomology n-manifold, this implies that

H∗(E(σ)) ≈ H∗(|Lk(σ,M)′|) ≈ H∗(Lk(σ,M)) ≈ H∗(Sn−k−1) .

The simplicial isomorphisms p : D(σ) → Lk(σ,M)′ and q : Lk(σ,M)′ → D(σ) aredefined as follows.

• Let τ ∈ V (D(σ)). This implies that σ ⊂ τ and τ ∈ S(M). Hence,κ = τ − σ ∈ S(Lk(σ,M)). We set p(τ ) = κ.• Let ω ∈ V (Lk(σ,M))′). Then, ω ∈ S(Lk(σ,M)), whence ω ∪ σ ∈ S(M).

We set q(ω) = ω ∪ σ.


We check that p and q are simplicial maps and are inverse of each other. Thisproves Point (a).

To prove Point (b), let σ ∈ Sk(M). We leave as an exercise to the reader thata simplicial complex K is an m-dimensional pseudomanifold if and only if K ′ is so.Therefore, by Point (a) above and its proof, it is enough to prove that L = Lk(σ,M)is (n− k − 1)-dimensional pseudomanifold.

Let τ ∈ S(L). By Lemma 31.2, the simplex σ ∗ τ is contained in some n-simplex of M , which is of the form σ ∗ τ ∗ κ. Therefore, τ ⊂ τ ∗ κ ∈ Sn−k−1(L).Now, if τ ∈ Sn−k−2(L), then σ ∗ τ is a common face of exactly two n-simplexesof M (by Lemma 31.2). Hence, τ is a common face of exactly two (n − k − 1)-simplexes of L. We have proven that L satisfies to Conditions (a) and (b) of thedefinition of an (n − k − 1)-dimensional pseudomanifold. By Proposition 4.9, L isan (n− k − 1)-dimensional pseudomanifold.

Lemma 31.9 permits us to see |M | as a homology-cell complex (see p. 92). Ther-skeleton |M |r is defined by

(31.10) |M |r =⋃

σ∈Ss(M)s≥n−r

E(σ) .

Indeed, the space |M ′| is is the disjoint union of its geometric open simplexes

|M ′| =⋃

t∈S(M ′)

(|t| − |t|

)

and each |t|−|t| is contained in a single open dual cell E(σ)−E(σ), the one associatedto σ for which σ = min t. This shows that |M |n = |M |. If σ ∈ Sn−r(M), then

E(σ) = E(σ) ∩ |M |r−1; if σ′ ∈ Sn−r(M) is distinct from σ, the open dual cells of σand σ′ are disjoint. This shows that |M |r+1 is obtained by from |M |r by adjunctionof the family of r-homology cells:

|M |r+1 = |M |r ∪ϕ(∪σ∈Sn−r(M) E(σ)

),

where ϕ is the attaching map

ϕ : ∪σ∈Sn−r(M) E(σ)→→ ∪σ∈Sn−r(M) E(σ) ⊂ |M |r .

We denote byX the space |M | endowed with this (regular) homology-cell struc-

ture. As noted in p. 93, the cellular homology H∗(X) (defined with the homologycells) is isomorphic to the singular homology H∗(|M |) of |M |. If σ ∈ Sk(M), then

E(σ) is the union of those E(τ) for which τ ∈ Sk+1(M) has σ as a face. UsingFormula (2.6), this amounts to

(31.11) E(σ) =⋃

τ∈δ(σ)

E(τ) .

On the other hand, since D(σ) is a (n − k − 1)-dimensional pseudomanifold by

Lemma 31.9, Proposition 4.6 tells us that Hn−k−1(D(σ)) = Z2 is generated by

[D(σ)] = Sn−k−1(D(σ)) and the generator Hn−k−1(D(σ)) = Z2 is represented by

any cochain formed by a single (n−k−1)-simplex. Hence, Hn−k(D(σ), D(σ)) = Z2

is generated by [D(σ)] = Sn−k(D(σ)) and the generator Hn−k(D(σ), D(σ)) = Z2 is


represented by any cochain formed by a single (n− k)-simplex of D(σ). The proof

of Lemma 16.5 thus works and, using (31.11), ∂ : Cn−k(X)→ Cn−k−1(X) satisfies

(31.12) ∂(D(σ)) =∑

τ∈δ(σ)

[D(τ)] .

As M is a finite simplicial complex, Ck(M) is isomorphic to the vector space gen-erated by Sk(M). Therefore, the correspondence σ 7→ E(σ) gives a linear map

Φ1 : Ck(M)→ Cn−k(X) which, by (31.12), satisfies

(31.13) ∂Φ1 = Φ1δ .

As Φ1 is bijective, the induced map

(31.14) Φ1 : Hk(M)≈−→ Hn−k(X)

is an isomorphism. Observe that this proves the weak form of Poincare duality ofCorollary 31.4.

To prove Theorem 31.3, we now need to identify Φ1 with a cap product. Thecorrespondence E(σ) 7→ [D(σ)] provides a linear map Φ2 : Cn−k(X) → Cn−k(M

′).

By (31.11), Φ2 is a chain map, thus inducing a linear map Φ2 : Hn−k(X) →Hn−k(M

′).

Lemma 31.15. Φ2 : Hn−k(X)→ Hn−k(M′) is an isomorphism.

Proof. The r-skeleton Xr of the homology-cell decomposition of X was givenin (31.10). Note that Xr = |Kr| where Kr is the subcomplex of M ′ given by

(31.16) Kr =⋃

σ∈Ss(M)s≥n−r

D(σ) .

Thus, Kr is a simplicial complex of dimension r. We can use the simplicialpairs (Kr,Kr−1) to compute the simplicial homology of M ′. Define Cr(M

′) =

Hr(Kr,Kr−1) with the boundary ∂ : Cr((M′)→ Cr−1((M

′) given by the composi-tion

Hr(Kr,Kr−1)→ Hr−1(Kr−1)→ Hr−1(Kr−1,Kr−2) .

One has ∂ ∂ = 0. Set H∗(M′) = ker ∂/Image∂. The correspondence E(σ) 7→ [D(σ)]

gives an isomorphism Φ′2 : Hr(X)≈−→ Hr(M

′). Note that

Cr(M′) = Hr(Kr,Kr−1)

= ker

(Cr(Kr)/Cr(Kr−1)︸︷︷︸

Cr(Kr)

∂−→ Cr−1(Kr)/Cr−1(Kr−1)

),

whence

Cr(M′) =

α ∈ Cr(Kr) | ∂α ∈ Cr−1(Kr−1)

⊂ Cr(Kr) ⊂ Cr(M ′) .


The inclusion Φ′′2 : C∗(M′) → C∗(M

′) is clearly a morphism of chain complexes.

It induces a homomorphism Φ′′2 : H∗(M′) → H∗(M

′). As in the proof of Theo-rem 16.6, we have the commutative diagram

(31.17)

Hr+1(Kr+1, Kr)

∂r+1

∂r+1

((QQQQQQQQQQQ

0 // Hr(Kr+1)

≈

Hr(Kr)vv

j

vvmmmmmmmmmmm

// //

µ77 77ooooooooo

0

Hr(M′)

Hr(Kr, Kr−1)

∂r

∂r

((QQQQQQQQQQQ

Hr−1(Kr−1)vv

j

vvmmmmmmmmmmm

Hr(Kr−1, Kr−2)

which permits us to compute H∗(M′). As Φ′′

2 is just the inclusion, the following diagram

Hr(Kr)/Im∂r+1

j ≈

µ

≈// ker∂r/Im∂r+1

= // ker ∂r/Im∂r+1

=

Hr(M

′) Hr(M′)

Φ′′2oo

is commutative, which proves that Φ′′2 is an isomorphism. Finally, the following commu-

tative diagram

Hr(X)

Φ′2

≈ $$JJJJ

JJJJ

Φ2 // Hr(M′)

Hr(M′)

Φ′′2

≈

99tttttttt

shows that Φ2 is an isomorphism.

We now need a good identification of the simplicial (co)homology of M withthat of M ′. Choose a simplicial order on M . One has a simplicial map g : M ′ →M given, for σ ∈ Sm(M), by

(31.18) g(σ) = max σ .

In the other direction, one has a chain map sd : Cm(M) → Cm(M ′) given, forσ ∈ Sm(M), by

(31.19) sd(σ) = Sm(σ′) .

(This chain map is in fact defined for any subdivision and is called the subdivi-sion!operator). Observe that, for any σ ∈ Sm(M), there exists a unique τ ∈ Sm(M ′)such that C∗g(τ) = σ. Indeed, if σ = v0, v1, . . . , vm with v0 v1 · · · , vm,then τ = σ0, σ1, . . . , σm, where σi is the barycenter of v0, v1, . . . , vi. The otherm-simplexes of σ′ are mapped to proper faces of σ. This defines a map

(31.20) s : S(M)→ S(M ′)


by s(σ) = τ . For σ ∈ S(M), one has

(31.21) C∗gC∗sd(σ) = C∗gs(σ) = σ ,

which proves that H∗gH∗sd = idH∗(M).On the other hand, if t = σ0, σ1, . . . , σm ∈ Sm(M ′), with σ0 ⊂ · · · ⊂ σm,

then sdg(t) ∈ C∗(σ′m). As, also t ∈ C∗(σ′m), the correspondence t 7→ C∗(σ′m) is

an acyclic carrier for both sdg and idC∗(M ′). By Proposition 9.1, this implies thatH∗sdH∗g = idH∗(M ′). Therefore, g and sd induce isomorphisms in (co)homologywhich are inverse of each other. In particular, H∗g et H∗g do not depend on theorder since this is the case for sd.

Is is straightforward that sd([M ]) = [M ′]. As g : M ′ →M is a simplicial map,Proposition 26.8 gives the formula

H∗g(H∗g(a) [M ′]

)= a [M ] ,

which is equivalent to the commutativity of the following diagram.

(31.22)

Hk(M)

≈H∗g

[M ]// Hn−k(M)

Hk(M ′)[M ′]// Hn−k(M

′)

≈ H∗ g

OO

The identification of the isomorphism Φ1 with the cap product with the fundamentalclass then follows from the following lemma.

Lemma 31.23. The following diagram

Hk(M)Φ1

≈//

≈ H∗g≈

Hn−k(X)

Φ2≈

Hk(M ′)−[M ′]// Hn−k(M

′)

is commutative.

Proof. Let σ ∈ Sk(M). The properties of the map s : S(M)→ S(M ′) definedin (31.20) imply that C∗g(σ) = s(σ) and max s(σ) = σ (for the natural simplicialorder ≤ on M ′ defined in (1.15)). The isomorphism Ψ∗ = Φ2Φ1 comes from themorphism of cochain-chains Ψ : C∗(M)→ Cn−∗(M

′) such that

Ψ(σ) = [D(σ)] = t ∈ Sn−k(M ′) | min t = σ= t ∈ Sn−k(M ′) | s(σ) ∪ t ∈ Sn(M ′) .

On the other hand, if τ = σ0, . . . σn ∈ Sn(M ′) with σ0 ⊂ σ1 ⊂ · · · ⊂ σn,Formula (26.1) gives

C∗g(σ) τ = s(σ) ≤ τ = 〈s(σ), σ0, . . . , σk〉 σk, . . . , σn .But

〈s(σ), σ0, . . . , σk〉 =

1 if s(σ) = σ0, . . . , σk0 otherwise

=

1 if s(σ) ∪ σk, . . . , σn ∈ Sn(M ′)0 otherwise.


ThereforeC∗g(σ) [M ′] = s(σ) [M ′] = Ψ(σ) .

The proof of Poincare Duality Theorem 31.3 is now complete.

32. Other forms of Poincare Duality

32.1. Relative manifolds. A topological pair (X,Y ) such that

Hj(X,X − x) =

Z2 j = n

0 j 6= n .

for any x ∈ X − Y is called a relative homology n-manifold. The condition is forinstance fullfiled if X − Y is n-dimensional topological manifold, by (14.5).

A simplicial pair (M,A) is a relative polyhedral homology n-manifold if, for eachσ ∈ Sk(M)−Sk(A), the link Lk(σ) of σ in M is a simplicial complex of dimensionn− k − 1 which has the homology of the sphere Sn−k−1. For instance, (M, ∅) is arelative polyhedral homology n-manifold if and only if M is a polyhedral homologyn-manifold.

The following result is proven in e.g. [152, Theorem 63.2]:

Proposition 32.1. If (K,L) is a simplicial pair such that (|K|, |L|) is a relativehomology n-manifold, then K is a relative polyhedral homology n-manifold.

A topological pair (X,Y ) is triangulable if there exists a simplicial pair (K,L)and a homeomorphism of pair h : (|K|, |L|)→ (X,Y ). Such a homeomorphism h iscalled a triangulation of (X,Y ).

Theorem 32.2 (Lefschetz duality). Let (X,Y ) be a compact relative homologyn-manifold which is triangulable. Then, for any integer k, there is an isomorphism

Φ: Hk(X,Y ) ≈ Hn−k(X − Y ) .

Proof. Let (M,A) be a simplicial pair such that (|M |, |A|) is homeomorphic to(X,Y ). By Proposition 32.1, (M,A) is a relative polyhedral homology n-manifold.We shall construct an isomorphism

(32.3) Φ0 : Hk(M,A)≈−→ Hn−k(|M | − |A|) ,

where Hk(M,A) is the simplicial cohomology. The proof is close to that of The-orem 31.3, so we just sketch the argument. For more details (see [152, Theo-rem 70.2]).

Let M∗ be the subcomplex of the first barycentric subdivision of M consistingof all simplexes of M ′ that are disjoint from A. As in the proof of Theorem 31.3,consider the dual cell D(σ) for each σ ∈ S(M)−S(A) and its geometric realizationE(σ) = |D(σ)|. Lemma 31.9 holds for these dual cells and, as in the proof ofTheorem 31.3, they provide a structure of a homology-cell complex on |M∗|. CallX∗ the space |M∗| endowed with this homology-cell decomposition. As M is afinite complex, then Ck(M,A) is the vector space with basis Sk(M) − Sk(A) (seep. 38). As in (31.14), the correspondence σ 7→ E(σ) produces an isomorphism

(32.4) Φ1 : Hk(M,A)≈−→ Hn−k(X

∗) .

To get the isomorphism Φ0 from Φ1, we use that |M∗| ≈ X∗ is a deformationretract of |M | − |A| (see [152, Lemma 70.1]).

32. OTHER FORMS OF POINCARE DUALITY 179

Corollary 32.5. Let (X,Y ) be a connected compact relative homology n-manifold which is triangulable. If Y 6= ∅, then Hn(X − Y ) ≈ Hn(X − Y ) = 0.

Proof. By Kronecker duality, it is enough to prove that Hn(X − Y ) = 0. ByTheorem 32.2, Hn(X−Y ) ≈ H0(X,Y ) and, as X is path-connected, H0(X,Y ) = 0if Y 6= ∅. (Corollary 32.5 may also be obtained using the cohomology with compactsupports: see [80, Theorem 3.35]).

The following consequence of Corollary 32.5 is often referred to as the Z2-orientability of finite polyhedral homology n-manifolds (see e.g. [80, pp. 235–236]).

Corollary 32.6. Let M be a finite polyhedral homology n-manifold and letx ∈M . We denote by j : (M, ∅)→ (M,M − x) the pair inclusion. Then

H∗j : Hm(M)→ Hm(M,M − x)sends [M ] onto the generator of Hn(M, ,M − x) ≈ Z2. In particular, if M isconnected, H∗j is an isomorphism.

Proof. The fundamental class of M being the sum of those of its connectedcomponents, it is enough to consider the case where M is connected. Corollary 32.6then follows from the exact sequences

Hm(M − x)→ Hm(M)H∗j−−→ Hm(M,M − x) ,

using that Hm(M − x)) = 0 by Corollary 32.5.

Let (X,Y ) be a compact triangulable relative homology n-manifold. Choosea simplicial pair (M,A) such that (|M |, |A|) is homeomorphic to (X,Y ). Then,(M,A) is a finite relative polyhedral homology n-manifold by Proposition 32.1.Lemma 31.2 holds true for the simplexes of M which are not in A. As a conse-quence, the n-chain Sn(M)−Sn(A) is a cycle relative to A and represent a homologyclass [M ] ∈ Hn(M,A) called the fundamental class of (M,A). Under the isomor-phism between simplicial and singular homology of Theorem 17.5, the class [M ]corresponds to a singular class [X ] ∈ Hn(X,Y ) called fundamental class of (X,Y ).Let i : X − Y → X denote the inclusion. The isomorphism Φ of Theorem 32.2 isrelated to the cap product with [X ] in the following way.

Proposition 32.7. Let (X,Y ) be a compact relative homology n-manifoldwhich is triangulable. Then the following diagram

Hk(X,Y )

[X] ''OOOOOOOOOO

Φ

≈// Hn−k(X − Y )

H∗i

Hn−k(X)

.

is commutative.

Proof. As in the proof of Theorem 32.2, we choose a finite relative polyhedralhomology n-manifold (M,A) such that such that (|M |, |A|) is homeomorphic to(X,Y ) and we use the same definitions and notations, such that X∗ ≈ M∗. The

isomorphism Φ2 : Hn−k(X∗)→ Hn−k(M

∗) may be established as in Lemma 31.15.The subdivision operator sd : Cm(M)→ Cm(M ′) of (31.19) is defined, as well as thesimplicial map g : M ′ →M of (31.18), choosing for the latter a simplicial order onM . They induced reciproqual isomorphisms on (co)homology. Ons has sd([M ]) =


[M ′], where [M ′] ∈ Hn(M′, A′) is the class of the relative cycle Sn(M ′)− Sm(A′).

The commutative diagram (31.22) becomes

(32.8)

Hk(M,A)

≈H∗g

[M ] // Hn−k(M)

Hk(M ′, A′)[M ′]// Hn−k(M

′)

≈ H∗ g

OO

If i : M∗ →M ′ denotes the simplicial map given by the inclusion, the commutativityof the diagram

Hk(M,A)Φ1

≈//

≈ H∗g≈

Hn−k(X∗)

Φ2

≈// Hn−k(M

∗)

H∗i≈

Hk(M ′, A′)−[M ′] // Hn−k(M

′)

is proven as in Lemma 31.23. Finally, as mentionned in the proof of Theorem 32.2,|M∗| ≈ X∗ is a deformation retract of as |M | − |A|, hence a commutative diagraminvolving simplicial and singular homology:

H∗(M∗)

≈ //

H∗i

H∗(|M | − |A|)H∗j

H∗(M′)

≈ // H∗(|M |)

.

In the definition of a relative homology n-manifold (X,Y ), it is not requiredthat X itself is a homology n-manifold. If this is the case (and if X is compact andtriangulable), the fundamental class [X ] ∈ Hn(X) is defined. To distinguish, call[X ]rel ∈ Hn(X,Y ) the class of Proposition 32.7. If (M,A) is a simplicial pair with(|M |, |A|) homeomorphic to (X,Y ), then H∗j([M ]) = [M ]rel, where j : (M, ∅) →(M,A) (or j : (X, ∅) → (X,Y )) denote the pair inclusion. Therefore H∗j([X ]) =[X ]rel.

Proposition 32.9. Let X be a compact homology n-manifold and let Y be aclosed subset of X. Assume that the pair (X,Y ) is triangulable. Then (X,Y ) is arelative homology n-manifold and the following diagram

Hk(X,Y )

[X]rel

''OOOOOOOOOO

Φ

≈//

H∗j

Hn−k(X − Y )

H∗i

Hk(X)[X] // Hn−k(X)

.

is commutative. Here, j : (X, ∅) → (X,Y ) is the inclusion and φ is the Lefschetzduality isomorphism of Theorem 32.2.

Proof. Only the commutativity of the diagram requires a proof. The com-mutativity of the upper triangle is established in Proposition 32.7. For the lower


triangle, let a ∈ Hk(X,Y ) and u ∈ Hn−k(X). One has

〈u, a [X ]rel〉 = 〈u a, [X ]rel〉= 〈u a,H∗j([X ])〉 as [X]rel = H∗j([X])

= 〈H∗j(u a), [X ]〉= 〈u H∗j(a), [X ]〉 by Lemma 22.11

= 〈u,H∗j(a) [X ]〉 ,which is, in formula, the commutativity of the lower triangle.

32.2. Manifolds with boundary. LetX be a compact topological n-manifoldwith boundary Y = BdX . Then (X,Y ) is a compact relative homology n-manifold.As seen in the previous subsection, if the pair (X,Y ) is triangulable, the funda-mental class [X ] ∈ Hn(X,Y ) is defined.

Theorem 32.10. Let X be a compact topological n-manifold with boundaryY = BdX. Suppose that the pair (X,Y ) is triangulable. Then, for any integer k,the linear maps

−[X ] : Hk(X) −→ Hn−k(X,Y )

and

−[X ] : Hk(X,Y ) −→ Hn−k(X)

given by the cap product with [X ] ∈ Hn(X,Y ) are isomorphisms.

Theorem 32.10 is also true without the hypothesis of the triangulability of(X,Y ), [80, Theorem 3.43].

Proof. We first establish the isomorphism.

−[X ] : Hk(X,Y ) −→ Hn−k(X) .

As X is a topological manifold, its boundary admits a collar neighbourhood, i.e.there exists a embedding h : Y × [0, 1) → X , extending the identity on Y (see,e.g. [80, Proposition 3.42]). Then, X − h(Y × [0, 1/2]) is a deformation retractof both X and X − Y . It follows that the inclusion X − Y → X is a homotopyequivalence. Hence, the result follows from Proposition 32.7.

The other isomorphism comes from the five lemma applied to the followingdiagram

// Hk(X,Y )

[X]≈

// Hk(X)

[X]

// Hk(Y )

[Y ]≈

δ∗ // Hk+1(X,Y )

[X]≈

// Hn−k(X) // Hn−k(X,Y )∂∗ // Hn−k−1(Y ) // Hn−k−1(X)

The commutativity of the above diagram comes from Lemma 26.16, since

(32.11) ∂∗([X ]) = [Y ] .

Indeed, if (M,N) be a finite a simplicial pair triangulating (X,Y ), the fundamentalclass [M ] is represented by the chain Sn(M) ∈ C∗(M) and ∂∗([M ]) is representedby ∂(Sn(M)) = Sn−1(N).


Corollary 32.12. Let X be a compact triangulable topological n-manifold withboundary BdX = Y . Suppose that is Y = Y1∪Y2 the union of two compact (n−1)-manifolds with common boundary Y1 ∩ Y2 = BdY1 = BdY2. Then, for any integerk, the linear map

−[X ] : Hk(X,Y1) −→ Hn−k(X,Y2)

given by the cap product with [X ] ∈ Hn(X,Y ) is an isomorphism.

Again, Corollary 32.12 is true without the hypothesis of triangulability (see[80, Theorem 3.43]).

Proof. Corollary 32.12 reduces to Theorem 32.10 by applying the five lemmato the following diagram

// Hk(X,Y )

[X]≈

// Hk(X,Y1)

[X]

// Hk(Y, Y1)

µ≈

// Hk+1(X,Y )

[X]≈

//

// Hn−k(X) // Hn−k(X,Y2) // Hn−k−1(Y2) // Hn−k−1(X) //

The top line is the cohomology exact sequence for the triple (X,Y, Y1) and thebottom line is the homology exact sequence for the pair (X,Y2). The isomorphismµ is the composition

µ : Hk(Y, Y1)≈−→ Hk(Y2,BdY2)

[Y2]−−−−→ Hn−k−1(Y2) .

The commutativity of the above diagram is obtained as for those in the proofof § 32.1.

Here below some applications of the Poincare duality for compact manifoldswith boundaries.

Proposition 32.13. Let X be a compact triangulable manifold of dimension2n+ 1, with boundary Y . Let B = Image

(Hn(X)→ Hn(Y )

). Then

(1) Let u ∈ Hn(Y ). Then

u ∈ B ⇐⇒ 〈u B, [Y ]〉 = 0 .

In particular, 〈B B, [Y ]〉 = 0.(2) dimHn(Y ) = 2 dimB.

For example, RP 2n is not the boundary of a compact manifold.

Proof. We follow the idea of [130, Lemma 4.7 and Corollary 4.8]. Let i : Y →X denote the inclusion and let a, b ∈ Hn(X). Then,

〈H∗i(a) H∗i(b), [Y ]〉 = 〈H∗i(a b), [Y ]〉 = 〈a b,H∗i([Y ])〉 = 0 ,

since H∗i([Y ]) = 0 by (32.11). This proves the implication ⇒ of (1). Conversely,suppose that 〈u B, [Y ]〉 = 0 for u ∈ Hn(Y ). Since B = ker(δ : Hn(Y ) →Hn+1(X,Y ), it suffices to prove that δ(u) = 0. Let v ∈ Hn(X). One has

0 = 〈u H∗i(v), [Y ]〉= 〈u H∗i(v), ∂[X ]〉 by (32.11).

= 〈δ(u H∗i(v)), [X ]〉= 〈δ(u) v), [X ]〉 by Lemma 22.12.


This equality, holding for any v ∈ Hn(X), implies, by Theorem 32.18 below, thatδ(u) = 0.

To prove (2), let us consider the linear map Φ : Hn(Y ) → Hn(Y )♯ given byΦ(a)(b) = 〈a b, [Y ]〉). Let ΦB be the restriction of Φ to B. The map Φ is anisomorphism by Theorem 32.19 below. By (1), ΦB(B) = A♯, where A = Hn(Y )/B.

Thus, there is a quotient map Φ fitting in the commutative diagram

(32.14)

0 // B //

ΦB≈

Hn(Y ) //

Φ≈

A //

Φ≈

0

0 // A♯ // Hn(Y )♯ // B♯ // 0

(whose rows are exact) and Φ is also an isomorphism. Therefore,

dimHn(Y ) = dimB + dimA = 2 dimB .

(Remark: the proof does not use the map φ, only that ΦB is an isomorphism; butDiagram (32.14) will be useful later.)

Proposition 32.13 and Corollary 31.5 have the following consequence on theEuler characteristic of bounding manifolds.

Corollary 32.15. Let Y be a closed triangulable n-manifold. If Y is is theboundary of a compact triangulable manifold, then χ(Y ) is even.

32.3. The intersection form. Let X be a compact topological n-manifoldwith boundary Y = BdX . We assume that the pair (X,Y ) is triangulable.From Theorem 32.10, the cap product with [X ] ∈ Hn(X,Y ) induces isomorphisms

Hq(X)≈−→ Hn−q(X,Y ) and Hq(X,Y )

≈−→ Hn−q(X). We denote by PD the inverseof these isomorphisms. Thus, if α ∈ Hq(X) and β ∈ Hq(X,Y ), their Poincare dualPD(α) ∈ Hn−q(X,Y ) and PD(β) ∈ Hn−q(X) are thus the classes determined bythe equations

PD(α) [X ] = α and PD(β) [X ] = β .

(The first equation uses the cap product of (26.21) and the second that of (26.19)).This permits us to define two intersection forms on the homology of X .

(1) If α ∈ Hq(X) and β ∈ Hn−q(X), we set

α ·a β = 〈PD(α) PD(β), [X ]〉 .

This defines the (absolute) intersection form H∗(X)⊕Hn−∗(X)·a−→ Z2.

(2) Similarly, if α ∈ Hq(X) and β ∈ Hn−q(X,Y ), the same formula defines

the (relative) intersection form H∗(X)⊕Hn−∗(X,Y )·r−→ Z2.

The name “intersection form” will be justified by Corollary 33.17 below.Let j : (X, ∅) → (X,Y ) denote the pair inclusion. For α ∈ Hq(X) and β ∈

Hn−q(X), the absolute and relative intersection forms are related by the formula

α ·a β = H∗j(α) ·r β = H∗j(β) ·r α .


Indeed:

H∗j(α) ·r β = 〈PD(H∗j(α)) PD(β), [X ]〉= 〈(H∗j(PD(α)) PD(β), [X ]〉 by Lemma 26.16

= 〈PD(α) PD(β), [X ]〉 by Lemma 22.8

= α ·a βand the other equality is proven the same way.

The absolute and relative intersection forms coincide when Y is empty. Evenwhen Y 6= ∅, we shall usually not distinguish between the two forms and just writeα · β when the context makes it clear. In both cases, since 〈a b, γ〉 = 〈a, b γ〉(see (26.2)), one has

(32.16) α · β = 〈PD(α), β〉 = 〈PD(β), α〉 .By Theorem 32.19 below, the relative intersection form is non-degenerate, i.e.

induces an isomorphism Hq(X)≈−→ Hn−q(X,Y )♯ for all q. If Y 6= ∅, the absolute

intersection form may be degenerate (example: X = S1 × D2). In fact, if X isconnected, it is always degenerate for q = 0, since Hn(X) = 0. However, one hasthe following proposition.

Proposition 32.17. Suppose that X is connected and that Y is not empty.Then, the following conditions are equivalent.

(a) The absolute intersection form induces an isomorphism Hq(X)≈−→ Hn−q(X)♯

for 1 ≤ q ≤ n− 1.(b) Y is a Z2-homology sphere.

Proof. Let j : (X, ∅) → (X,Y ) denote the pair inclusion. By (32.16), thecomposed homomorphism

Hq(X)Hqj // Hq(X,Y )

PD

≈// Hn−q(X)

k

≈// Hn−q(X)♯

is just the absolute intersection form of X . Thus, (a) is equivalent to Hqj being anisomorphism for 1 ≤ q ≤ n− 1. By the exact homology sequence of (X,Y ) this isequivalent to (b) if X is connected.

32.4. Non degeneracy of the cup product.

Theorem 32.18. Let M be a finite polyhedral homology n-manifold. Then, forany integer k, the bilinear map

Hk(M)×Hn−k(M)−→ Hn(M)

〈−,[M ]〉−→ Z2

induces an isomorphism Hk(M)≈−→ Hn−k(M)♯.

Proof. By Corollary 31.4, it suffices to prove that the linear map Φ : Hk(M)→Hn−k(M)♯ given by

aΦ7→

(b 7→ 〈a b, [M ]〉

)

is injective. Suppose that a ∈ kerΦ. Then

0 = 〈a b, [M ]〉 = 〈b, a [M ]〉for all b ∈ Hn−k(M). By Point (a) of Lemma 3.4, we deduce that a [M ] = 0,which implies that a = 0 by Theorem 31.3.


The same proof, using Corollary 32.12, gives the following result.

Theorem 32.19. Let X be a compact triangulable topological n-manifold withboundary BdX = Y . Suppose that is Y = Y1∪Y2 the union of two compact (n−1)-manifolds with common boundary Y1 ∩ Y2 = BdY1 = BdY2. Then, for any integerk, the bilinear map

Hk(X,Y1)×Hn−k(X,Y2)−→ Hn(X,Y )

〈−,[X]〉−→ Z2

induces an isomorphism Hk(X,Y1)≈−→ Hn−k(X,Y2)

♯.

32.5. Alexander Duality. The first version of Alexander Duality was provenin a paper [6] of James Waddell Alexander II (1888–1971). This article pioneeredseveral new methods and was very influential at the time (see [38, p. 56]). In hispaper, Alexander used the homology mod 2. Classical Alexander duality relatesthe cohomology of a closed subset A or Sn to the homology of Sn − A. We givebelow a version where Sn is replaced by a homology sphere (for instance a lensspace with odd fundamental group).

Theorem 32.20 (Alexander Duality). Let (X,A) be a compact triangulable pairwith ∅ 6= A 6= X. Suppose that X is a relative homology n-manifold and has itshomology isomorphic to that of Sn. Then, for all integer k, there is an isomorphism

Hk(A) ≈ Hn−k−1(X −A) .

Particular case of Alexander duality were encountered in Proposition 14.7 andCorollary 14.8. For a vesion of Theorem 32.20 without the assumption of triangu-lability (see [80, Theorem 3.44]).

Proof. The case n = 0 being trivial, we assume n > 0. The pair (X,A) satis-fies the hypotheses of Lefschetz duality Theorem 32.2. This gives an isomorphism

Φ: Hk+1(X,A) ≈ Hn−k−1(X −A) .

Suppose that k 6= n, n−1. Since H∗(X) ≈ H∗(Sn), the connecting homomorphism

δ∗ : Hk(A)→ Hk+1(X,A) is an isomorphism and Hn−k−1(X −A) ≈ Hn−k−1(X −A). This proves the result in this case.

Let (M,L) be a simplicial pair such that (|M |, |L|) is homeomorphic to (X,A).As M is a relative polyhedral homology n-manifold by Proposition 32.1. As Lis a proper subcomplex of M , one has Hn(L) = 0, since Sn(M) is the only non-

vanishing n-cycle of M . Hence, Hn(A) ≈ Hn(A) = 0 by Kronecker duality. As,

Hn−k−1(X −A) = H−1(X −A) = 0, the theorem is true for k = n.When k = n− 1, consider the diagram

Hn−1(X)H∗i // Hn−1(A)

Φ

// Hn(X,A)

Φ≈

// Hn(X)

[X]≈

// 0

0 // H0(X −A) // H0(X −A)H∗j // H0(X) // 0

where i : A→ X and j : (X−A)→ X denote the inclusions. The bottom line is theexact sequence of Lemma 12.16 and the commutativity of the right hand square is

the contents of Proposition 32.9. Then the homomorphism φ : Hn−1(A)→ H0(X−A) exists, making the diagram commutative. If n > 1, Hn−1(A) = Hn−1(A) and,

as Hn−1(X) = 0, the map φ is an isomorphism by the five lemma. Finally, when


n = 1, then cokerH∗i = H0(A) by Lemma 12.16 and φ induces an isomorphism

from Hn−1(A) to H0(X −A).

33. Poincare duality and submanifolds

In this section, we assume some familiarity of the reader with standard tech-niques of smooth manifolds, as exposed in e.g. [93].

33.1. The Poincare dual of a submanifold. Let M be a smooth compactn-manifold and letQ ⊂M be a closed smooth submanifold of codimension r. Recallthat smooth manifolds admit PL-triangulations [204], so the fundamental classes[Q] ∈ Hn−r(Q) and [M ] ∈ Hn(M,BdM) do exist. We are interested in the Poincaredual PD(H∗i([Q])) ∈ Hr(M,BdM) (see § 32.3) of the class H∗i([Q]) ∈ Hn−r(M),where i : Q→M denotes the inclusion. We write PD(Q) for PD(H∗i([Q])) and callit the Poincare dual of Q. It is thus characterized by the equation

PD(Q) [M ] = H∗i([Q]) ,

Two simple examples are given in Figure 2.

1 2 3 1

4

7

1 2 3 1

4

7

5

8

6

9PD(Q)

Q

W

1 2 3 1

4

7

1 2 3 1

7

4

5

8

6

9

Q

W

PD(Q)

Figure 2. The Poincare dual PD(Q) of a circle Q in the torus(left) or the Klein bottle (right). This illustrates the localizationprinciple of Remark 33.6: PD(Q) is supported in a tubular neigh-bourhood W of Q.

Example 33.1. For a more elaborated example, let Q be a smooth closedconnected manifold, seen as the diagonal submanifold of M = Q × Q. Let A =a1, a2, . . . ⊂ H∗(Q) be an additive basis of H∗(Q). By Theorem 32.18, there isa basis B = b1, b2, . . . of H∗(Q) which is dual to A for the Poincare duality, i.e.〈ai bj, [Q]〉 = δij . We claim that

(33.2) PD(Q) =∑

i

ai × bi .

Indeed, by the Kunneth theorem, ai × bj is a basis of H∗(M), so there are uniquecoefficients γij ∈ Z2 such that

PD(Q) =∑

i,j

γij ai × bj .

Let ∆: Q→M be the diagonal inclusion, ap ∈ A and bq ∈ B. As H∗∆(bp × aq) =bp aq (see Remark 27.5), one has

(33.3) 〈bp × aq, H∗∆([Q])〉 = 〈H∗∆(bp × aq), [Q]〉 = 〈bp aq), [Q]〉 = δpq .

33. POINCARE DUALITY AND SUBMANIFOLDS 187

Without loss of generality, we may suppose that Q is connected. Let [Q]∗ be thenon zero element of HdimQ(Q). One has 〈[Q]∗ × [Q]∗, [M ]〉 = 1 and(33.4)〈bp × aq, H∗∆([Q])〉 = 〈(bp × aq) q, [M ]〉

= 〈(bp × aq) ∑i,j γij ai × bi, [M ]〉

=∑

i,j γij 〈(bp × aq) (ai × bi), [M ]〉=

∑i,j γij 〈(bp ai)× (aq bi), [M ]〉 by Remark 27.10

=∑

i,j γij 〈δpi[Q]∗ × δqj [Q]∗, [M ]〉= γpq .

Thus, Equation (33.2) follows from (33.3) and (33.4).

Let us denote by ν = ν(M,Q) the normal bundle of Q in M . A Riemannianmetric provides a smooth bundle pair (D(ν), S(ν)) with fiber (Dr, Sr−1) and thereis a diffeomorphism from D(ν) to a closed tubular neighbourhood W of Q in M .By excision,

H∗(M,M −Q)≈−→ H∗(W,BdW ) ≈ H∗(D(ν), S(ν)) .

Hence, the Thom class U(ν) ∈ Hr(D(ν), S(ν)) determines an element U(M,Q) ∈Hr(M,M −Q). Let j : (M, ∅)→ (M,M −Q) denote the pair inclusion.

Lemma 33.5. PD(Q) = H∗j(U(M,Q)).

Proof. We first reduce to the case where Q is connected. Indeed, as Q is thefinite union of components Qi, with tubular neighbourhood Wi, then

Hr(M,M −Q)≈−→ Hr(W,BdW ) ≈

⊕Hr(Wi,BdWi) ≈

⊕Hr(M,M −Qi)

and U(M,Q) =∑U(M,Qi). On the other hand, PD(Q) =

∑PD(Qi). Thus, we

shall assume that Q is connected.Let us consider the case M = D(ν) and Q is the image of the zero section. As

Hn−r(Q) = Z2 = Hn(D(ν), S(ν)), the the Thom isomorphism of Theorem 28.49says that U(ν) [D(ν)] = [Q]. This proves the lemma for any tubular neigh-bourhood of Q, for instance W or a smaller tube W ′ contained in the interior ofW .

Let us choose a triangulation of M for which W and W ′ are subcomplexes. Theclass H∗j(U(M,Q)) ∈ Hr(M) may then be represented by a simplicial cocycle q ⊂Sr(W ′). The n-simplexes of M involved in the computation of H∗j(U(M,Q)) [M ] are then all simplexes of W . Therefore

H∗j(U(M,Q)) [M ] = H∗i(U(W,Q)) [W ]

)= H∗i([Q]) .

.

Remark 33.6. We see in the proof of Lemma 33.5 that the Poincare dualPD(Q) of a submanifold Q ⊂ M is supported in an arbitrary small tubular neigh-bourhood of Q. This localization principle is illustrated in Figure 2 for Q a circle inthe torus or the Klein bottle. For the analogous localization principle in de Rhamcohomology, see [18, Proposition 6.25].

Lemma 33.7. With the notations of Lemma 33.5, the restriction ofPD(Q) ∈ Hr(M,BdM) to H∗(Q) is equal to the Euler class of the normal bundleν = ν(M,Q).


Proof. We use the notations of the proof of Lemma 33.5. Let σ0 : Q→ D(ν).The various inclusions give rise to the following commutative diagram

Hr(M,M −Q)

≈

H∗j // Hr(M)

H∗i

%%LLLLLLLL

Hr(W,BdW )

≈

Hr(W )

≈

Hr(Q)

Hr(D(ν), S(ν))H∗jν // Hr(D(ν))

H∗σ0

99rrrrrrrr

The Euler class e(ν) ∈ Hr(Q) is characterised by the equation H∗jν(U(ν)) =H∗p(e(ν)), where p : D(ν)→ Q is the bundle projection (see p. 154). Since pσ0 =idQ, the previous diagram and Lemma 33.5 yield

H∗i(PD(Q)) = H∗iH∗j(U(M, Q)) = H∗σ0 H∗j(U(ν)) = H∗σ0 H∗p(e(ν)) = e(ν) .

Proposition 33.8. Let M be a smooth compact n-manifold and let i : Q →Mbe the inclusion of a closed smooth submanifold Q of codimension r. Then

kerH∗i ⊂ Ann (PD(Q)) = x ∈ H∗(M) | x PD(Q) = 0(for a class b, the ideal Ann (b) is called the annihilator of b). The above inclusionis an equality if and only if H∗i is surjective.

Proof. Let a ∈ H∗(M) and let q = PD(Q). Then

H∗i(H∗i(a) q)

)= a H∗i([Q]) by (26.7)

= a (q [M ]) by definition of q

= (a q) [M ] by Proposition 26.8.

Therefore, if H∗i(a) = 0, then a ∈ Ann (q) and the converse is true if and only ifH∗i is injective. By Kronecker duality (see Corollary 3.17), the latter is equivalentto H∗i being surjective.

Example 33.9. Let i be the standard inclusion of Q = RP k in M = RP k+r.By Proposition 24.21, one has a commutative diagram

Z2[a]/(ak+r+1)

≈

// // Z2[a]/(ak+1)

≈

H∗(RP k+r)H∗i // // H∗(RP k)

where a is of degree 1. Then, via the above vertical isomorphisms,

kerH∗i = (ak+1) = Ann (ar) = Ann (PD(RP k)) .

The following proposition states the functoriality of the Poincare dual fortransversal maps.

Proposition 33.10. Let f : M → N be a smooth map between smooth closedmanifolds. Suppose that f is transversal to a closed submanifold Q of N . Then

PD(f−1(Q)) = H∗f(PD(Q)) .


Proof. Let P = f−1(Q). We consider the commutative diagram

H∗(N,N −Q)H∗f //

H∗i

H∗(M,M − P )

H∗j

H∗(N)H∗f // H∗(M)

where the vertical arrow are induced by the inclusions i : (N, ∅)→ (N,N −Q) andj : (M, ∅)→ (M,M − P ). Then,

PD(P ) = H∗j(U(M,P )) by Lemma 33.5

= H∗j H∗f(U(N,Q)) by transversality and Lemma 28.50

= H∗f H∗i(U(N,Q))

= H∗f(PD(Q)) by Lemma 33.5.

33.2. The Gysin Homomorphism. Let (M,Q) be a pair of smooth compactmanifolds, with Q closed. Let i : Q→ M denote the inclusion. Set q = dimQ andm = dimM = q + r. The Gysin homomorphism Gys: Hp(Q) → Hp+r(M,BdM)is defined for all p ∈ N by the composed homomorphism

Hp(Q)[Q]

≈// Hq−p(Q)

H∗i // Hq−p(M) oo [M ]

≈Hp+r(M,BdM) .

The notation i! and the terminology umkehr homomorphism are also used in theliterature.

For example, Gys(1) = PD(Q), the Poincare dual of Q. More generally:

Lemma 33.11. For a ∈ Hp(M), one has Gys(H∗i(a)

)= a PD(Q).

Proof.

(a PD(Q)) [M ] = a (PD(Q)) [M ])

= a H∗i([Q])

= H∗i(H∗i(a) [Q]

)by Proposition 26.8

= Gys(H∗i(a)

) [M ]

As − [M ] is an isomorphism, this proves the lemma.

Example 33.12. Let M be the total space of an r-disk bundle π : M → Q. Wesee Q as a submanifold of M via there 0-section i : Q→M . Let U ∈ Hr(M,BdM)be the Thom class. Since U [M ] = H∗i([Q]), we see that U = PD(Q). Forb ∈ Hp(Q), one has

Gys(b) [M ] = Gys(H∗iH∗π(b)

) [M ] since π i = idQ

=(H∗π(b) PD(Q)

) [M ] by Lemma 33.11

=(H∗π(b) U

) [M ] since U = PD(Q)

= Thom(b) [M ] .

Since − [M ] is an isomorphism, we see in this example that the Gysin homo-morphism is identified with the Thom isomorphism.


Proposition 33.13. Let (M,Q) be a pair of smooth closed manifolds, with Qof codimension r. Let W be a closed tubular neighbourhood of Q in M . There is acommutative diagram

// Hp−1(M −Q) //

Hp−r(Q)Gys //

=

Hp(M) //

Hp(M −Q) //

// Hp−1(BdW ) // Hp−r(Q) // Hp(Q) // Hp(BdW ) //

where the vertical arrows are induced by inclusions. The horizontal lines are exactsequences and the bottom one is the Gysin exact sequence of the sphere bundleBdW → Q.

Proof. We start with the commutative diagram

// Hp−1(M −Q) //

Hp(M, M −Q) //

Hp(M) //

Hp(M −Q) //

// Hp−1(BdW ) // Hp(W,Bd W ) // Hp(W ) // Hp(Bd W ) //

using the cohomology exact sequences of the pairs (M,M −Q) and (W,BdW ). Toget the diagram of the proposition, we use the identification

Hp−r(Q)Thom

≈// Hp(W,BdW ) oo

≈Hp(M,M −Q)

and H∗(W ) ≈ H∗(Q). Thus the bottom line is the Gysin sequence of BdW → Q.It remains to identify the homomorphism Hp−r(Q) → Hp(M) with the Gysinhomomorphism. This amounts to the commutativity of the following diagram.

Hp−r(Q)Thom

≈//

−[Q]≈

Hp(W,BdW ) oo≈

−[W ]≈

Hp(M,M − intW ) // Hp(M)

−[M ]≈

Hq−p+r(Q) // Hq−p+r(W ) // Hq−p+r(M)

The commutativity of the left square was observed in Example 33.12. That of theright square may be checked using simplicial (co)homology for a triangulation ofM extending one of W .

Proposition 33.14. Let f : M ′ → M be a smooth map between closed man-ifolds. Let Q be a closed submanifold of codimension r in M . Suppose that f istransversal to Q. Then, for all p ∈ N, the diagram

Hp(Q)Gys //

H∗f

Hp+r(M)

H∗f

Hp(f−1(Q))Gys // Hp+r(M ′)

is commutative.

Proof. Let Q′ = f−1(Q). By transversality, f : Q′ → Q is covered by a

morphism of vector bundle f : ν(M ′, Q′)→ ν(M,Q). Put a Riemannian metric on

ν(M,Q) and pull it back on ν(M ′, Q′), so that f is an isometry on each fiber. By


standard technique of Riemannian geometry, one can find a tubular neighbourhoodW ofQ and a tubular neighbourhoodW ′ ofQ′ and modify f by a homotopy relativeto Q′ so that f(W ′) ⊂ W , f(BdW ′) ⊂ BdW , f(M ′ − intW ′) ⊂ M − intW and

f : W ′ → W coincides with f via the exponential maps of W and W ′. We thus geta diagram.

Hp−r(Q)Thom

≈//

H∗f

Hp(W,BdW ) oo≈

H∗f

Hp(M,M − intW ) //

H∗f

Hp(M)

H∗f

Hp−r(Q′)Thom

≈// Hp(W ′,BdW ′) oo

≈Hp(M ′,M ′ − intW ′) // Hp(M ′)

.

The left square is commutative by construction and the functoriality of the Thomisomorphism (coming from Lemma 28.50). The other squares are obviously com-mutative. But, as seen in the proof of Proposition 33.13, the compositions from theleft end to the right end of the horizontal lines are the Gysin homomorphisms.

33.3. Intersections of submanifolds. Consider two closed submanifolds Qi(i = 1, 2) of the compact smooth n-manifold M , Qi being of codimension ri. Wesuppose that Q1 and Q2 intersect transversally. Then, Q = Q1 ∩ Q2 is a closedsubmanifold of codimension r = r1 + r2.

Proposition 33.15. Under the above hypotheses

PD(Q) = PD(Q1) PD(Q2) .

Proof. As (M−Q1)∪(M−Q2) = M−Q, the cup product provides a bilinearmap

Hr1(M,M −Q1)×Hr2(M,M −Q2)−→ Hr(M,M −Q) .

In virtue of Lemma 33.5, it suffices to prove that

(33.16) U(M,Q1) U(M,Q2) = U(M,Q) .

We may suppose that Q 6= ∅ for, otherwise, the proposition is trivially true, sinceq = 0 and r > n.

If A is a submanifold of B, we denote by ν(B,A) the normal bundle of A in B.Choose an embedding µ : D(ν(B,A)) → B parametrising a tubular neighbourhoodW (B,A). If V ⊂ B the notation W (A,B)V means µ(D(ν(B,A)V )). As Q1 andand Q2 intersect transversally, one has

ν(M,Q) = ν(Q1, Q)|Q ⊕ ν(Q2, Q)|Q .

Let b ∈ Q. One may choose convenient tubular neighbourhood parametrisationsso that W (M,Q)b ∩ Qj = W (Qj , Q)b. Let W1 = W (Q1, Q)b ≈ Dr2 , W2 =W (Q2, Q)b ≈ Dr1 and W = W (M,Q)b ≈W1 ×W2 ≈ Dr. Let πi : W →Wi bethe projection. By Lemma 33.5:

• the class U(M,Q1) ∈ Hr1(M,M − Q1) restricts to the non-zero elementa1 ∈ Hr1(W,W1 × BdW2) = Z2;• the class U(M,Q2) ∈ Hr2(M,M − Q2) restricts to the non-zero elementa2 ∈ Hr2(W,BdW1 ×W2) = Z2.

Hence, U(M,Q1) U(M,Q2) restricts to a1 a2 ∈ Hr(W,BdW ) = Z2. Wehave to prove that a1 a2 6= 0. Let 0 6= ai ∈ Hri(Wj ,BdWj) (i 6= j). Then

a1 a2 = H∗π1(a1) H∗π2(a2) = a1 × a2 .


By the relative Kunneth theorem 27.25, a1 × a2 6= 0 in Hr(W,BdW ). Hence,U(M,Q1) U(M,Q2) restricts to the non-zero element ofHr(W (M,Q)b,BdW (M,Q)b) for all b ∈ Q. By Lemma 33.5, this proves(33.16).

An interesting case is when dimQ1 + dimQ2 = dimM . If Q1 and Q2 intersecttransversally, then Q1 ∩Q2 is a finite collection of points. Let ji : Qi →M denotethe inclusion. The following result says that the parity of this number of pointsdepends only on [Qi]M = H∗ji([Qi]) and justifies the terminology of intersectionform.

Corollary 33.17. Let Qi (i = 1, 2) be two closed submanifolds of the compactsmooth n-manifold M , with dimQ1 + dimQ2 = n. Let qi = PD(Qi). Suppose thatQ1 and Q2 intersect transversally. Then

♯ (Q1 ∩Q2) ≡ 〈q1 q2, [M ]〉 mod 2 .

In other words,

♯ (Q1 ∩Q2) ≡ H∗j1([Q1)] ·H∗j2([Q2)] mod 2 ,

where “·” denotes the (absolute) intersection form (see § 32.3).

Proof. One has

〈q1 q2, [M ]〉 = 〈1, (q1 q2) [M ]〉 = 〈1, [Q1 ∩Q2]〉 ≡ ♯ (Q1 ∩Q2) mod 2 .

Lemma 33.18. Let ξ = (p : E → N) be a smooth vector bundle over a closedsmooth manifold N . Let σ, σ′ : N → E be two smooth sections of ξ which aretransversal. Let Q be the submanifold of N defined by Q = σ−1

(σ(N) ∩ σ′(N)

).

Then, the Poincare dual of Q in N is the Euler class e(ξ) of ξ.

Proof. We us the following notation: if λ : Y → X is a continuous map andY a closed manifold, we write [Y ]X = H∗λ([Y ]) ∈ HdimY (X); the map λ is usuallyimplicit, being an inclusion or an embedding obvious from the context.

Endow ξ with an Euclidean structure and consider the pair (D,S) = (D(ξ), S(ξ))of the associated unit disk and sphere bundle. Using a homotopy in each fiber, wecan assume that σ(N) and σ′(N) are contained in the interior ofD. All the sectionsof a bundle are homotopic. By Lemma 33.5 and its proof, the Thom class U of ξis the Poincare dual in D of [N ]D = H∗σ([N ]) = H∗σ

′([N ]). By Proposition 33.15,U U is the Poincare dual in D of [Q]D. Let j : (D, ∅)→ (D,S) denote the pairinclusion. As pσ = idN , one has

[Q]N = H∗p([Q]D)

= H∗p((U U) [D]

)cap product of (26.12)

= H∗p(U (U [D])

)by Formula (26.13)

= H∗p(U [N ]D

)

= H∗p(H∗j(U) [N ]D

)by definition of the cap product (26.10)

= H∗p(H∗p(e(ξ)) [N ]D

)by definition of the Euler class

= e(ξ) H∗p([N ]D)

= e(ξ) [N ] .

which proves the lemma.


When, in Lemma 33.18, the rank of ξ is equal to the dimension of of themanifold N , then σ(N) ∩ σ′(N) is a finite collection of point and one gets thefollowing corollary.

Corollary 33.19. Let ξ = (p : E → N) be a smooth vector bundle of rank nover a closed smooth n-manifold N . Let σ, σ′ : N → E be two smooth sections of ξwhich are transversal. Then

♯ (σ(N) ∩ σ′(N)) ≡ 〈e(ξ), [N ]〉 mod 2 .

The following corollary is a justification for the name Euler class.

Corollary 33.20. Let N be a smooth closed manifold. Then the followingcongruences mod 2 hold:

〈e(TN), [N ]〉 ≡ χ(N) ≡ dimH∗(N) mod 2 .

Proof. As χ(N) =∑

i(−1)i dimHi(N), the second congruence is obvious.Let σ0 : N → D(TN) be the zero section and let σ : N → D(TN) be another smoothsection (i.e. a vector field on N) which is transversal to σ0. By Corollary 33.19, thenumber of zeros of σ is congruent mod 2 to 〈e(TN), [N ]〉. It then suffices to findsome vector field transversal to σ0 for which we know that its number of zeros iscongruent mod 2 to χ(N). Observe that, for a finite CW-complex X , the followingcongruence mod 2 holds

χ(X) ≡ dimH∗(X) ≡ ♯Λ(X) mod 2 ,

where Λ(X) is set of cells of X . For the required vector field, one can take thegradient vector field σ = gradf of a Morse function f : N → R. Then σ−1

0

(σ(N) ∩

σ0(N))

= Crit f , the set of critical points of f . The transversality of σ with σ0

is equivalent to f being a Morse function (see [93, Chapter 6]). By Morse theory,N has then the homotopy type of a CW-complex X with ♯Λ(X) = ♯Crit f , [93,Chapter 6, Theorem 4.1]. One can also use the classical vector field associated toa C1-triangulation of N , with one zero at the barycenter of each simplex (see, e.g.[176, pp. 611–612]).

We give below a second proof of Corollary 33.20, using the following lemma.

Lemma 33.21. Let ∆N be the diagonal submanifold of M = N×N , with normalbundle ν(M,∆N ). Then, there is a canonical isomorphism of vector bundles

ν(M,∆N ) ≈ TN .

Proof. Let p1, p2 : N × N → N be the projections onto the first and secondfactor. For x ∈ N , consider the commutative diagram in the category of real vectorspaces

0 // T(x,x)(∆N ) //

T(x,x)(N ×N) //

φ

ν(x,x)(M,∆N )

φ

// 0

0 // ∆(TxN) // TxN × TxN− // TxN // 0

where the rows are exact and φ(v) = (Tp1(v), T p2(v)). The map φ is an isomor-phism and sends T(x,x)(∆N ) onto ∆(TxN). Hence, φ descends to the isomorphism

φ : ν(M,∆N )≈−→ TxN which, of course, depends continuously on x.


Second proof of Corollary 33.20. We consider N as the diagonal sub-manifold of M = N × N , with the diagonal inclusion ∆: N → M . The normalbundle ν(M,N) is isomorphic to the tangent bundle of M by Lemma 33.21.

By (33.2), the Poincare dual of N is equal to∑

i ai×bi, where A = a1, a2, . . . and B = b1, b2, . . . are bases of H∗(N) dual one to the other for the Poincareduality.

〈e(TM), [N ]〉 = 〈e(ν(M,N)), [N ]〉= 〈H∗∆(

∑i ai × bi), [N ]〉 by Lemma 33.7

= 〈∑i ai bi, [N ]〉 ≡ dimH∗(N) mod 2 .

Example 33.22. As χ(Sn) ≡ 0 mod 2, the Euler class of TSn vanishes byCorollary 33.20. Let T 1Sn be the associated sphere bundle. By Proposition 28.57and the Leray-Hirsch theorem, we get an isomorphism of H∗(Sn)-module

H∗(T 1Sn) ≈ H∗(Sn)⊗H∗(Sn−1) .

If n ≥ 3, Poincare duality implies that this isomorphism is a ring-isomorphism.This is not true if n = 2 (see Remark 28.56).

Thus, for n ≥ 3, T 1Sn has the same cohomology ring as Sn × Sn−1. However,by [107, Theorem 1.12], these two spaces have the same homotopy type if and onlyif there exists a map f : S2n+1 → Sn+1 with Hopf invariant one (see § 37). ByTheorem 50.6, such an f exists if and only if n = 1, 3, 7.

33.4. The linking number. Let Q and Q′ be two disjoint closed submanifoldof a closed manifold Σ (say, in the smooth category), with q = dimQ, q′ = dimQ′

and s = dimΣ. We assume that

(1) q + q′ = s− 1.(2) Σ is a Z2-homology sphere, i.e. H∗(Σ) ≈ H∗(Ss).(3) 〈1Q, [Q]〉 = 〈1Q′ , [Q′]〉 = 0. This condition is always satisfied when q and

q′ are not zero. If, say, q = 0, it means that Q has an even number ofpoints, so that [Q] ∈ H0(Q) = ker〈1Q, 〉.

Thanks to (2), Alexander duality (see Theorem 32.20) provides an isomorphism

A : Hq(Q)≈−→ Hs−q−1(Σ−Q) .

Note that s− q− 1 = q′. By Condition (3), [Q] ∈ Hq(Q) and [Q′] ∈ Hq′ (Q′), so we

can define the linking number (sometimes called the linking coefficient) l(Q,Q′) ofQ and Q′ in Σ by

(33.23) l(Q,Q′) = 〈A[Q], H∗i([Q′])〉 ∈ Z2 ,

where i : Q′ → Σ denotes the inclusion. Although the asymmetry of the definition,the equality l(Q,Q′) = l(Q′, Q) will be proven in Proposition 33.32 below.

The linking number l(Q,Q′) was introduced in 1911 by Lebesgue [128, pp. 173–175], with a definition in the spirit of Proposition 33.29 below. Lebesgue called Qand Q′ “enlacees” if l(Q,Q′) = 1. One year later, Brouwer [21, pp. 511–520]refined the idea when Σ, Q and Q′ are oriented, defining an integral linking numberwhose reduction mod 2 is l(Q,Q′) (for the philosophy of Brouwer’s definition, seeExercise 4.16). More history and references about the linking numbers may befound in [38, pp. 176–179 and 185].


As Q is a submanifold of Σ, the isomorphism A may be described in the fol-lowing way, which will be useful for computations. Let V be a closed tubularneighbourhood of Q in Σ−Q′ and let X = Σ− intV . The cohomology sequence of(Σ, X)

Hs−q−1(Σ) // Hs−q−1(X)δ // Hs−q(Σ, X) // Hs−q(Σ)

shows that the connecting homomorphism δ descends to an injection δ of

coker (Hs−q−1(Σ)→ Hs−q−1(X)) ≈ Hs−q−1(X)

into Hs−q(Σ, X). Let j : (V,BdV )→ (Σ, X) denote the pair inclusion. IdentifyingH∗(V ) with H∗(Q), one has the following diagram

(33.24)

Hs−q−1(X)δ

oo A

≈ Hq(Q)

Hs−q(Σ, X)

H∗j

≈//

Hs−q(V,BdV )[V ]

≈// Hq(Q)

Hs−q(Σ)

[Σ]

≈// Hq(Σ)

whose columns are exact (the right hand one by Lemma 12.16) and whose bottomrectangle is commutative by Proposition 32.9. Then A is the unique isomorphismmaking the top rectangle commutative (compare the proof of Theorem 32.20).

Remark 33.25. Diagram (33.24) uses the singular (co)homology. But, via atriangulation of Σ, the various spaces may be the geometric realizations of simplicialcomplexes (by abuse of notations we use the same letters, i.e. Σ = |Σ|, etc).Then, Diagram (33.24) makes sense for simplicial (co)homology; the isomorphismHs−q(Σ, X) ≈ Hs−q(V,BdV ) is just the simplicial excision (see Exercise 1.17).

Remark 33.26. Suppose that Q is connected or consists of two points. ThenHq′(Σ−Q) ≈ Hq(Q) ≈ Z2. Therefore, l(Q,Q′) = 1 if and only if H∗i([Q

′]) 6= 0 inHq′(Σ−Q). Also, l(Q,Q′) determines H∗i([Q

′]).

The following lemma shows that l(Q,Q′) is not always zero. We say that Q′ isa meridian sphere for Q if Q′ is the boundary of a (s− q)-disk ∆ in Σ intersectingQ transversally in one point.

Lemma 33.27. Let Q, Q′ and Σ satisfying (1)–(3) above. Suppose that Q′ is ameridian sphere for Q. Then l(Q,Q′) = 1.

Proof. If Q = Q1 ∪Q2, then l(Q,Q′) = l(Q1, Q′) + l(Q2, Q

′) since A is ad-ditive. We can thus restrict ourselves to the case where Q is connected or con-sists of two points. By Remark 33.26, it suffices to prove that H∗i([Q

′]) 6= 0 inHs−q−1(Σ−Q).

Let ν(Q,Σ) be the normal bundle of Q in Σ. A Riemannian metric provides asmooth bundle pair (D(ν), S(ν)) with fiber (Ds−q, Ss−q−1) and a diffeomorphism

φ : (D(ν), S(ν))≈−→ (V,BdV ), where V is a tubular neighbourhood of Q in Σ.

By choosing the Riemannian metric conveniently, we may assume that (B,S) =(∆∩V,∆∩BdV ) is the image by φ of a fiber of (D(ν), S(ν)). Then [Q′] represents


the same class as [S] in Hs−q−1(Σ − Q). We must thus prove that Hs−1(S) →Hs−1(Σ−Q) is injective. Consider the following commutative diagram

(33.28)

Hs−q(B,S) // //

≈

Hs−q(V,BdV )≈ //

Hs−q(Σ,Σ−Q)

∂

Hs−q−1(S) // Hs−q−1(BdV ) // Hs−q−1(Σ−Q)

.

The top-left horizontal arrow in injective by Lemma 28.48. If q 6= 0, Hs−q(Σ) = 0and ∂ is injective, proving that Hs−q−1(S)→ Hs−q−1(Σ−Q) is injective. If q = 0,so Q = q1, q2 and V = V1 ∪V2. This provides an isomorphism

Z2 ⊕ Z2 ≈ Hs(V,BdV ) ≈ Hs(Σ,Σ−Q)

under which the generator of Hs(B,S) goes to (1, 0) or (0, 1) while, by Corol-lary 32.6, the image of [M ] into Hs(Σ,Σ − Q) is (1, 1). Hence, Hs−1(S) →Hs−1(Σ−Q) is again injective.

The following proposition gives a common way to compute a linking number,related to the original definition of Lebesgue [128, pp. 173–175]. Let Q, Q′ andΣ satisfying (1)–(3) above. Suppose that there exists a compact manifold W withBdW = Q′ and that the inclusion of Q′ into Σ extends to a map j : W → Σ which istransverse to Q (j needs not to be an embedding). Then j−1(Q) is a finite numberof points in W .

Proposition 33.29. l(Q,Q′) = ♯ (j−1(Q)) mod 2 .

Proof. Let k = ♯ (j−1(Q)) ∈ N. Let W0 be the manifold W minus an opentubular neighbourhood of j−1(Q). By (33.23), l(Q,Q′) depends only on the homol-ogy classH∗i([Q

′]) ∈ Hq′(Σ−Q) which, thanks to the map j (see Exercise 4.8), is thesame as that of k meridian spheres. The result then follows from Lemma 33.27.

We now introduce some material for Lemma 33.30 below, which will enable usto compute linking numbers using convenient singular cochains. Let Q, Q′ and Σsatisfying (1)–(3) above. Let W be a closed tubular neighbourhood of Q in Σ−Q′and let V be a closed tubular neighbourhood of Q in intW . We also considerthe symmetric data Q′ ⊂ V ′ ⊂ W ′ ⊂ Σ − Q, assuming that W ∩ W ′ = ∅. LetB = W,W ′,Σ− (V ∪ V ′); note that the small simplex theorem 12.56 holds truefor B.

Let c ∈ Zs−q(W,W − intV ) be a singular cocycle representing the Poincaredual class of Q in Hs−q(W,W − intV ) ≈ Hs−q(V,BdV ). We can see c as a cocycle

of W and take its zero extension c ∈ Cs−qB (Σ), i.e.

〈c, σ〉 =〈c, σ〉 if σ ∈ Ss−q(W )

0 otherwise.

Since c vanishes on Ss−q(W,W − intV ), the cochain c is a B-small cocycle, i.e.

c ∈ Zs−qB (Σ). We claim that we can chose a ∈ Cs−q−1B (Σ) such that δa = c.

Indeed, by (2) and the small simplex theorem 12.56, 0 = Hs−q(Σ) ≈ Hs−qB (Σ) when

q > 0. When q = 0, c represents the Poincare dual class of Q in HsB(Σ) ≈ Hs(Σ).

But, by (3), [Q] represents 0 in H0(Σ), so c represents 0 in HsB(Σ). A cochain


a′ ∈ Cs−q′−1

B (Σ) with δ(a′) = c′ may be also chosen, using the symmetric dataQ′ ⊂ V ′ ⊂W ′ ⊂ Σ−Q.

Finally, let µ′ ∈ Zq′(Σ − intV ) representing H∗i([Q′]) and let ν ∈ ZBs (Σ)

representing [Σ] in HBs (Σ) ≈ Hs(Σ).

Lemma 33.30. The following equalities hold true.

(a) l(Q,Q′) = 〈a, µ′〉.(b) l(Q,Q′) = 〈a c′, ν〉.

Proof. We first establish some preliminary steps.

Step 1: If a1, a2 ∈ Cs−q−1(Σ) satisfy δ(a1) = δ(a2), then 〈a1, µ′〉 = 〈a2, µ

′〉. In-deed, one has then δ(a1 + a2) = 0. If s− q− 1 = q′ > 0, Condition (2) implies thatthere exists b ∈ Cs−q−2(Σ) such that δb = a+ a. Hence,

〈a1, µ′〉+ 〈a2, µ

′〉 = 〈a1 + a2, µ′〉 = 〈δb, µ′〉 = 〈b, ∂µ′〉 = 0 .

If q′ = 0, then δ(a1 + a2) = 0 implies that a1 + a2 = 1 by Proposition 12.12, thus〈a1 + a2, µ

′〉 = 0 by Condition (3).

Step 2: Let ci ∈ Zs−q(W,W − intV ) (i = 1, 2) be singular cocycles both representing

the Poincare dual class of Q in Hs−q(W,W − intV ). Let ai ∈ Cs−q−1B (Σ) such that

δai = ci as above. Then 〈a1, µ′〉 = 〈a2, µ

′〉. Indeed, there exists b ∈ Cs−q−1(W,W−intV ) such that δ(b) = c1 + c2. Its zero extension b ∈ Cs−q−1

B (Σ) then satisfiesδ(b) = c1 + c2 and then δ(a1 + b) = c2. Thus

〈a2, µ′〉 = 〈a1 + b, µ′〉 by Step 1

= 〈a1, µ′〉 since 〈b, µ′〉 = 0 .

We can now start the proof of Lemma 33.30. Given Steps 1 and 2, it is enoughto prove (a) for a particular choice of c and a. We use Diagram (33.24) and see A

as an isomorphism from Hq(Q) onto Hs−q−1(Σ− intV ). Let a ∈ Zs−q−1(Σ− intV )

representing A([Q]). Let a ∈ Cs−q−1B (Σ) be its zero extension and let c = δ(a) ∈

Zs−qB (Σ). By Lemma 12.25, c represents δ(A([Q]). Also, c is the zero extension ofthe cocycle c ∈ Zs−q(W,W − intV ) which, by definition of A and Diagram (33.24),represents the Poincare dual class of Q in Hs−q(W,W − intV ). Therefore, since arepresents A([Q]) and µ′ ∈ Zq′(Σ− intV ) represents H∗i([Q

′]), one has l(Q,Q′) =〈a, µ′〉 = 〈a, µ′〉.

To prove (b), consider the pair inclusions j1 : (Σ, ∅) → (Σ,Σ − intV ′) andj2 : (W ′,W ′ − intV ′) → (Σ,Σ − intV ′). Since ν ∈ ZBs (Σ), there exists a (unique)ν′ ∈ Zs(W

′,W ′ − intV ′) such that C∗j2(ν′) = C∗j1(ν). As H∗j1 and H∗j2 are

isomorphisms, ν′ represents the generator of Hs(W′,W ′− intV ′) = Z2. Therefore,

c′ ν = c′ ν′ represents H∗i([Q′]) and, by (a),

l(Q,Q′) = 〈a, c′ ν〉 = 〈a c′, ν〉 .

Remark 33.31. The proof of Lemma 33.30 in the simplicial category (seeRemark 33.25) is somewhat simpler. It uses only the tubular neighbourhoods Vi andnot the Wi’s, and, of course, does not require the use of small simplex techniques.Also, ν may be taken explicitely as Ss(Σ). Writing the details is left to the readeras an exercise.


Lemma 33.30 will be used for the Hopf invariant (see § 37.4). For the moment,its main consequence is the following proposition.

Proposition 33.32. Let Q, Q′ and Σ satisfying (1)–(3) above. Then

l(Q,Q′) = l(Q′, Q) .

Proof. By Part (b) of Lemma 33.30, one has l(Q,Q′) = 〈a c′, ν〉 andl(Q′, Q) = 〈a′ c, ν〉. Then

l(Q,Q′) + l(Q′, Q) = 〈a c′ + a′ c, ν〉 = 〈δ(a a′), ν〉 = 〈a a′, ∂(ν)〉 = 0 .


4.1. Prove that the product of two homology manifolds is a homology manifold.

4.2. Let p : X → X be an odd covering, where X is a two-torus or a Klein bottle.Prove that H∗p is an isomorphism.

4.3. Let S be a smooth k-sphere embedded in a closed 2k-manifold which is aZ2-homology sphere. Show that the Euler class of the normal bundle vanishes.

4.4. Let M be a manifold with boundary such that H∗(M) = 0. Show that theboundary of M is a homology sphere.

4.5. Check the Poincare duality (Theorem 32.10) for the manifolds S1× I and theMobius band.

4.6. Show that there is no continuous retraction of a manifold onto its boundary.

4.7. Let M be a closed triangulable manifold of dimension n. Prove that thehomomorphismHk(M−pt)→ Hk(M) induced by the inclusion is an isomorphismfor k < n.

4.8. LetM be a compact triangulable topological n-manifold with boundary BdM =N . Suppose that is N = N1∪N2 the union of two closed (n − 1)-manifolds. Letf : M → X be a continuous map. Show that H∗f([M1]) = H∗f([M2) in Hn−1(X).What happens if N2 = ∅?4.9. Let f : M → N be a map between closed n-dimensional manifolds of the samedimension. Show that the degree of f may be computed locally, using a topologicalregular value, like in Proposition 13.7.

4.10. Let Σm be the orientable surface of genus m and let Σn be the nonorientablesurface of genus n. For which m and n does there exist a continuous map of degreeone from Σm to Σn or from Σn to Σn?

4.11. Let Q1 and Q2 be closed submanifolds of a closed manifold M (in the smoothcategory). Suppose that Q1 and Q2 intersect transversally in an odd number ofpoints. Show that [Q1] and [Q2] represent non-zero classes in H∗(M).

4.12. For A ⊂ 0, 1, . . . , n, let PA = [x0 : · · · : xn] ∈ RPn | xi = 0 when i /∈ A.What is the diffeomorphism type of PA? Show that, if A ∪ B = 0, 1, . . . , n,then PA and PB intersect transversally (what is the intersection?). How doesProposition 33.15 apply in this example?


4.13. Poincare dual classes in a product. Let M1 and M2 be smooth compactmanifolds. Let Qi be a closed submanifold of Mi (i = 1, 2). Then Q1 × Q2 is aclosed submanifold of M1 ×M2. Prove that

(34.1) PD(Q1 ×Q2) = PD(Q1)× PD(Q2)

in H∗(Q1 ×Q2).

4.14. Poincare dual classes in a product II. Let M and M ′ be smooth closedmanifolds and let x ∈M and x′ ∈M ′. What are PD(x×M ′) and PD(M ×x′)in H∗(M ×M ′)? Check that PD(x ×M ′) PD(M × x′) = PD((x, x′)).4.15. Let Q and Q′ be disjoint submanifolds of S2, where Q consists of two circlesand Q′ of four points. Using Proposition 33.29, compute the linking numbersl(Q,Q′) and l(Q′, Q) for the various possibilities.

4.16. Brouwer’s definition of the linking number. Let Q and Q′ be two disjointclosed submanifolds of Sn, of dimension respectively q and q′ satisfying p+q = n−1.When of dimension 0, the manifolds Q or Q′ should consist of an even number ofpoints. We see Q and Q′ as submanifolds of Rn via a stereographic projectionof Sn − pt onto Rn. Consider the Gauss map λ : P × Q → Sn−1 given byλ(x, y) = x−y

||x−y|| . Show that the degree of λ is equal to the linking number l(Q,Q′)

(see § 33.4). [Hint: use Proposition 33.29.]

4.17. Write the proof of Lemma 33.30 in the simplicial category (see Remarks 33.25and 33.31).

4.18. Let Σ be the unit sphere in Rq+1 × Rq′+1. Let Q = Sq × 0 ⊂ Σ and

Q′ = 0 × Sq′ ⊂ Σ. Compute the linking number l(Q,Q′) in Σ.

CHAPTER 5

Projective spaces

35. The cohomology ring of projective spaces - Hopf bundles

The cohomology ring of RPn for n ≤ ∞ was established in Proposition 24.21,using the transfer (or Gysin) exact sequence for the double cover (S0-bundle) Sn →RPn. It gives a GrA-isomorphism Z2[a]/(a

n+1)→ H∗(RPn). Here below, we givea completely different proof of this fact, which is based on Poincare duality (asRPn is a smooth closed manifold, it can be triangulated as a polyhedral homologyn-manifold: see p 170). We shall also discuss the cases of complex and quaternionicprojective spaces CPn and HPn, and of the octonionic projective plane OP 2.

Proposition 35.1. The cohomology algebra of RPn (n ≤ ∞) is given by

H∗(RPn) ≈ Z2[a]/(an+1) , H∗(RP∞) ≈ Z2[a] ,

with a ∈ H1(RPn).

Proof. We prove the first statement by induction on n. It is true for n = 1since RP 1 = S1 (and also proven for n = 2 at p. 116). Suppose, by induction, thatit is true in for RPn−1.

In Example 15.6 is given the standard CW-structure of RPn, with one k-cellfor each k = 0, 1 . . . ,m. It follows that Hk(RPn,RPn−1) = 0 for k ≤ n − 1and Hn(RPn,RPn−1) = Z2. By Poincare duality, Hn(RPn) = Z2, so the exactsequence for the pair (RPn,RPn−1) gives

0→ Hn−1(RPn)→ Hn−1(RPn−1)︸︷︷︸Z2

→ Hn(RPn,RPn−1)︸︷︷︸Z2

→ Hn(RPn)︸︷︷︸Z2

→ 0 .

All that implies that the inclusion induces an isomorphismHk(RPn)→ Hk(RPn−1)for k ≤ n − 1. By the induction hypothesis and functoriality of the cup product,Hk(RPn) = Z2 for k ≤ n− 1, generated by ak.

By Poincare duality, Hn(RPn) = Z2 with generator [RPn] and, by Theo-rem 32.18, the bilinear map

H1(RPn)×Hn−1(RPn)→ Hn(RPn)[RPn]−−−−−→ Z2

is not degenerated. Therefore, a an−1 6= 0, which proves that H∗(RPn) ≈Z2[a]/(a

n+1).Finally, by the standard CW-structure of RP∞, one has, for all integer m,

that Hk(RP∞,RPn) = 0 for k < m. The first statement then implies the second.Note that we have also proven that the standard CW-structure of RPn (n ≤ ∞) isperfect

201

202 5. PROJECTIVE SPACES

Corollary 35.2. The Poincare series of RPn (n ≤ ∞) are

Pt(RPn) = 1 + t+ · · ·+ tn =

1− tn+1

1− t and Pt(RP∞) =

1

1− t .

Remark 35.3. Proposition 35.1 and its proof show that the GrA-homomorphismHj(RPn+k) → Hj(RPn) induced by the inclusion RPn → RPn+k is surjective(k ≤ ∞). In particular, it is an isomorphism for j ≤ n.

Corollary 35.4. The Lusternik-Schnirelmann category cat (RPn) of RPn isequal to n+ 1.

The topological complexity of RPn is much more complicated to compute (seeTheorem 36.13).

Proof. As a smooth manifold, RPn admits a triangulation [204, Theorem 7],by a connected simplicial complex of dimension n. Therefore, cat (|K|) ≤ n +1 by Proposition 25.1. On the other hand, by Proposition 25.3, cat (RPn) ≥nilH>0(RPn) and nilH>0(RPn) = n+ 1 by Proposition 35.1.

Remark 35.5. The polynomial structure on H∗(RP∞) implies the followingfact: if f : RP∞ → X is a continuous map withX a finite dimensional CW-complex,then H∗f = 0. Actually, f is homotopic to a constant map. This result is a weakversion of the original Sullivan conjecture [184, p. 180], which lead to importantresearches in homotopy theory (see, e.g. [168]) and was finally proven, in a moregeneral form, by H. Miller [140].

We now pass to the complex projective space CPn, the space of complex lines inCn+1. Such a line is represented by a non-zero vector z = (z0, . . . , zn) ∈ Cn+1−0,and two such vectors z and z′ are in the same line if and only if z′ = λz withλ ∈ C∗ = C− 0. If |z| = |z′| = 1, then λ ∈ S1. Thus

CPn = (Cn+1 − 0)/C∗ = S2n+1

/S1 .

The image of (z0, . . . , zn) in CPn is denoted by [z0 :z1 : . . . :zn]. As S1 acts smoothlyon S2n+1, the quotient CPn is a closed smooth manifold and the quotient map

p : S2n+1 → CPn

is a principal S1-bundle [80, Example 4.44], called the Hopf bundle. In this simpleexample, this can be proved directly. Consider the open set Vk ⊂ Cn+1−0 givenby Vk = (z0, . . . , zn) ∈ Cn+1 | zk 6= 0. Its image in CPn is an open set Uk,

domain of the chart ϕk : Cn≈−→ Uk given by

ϕk(z0, . . . , zn−1) = [z0 :z1 : . . . :zk−1 : 1 : zk : . . . : zn−1] .

On the other hand, a trivialization ϕk : Uk × S1 ≈−→ p−1(Uk) is given by(35.6)

ϕk(ϕk(z0, . . . , zn−1), g) =1√

1 +∑n−1i=0 |zi|2

(z0, z1, . . . , zk−1, g, zk, . . . , zn−1) .

It is also classical that CPn is obtained from CPn−1 by attaching one cell ofdimension 2n,

CPn = CPn−1 ∪p D2n ,

with the attaching map p : S2n−1 → CPn−1 being the quotient map (see e.g. [80,Example 0.6] or [152, Theorem 40.2]). This gives a standard CW-structure on

35. THE COHOMOLOGY RING OF PROJECTIVE SPACES - HOPF BUNDLES 203

CPn with one cell in each even dimension ≤ 2n. For the direct limit CP∞, weget CW-structure with one cell in each even dimension. For these CW-structure,the vector space of cellular chains vanish in odd degree, so the cellular boundary isidentically zero. Therefore,

Pt(CPn) = 1 + t2 + · · ·+ t2n =

1− t2(n+1)

1− t2 and Pt(CP∞) =

1

1− t2 .

As CPn is a smooth manifold, the same proof as for Proposition 35.1, usingPoincare duality, gives Proposition 35.7 below. One can also adapt the proof ofProposition 24.21, using the Gysin exact sequence of the Hopf bundle (see Exer-cise 5.2):

· · ·Hk−1(S2n+1)→ Hk−2(CPn)−e(ξ)−−−−−→ Hk(CPn)

H∗p−−−→ Hk−1(S2n+1)→ · · · .Proposition 35.7. The cohomology algebra of CPn (n ≤ ∞) is given by

H∗(CPn) ≈ Z2[a]/(an+1) , H∗(CP∞) ≈ Z2[a] ,

with a ∈ H2(CPn). The class a is the Euler class of the Hopf bundle S2n+1 → CPn.

If we replace the field of complex numbers by that of quaternions H, we getquaternionic projective space HPn:

HPn = (Hn+1 − 0)/H∗ = S4n+3

/S3

(it is usual to take the right H-vector space structure on Hn+1). The space HPn

is obtained from HPn−1 by attaching one cell of dimension 4n, with the attachingmap p : S4n−1 → HPn−1 being the quotient map. The map p is an S3-bundle calledthe Hopf bundle.This gives a standard CW-structure on HPn with one cell in eachdimension 4k ≤ 4n. For the direct limit HP∞, we get CW-structure with one cellin each dimension 4k and

Pt(HPn) = 1 + t4 + · · ·+ t4n =

1− t4n+1)

1− t4 and Pt(HP∞) =

1

1− t4 .

Proposition 35.8 below is proven as Proposition 35.7, either using Poincareduality (HPn is a smooth 4n-manifold), or the Gysin exact sequence of the Hopfbundle (see Exercise 5.2).

Proposition 35.8. The cohomology algebra of HPn (n ≤ ∞) is given by

H∗(HPn) ≈ Z2[a]/(an+1) , H∗(HP∞) ≈ Z2[a] ,

with a ∈ H4(HPn). The class a is the Euler class of the Hopf bundle S4n+3 → HPn.

Let K = R, C or H and let d = d(K) = dimR K. The space KP 1 has a CW-structure with one 0-cell and one d-cell and is then homeomorphic to Sd. Thequotient maps S2d−1 →→ KP 1 thus give maps

h1,1 : S1 →→ S1 , h3,2 : S3 →→ S2 and h7,4 : S7 →→ S4

called the Hopf maps. Note that h1,1 is just a 2-covering. Using the homeomorphism

Sd ≈ K = K ∪ ∞ given by a stereographic projection, these Hopf maps admitthe formula

(35.9) hi,j(v, w) =

vw−1 if w 6= 0

∞ otherwise.


This formula also makes sense for K = O, the octonions, whose multiplicationadmits inverses for non zero elements. This gives one more Hopf map h15,8 : S15 →S8. One can prove that h15,8 is an S7-bundle (see [80, Example 4.47]), so also calledthe Hopf bundle. Attaching a 16-cell to S8 using h15,8 produces the octonionicprojective plane OP 2 (because of non-associativity of the octonic multiplication,there are no higher dimensional octonionic projective spaces).

Proposition 35.10. The cohomology algebra of OP 2 is given by

H∗(OP 2) ≈ Z2[a]/(a3)

with a ∈ H8(OP 2) = Z2. In particular, Pt(OP 2) = 1 + t8 + t16.

Proof. By its cellular decomposition, Hk(OP 2) = Z2 for k = 0, 8, 16 andzero otherwise. Let a ∈ H8(OP 2) and b ∈ H8(S8) be the non-zero elements. The

mapping cylinder E of h15,8 is the disk bundle associated to the Hopf bundle and

OP 2 has the homotopy type of E ∪D16, with E ∩D16 = S15. The Thom class ofthe Hopf bundle h15,8

U ∈ H8(E, S15) ≈ H8(OP 2, E) ≈ H8(OP 2, intD16) ≈ H8(OP 2) ,

is not zero, so corresponds to a ∈ H8(OP 2). The following diagram

H8(E)−U

≈// H16(E, S15)

H8(OP 2) //

−a

33

≈

OO

H16(OP 2, intD16)

≈

OO

≈ // H16(OP 2)

is then commutative by the analogue in singular cohomology of Lemma 22.11. Thisproves that − a : H8(OP 2)→ H16(OP 2) is bijective.

Remark 35.11. Proposition 35.10 may also be proved using Poincare duality,since OP 2 has the homotopy type of a closed smooth 16-manifold, in fact a homo-geneous space of the exceptional Lie group F4 (see [203, Theorem 7.21, p 707]).

The computations of the cohomology algebra H∗(KP 2) have the following con-sequence.

Corollary 35.12. The Hopf maps

h1,1 : S1 → S1 , h3,2 : S3 → S2 , h7,4 : S7 → S4 and h15,8 : S15 → S8

are not homotopic to constant maps.

We shall prove later that no suspension of these Hopf maps is homotopic to aconstant map (see Proposition 50.1).

Proof. One has S1 ∪h1,1 D2 ≈ RP 2. If h1,1 were null-homotopic, RP 2 would

have the homotopy type of S1∨S2 (see [80, Proposition 0.18]). But, in H∗(S1∨S2),the cup-square map vanishes by (23.1), which is not the case in H∗(RP 2). The sameproof works for the other Hopf maps.

36. APPLICATIONS 205

We now consider the vector bundle γK over KPn associated to the Hopf bundle,where K = R,C,H and n ≤ ∞. For n ∈ N, total space of γK is

E(γK) = (a, v) ∈ KPn ×Kn+1 | v ∈ a ,with bundle projection (a, v) 7→ a. For instance, for n = 1, E(γR) is the Mobiusband. The correspondence (a, v) → V defines a map E(γK) → Kn+1 whose re-striction to each fiber is linear and injective. This endows γK with an Euclideanstructure, whose unit sphere bundle is the Hopf bundle. The vector bundle γK iscalled the Hopf vector bundle or the tautological bundle over KPn. Its (real) rankis d = d(K) = dimR K. By passing to the direct limit when n → ∞, we get atautological bundle γK over KP∞. Propositions 35.7, 35.8 (and 28.58 for K = R)gives the following result.

Proposition 35.13. For n ≤ ∞, the Euler class e(γK) is the non-zero elementad ∈ Hd(KPn) = Z2.

The inclusions R ⊂ C ⊂ H induce inclusions

RPnj1−→ CPn

j2−→ HPn , n ≤ ∞ .

The above proposition permits us to determine the GrA-homomorphism inducedin cohomology by these inclusions.

Proposition 35.14. H∗jd (a2d) = a2d.

Proof. Observe that γC is a complex vector bundle, so the multiplication byi ∈ C is defined on each fiber. We notice that

(35.15) i∗1γC = γR ⊕ i γR ≈ γR ⊕ γR .

ThenH∗j1 (a2) = H∗j1 (e(γC)) by Proposition 35.14

= e(j∗1γC)

= e(γR ⊕ γR) by (35.15)

= e(γR) e(γR) by Proposition 28.62

= a21 by Proposition 35.14.

The proof that H∗j2 (a4) = a22 is the same, using the multiplication by j ∈ H on

the fiber of γH which is a quaternionic vector bundle.

The Hopf bundles are sphere bundles over Sp such that the total space is also asphere. We shall see in Proposition 37.6 that p = 1, 2, 4, 8 are the only dimensionswhere such examples may occur.

36. Applications

36.1. The Borsuk-Ulam theorem. A (continuous) map f : Rm → Rn orf : Sm → Sn such that f(−x) = −f(x) is called an odd map.

Theorem 36.1. Let f : Sm → Sn be an odd map. Then:

(1) n ≥ m.(2) if m = n, deg f = 1.


Proof. If f is odd, it descends to a map f : RPm → RPn with a commutativediagram

Sm

pm

f // Sn

pn

RPm

f // RPn

.

The two-fold covering pn is induced from p∞ : S∞ → RP∞ by the inclusion RPn →RP∞. By Lemma 24.11 and Proposition 24.21, the characteristic classes w(pm) ∈H1(RPm) and w(pn) ∈ H1(RPn) are the generators of these cohomology groupsandH∗f(w(pn)) = w(pm). By Proposition 24.21 again, one has that 0 = H∗f(w(pn)n) =(w(pn)n which implies that n ≥ m.

Now, if m = n, observe that H∗pn : Hn(RPn)→ Hn(Sn) is the zero homomor-phism since pn is of local degree 2. The transfer exact sequence of (24.17), whichis functorial, gives the following commutative diagram

Hn(Sn)

H∗f

tr∗

≈// Hn(RPn)

H∗ f≈

Hn(Sn)tr∗

≈// Hn(RPn)

,

proving that deg f = 1.

As a corollary, we get the theorem. of Borsuk-Ulam.

Corollary 36.2 (Borsuk-Ulam theorem). Let g : Sn → Rn be a continuousmap. Then, there exists z ∈ Sn such that g(z) = g(−z).

Proof. Otherwise, the map f : Sn → Sn−1 defined by

f(z) =g(z)− g(−z)|g(z)− g(−z)|

is continuous and odd, which contradicts Theorem 36.1.

A famous consequence is the ham sandwich theorem. For an early history ofthis theorem, see [13].

Corollary 36.3. Let A1, . . . , An be n bounded Lebesgue measurable subsetsof Rn. Then, there exists a hyperplane which bisects each Ai.

Proof. Identify Rn by an isometry with an affine n-subspace W of Rn+1

not passing through the origin, and thus see A1, . . . , An ⊂ W . For each unitvector v ∈ Rn+1, consider the half-space Q(v) = x ∈ Rn+1 | 〈v, x〉 > 0. Letgi : S

n → R defined by gi(v) = measure(Ai∩Q(v)). The maps gi are the coordinatesof a continuous map g : Sn → Rn. By Corollary 36.2, there is z ∈ Sn such thatg(z) = g(−z), which means that gi(z) = 1

2 measure(Ai). Then, P (z) ∩W is thedesired bisecting hyperplane.

36.2. Non-singular and axial maps. A continuous map µ : Rm×Rm → Rk

is called non-singular if

(1) µ(αx, βy) = αβ µ(x, y) for all x, y ∈ Rm and all α, β ∈ R, and(2) µ(x, y) = 0 implies that x = 0 or y = 0.


Non-singular maps generalize bilinear maps without zero divisors. They wereintroduced in [59], from where the results of this section are extracted. For refer-ences about the earlier literature, see also [59].

Non-singular maps are related to axial maps. A continuous map g : RPm ×RPm → RP ℓ, with ℓ ≥ m, is called axial if the restriction of g to each slice is nothomotopic to a constant map. By Corollary 19.7, this is equivalent to ask that theserestrictions g1

x : x × RPm → RP ℓ or g2x : RPm × x → RP ℓ to be homotopic to

the inclusion RPm → RP ℓ. Using Corollary 19.7, this is equivalent to ask thatH∗gix(aℓ) = am, where aj is the generator of H1(RP j). By Corollary 28.6, wededuce that a continuous map g : RPm × RPm → RP ℓ is axial if and only if

(36.4) H∗g(aℓ) = 1× am + am × 1 .

The name of axial map appeared in [5] where references about the earlierliterature on the subject may be found. It started with the work of Stiefel andHopf [99].

Let µ : Rm × Rm → Rk be a non-singular map. By Point (2) of the definition,we get a continuous map µ : Sm−1 × Sm−1 → Sk−1 defined by

(36.5) µ(x, y) =µ(x, y)

|µ(x, y)| .

Point (1) above implies that µ descends to a map

(36.6) µ : RPm−1 × RPm−1 → RP k−1 .

For x ∈ RPm−1, the restriction of µx to the slice x × RPm−1 is covered bytwo-fold covering maps:

x × Sm−1

pm−1

// RPm−1 × Sm−1

// Sk−1

pk−1

x × RPm−1 //

µx

11RPm−1 × RPm−1µ // RP k−1

.

By Lemma 24.11 and Proposition 24.21, the characteristic classes w(pm−1) ∈H1(RPm−1) and w(pk−1) ∈ H1(RP k−1) are the generators of these cohomologygroups and H∗µx(w(pk−1)) = w(pm−1). Hence, µx is not homotopic to a constantmap. The same reasoning holds for the slices RPm−1 × x. Therefore, µ is axial.Conversely, if g : RPm−1 × RPm−1 → RP k−1 is an axial map induces on universalcovers a map g : Sm−1 × Sm−1 → Sk−1 satisfying g(−x, y) = g(x,−y) = −g(x, y).The map µ : Rm × Rm → Rk defined by

µ(x, y) = |x| · |y| · g( x|x| ,y

|y| )

is a non-singular map. This proves the following lemma.

Lemma 36.7. The correspondence µ 7→ µ provides a bijection between non-singular maps Rm × Rm → Rk (up to multiplication by constants) and axial mapsRPm−1 × RPm−1 → RP k−1.

Let µ : Rm × Rm → Rk be a non-singular map. The restriction of µ to eachslice is odd. Hence, if a non-singular map µ : Rm×Rm → Rk exists, it follows formTheorem 36.1 that k ≥ m. When m = k, the following theorem is attributed toStiefel. For other proofs (see [150, Theorem 4.7] or Remark 50.7).


Proposition 36.8. Let µ : Rm × Rm → Rm be a non-singular map. Thenm = 2r.

In fact, by a famous result of J.F. Adams (see Remark 50.7), non-singular mapsRm × Rm → Rm exist only if m = 1, 2, 4, 8.

Proof. We consider the associated axial map µ : RPm−1×RPm−1 → RPm−1

and denote by a the generator of Hn−1(RPm−1). The Kunneth theorem impliesthat the correspondence x 7→ 1× a and y 7→ a× 1 provides a GrA-isomorphism

Z2[x, y]/(xm, ym)

≈−→ H∗(RPm−1 × RPm−1) .

By (36.4), H∗µ(a) = x + y. Therefore, (x + y)m = 0. As xm and ym also vanish,one has

(x + y)m =

m∑

i=0

(mi

)xiym−i =

m−1∑

i=1

(mi

)xiym−i

This implies that(mi

)≡ 0 mod 2 for all i = 1, . . . ,m − 1 which, by Lemma 36.9

below, happens only if m = 2r.

For n ∈ N, denote its dyadic expansion in the form n =∑j∈J(n) 2j where

J(n) ⊂ N.

Lemma 36.9 (Binomial coefficients mod 2). Let m, r ∈ N. Then(mr

)≡ 1 mod 2 ⇐⇒ J(r) ⊂ J(m) .

In other words,(mr

)≡ 1 mod 2 if and only if the dyadic expansion of r is a

subsum of that of m.

Proof. In Z2[x], the equation (1 + x)2 = 1 + x2 holds, whence (1 + x)2j

=

1 + x2j

. Thereforem∑

r=0

(mr

)xr = (1 + x)m =

∏

j∈J(m)

(1 + x)2j

=∏

j∈J(m)

(1 + x2j

) .

The identification of the coefficient of xr gives the lemma.

The technique of the proof of Proposition 36.8 also gives a result of H. Hopf,[99, Satz I.e].

Proposition 36.10. Let µ : Rm×Rm → Rk be a non-singular map. If m > 2r,then k ≥ 2r+1.

Proof. We already know that k > m. As in the proof of Proposition 36.8,we consider the associated axial map µ : RPm−1 × RPm−1 → RP k−1 and use thesame notations. We get the equation (x + y)k = 0 in Z2[x, y]

/(xm, ym), which,

as m > 2r, implies that(ki

)= 0 for all 1 ≤ i ≤ 2r. By Lemma 36.9, the dyadic

expansion k =∑

j kj2j must satisfy kj = 0 for j ≤ r, which is equivalent to

k ≥ 2r+1.

Remark 36.11. There always exists a non-singular map µ : Rm×Rm → R2m−1

(see [59, § 5]).

We finish this section by mentioning two results connecting non-singular oraxial maps to the immersion problem and the topological complexity of projectivespaces. The following proposition was proven in [5].

37. THE HOPF INVARIANT 209

Proposition 36.12. There exists an axial map g : RPn×RPn → RP k (k > n)if and only if there exists an immersion of RPn in Rk.

We shall not talk about the large literature and the many results on the problemimmersing or embedding RPm in Rq (see however Proposition 55.38). Tables andreferences are available in [34]. The existence of non-singular maps is also relatedto the topological complexity of the projective space. The following is proven in[59, Theorem 6.1].

Theorem 36.13. The topological complexity TC (RPn) is equal to the smallestinteger k such that there is a non-singular map µ : Rn+1 × Rn+1 → Rk.

Symmetric non-singular maps (i.e. µ(x, y) = µ(y, x) are, in some range, re-lated to embeddings of RPn in Euclidean spaces or to the symmetric topologicalcomplexity. For results and references, see [66].

37. The Hopf invariant

37.1. Definition. Let f : S2m−1 → Sm be a continuous map. The space Cf =D2m ∪f Sm is a CW-complex with one cell in dimension 0, m and 2m. Considerthe cup-square map 2

m : Hm(Cf ) → H2m(Cf ), given by 2m (x) = x x. The

Hopf invariant Hopf (f) ∈ Z2 is defined by

Hopf (f) =

1 if 2

m is surjective for Cf .

0 otherwise.

The space Cf depends only on the homotopy class of f (see, e.g. [80, Proposi-tion 0.18]), then so does the Hopf invariant. The trivial map has Hopf invariant 0.The computation of the cohomology ring of the various projective planes in § 35shows that the 2-fold cover S1 → S1 as well as the other Hopf maps h3,2 : S3 → S2,h7,4 : S7 → S4 and h15,8 : S15 → S8 have Hopf invariant 1.

Our Hopf invariant is just the reduction mod 2 of the classical integral Hopfinvariant defined in e.g. [80, § 4.B]. The form of our definition is motivated byextending the statements to the case m = 1, usually not considered by authors.

Note that Hopf defined his invariant in 1931–35 [97, 98], before the inventionof the cup product. He used linking numbers (see § 37.4 below).

For m = 1, recall the bijection DEG : [S1, S1]≈−→ Z given in (13.11).

Proposition 37.1. Let f : S1 → S1. Then

Hopf (f) =

0 if DEG (f) ≡ 0 mod 4

1 otherwise.

Proof. Let C = Cf . If DEG (f) is odd, then deg(f) = 1 by Proposition 13.12.The computation of the cellular cohomology of C using Lemma 16.3 shows thatH∗(C) = 0, so 2

1 is surjective and Hopf (f) = 1. If DEG (f) = 2k, then, H1(C) ≈Z2 ≈ H2(C). Consider the 2-fold covering p : C → C whose characteristic class isthe non zero element a ∈ H1(C). Its transfer exact sequence looks like

0→ H0(C)︸︷︷︸Z2

→ H1(C)︸︷︷︸Z2

H∗p−−−→ H1(C)tr∗−−→ H1(C)︸︷︷︸

Z2

−a−−−→ H2(C)︸︷︷︸Z2


By Van Kampen’s theorem, π1(C) is cyclic of order 2k. Thus, π1(C) is cyclic of

order k and H1(C) = Z2 if k is even while it vanishes if k is odd. The propositionthus follows from the above transfer exact sequence.

37.2. The Hopf invariant and continuous multiplications. A classicalconstruction associates a map fκ : S2m−1 → Sm to a “continuous multiplication”κ : Sm−1 × Sm−1 → Sm−1. Let D = Dm and S = Sm−1 = ∂D. Divide Sm intothe upper and lower hemisphere: Sm = B+ ∪ B− ⊂ Rm × R, with B+ ∩ B− =Sm ∩ Rm × 0 = S. Using the decomposition

∂(D ×D) = ∂D ×D ∪ D × ∂D = S ×D ∪ D × S ,the map fκ : ∂(D ×D)→ Sm is defined, for x, y ∈ S et t ∈ [0, 1], by

fκ(tx, y) = (t κ(x, y),√

1− t2) and fκ(x, ty) = (t κ(x, y),−√

1− t2) .For u, v,∈ Sm−1, we consider the hypothesis H(u, v) on κ:

H(u, v): κ(u,−x) = −κ(u, x) and κ(−x, v) = −κ(x, v) for all x ∈ Sm−1.

For example, H(u, v) holds for all u, v ∈ Sm−1 if κ = µ, the map associatedusing (36.5) to a non-singular map µ : Rm ×Rm → Rm. Also, H(e, e) is satisfied ife is a neutral element for κ.

Example 37.2. Let κ : S0× S0 → S0 be the usual sign rule (S0 = S = ±1).Then D = [−1, 1] and the map fκ : ∂(D ×D)→ S1 is pictured in Figure 1.

(−1,−1) (1,−1)

(1, 1)(−1, 1) A

C

B D

fκ1−1

fκ(A) = fκ(C)

fκ(B) = fκ(D)

Figure 1. The map fκ for the usual sign rule.One sees that fκ has degree 2. By Proposition 37.1, this implies that Hopf (fκ) =

1. Actually, the map fκ is topologically conjugate to the projection S1 →→ S1/x ∼−x, so Cfκ

is homeomorphic to RP 2.

The same exercise may be done for the other possible multiplications on S0.By changing the sign of κ if necessary, we may assume that κ(1, 1) = 1. There arethen 8 cases.

κ(1, 1) κ(−1, 1) κ(−1,−1) κ(1,−1) DEG(fκ) Hopf (fκ) satisfies

1 1 1 1 1 0 0

2 1 1 1 -1 1 1 H(1,−1)

3 1 1 -1 1 -1 1 H(−1,−1)

4 1 1 -1 -1 0 0

5 1 -1 1 1 1 1 H(−1, 1)

6 1 -1 1 -1 2 1 H(u, v) ∀ u, v

7 1 -1 -1 1 0 0

8 1 -1 -1 -1 1 1 H(1, 1)

One sees that Hopf (fκ) = 1 if and only if H(u, v) is satisfied for some u, v ∈ S0.This is partially generalized in the following result.


Proposition 37.3. Let κ : Sm−1 × Sm−1 → Sm−1 be a continuous multiplica-tion. Suppose that H(u, v) is satisfied for some u, v ∈ Sm−1. Then Hopf (fκ) = 1.

Proof. The case m = 1 was done in Example 37.2. We may thus assume thatm > 1. The following proof is inspired by that of [79, Lemma 2.18]. Let f = fκ.Consider the commutative diagram

Hm(Cf )⊗Hm(Cf ) // H2m(Cf )

Hm(Cf , B+)⊗Hm(Cf , B−)

≈

OO

φ∗⊗φ∗

// H2m(Cf , Sm)

≈

OO

≈ Φ∗

Hm(D ×D,S ×D)⊗Hm(D ×D,D × S)

// H2m(D ×D, ∂(D ×D))

Hm(D,S)⊗Hm(D,S)

×

≈

33ffffffffffffffffffπ∗1⊗π

∗2 ≈

OO

where φ : D × D → Cf is the characteristic map for the 2m-cell of Cf and φ∗ =H∗φ. The cross-product map at the bottom of the diagram is an isomorphismby the relative Kunneth theorem 27.25. Hence, Hopf (fκ) = 1 if and only if thehomomorphism φ∗ ⊗ φ∗ in the left column is an isomorphism. By symmetry, it isenough to prove that φ∗ : Hm(Cf , B+)→ Hm(D×D,S×D) is not zero. Considerthe commutative diagram

Hm(Cf , B+)

φ∗

≈ // Hm(Sm, B+)

f∗κ

≈ // Hm(B−, S)

f∗κ

Hm(D ×D,S ×D)≈ // Hm(∂(D ×D), S ×D)

≈ // Hm(D × S, S × S) .

The left horizontal maps are isomorphism since m > 1 and the right ones byexcision. It then suffices to prove that f∗κ : Hm(B−, S)→ Hm(D× S, S × S) is notzero.

As the restriction of fκ to S ×S is equal to κ, one has a commutative diagram

Hm−1(S)

κ∗

≈ // Hm(B−, S)

f∗κ

Hm−1(D × S) // Hm−1(S × S)δ∗ // Hm(D × S, S × S)

Hm−1(S)

π∗2≈

OOπ∗2

66mmmmmmmmmm

where the second line is the exact sequence of the pair (D × S, S × S). Let u, v ∈Sm−1 such that H(u, v) is satisfied. Let s1, s2 : S → S × S be the slice inclusionsgiven by s1(x) = (x, v) and s2(x) = (u, x). The composition κsi is thus an oddmap. Therefore, Theorem 36.1 implies that H∗(κsi)(a) = a. By Lemma 28.5, one


deduces that

(37.4) H∗κ(a) = 1× a+ a× 1 .

On the other hand, ker δ∗ = Imageπ∗2 = 0,1× a. Therefore, f∗κ does not vanish.

In the proof of Proposition 37.3, the hypothesis on H(u, v) is only used toobtain Equation (37.4). Therefore, one has the following proposition.

Proposition 37.5. Let κ : Sm−1 × Sm−1 → Sm−1 be a continuous multi-plication. Suppose that H∗κ(a) = 1 × a + a × 1 for a ∈ Hm−1(Sm−1). ThenHopf (fκ) = 1.

37.3. Dimension restrictions. We shall prove in Corollary 50.4 that, if thereexists a map f : S2m−1 → Sm with Hopf invariant 1, then m = 2r. Actually,m = 1, 2, 4, 8 by a famous theorem of Adams (see Theorem 50.6). This theoremimplies the following result.

Proposition 37.6. Let Sqi−→ E

π−→ Sp be a locally trivial bundle. Suppose thatH∗(E) ≈ H∗(Sp+q). Then q = p− 1 and p = 1, 2, 4 or 8.

Proof. If p = 1 and q > 0, then H∗(E) is GrV-isomorphic to H∗(S1) ⊗H∗(Sq) by the argument of Example 28.72. Thus, we must have q = 0 and π is thenon-trivial double cover of S1.

Let us suppose that p ≥ 2. If H∗(E) ≈ H∗(Sp+q), then H∗i is not surjective;otherwise, H∗(E) is GrV-isomorphic to H∗(Sp) ⊗ H∗(Sq) by the Leray-Hirschtheorem. The Wang exact sequences (see Proposition 28.68)

· · · → Hq(E)H∗i−−→ Hq(Sq)

Θ−→ Hq+1−p(Sq)→ · · · .then implies that q + 1− p = 0 (since p > 1). Therefore, q = p− 1.

The bundle gluing map φ : Sq × Sq → Sq (see p 159) may thus be seen asa continuous multiplication, to which a map fφ : S2p−1 → Sp may be associated

using (35.6). We shall prove that Hopf (fφ) = 1. By Theorem 50.6, this impliesthat p = 1, 2, 4 or 8.

Let a ∈ Hq(Sq) be the generator. The restriction of φ to a slice x×Sq beinga homeomorphism, one has, using Lemma 28.4, that

H∗φ(a) = 1× a+ λ(1× a)for some λ ∈ Z2. As Θ 6= 0 and p > 1, one gets from Proposition 28.73 that

0 6= e×Θ(a) = H∗φ(a) + 1× a ,where e ∈ Hp(Sp) is the generator. Therefore, H∗φ(a) = a × 1 + 1 × a. ByProposition 37.5, this implies that Hopf (fφ) = 1.

Examples 37.7. Consider the bundle S1 → E → S2 with gluing map φ(u, z) =ukz. The total space E, obtained by gluing two copies of D2 × S1 using the mapφ, is then a lens space with fundamental group of order k. Thus, if k is odd, Esatisfies the hypotheses of Proposition 37.6. Other famous examples are the bundlesS3 → E → S4 which were used by J. Milnor to produce his exotic 7-spheres [142,§ 3]. Indeed, with a well chosen gluing map, the total spaceE is a smooth 7-manifoldhomeomorphic but not diffeomorphic to S7.


37.4. The Hopf invariant and linking numbers. Let f : S2m−1 → Sm

be a smooth map. Let y, y′ ∈ Sm be two distinct regular values of f . ThenQ = f−1(y) and Q′ = f−1(y′) are two disjoint closed submanifolds of S2m−1,both of dimension m−1. Therefore, their linking number l(Q,Q′) ∈ Z2 (see § 33.4)is defined, at least if m > 1 (see also Remark 37.19 below).

Proposition 37.8. If m > 1, then Hopf (f) = l(Q,Q′).

Actually, l(Q,Q′) is the original definition by H. Hopf of his invariant [97,98]. Proposition 37.8 goes back to the work of N. Steenrod [178], after which thedefinition of Hopf (f) with the cup product in Cf was gradually adopted.

Proof. Let Σ = S2m−1. We consider the mapping cylinder Mf of f

Mf = [(S2m−1×I) ∪Sm]/(x, 0) ∼ f(x) | x ∈ S2m−1 .

The correspondence (x, t) 7→ f(x) descends to a retraction by deformation ofρ : Mf → Sm. we identify Σ with the subspace Σ × 1 of Mf . The mapping

cone Mf of f , defined as

Mf = Mf ∪Σ CΣ ≈Mf ∪Σ D2m ,

whereCΣ ≈ D2m is the cone over Σ, is homotopy equivalent to the CW-complex Cf .We first introduce some material in order to compute l(Q,Q′) using Lemma 33.30.

Let W0 be a closed tubular neighbourhood of y in Sm−y′ and let V0 be a closedtubular neighbourhood of y in intW0 (W0 and V0 are just m-disks). Let y′ ∈ V ′0 ⊂W ′0 be a symmetric data for y′ withW0∩W ′0 = ∅. Let B0 = W0,W

′0, S

m−(V0∪V ′0 ).As y and y′ are regular values of f , we may assume, provided W0 and W ′0 are smallenough, that W = f−1(W0) and V = f−1(V0) are nested tubular neighbourhoodsof Q and that W ′ = f−1(W ′0) and V ′ = f−1(V ′0 ) are nested tubular neighbourhoodsof Q′. Let B = f−1(B0) = W,W ′,Σ− (V ∪ V ′).

Let us briefly repeat the preliminary constructions for Lemma 33.30 (see p. 196for more details), in our context, with the dimensions q = q′ = m − 1 and s =2m− 1. Let c0 ∈ Zm(W0,W0 − intV0) represent the Poincare dual class of y inHm(W0,W0 − intV0) ≈ Hm(V0,BdV0) and let c′0 ∈ Zm(W ′0,W

′0 − intV ′0) represent

the Poincare dual class of y′ in Hm(W ′0,W′0 − intV ′0) ≈ Hm(V ′0 ,BdV ′0). Let

c0, c′0 ∈ ZmB0

(Sm) be their zero extensions. Then c = C∗f(c0) ∈ Zm(W,W − intV )represent the Poincare dual class of Q in Hm(W,W − intV ) ≈ Hm(V,BdV ) andc′ = C∗f(c′0) ∈ Zm(W ′,W ′ − intV ′) represent the Poincare dual class of Q′ inHm(W ′,W ′ − intV ) ≈ Hm(V ′,BdV ′). Also, c = C∗f(c0) ∈ ZmB (Σ) and c′ =

C∗f(c′0) ∈ ZmB (Σ) are the zero extensions of c and c′. Chose a ∈ Cm−1B (Σ) such that

δa = c. Let ν ∈ ZB2m−1(Σ) representing [Σ]B in HB2m−1(Σ) ≈ H2m−1(Σ). According

to Lemma 33.30, one has l(Q,Q′) = 〈a c′, ν〉. We note that a c′ ∈ Z2m−1B (Σ).

Indeed, for any σ ∈ S2m−1(Σ), one has

(37.9) 〈δ(a c′), σ〉 = 〈c c′, σ〉 = 0

since the support of c is in W and that of c′ is in W ′. Therefore, δ(a c′) = 0and a c′ represents a cohomology classes |a c′| ∈ H2m−1

B (Σ) (in this proof, weuse the notation | | for the cohomology class of a cocycle). The equality l(Q,Q′) =〈a c′, ν〉 is equivalent to

(37.10) |a c′| = l(Q,Q′) [Σ]♯B ,


an equality holding in H2m−1B (Σ), where [Σ]♯B is the generator of H2m−1

B (Σ) ≈H2m−1(Σ) = Z2.

Let B1 = ρ−1(B). The inclusion i : Σ → Mf induces a morphism of cochaincomplexes C∗i : C∗B1

(Mf ) → C∗B(Σ) whose kernel is denoted by C∗B1(Mf ,Σ). Note

that C∗B1(Mf , ∅) = C∗B1

(Mf ) and so the inclusion C∗B1(Mf ,Σ) → C∗B1

(Mf) coincideswith C∗j, the morphism induced by the pair inclusion j : (Mf , ∅) → (Mf ,Σ) (seeRemark 12.41). One has the commutative diagram

(37.11)

0 // C∗B1(Mf ,Σ)

C∗j // C∗B1(Mf )

C∗i //OO

C∗ρ

C∗B(Σ) // 0

C∗B0(Sm)

C∗f

99ssssssss

where the top row is an exact sequence of cochain complexes. This sequence givesrise to a connecting homomorphism δ sitting in the exact sequence

H2m−1B1

(Mf )H∗i // H2m−1

B (Σ)δ // H2m

B1(Mf ,Σ)

H∗j // H2mB1

(Mf ) .

As m > 1, one has H2m−1B1

(Mf ) ≈ H2m−1(Mf ) = 0 and H2mB1

(Mf ) ≈ H2m(Mf ) =

0. Therefore δ : H2m−1B (Σ) → H2m

B1(Mf ,Σ) is an isomorphism. Let b = δ([Σ]♯B)

be the generator of H2mB1

(Mf ,Σ). By (37.10), the linking number l(Q,Q′) is thendetermined by the equation

(37.12) δ(|a c′|) = l(Q,Q′) b .

To compute δ(|a c′|), write δM for the coboundary operator in C∗B1(Mf ).

Let c1 = C∗ρ(c0) and c′1 = C∗ρ(c′0), both in ZmB1(Mf ). Let a1 ∈ Cm−1(Mf) such

that C∗i(a1) = a. By the commutativity of diagram (37.11), one has C∗i(a1 c′1) = a c′. Then δM (a1 c′1) is a cocycle in kerC∗i, so there is a uniqueu ∈ Z2m

B1(Mf ,Σ) such that C∗j(u) = δM (a1 c′1). As in Lemma 7.2, one has

(37.13) |u| = δ(|a c′|) .The cohomology class |u| may be described in another way. As for (37.9), one hasc1 c′1 = 0 for support reasons. Therefore,

(37.14) (δM (a1) + c1) c′1 = δM (a1) c′1 = δM (a1 c′1) .

Now, C∗i(δM (a1)+ c1) = 0, thus there is a unique w ∈ ZmB1(Mf ,Σ) with C∗j(w) =

δM (a1)+ c1. The first cup product of (37.14) may be understood as relative cochaincup product giving rise to a relative cohomology cup product

H∗B1(Mf ,Σ)×H∗B1

(Mf )−→ H∗B1

(Mf ,Σ)

analogous to that of Lemma 22.19 (in the case Y2 = ∅). Equation (37.13) is equiv-alent to

(37.15) |w| |c′1| = δ(|a c′|)and, using (37.13), we get the equality

(37.16) |w| |c′1| = l(Q,Q′) b

holding in H2mB1

(Mf ,Σ).

Now, under the isomorphism Hm(Sm)≈−→ Hm

B0(Sm) (due to the small sim-

plex theorem 12.56), the cohomology class |c0| corresponds to the Poincare dual


PD(y) ∈ Hm(Sm), which is [Sm]♯. Anagously, |c′0| also corresponds to [Sm]♯.

Hence, under the isomorphism Hm(Mf )≈−→ Hm

B1(Mf ), |c1|, |c′1| and H∗j(|w|) all

correspond to H∗ρ([Sm]♯), that is the generator e of Hm(Mf ) ≈ Z2. The followingdiagram

(37.17)

HmB1

(Mf )×HmB1

(Mf ,Σ) //OO≈

H2mB1

(Mf ,Σ)OO≈

Hm(Mf )×Hm(Mf ,Σ) //

OO≈

H2m(Mf ,Σ)OO≈

Hm(Mf )×Hm(Mf , CΣ) //

≈

H2m(Mf , CΣ)

≈

Hm(Mf )×Hm(Mf ) // H2m(Mf )

is commutative, where the vertical arrows are the obvious ones or induced by theinclusions (the commutativity of the bottom square is the content of the singular

analogue of Lemma 22.11). Let k : Mf → Mf denote the inclusion. Then H∗k (e) =

e, the generator of Hm(Mf ) ≈ Z2. Equation (37.16), obtained using the top lineof (37.17), becomes, using the bottom line

(37.18) e e = l(Q,Q′) b

where b is the generator of H2m(Mf ) ≈ Z2. But, as m > 1, the equality e e =

Hopf (f) b holds true, by definition of the Hopf invariant. Therefore, Hopf (f) =l(Q,Q′).

Remark 37.19. For a map f : S1 → S1 with even degree, the equality Hopf (f) =l(Q,Q′) holds true, using Proposition 37.1 (see Exercise 5.9). When deg f is odd,the linking number l(Q,Q′) is not defined. Indeed, both Q and Q′ have an oddnumber of points and condition (3) of p. 194 is not satisfied.


5.1. What is the Lusternik-Schnirelmann category of KPn for K = R,C and H.

5.2. Compute the cohomology ring of CPn and HPn, using the Gysin exact se-quence for the Hopf bundles. [Hint: like in § 24.4.]

5.3. Compute the cohomology algebra of S4 × S4 and of HP 2 ♯HP 2. Are theyGrA-isomorphic?

5.4. For any positive integer n, construct a vector bundle ξ of rank n over a closedn-dimensional manifold such that e(ξ) 6= 0.

5.5. Let X be a CW-complex of dimension n = 2k with n = 1, 2, 4, 8 and leta ∈ Hn(X). Prove that there exists a vector bundle ξ over X with e(ξ) = a.

5.6. Prove that there is no continuous injective map f : Rn → Rk if n > k. [Hint:use the Borsuk-Ulam theorem.]

5.7. Check the table of p. 210.


5.8. Show that the Hopf vector bundle over KP 1 ≈ Sd (d = dimR K) cannot be thenormal bundle of an embedding of Sd into a Z2-homology sphere of dimension 2d.

5.9. Let f : S1 → S1 be a smooth map with even degree. Show that the Hopfinvariant of f is equal to the linking number of the inverse image of two regularvalues of f , as in Proposition 37.8. [Hint: use Proposition 37.1.]

5.10. Using the linking numbers and Proposition 37.8, show that the various Hopfmaps have Hopf invariant 1. [Hint: use Formula (35.9) and Exercise 4.18.]

5.11. Let g : S2m+1 → Sm be a continuous map, as well as f : Sm → Sm andh : S2m+1 → S2m+1. Prove that Hopf (f gh) = deg(h) deg(f)2 Hopf (g). (Re-mark: of course, deg(f)2 ≡ deg(f) mod 2 but the formula is the one which is validfor the cohomology with any coefficients.)

CHAPTER 6

Equivariant cohomology

Equivariant cohomology being a huge subject, our aim here is mostly to developsome material needed in the forthcoming chapters. For instance, the definition andmost properties of the Steenrod squares use equivariant cohomology for spaceswith involution. This case is treated in detail in § 39, at an elementary level andwith ad hoc techniques. A second section deals with Γ-spaces for any topologicalgroup Γ (the proof of the Adem relations requires Γ-equivariant cohomology withΓ the symmetric group Sym4). Equivariant cross products, done in § 42, will bealso used. Only § 41 is written uniquely for its own interest, devoted to somesimple form of localization theorems and Smith theory. A final section presents theequivariant Morse-Bott theory, used in § 55 to compute the cohomology of the flagmanifolds (see also § 62.5). For further readings on equivariant cohomology, seee.g. [35, 8, 101]

39. Spaces with involution

An involution on a topological space X is a continuous map τ : X → X suchthat τ τ = id. The letter τ is usually used for all encountered involutions. Wealso use the symbol τ for the non-trivial element of the cyclic group G = id, τ oforder 2; thus an involution on X is equivalent to a continuous action of G on Xand a space with involution is equivalent to a G-space, i.e. a space together withan action of G. We often pass from one language to the other. If X is a G-space,its fixed point subspace XG is defined by

XG = x ∈ X | τ(x) = x .As G has only two elements, the complement of XG is the subspace where theaction is free.

A continuous map f : X → Y between G-spaces is a G-equivariant map, orjust a G-map if it commutes with the involutions: f τ = τ f . Two G-mapsf0, f1 : X → Y are G-homotopic if there is a homotopy F : X × I → Y connectingthem which is a G-map. Here, the involution on X × I is τ(x, t) = (τ(x), t). Thispermits us to define the notion of G-homotopy equivalence and of G-homotopy type.For instance, a G-space is G-contractible if it has the G-homotopy type of a point.

Let X be a G-space. A CW-structure on X with set of n-cells Λn is a G-CW-structure if the following condition is satisfied: for each integer n, there is aG-action on Λn and a G-equivariant global characteristic map ϕn : Λn ×Dn → X,where the G-action on Λn × Dn is given by τ(λ, x) = (τ(λ), x). In particular, ifλ ∈ Λn satisfies τ(λ) = λ, then τ restricted to λ×Dn is the identity. These cellsare called the isotropic cells; they form a G-CW-structure for XG. The other cells,the free cells, come in pairs (λ, τ(λ)). A G-space endowed with a G-CW-structureis a G-CW-complex, or just a G-complex. Observe that, if X is a G-complex,

217

218 6. EQUIVARIANT COHOMOLOGY

then the quotient space X/G inherits a CW -structure (with set of n-cells equalto Λn/G) for which the quotient map is cellular. A smooth G-manifold admits aG-CW-structure, in fact a G-triangulation (see [104]).

Example 39.1. Let X = Sn (n ≤ ∞) be the standard sphere endowed with theCW-structure where the m-skeleton is Sm and having twom-cells in each dimensionm ≤ n (see Example 15.5). This is a G-CW-structure for the free involution givenby the antipodal map z 7→ −z. The quotient space X/G is RPn with its standardCW-structure.

Let X be a space with an involution τ . The Borel construction XG, also knownas the homotopy quotient, is the quotient space

(39.2) XG = S∞ ×G X = (S∞ ×X)/∼

where∼ is the equivalence relation (z, τ(x)) ∼ (−z, x). IfX and Y areG-spaces andif f : Y → X is a continuous G-equivariant map, the map id×f : S∞×Y → S∞×Xdescends to a map fG : YG → XG. This makes the Borel construction a covariantfunctor from the category TopG to Top, where TopG is the category of G-spacesand G-equivariant maps. Using the obvious homeomorphism between (X × I)Gand XG× I, a G-homotopy between two G-maps f0 and f1 : X → Y descends to ahomotopy between f0

G and f1G. Hence, XG and YG have the same homotopy type

if X and Y have the same G-homotopy type.Let p : S∞ → RP∞ be the quotient map (this is a 2-fold covering projection).

A map p : XG → RP∞ is then given by p(z, x) = p(z). Observe that p coincideswith the map fG : XG → ptG = RP∞ induced by the constant map X → pt.

Example 39.3. Suppose that the involution τ is trivial, i.e. τ(x) = x for allx ∈ X . The projection S∞ ×X → X then descends to XG → X . Together with

the map p, this gives a homeomorphism XG≈−→ RP∞ ×X .

Lemma 39.4. (1) The map p : XG → RP∞ is a locally trivial fiber bundlewith fiber homeomorphic to X.

(2) If f : Y → X is a G-equivariant map, then the following diagram

YGp

##FFF

FFFF

fG // XG

p

wwwww

ww

RP∞

is commutative.(3) If τ has a fixed point, then p admits a section. More precisely, the choice

of a point v ∈ XG provides a section sv : RP∞ → XG of p.(4) The quotient map S∞ × X → XG is a 2-fold covering admitting p as a

characteristic map.

Proof. We use that p : S∞ → RP∞ is a principal G-bundle, i.e. a 2-foldcovering. Denote by z = (z0, z1, . . . ) the elements of S∞. The set Vi = z ∈ S∞ |zi 6= 0 is an open set of S∞. As p is an open map, Ui = p(Vi) is an open setof RP∞. A trivialization ψi : Vi → Ui × ±1 is given by ψi(z) = (p(z), zi/|zi|).Using the group isomorphism ±1 ≈−→ G, this gives a trivialization ψi : Vi →Ui × G. Now, ψi × id : Vi × X

≈−→ Ui × G × X descends to a homeomorphism

39. SPACES WITH INVOLUTION 219

p−1(Ui)≈−→ Ui × (G ×G X). Here, G ×G X denotes the quotient of G × X by

the equivalence relation (g, τ(x)) ∼ (gτ, x). But the map x 7→ (id, x) provides ahomeomorphism from X onto G×GX . This proves Point (1). This also shows that,over p−1(Ui), the map S∞ × X → XG looks like the projection G × p−1(Ui) →p−1(Ui). Therefore, S∞×X → XG is a 2-fold covering, with the product involutionτ×(z, x) = (−z, τ(x)) as deck transformation. The following diagram

S∞ ×X

projS∞ // S∞

XG

p // RP∞

is commutative and projS∞(τ×(y)) = −projS∞(y). By Lemma 24.8, this impliesthat p is a characteristic map for the covering S∞ × X → XG. Point (4) is thusestablished.

Point (2) is obvious from the definitions. For Point (3), let v ∈ XG. ByPoint (2), the inclusion i : v → X gives rise to a commutative diagram

vG≈

$$HHHHHHH

iG // XG

p

wwwwwww

RP∞

.

Hence, iG provides a section sv : RP∞ → XG of p, depending on the choice of thefixed point v.

The projection q : S∞ × X → X seen in Example 39.3 descends to q : XG →X/G.

Lemma 39.5. Let X be a free G-space such that X is Hausdorff. Then, theGrA-morphism H∗q : H∗(X/G)→ H∗(XG) is an isomorphism. Moreover, if X isa free G-complex, then the map q : XG → X/G is a homotopy equivalence and themap p : XG → RP∞ is homotopic to the composition of q with a characteristic mapfor the covering X → X/G.

Proof. If X is Hausdorff, such a projection X → X/G is a 2-fold coveringand X/G is Hausdorff. Over a trivializing open set U ⊂ X/G, this covering isequivalent to G× U → U . Then (G× U)G ≈ S∞ × U since any class has a unique

representative of the form (z, id, u) ∈ S∞ ×G× U . Hence, XGq−→ X/G is a locally

trivial bundle with fiber S∞. As H∗(S∞) = 0, the map q∗ : H∗(X/G)→ H∗(XG)is a GrA-isomorphism by Corollary 28.34. Actually. as S∞ is contractible [80,example 1.B.3 p. 88], the homotopy exact sequence of the bundle [80, Thoerem 4.41and Proposition 4.48] implies that q is a weak homotopy equivalence. If X is a G-complex, then X/G is a CW -complex. Also, S∞ × X is a free G-complex andthus XG is a CW-complex. Therefore, a weak homotopy equivalence is a homotopyequivalence by the Whitehead theorem [80, Theorem 4.5]. Also, again since S∞

is contractible and X/G is a CW-complex, a direct proof that q is a homotopyequivalence is available using [40, Theorem 6.3].


Let f : X/G→ RP∞ be a characteristic map for the covering X → X/G. Thefollowing diagram

S∞ ×Xcovering

// X

covering

f // S∞

XG

q // X/Gf // RP∞

is commutative and the upper horizontal arrows commute with the deck involutions.By Lemma 24.8, this implies that f q is a characteristic map for the coveringS∞×X → XG. By Lemma 39.4, so is p. By Corollary 24.7, two characteristic mapsof a covering are homotopic. Therefore, the maps p and f q are homotopic.

Corollary 39.6. Let X be a finite dimensional G-complex. Then, the follow-ing conditions are equivalent.

(1) X has a fixed point.(2) The morphism H∗p : H∗(RP∞)→ H∗(XG) is injective.

Proof. If X has a fixed point, then p admits a section by Lemma 39.4, so H∗pis injective. IfX has no fixed point, thenX is a freeG-complex and, by Lemma 39.5,H∗(XG) ≈ H∗(X/G). Also, X/G is a finite dimensional CW-complex, so H∗p isnot injective.

Let X be a space with an involution τ and let Y ⊂ X be an invariant sub-space. Then YG ⊂ XG. The (relative) G-equivariant cohomology H∗G(X,Y ) is thecohomology algebra

H∗G(X,Y ) = H∗(XG, YG) .

We shall mostly concentrate on the absolute case H∗G(X) = H∗(XG) = H∗G(X, ∅).The map p : XG → RP∞induces a GrA-homomorphism p∗ : H∗(RP∞)→ HG(X).By Proposition 35.1, H∗(RP∞) is GrA-isomorphic to the polynomial ring Z2[u],where u is a formal variable in degree 1. Hence, the GrA-homomorphism p∗ giveson H∗G(X) a structure of Z2[u]-algebra. In particular, H∗G(pt) = Z2[u].

As an important example, let us consider the case of a G-space Y with Y = Y G,i.e. the involution τ is trivial. As seen in Example 39.3, we get an identificationYG = RP∞ × Y . By the Kunneth theorem,

(39.7) H∗G(Y ) ≈ Z2[u]⊗H∗(Y ) ≈ H∗(Y )[u] .

The GrA-homomorphismH∗(Y )→ H∗G(Y ) induced by the projection RP∞×Y →Y corresponds to the inclusion of the “ring of constants” H∗(Y ) into H∗(Y )[u].

The functoriality of the Borel construction and of the cohomology algebra, to-gether with Point (2) of Lemma 39.4, says that, if f : Y → X is a G-equivariantmap between G-spaces, then H∗fG : H∗G(X) → H∗G(Y ) is a GrA-homomorphismcommuting with the multiplication by u. We are then driven to consider the cat-egory GrA[u] whose objects are graded Z2[u]-algebras and whose morphisms areGrA-homomorphism commuting with the multiplication by u. Hence, the cor-respondence X 7→ H∗G(X) is a contravariant functor from TopG to GrA[u]. Iff : Y → Y is a G-equivariant map between trivial G-spaces (i.e., any continuous


map), the following diagram is commutative:

(39.8)

H∗(Y )[u]

H∗f [u]

≈ // H∗G(Y )

H∗Gf

H∗(Y )[u]≈ // H∗G(Y ) .

Choosing a point z ∈ S∞ provides, for each G-space X , a map iz : X → XG

defined by iz(x) = [z, x]. As S∞ is path-connected, the homotopy class of iz doesnot depend on z. Therefore, we get a well defined GrA-homomorphism

(39.9) ρ : H∗G(X)→ H∗(X)

given by ρ = H∗iz for some z ∈ S∞. We can call ρ the forgetful homomorphism(it frogets the G-action). Observe that ρ is functorial. Indeed, if f : X → X be aG-equivariant map, the following diagram

X

f

iz // XG

fG

Xiz // XG

is commutative, and so is the following diagram

(39.10)

H∗(X)

H∗f

H∗G(X)ρXoo

H∗Gf

H∗(X) H∗G(X)ρXoo .

If Y is a G-space with Y = Y G, we get, using the GrA[u]-isomorphismof (39.7), the commutative diagram

(39.11)

H∗(Y )[u]

ev0 %%LLLLLLLL

≈ // H∗G(Y )

ρzzttttt

ttt

H∗(Y )

where ev0 is the evaluation of a polynomial at u = 0, i.e. the unique algebrahomomorphism extending the identity on H∗(Y ) and sending u to 0.

Observe that ρ = H∗ρ where ρ : X → XG is the composition

ρ : Xslice−−−→ X × S∞ → XG .

As S∞ is contractible [80, example 1.B.3 p. 88], the slice inclusion is a homotopyequivalence. By Lemma 39.4, the map X × S∞ → XG is then a a 2-fold coveringwith characteristic map p : XG → RP∞. Therefore, the transfer exact sequenceof the covering and its naturality (see Proposition 24.17) gives the following exactsequence.(39.12)

· · · → Hm−1G (X)

−·u−−→ HmG (X)

ρ−→ Hm(X)tr∗−−→ Hm

G (X)−·u−−→ Hm+1

G (X)→ · · ·


Denote by (u) the ideal of H∗G(X) generated by u and by

Ann (u) = x ∈ H∗G(X) | ux = 0the annihilator of u. The information carried by exact Sequence (39.12) may beconcentrated in the following short exact sequence of graded Z2[u]-modules.

(39.13) 0→ H∗G(X)/(u)ρ−→ H∗(X)

tr∗−−→ Ann (u)→ 0 .

A G-space X is called equivariantly formal if ρ : H∗G(X)→ H∗(X) is surjective.For instance, X is equivariantly formal if the G-action is trivial. See 40.15 for adiscussion of this definition in a more general setting.

Proposition 39.14. For a G-space X, the following conditions are equivalent.

(1) X is equivariantly formal.(2) H∗G(X) is a free Z2[u]-module.(3) Ann (u) = 0.

Proof. That (2) ⇒ (3) is obvious and (3) ⇔ (1) follows from (39.13). Now,if ρ is surjective, chose a GrV-section θ : H∗(X) → H∗G(X) of ρ. Then θ is a

cohomology extension of the fiber for the fiber bundle X → XGp−→ RP∞. As

RP∞ is path-connected and of finite cohomology type, the Leray-Hirsch theoremII (Theorem 28.30) implies that H∗G(X) is a free Z2[u]-module generated by θ(B),where B is a Z2-basis of H∗(X).

Remark 39.15. As noted before, H∗(X) ⊗ Z2[u] is isomorphic, as a Z2[u]-algebra, to H∗(X)[u]. If X is equivariantly formal, the Leray-Hirsch theorem II(Theorem 28.30) thus provides an isomorphism of Z2[u]-modules between H∗G(X)andH∗(X)[u]. This isomorphism depends on the choice of a GrV-section θ : H∗(X)→H∗G(X) of ρ : H∗G(X) → H∗(X) and is not, in general, an isomorphism of algebra.However, as in the case of a trivial G-action, Diagram (39.11) is commutative.

For X a G-space, let r : H∗G(X) → H∗G(XG) and r : H∗(X) → H∗(XG) bethe GrA[u]-momorphisms induced by the inclusion XG → X . One can composetr∗ : H∗(X)→ H∗G(X) with r.

Proposition 39.16. rtr∗ = 0.

Proof. As G acts trivially on XG, one has H∗G(XG) ≈ H∗(RP∞)⊗H∗(XG).The following commutative diagram

H∗(S∞ ×X)

tr∗

r // H∗(S∞ ×XG)

tr∗

H∗(S∞)⊗H∗(XG)

tr∗⊗id

≈oo

H∗(S∞ ×G X)r // H∗(RP∞ ×XG) H∗(RP∞)⊗H∗(XG)

≈oo

proves the proposition since tr∗ : H∗(S∞) → H∗(RP∞) is the zero map (even indegree 0, by the transfer exact sequence).

To say more about the image of ρ : H∗G(X)→ H∗(X), define

H∗(X)G = a ∈ H∗(X) | H∗τ(a) = a ⊂ H∗(X) .

As H∗τ is a GrA-morphism, H∗(X)G is a Z2-graded subalgebra of H∗(X).

Lemma 39.17. ρ(H∗G(X)) ⊂ H∗(X)G.


Proof. Let z ∈ S∞ and b ∈ H∗G(X). Then,

H∗τ ρ(b) = H∗τ H∗iz(B) = H∗(iz τ)(b) = H∗i−z(b) = ρ(b) .

The reverse inclusion in Lemma 39.17 is wrong, as shown by the followingexample.

Example 39.18. Let τ be the antipodal map on the sphere X = Sn, makingX a free G-complex, as seen in Example 39.1. The equality Hn(X) = Hn(X)G

holds true since Hn(X) = Z2. For any z ∈ S∞, the composition Xiz−→ XG

q−→ X/Gcoincides with the quotient map X →→ X/G = RPn. By Lemma 39.5, the mapq is a homotopy equivalence. But, since Sn →→ RPn is of local degree 2, thehomomorphism Hn(RPn) → Hn(Sn) vanishes. Thus ρ = 0. In this example, thenon-existence of fixed point is important (see Proposition 39.21 below).

LetX be aG-space. The reduced equivariant cohomology H∗G(X) is the GrA[u]-algebra defined by

(39.19) H∗G(X) = coker(H∗pG : H∗G(pt)→ H∗G(X)

)

where pG : X → pt denotes the constant map to a point (which is G-equivariant).

Warning: H∗G(X) 6= H∗(XG). Here below some examples.

(1) H∗G(pt) = 0.(2) Let X = Sn with the antipodal involution. By Lemma 39.5, XG has

the homotopy type of RPn and H∗p : H∗(RP∞)→ H∗G(X) is surjective.

Therefore, H∗G(X) = 0.(3) If Y is a space with trivialG-action, one has a natural GrA[u]-isomorphism

(39.20) H∗G(Y ) = H∗(Y )[u]/Z2[u] ≈ H∗(Y )[u]

(4) IfX is equivariantly formal, we get, as in Remark 39.15, an isomorphism of

Z2[u]-modules between H∗G(X) and H∗(X)[u]. This isomorphism dependson the choice of a section of ρ : H∗G(X) → H∗(X) and is not, in general,an isomorphism of algebra.

Any G-equivariant map f : Y → X satisfies pf = p, so H∗G is a contravariantfunctor from TopG to GrA[u]. One checks that the homomorphisms ρ : H∗G(X)→H∗(X)G and tr∗ : H∗(X) → H∗G(X) descend to ρ : H∗G(X) → H∗(X)G and to

tr∗: H∗(X)→ H∗G(X).The equivariant reduced cohomology will be further developed in a more general

setting (see 40.17 in the next section). Here, we shall only prove the followingproposition, which plays an important role in the construction of the Steenrodsquares in Chapter 7. Note that H∗(X)G contains the classes a + τ∗(a) for alla ∈ H∗(X).

Proposition 39.21. Let X be a G-space with XG 6= ∅. Suppose that Hj(X) =

0 for 0 ≤ j < r. Then, ρ : HrG(X) → Hr(X)G is an isomorphism. Moreover,

ρ−1(a+ τ∗(a)) = tr∗(a) for all a ∈ Hr(X).

Proof. The last assertion follows from the main one since ρ tr∗(a) = a+τ∗(a)

by definition of the transfer. We shall prove that the following sequence

(39.22) 0→ HrG(pt)

H∗p−−−→ HrG(X)

ρ−→ Hr(X)G → 0


is exact. This will prove the main assertion.For 0 ≤ k ≤ ∞, let

Zk = Sk ×G X .

Thus, XG = Z∞ and, as in Lemma 39.4, there is a natural locally trivial bundlep : Zk → RP k with fiber X .

Choosing a point z ∈ S1 provides a map iz : X → Z1 ⊂ Zk, defined by iz(x) =[z, x], which induces a GrA-homomorphism ρ : H∗(Zk)→ H∗(X) (independent ofz) given by ρ = H∗iz. As in Lemma 39.17, one proves that ρ(H∗(Zk)) ⊂ H∗(X)G.We shall prove, by induction on k, that the following sequence

(39.23) 0→ Hr(RP k)H∗p−−−→ Hr(Zk)

ρ−→ Hr(X)G → 0

is exact for each k ≥ 1. Since any compact subset of XG = Y∞ is contained in Zkfor some k, the exactness of (39.22) will follow, using Corollary 12.22.

Observe that, in Sequence (39.23), the homomorphism H∗p is injective sincethe choice of a G-fixed point in X provides a section of p. It is also clear thatρH∗p = 0.

We start with k = 1. The space Z1 is the mapping torus of τ . The mappingtorus exact sequence of Proposition 28.71 is of the form

· · · → Hr−1(X)Θ−→ Hr−1(X)

J−→ Hr(Z1)ρ−→ Hr(X)

Θ−→ Hr(X)→ · · · ,

where Θ = id + H∗τ . Hence, ker(Hr(X)Θ−→ Hr(X)) = Hr(X)G. If r ≥ 2, then

H1(X) = 0, which proves that ρ : Hr(Z1)→ Hr(X)G is an isomorphism and thusSequence (39.23) is exact (for k = 1). If r = 1, then Θ: H0(X)→ H0(X) is the null-

homomorphism, since X is path-connected. Then, ker(H1(Z1)ρ−→ H1(X)G) ≈ Z2

which implies that Sequence (39.23) is also exact when k = r = 1.Take, as induction hypothesis, that Sequence (39.23) is exact for k = ℓ−1 ≥ 1.

We have to prove that it is exact for k = ℓ. The space Zℓ is obtained from Zℓ−1 bygluing Dℓ ×X using the projection Sℓ−1 ×X →→ Zℓ−1. Let e be the generator ofHℓ(Dℓ, Sℓ−1) = Z2. Using excision and the relative Kunneth theorem, we get thefollowing commutative diagram

H∗(RP ℓ,RP ℓ−1)

≈// H∗(Dℓ, Sℓ−1)

H∗−ℓ(pt)≈

e×−oo

H∗(Zℓ, Zℓ−1) ≈

// H∗(Dℓ ×X,Sℓ−1 ×X) H∗−ℓ(X)≈

e×−oo

This diagram, together with the cohomology exact sequences for the pairs (RP ℓ,RP ℓ−1)and (Zℓ, Zℓ−1) gives the commutative diagram:

(39.24)

Hr−ℓ(pt)

// Hr(RP ℓ)H∗p

// Hr(RP ℓ−1)H∗p

// Hr+1−ℓ(pt)

Hr−ℓ(X) // Hr(Zℓ)

ρ

// Hr(Zℓ−1)

ρ

// Hr+1−ℓ(X)

Hr(X)G= // Hr(X)G

.


where the two long lines are exact. The induction step follows by comparing thetwo middle columns. The argument divides into four cases.Case 1: ℓ < r. As ℓ ≥ 2, one has 0 < r − ℓ ≥ r − 2. By hypothesis, Hj(X) = 0 for1 ≤ j < r. Therefore the left and right columns vanish and the two middle columnsare isomorphic.Case 2: ℓ = r. Since ℓ ≥ 2, one hasH1(X) = 0 and the right column vanishes. Also,

Hr(RP ℓ−1) = 0 and the left vertical arrow is an isomorphism (since H0(X) = 0).The induction step follows. Observe that H0(X)→ Hr(Zℓ) is injective.Case 3: ℓ = r+ 1. The left column vanishes. By step 2, Diagram (39.24) continueson the right by injections

H0(pt)

≈

// // Hr+1(RP ℓ)H∗p

H0(X) // // Hr+1(Zℓ)

.

Hence, the two middle columns are isomorphic.Case 4: ℓ > r + 1. The left and right columns vanish for dimensional reasons, sothe two middle columns are isomorphic.

Remark 39.25. The Serre spectral sequence for the bundle X → XG → RP∞

provides a shorter proof of the exactness of sequences (39.22) and (39.23). Thiswill be used to prove the more general Proposition 40.28 in the next section.

Example 39.26. Linear involution on spheres. Let Sn be the standard sphereequipped with an involution τ ∈ O(n+ 1). In Rn+1, the equality

x =x+ τ(x)

2+x− τ(x)

2

gives the decomposition Rn+1 = V+ ⊕ V− with V± being the eigenspace for theeigenvalue ±1. As τ is an isometry, the vector spaces V+ and V− are orthogonal.Therefore, two elements τ, τ ′ ∈ O(n + 1) of order 2 are conjugate in O(n + 1) if

and only if dim(Sn)τ = dim(Sn)τ′

. We write Snp , (−1 ≤ p ≤ n) for the sphereSn equipped with an involution τ ∈ O(n + 1) such that dim(Sn)τ = p. Hence,(Snp )τ ≈ Sp. Being a smooth G-space, Snp admits a G-CW-structure [104].

The involution on Sn−1 is just the antipodal map and, by Lemma 39.5, (Sn−1)G ≈RPn. For p ≥ 0, the inclusion Sp = (Snp )G → Snp gives rise to GrA[u]-morphisms

r : H∗G(Snp )→ H∗G((Snp )G) = H∗G(Sp) ≈ H∗(Sp)[u]and

r : H∗G(Snp )→ H∗G((Snp )G) = H∗G(Sp) ≈ H∗(Sp)[u] .If n ≥ 1, then Hj(Snp ) = 0 for 0 ≤ j < n and Proposition 39.21 asserts that

ρ : HnG(Snp ) → Hn(Snp )G = Hn(Snp ) is an isomorphism (this is also true if n = p =

0). Let a ∈ Hn(Sn) and b ∈ Hp(Sp) be the generators.

Proposition 39.27. When p ≥ 0 the GrA[u]-morphisms r and r are injective.Moreover, r ρ−1(a) = b un−p.


Proof. The proposition is trivial if n = p, so we can suppose that n > p ≥ 0.Using the following commutative diagram

0 // H∗G(pt)

=

// H∗G(Snp )

r

// H∗G(Snp )

r

// 0

0 // H∗G(pt) // H∗G(Sp) // H∗G(Sp) // 0

,

the five-lemma technique show that r is injective if and only if r is injective. Thuswe shall prove that r is injective.

We first prove that r is injective when p = 0. One can see Sn0 as the suspensionof Sn−1, with (Sn0 )G = ω+, ω− ≈ S0. Then, X = Sn0 is the union of the G-equivariantly contractible open sets X+ = Sn0 − ω− and X− = Sn0 − ω+, withintersection X0 having the G-equivariant homotopy type of Sn−1. Hence, X±G hasthe homotopy type of ω±G and

H∗G(S0) ≈ HkG(ω−)⊕Hk

G(ω+) ≈ HkG(X−)⊕Hk

G(X+) .

By Lemma 39.5, X0G has the homotopy type of RPn−1. By Proposition 12.82, the

Mayer-Vietoris data (XG, X+G , X

−G , X

0G) gives rise to the long exact sequence

(39.28)

Hk−1(RPn)δMV // Hk

G(Sn0 )r // H∗G(S0)

J // Hk(RPn)δMV // · · ·

with J = H∗Gj+ + H∗Gj

−, where j± : X0 → X± denotes the inclusion. The mapj± is G-homotopy equivalent to the constant map Sn−1 7→ ω±. As noted be-fore Example 39.3, the induced map H∗G(ω±) → H∗G(X0) is the GrA-morphismH∗p : H∗(RP∞) → H∗(RPn−1). By Lemma 39.5, p is the characteristic map forthe covering Sn−1

−1 → (Sn−1−1 )G = RPn−1. As noticed in Example 24.9, this map

is just the inclusion RPn−1 → RP∞ and, then, H∗Gj± is surjective by Proposi-

tion 24.21. Hence, J is surjective and the exact sequence (39.28) splits. This provesthat r : H∗G(Sn0 )→ H∗G(S0) is injective. For a more precise analysis of H∗G(Sn0 ) (seeexamples 39.30 or 44.27).

Suppose, by induction on p ≥ 1, that r : HiG(Smp−1) → Hi

G(Sp−1) is injective

for all i ∈ N and all m ≥ p − 1. We have to prove that r : HkG(Snp ) → Hk

G(Sp)is injective for all k ∈ N and all n ≥ p. As (Snp )G and (Sp)G are path-connected,the required assertion is true for k = 0 by Lemma 12.15. Thus, we may supposethat k ≥ 1. As n ≥ p ≥ 1, the G-sphere Sn−1

p−1 exists and Snp is the suspension

of Sn−1p−1 , with (Snp )G being the suspension of (Sn−1

p−1 )G. Let Y = Y G = ω−, ω+be the suspension points. As above, we decompose the X = Snp = X− ∪X+ with

X− ∩X+ = X − Y . The maps r sit in a commutative diagram

(39.29)

Hk−1G (Y )

≈ r

// Hk−1G (Sn−1

p−1 )δMV //

r

HkG(Snp )

r

// HkG(Y )

≈ r

Hk−1G (Y G) // Hk−1

G (Sp−1)δMV // Hk

G(Sp) // HkG(Y G)


where the horizontal line are the Mayer-Vietoris sequences for the data(X,X+, X−, X − Y ) and (XG, (X+)G, (X−)G, (X − Y )G). Hence, r : Hk

G(Snp ) →HkG(Sp) is injective by the proof of the five lemma (see [80, p. 129]).

The last assertion is now obvious, since r ρ−1 is injective and, as H∗G(Sp) ≈H∗(Sp)[u], one has Hn

G(Sp) = Z2un−p.

Example 39.30. We use the notations of the proof of Proposition 39.27 in thecase p = 0, with (Sn0 )G = S0 = ω±. The isomorphism σ− : Hn(Sn0 ) → Hn

G(Sn0 )defined by the following commutative diagram

Hn(Sn0 )ρ−1

≈// Hn

G(Sn0 ) // r // HnG(S0)

HnG(Sn0 , ω−)

≈

OO

// r //

HnG(S0, ω−)

≈

OO

Hn(Sn0 )σ− //

≈

OO

HnG(Sn0 ) // r // Hn

G(S0) oo ≈ // HnG(ω−)⊕Hn

G(ω+)

is an extension of the fiber for the bundle Sn0 → (Sn0 )G → RP∞. Another one,σ+, is obtained using ω+ (there are two of them by the exact sequence (39.22) andσ+(a) = σ−(a) + un). Then, rσ−(a) = (un, 0) and rσ+(a) = (0, un). Hence,neither σ− nor σ+ is multiplicative. We see the relation rσ±(a)2 = un rσ±(a).Hence, as r is a monomorphism of Z2[u]-module, the relation σ±(a)2 = un σ±(a)holds in H∗G(Sn0 ). By the Leray-Hirsch Theorem 28.26, H∗G(Sn0 ) is a free Z2[u]-module generated by A = σ+(a) (or, by B = σ−(a)). By dimension counting, wecheck that H∗G(Sn0 ) admits, as a Z2[u]-algebra, the presentation

(39.31) H∗G(Sn0 ) ≈ Z2[u][A]/

(A2 + unA)

As σ−(a)σ+(a) = 0, a more symmetric presentation is obtained using the twogenerators A and B:

H∗G(Sn0 ) ≈ Z2[u][A,B]/I

where I is the ideal generated by

A+B + un , and A2 + unA .

Note that AB and B2 + unB are in I. Indeed, mod I, one has

AB = A(A+ un) = A2 + unA = 0

and

B2 = (A+ un)2 = A2 + u2n = unA+ u2n = un(A+ un) = unB .

Corollary 39.32. If p ≥ 0, then Snp is equivariantly formal. There is a section

σ : Hn(Snp ) → HnG(Snp ) of ρ such that rσ : Hn(Snp ) → Hn

G((Snp )G) → H∗(Sp)[u]satisfies

(39.33) rσ(a) = b un−p .


Proof. By Proposition 39.27 ρ : HnG(Snp )→ Hn(Snp ) is an isomorphism, so the

commutative diagram

HnG(Snp )

ρ //

Hn(Snp )

≈

HnG(Snp )

ρ

≈// Hn(Snp )

shows that ρ is surjective and thus Snp is equivariantly formal. Choose a section σ

of ρ. By Proposition 39.27, Equation (39.33) holds true modulo ker(HnG(Snp ) →

HnG(Snp )

)= Z2u

n. By changing σ(a) by σ(a) + un if necessary, (39.33) will holdtrue strictly.

As another example, we consider CPn as a G-space with the involution τ beingthe complex conjugation. Thus, (CPn)G = RPn. Let 0 6= a ∈ H2(CPn) and0 6= b ∈ H1(RPn).

Proposition 39.34. For n ≤ ∞, CPn is equivariantly formal. Moreover,there is a section σ : H∗(CPn) → H∗G(CPn) which is multiplicative and satisfiesrσ(a) = bu+ b2.

Proof. As Hi(CPn) = 0 for i ≤ 1, Proposition 39.21 implies that ρ : H2G(CPn)→

H2(CPn) is an isomorphism. As in the proof of Corollary 39.32, this implies thatρ : H2

G(CPn) → H2(CPn) is surjective. As H∗(CPn) is generated by a as an al-gebra and ρ is a GrA-morphism, we deduce that ρ : H∗G(CPn) → H∗(CPn) issurjective. Thus CPn is equivariantly formal.

Choose a section σ2 : H2(CPn)→ H2G(CPn) of ρ. AsH∗G(RPn) ≈ H∗(RPn)[u],

there exists λ, µ and ν in Z2 such that

rσ2 = λu2 + µbu+ νb2 .

By changing σ2(a) by σ′2(a) = σ(a) + λu2, we may assume that λ = 0. We mustprove that µ = ν = 1. The inclusions irpn→ CPn and j : CP 1 → CPn provide commutative diagrams

H2G(CPn)

r //

ρ

H2G(RPn)

ρG

H2(CPn)

H∗i // H2(RPn)

H2G(CPn)

ρ //

H∗Gj

H2(CPn)

H∗j≈

σmm

H2G(CP 1)

ρ1 // H2(CP 1)σ1

mm

with r = H∗Gi. By Proposition 35.14, H∗i(a) = b2, so ν = 1. Note that σ1 =H∗Gj σ(H

∗j)−1 is a section of ρ1. As CP 1 with the complex conjugation is G-diffeomorphic to S2

1 (via the stereographic projection of S2 onto C ∪ ∞ ≈ CP 1),Corollary 39.32 implies that r1 σ1(a) = au, which proves that µ = 1.

For n ≤ ∞, we now define σ[n] : H∗(CPn)→ H∗G(CPn) by σ[n](ak) = σ2(a)k.

This is a section of ρ and σ[∞] is clearly multiplicative. As σ[n] is the composi-tion of σ[∞] with the morphism H∗G(RP∞)→ H∗G(RPn) induced by the inclusion,the section σ = σn of ρ is multiplicative and satisfies the requirements of Proposi-tion 39.34.

40. THE GENERAL CASE 229

Corollary 39.35. For n ≤ ∞, let σ : H∗(CPn) → H∗G(CPn) be given byProposition 39.35. Then, the correspondence a 7→ σ(a) provides a GrA[u]-isomorphism

Z2[u, a]/(an+1)

≈−→ H∗G(CPn) .

Corollary 39.36. For n ≤ ∞, the restriction morphism r : HG(CPn) →HG(RPn) is injective.

Proof. Let x ∈ Hm(CPn) with x ∈ ker r. Write x under the form x =σ(ak)ur + ℓtr (k + r = m), where σ is given by Proposition 39.35 and ℓtr denotessome polynomial in the variable u of degree less than r. Then, the equation

(39.37) 0 = r(x) = bkur+k + ℓtr+k

holds in H∗G(RPn) = H∗(RPn)[u]. This first proves that k > 0. Choose x so thatk is minimal. Then, (39.37) again implies that bk = 0. Hence, n < ∞ and k > n.As σ is multiplicative, one has σ(ak) = 0 and x = ℓtr, contradicting the minimalityof k.

The proof of Corollary 39.36 generalizes for conjugation spaces (see Lemma 61.16).For n < ∞, Corollary 39.36 is a consequence of the equivariant formality of CPn

(see Proposition 41.12).

Remark 39.38. As an exercise, the reader may develop the analogous of Propo-sition 39.34 and Corollaries 39.35 and 39.36 for the G-space X = HPn, where Gacts via the H-involutions τ(x+ iy+ jz+ kt) = x+ iy− jz− kt (thus XG ≈ CPn),or τ(x+ iy+ jz+ kt) = x− iy− jz− kt (thus XG ≈ RPn). The same work may bedone with X = OP 2 with various R-linear O-involutions so that XG ≈ HP 2, CP 2

or RP 2.

40. The general case

Let Γ be a topological group. Let p : EΓ → BΓ be the universal principalΓ-bundle constructed by Milnor [141]. The space EΓ is contractible, being thejoin of infinitely many copies of Γ with a convenient topology. An element of EΓis represented by a sequence (tiγi) (i ∈ N) with (ti) ∈ ∆∞ and γi ∈ Γ; two suchsequences (tiγi) and (t′iγ

′i) represent the same class in EΓ if ti = t′i and γi = γ′i

whenever ti 6= 0. There is a free right action of Γ on EΓ given by (tiγi) g =(tiγi g). One defines BΓ = EΓ/Γ. The quotient map p : EΓ → BΓ enjoys localtriviality, in other words is a principal Γ-bundle. These constructions are functorial:a continuous homomorphism α : Γ′ → Γ induces a continuous map Eα : EΓ′ → EΓ,defined by Eα(tiγi) = (tiα(γi)), which descends to a continuous map Bα : BΓ′ →BΓ.

Example 40.1. Consider the case Γ = G = I, τ. Then, EG → BG ishomotopy equivalent to S∞ → RP∞. This is because the join of a space Y withG ≈ S0 is homeomorphic to the suspension of Y . In the same way, ES1 → BS1 ishomotopy equivalent to S∞ → CP∞.

Lemma 40.2. Let Γ be a finite group of odd order. Then H∗(BΓ) ≈ H∗(pt).Proof. By Kronecker duality, it is equivalent to prove that H∗(BΓ) ≈ H∗(pt).

When Γ is a discrete group, the principal Γ-bundle p : EΓ → BΓ is the universalcovering of BΓ. One has the transfer chain map tr∗ : Cm(BΓ) → Cm(EΓ) as in


§ 24.3, sending a singular simplex σ : ∆m → BΓ to the set of its liftings in EΓ. Ifthe number of sheets is odd, the composition

H∗(BΓ)tr−→ H∗(EΓ)

H∗p−−−→ H∗(BΓ)

is the identity. Since EΓ is contractible, this proves the lemma.

When Γ is a discrete group, the cohomology of BΓ is isomorphic to the co-homology H∗(Γ; Z2) of the group Γ in the sense of [25, 2]. The isomorphismH∗(BΓ) ≈ H∗(Γ; Z2) is proven in e.g. [2, § II.2]. The following proposition,proven in [25, Proposition III.8.3] will be useful.

Proposition 40.3. Let Γ be a discrete group. Let α be an inner automorphismof Γ, i.e. α(g) = g0gg

−10 for some g0 ∈ Γ. Then H∗Bα(a) = a for all a ∈ H∗(BΓ).

Let X be a left Γ-space. The Borel construction XΓ or homotopy quotient, isthe quotient space

XΓ = EΓ×Γ X = (EΓ×X)/∼

where ∼ is the equivalence relation (z, γx) ∼ (zγ, x) for all x ∈ X , z ∈ EΓ andγ ∈ Γ. A map p : XΓ → BΓ is then given by p(z, x) = p(z). If Γ = G = I, τ, thisBorel construction coincides with that defined in (39.2).

As in § 39, one proves the following statements.

(1) The Borel construction is a covariant functor from the category TopΓ toTop, where TopΓ is the category of Γ-spaces and Γ-equivariant maps.The map p : XΓ → BΓ coincides with the map XΓ → ptΓ = BΓ inducedby the constant map X → pt.

(2) XΓ and YΓ have the same homotopy type if X and Y have the sameΓ-homotopy type.

(3) The map p : XΓ → BΓ is a locally trivial fiber bundle with fiber homeo-morphic to X .

(4) If f : Y → X is a G-equivariant map, then the following diagram

YΓ

p

!!DDD

DDDD

fΓ // XΓ

p

||zzzz

zzz

BΓ

is commutative.(5) If the Γ action on X has a fixed point, then p admits a section. More

precisely, the choice of a point v ∈ XΓ provides a section sv : BΓ → XΓ

of p.(6) If Γ acts trivially on X , then XΓ has the homotopy type of BΓ×X (see

the proof of Point (3) of Lemma 39.4).(7) The projection EΓ × X → XΓ is a Γ-principal bundle induced from the

universal bundle by p.

Example 40.4. Let i : Γ0 → Γ denote the inclusion of a closed subgroup Γ0 ofΓ. We consider the homogeneous Γ-space X = Γ/Γ0. Then, the map h : EΓ×X →EΓ/Γ0 given by h(z, [γ]) = zγ descends to a homeomorphism h : EΓ ×Γ X

≈−→


EΓ/Γ0. Consider the following commutative diagram

Γ0//

=

EΓ0//

Ei

EΓ0/Γ0oo = //

Ei

BΓ0

Bi

gzzuuuuu

uuu

Γ0// EΓ //

OO=

EΓ/Γ0

EΓ // EΓ/Γ oo = // BΓ

The two upper lines are Γ0-principal bundles. As both EΓ0 and EΓ have vanishinghomotopy groups, the map Ei is a weak homotopy equivalence, and so is g. HenceXΓ has the weak homotopy type of BΓ0. In addition, the map Bi : BΓ0 → BΓis weakly homotopy equivalent to the locally trivial bundle XΓ → BΓ with fiberX . More generally, let Y is a Γ-space and consider Γ/Γ0 × Y endowed with thediagonal Γ-action; then H∗Γ(Γ/Γ0×Y ) ≈ H∗Γ0

(Y ) (see the proof of Theorem 42.13).

For a general Γ-space Y , the quotient map q : Y → Γ\Y descends to a surjectivemap q : YΓ → Γ\Y such that q−1([y]) has the weak homotopy type of BΓy for ally ∈ Y , where Γy is the stabilizer of y.

Let (X,Y ) ba a Γ-pair, i.e. a Γ-space X with an Γ-invariant subspace Y . TheΓ-equivariant cohomology H∗Γ(X,Y ) is the cohomology algebra

H∗Γ(X,Y ) = H∗(XΓ, YΓ) and H∗Γ(X) = H∗(XΓ) = HΓ(X, ∅) .In particular, H∗Γ(pt) = H∗(BΓ). The map p : XΓ → BΓ induces an GrA-homomorphism H∗p : H∗(BΓ) → H∗Γ(X), endowing the latter with a structureof H∗Γ(pt)-algebra.

40.5. Changing spaces and groups. Let α : Γ′ → Γ ba a continuous homomor-phism. Let X be a Γ-space and X ′ be a Γ′-space. A continuous map f : X ′ → Xsatisfying

f(γx) = α(γ)f(x)

is called equivariant with respect to α. The continuous map EΓ′×X ′ Eα×f−−−−→ EΓ×Xthen descends to a continuous map fΓ′,Γ : XΓ′ → XΓ (depending on α). There is acommutative diagram

(40.6)

XΓ′fΓ′,Γ //

XΓ

BΓ′

Bα // BΓ

.

The map fΓ′,Γ induces a GrA-homomorphism

(40.7) f∗Γ′,Γ : H∗Γ(X)→ H∗Γ′(X′) .

By commutativity of the Diagram (40.6), one has

f∗Γ′,Γ(av) = f∗Γ′,Γ(a)(Bα)∗(v) ∀ a ∈ H∗Γ(X) and v ∈ H∗Γ(pt)

for all a ∈ H∗Γ(X) and v ∈ H∗Γ(pt). We say that f∗Γ′,Γ preserves the module structures

via α. More simply, the Γ-spaceX becomes a Γ′-space via α, thusH∗Γ(X) becomes aH∗(BΓ′) module and f∗Γ′,Γ is a morphism of H∗(BΓ′)-algebra. An important case is


given by f = id: X → X , Setting id∗Γ,Γ′ = α∗, we get a map α∗ : H∗Γ(X)→ H∗Γ′(X)which is a morphism H∗(BΓ′)-algebra.

40.8. Free actions. Let Γ0 be a closed normal subgroup of Γ and let X be aΓ-space. For x ∈ X , γ ∈ Γ and γ0, γ

′0 ∈ Γ0, the equation

(γγ0) (γ′0x) = (γγ0γ′0γ−1) γx

shows that the Γ-action on X descends to a (Γ/Γ0)-action on Γ0\X . By the func-toriality of the equivariant cohomology (see (40.5)), we get a map

(40.9) H∗Γ/Γ0(Γ0\X)→ H∗Γ(X)

which is a homomorphism of H∗(B(Γ/Γ0))-algebras. The following lemma general-izes Lemma 39.5. To avoid point-set topology complications, we restrict ourselvesto the smooth action of a Lie group.

Lemma 40.10. Let Γ0 be a compact normal subgroup of of a Lie group Γ. LetX be a smooth Γ-manifold on which Γ0 acts freely. Then, the map (40.9) is anisomorphism of H∗(B(Γ/Γ0))-algebras.

Proof. Let Y = E(Γ/Γ0)×Γ/Γ0(Γ0\X). Consider the commutative diagram

EΓ×X q //

p

EΓ×Γ X

p

E(Γ/Γ0)× (Γ0\X)

q // Y

.

Let a ∈ Y represented by ((tiγi), x) in EΓ×X . Then,

(qp)−1(a) = ((tiγiδi), δx

)| δi, δ ∈ Γ0 .

Therefore,

(40.11) p−1(a) = q((qp)−1(a)) ≈ ((tiγiδi), x

)| δi ∈ Γ0 ≈−→ EΓ0 ,

the last homeomorphism being given by((tiγiδi), x

)7→ (tiδi).

As Γ0 acts smoothly and freely on X , the quotient map X → Γ0\X is a locallytrivial bundle (this follows from the slice theorem, see [11, Theorem 2.2.1]). Hence,p is homotopy equivalent to a locally trivial bundle, which is numerable ([175,p. 94]) since Γ0\X is paracompact. The map q is also a numerable locally trivialbundle (see (7) on p. 230). Therefore, qp is a fibration (i.e. satisfies the homotopycovering property for any space: see [175, Theorem 12, p. 95]), and so does p. AsΓ0\X is a manifold and E(Γ/Γ0) is contractible, the space Y admits a numerablecovering Vλλ∈Λ such that each inclusion Vλ → Y is null-homotopic. As eachfiber of p is contractible by (40.11), [40, Theorem 6.3] implies that p is a homotopyequivalence, which proves the lemma.

40.12. The forgetful homomorphism. Choosing a point ζ ∈ EΓ provides a mapiζ : X → XΓ defined by iζ(x) = [ζ, x]. As EΓ is path-connected, the homotopy classof iζ does not depend on ζ. For instance, we can take ζ = ζ0 = (1e, 0, 0, . . . ) wheree ∈ Γ is the unit element. Therefore, we get a well defined GrA-homomorphism

(40.13) ρ : H∗Γ(X)→ H∗(X)


given by ρ = H∗iζ for some ζ ∈ EΓ. As in (39.10), one proves that ρ is functorial.In fact, using 40.5, the homomorphism ρ coincides with the homomorphism ide,Γinduced by inclusion of the trivial group e into Γ:

(40.14) ρ = ide,Γ : H∗Γ(X)→ H∗e(X) = H∗(X) .

Indeed, iζ0 factors through X → Xe ≈ X . Hence, ρ may be seen as a forgetfulhomomorphism (one forgets the Γ-action).

A consequence of (40.13) is that ρ is functorial for the changing of groups: ifα : Γ′ → Γ is a continuous homomorphism and X a Γ-space, the following diagram

H∗Γ(X)α∗ //

ρ %%JJJJJ

JJJ

H∗Γ′(X)

ρyyssssssss

H∗(X)

is commutative.

40.15. Equivariant formality. A Γ-space X is called equivariantly formal ifρ : HΓ(X) → H∗(X) is surjective. For instance, X is equivariantly formal if theΓ-action is trivial. For relationships with other kind of “formal” spaces, see [169].If X is equivariantly formal, one can chose, as in the proof of Proposition 39.14,a GrV-section θ : H∗(X) → H∗Γ(X) of ρ. Then θ is a cohomology extension of

the fiber for the fiber bundle XΓp−→ BΓ. If X is of finite cohomology type, the

Leray-Hirsch theorem 28.26 then gives an map (depending on θ)

(40.16) H∗Γ(pt)⊗H∗(X)≈−→ H∗Γ(X)

which is an isomorphism of H∗Γ(pt)-module.

40.17. Reduced cohomology. Let X be a Γ-space. The reduced equivariantcohomology H∗Γ(X) is the H∗Γ(pt)-algebra defined by

(40.18) H∗Γ(X) = coker(H∗pΓ : H∗Γ(pt)→ H∗Γ(X)

)

where p : X → pt denotes the constant map to a point (which is Γ-equivariant).

Warning: H∗Γ(X) 6= H∗(XΓ). Examples:

(1) H∗Γ(pt) = 0.(2) If Y is a space with trivial Γ-action, there is an isomorphism of H∗Γ(pt)-

algebra

(40.19) H∗Γ(Y ) =(H∗(Y )⊗H∗Γ(pt)

)/(1⊗H∗Γ(pt)) ≈ H∗(Y )⊗H∗Γ(pt) .

(3) IfX is equivariantly formal and is of finite cohomology type, one uses (40.16)

to provides an isomorphism ofH∗Γ(pt)-modules between H∗Γ(X) and H∗(X)⊗H∗Γ(pt). This isomorphism depends on the choice of a section of ρ : H∗(X)→H∗Γ(X) and is not, in-general, an isomorphism of algebra.

Any Γ-equivariant map f : Y → X satisfies pf = p, so H∗Γ is a contravariantfunctor from TopΓ to the category of H∗Γ(pt)-algebra.


Let v ∈ XΓ. As for (12.32), one has the following diagram.

(40.20)

H∗Γ(pt)

p∗Γ

≈

$$IIII

IIII

H∗(XΓ, vΓ)j∗ //

≈

&&NNNNNNNNNH∗Γ(X)

i∗Γ //

H∗Γ(v)

H∗Γ(X)

where the line and the column are exact. This proves that

(40.21) H∗(XΓ, vΓ)≈−→ H∗Γ(X) .

Observe that, in (40.20) , i∗Γ

coincides with the section sv of p∗Γ. We see that

the choice of v ∈ XΓ produces a supplementary vector subspace to p∗Γ(H∗Γ(pt)) in

H∗Γ(X).

A pair (X,A) of Γ-spaces is called equivariantly well cofibrant if it admits apresentation (u, h) as a well cofibrant pair which is Γ-equivariant, i.e. u(γx) = u(x)and h(γx, t) = γ h(x, t) for all γ ∈ Γ, x ∈ X and t ∈ I.

Lemma 40.22. Let (X,A) be a pair of Γ-spaces which is equivariantly wellcofibrant. Then (XΓ, AΓ) is well cofibrant.

Proof. Let (u, h) be a presentation of (X,A) as an equivariantly well cofibrant

pair. Define u : EΓ × X → I and h : EΓ × X × I → EΓ × X by u(z, x) = u(x)

and h(z, x, t) = (z, h(x, t)). We check that these maps descend to uΓ : XΓ → I andhΓ : XΓ× I → XΓ and that (uΓ, hΓ) is a presentation of (XΓ, AΓ) as a well cofibrantpair.

Lemma 40.23. Let (X,A) be a Γ1-equivariantly well cofibrant pair of Γ1-spaces.Let (Y,B) be a Γ2-equivariantly well cofibrant pair of Γ2-spaces. Then, (X×Y,A×Y ∪ X × B) is a Γ12-equivariantly well cofibrant pair of Γ12-spaces, where Γ12 =Γ1 × Γ2.

Proof. One checks that the proof of Lemma 12.65 works equivariantly.

If (X,A) is a pair of Γ-spaces, the quotient space X/A inherits a Γ-action, with[A] ∈ (X/A)Γ, where [A] denotes the set A as a class in X/A. The proof of thefollowing lemma is the same as that of Lemma 12.69.

Lemma 40.24. If (X,A) is an equivariantly well cofibrant pair of Γ-spaces, sois the pair (X/A,A/A).

Example 40.25. Let (X,x) be a pointed space. The group G = I, τ acts onX ×X by exchanging the coordinates and this action descends to X ∧X . If (X,x)is well pointed, the proof of Lemma 12.65 shows that the pair (X ×X,X ∨ X) isG-equivariantly well cofibrant. By Lemma 40.24, (X ∧X,x∧ x) is G-equivariantlywell pointed.

The quotient map π : (X,A) → (X/A,A/A) is a Γ-equivariant map of pairswhich induces πΓ : (XΓ, AΓ)→ ((X/A)Γ, (A/A)Γ).


Proposition 40.26. Let (X,A) be a pair of Γ-spaces which is equivariantlywell cofibrant. Then,

π∗Γ: H∗((X/A)Γ, (A/A)Γ)

≈−→ H∗(XΓ, AΓ)

is an isomorphism.

Proof. Let (K,L) = (X/A,A/A). Let (u, h) be a presentation of (X,A) asan equivariant well cofibrant pair and let (u, h) be the induced presentation of(K,L). Let V = u−1([0, 1/2]) and W = u−1([0, 1/2]) = π(V ). As noticed in theproof of Lemma 12.66, the condition u(h(x)) ≤ u(x) implies that h and h restrictto Γ-equivariant deformation retractions from V to A and from W to L. Thetautological homeomorphism from EΓ× (V × I) onto (EΓ× V )× I descends to a

homeomorphism (V × I)Γ ≈−→ VΓ× I. Using this, h and h descend to a deformationretractions hΓ : VΓ×I → VΓ, and hΓ : WΓ×I →WΓ onto A and L (making (XΓ, AΓ)and (KΓ, LΓ) good pairs).

The inclusion (K,L)→ (K,W ) gives rise to a morphism of exact sequences

Hk−1(KΓ) //

=

Hk−1(LΓ) //

≈homotpy

Hk(KΓ, LΓ) //

Hk(KΓ) //

=

Hk(LΓ)

≈homotpy

Hk−1(KΓ) // Hk−1(WΓ) // Hk(KΓ,WΓ) // Hk(KΓ) // Hk(WΓ)

which, by the five lemma, implies that Hk(KΓ, LΓ)≈−→ Hk(KΓ,WΓ) is an iso-

morphism. The same proof gives the isomorphism Hk(XΓ, AΓ)≈−→ Hk(XΓ, VΓ).

Proposition 40.26 then comes from the commutativity of the following diagram(where the vertical arrows are induced by inclusions)

H∗(KΓ, LΓ)OO

≈

π∗Γ // H∗(XΓ, AΓ)OO

≈

H∗(KΓ,WΓ)π∗Γ // H∗(XΓ, VΓ)

H∗(KΓ − LΓ,WΓ − LΓ)π∗Γ

≈//

≈ excision

H∗(XΓ −AΓ, VΓ −AΓ) .

≈ excision

The bottom horizontal arrow is indeed an isomorphism since π : (X −A, V −A)→(K − L,W − L) is a Γ-equivariant homeomorphism.

Corollary 40.27. Let (X,A) be a pair of Γ-spaces which is equivariantly wellcofibrant. If A is non-empty, there is a functorial isomorphism of H∗Γ(pt)-algebras

H∗(XΓ/AΓ)≈−→ H∗Γ(X/A) .

The hypothesis A 6= ∅ is necessary since H∗(XΓ) is not isomorphic to H∗Γ(X).


Proof. This follows from the following diagram.

H∗((X/A)Γ, (A/A)Γ)

π∗Γ≈

≈ // H∗Γ(X/A)

H∗(XΓ, AΓ)≈ // H∗(XΓ/AΓ)

.

The bijectivities come

• from (40.21) for the top horizontal arrow, since, as A 6= ∅, A/A is a point.• from Lemma 40.22 and Proposition 12.71 for the bottom horizontal arrow.• from Proposition 40.26 for the vertical arrow.

As in 40.5, the reduced equivariant cohomology is functorial for changinggroups. In particular, as in 40.12, the inclusion of the trivial group e into Γprovides the forgetful homomorphism

ρ : H∗Γ(X)→ H∗e(X) ≈ H∗(X)

which is functorial.

As in Lemma 39.17, one proves that ρ(H∗Γ(X)) ⊂ H∗(X)Γ. The followingstatement generalizes Proposition 39.21.

Proposition 40.28. Let X be a Γ-space with XΓ 6= ∅. Suppose that Hj(X) = 0

for 0 ≤ j < r. Then, ρ : HrΓ(X)→ Hr(X)Γ is an isomorphism.

Proof. (Using a spectral sequence). In the E2-term of the Serre spectralsequence of the bundle X → XΓ → BΓ, the lines from 1 to n− 1 vanish:

H0(BΓ) H1(BΓ) H2(BΓ)· · ·0

...

0

Hr(X)Γ

0

Therefore, it gives rise to the following edge exact sequence:

(40.29) 0→ Hr(BΓ)H∗p−−−→ Hr

Γ(X)ρ−→ Hr(X)Γ −→ Hr+1(BΓ)

H∗p−−−→ Hr+1Γ (X) .

The choice of a fixed point v ∈ XΓ provides a section sv : BΓ → XΓ, as seen

in Lemma 39.4. Therefore, H∗p is injective and HrΓ(X)

ρ−→ Hr(X)Γ is surjective.Proposition 40.28 follows form this, as in the proof of Proposition 39.21.

Remark 40.30. When Γ is discrete, a proof of Proposition 40.28 without spec-tral sequence is possible, following the pattern of that of Proposition 39.21. Therole of RP k is played by BkΓ, the quotient by Γ of the join EkΓ = Γ ∗ · · · ∗Γ (k+ 1times) (see [141, § 3]). The space Yk is defined to be EkΓ ×Γ X . We leave thisproof as an exercise to the reader.

41. LOCALIZATION THEOREMS AND SMITH THEORY 237

41. Localization theorems and Smith theory

As in § 39, we consider in this section the group G = I, τ of order 2, soBG ≈ RP∞ and H∗G(pt) = Z2[u] with u of degree one. Strong result come outif we invert u, namely if we tensor the Z2[u]-modules with the ring of Laurentpolynomials Z2[u, u

−1]. For a pair (X,Y ) of G-spaces, we thus define

h∗G(X,Y ) = Z2[u, u−1]⊗Z2[u] H

∗G(X,Y ) ,

with the notation h∗G(X) = h∗G(X, ∅). Note that Z2[u, u−1] is Z-graded, with

Z2[u, u−1]k = Z2u

k and we use the graded tensor product. Hence, h∗G(X,Y ) is aZ-graded Z2[u, u

−1]-algebra, with

(41.1) hkG(X,Y ) =⊕

i+j=k

Z2ui ⊗Hj

G(X,Y ) ≈⊕

ℓ∈Z

Hk−ℓG (X,Y ) .

The theorem below is an example of the so called localization theorems. Formore general statements (see, e.g. [35, Chapter 3] or [8, Chapter 3]).

Theorem 41.2. Let X be a finite dimensional G-complex. Then, the inclusionXG ⊂ X induces an isomorphism

h∗G(X)≈−→ h∗G(XG)

of Z-graded Z2[u, u−1]-algebras.

Before proving Theorem 41.2, we discuss a few examples.

Example 41.3. Suppose, in Theorem 41.2, that X is a free G-complex. ByLemma 39.5, H∗G(X) ≈ H∗(X/G). As X/G is a finite dimensional CW-complex,there exists an integer m such that um ·H∗G(X) = 0. As u in invertible in Z2[u, u

−1],this proves that h∗G(X) = 0, as predicted by Theorem 41.2, since XG = ∅. We seehere that the finite dimensional hypothesis is necessary in Theorem 41.2. Indeed,the free G-complex S∞ = EG satisfies H∗G(EG) = H∗(BG) = Z2[u], so h∗G(EG) =Z2[u, u

−1].

Example 41.4. Consider the G-space Snp of Example 39.26, i.e. the sphere

Sn endowed with a linear G-action with (Snp )G ≈ Sp. We assume that 0 ≤ p ≤n. Using Corollary 39.32, there are elements a ∈ Hn

G(Snp ) and b ∈ HpG((Snp )G)

generating respectively H∗G(Snp ) and H∗G((Snp )G) as free Z2[u]-modules and r(a) =

bun−p. Then, as predicted by Theorem 41.2, r : h∗G(Snp ) → h∗G((Snp )G) admits an

inverse, sending b to aup−n.

Example 41.5. Consider the G-space CPn (n ≥ 1), where G acts via the com-plex conjugation, with (CPn)G = RPn. By Proposition 39.34 and Corollary 39.35,there is a commutative diagram

Z2[u, u−1, a]

/(an+1)

r //

≈

Z2[u, u−1, b]

/(bn+1)

≈

h∗G(CPn)r // h∗G(RPn)

with a of degree 2, b of degree 1 and r(a) = bu+ b2. If n <∞, the correspondence

(41.6) b 7→ au−1 + a2u−3 + a4u−7 + · · · =∑

i≥0

a2i

u2i+1−1


extends to a GrA[u]-isomorphism r−1 : Z2[u, u−1, b]

/(bn+1)→ Z2[u, u

−1, a]/(an+1)

which is the inverse of r.

Example 41.7. If n =∞, the right hand member of (41.6) is not a polynomialand no inverse of r may be defined this way. In fact, r (and then r) is not anisomorphism. Indeed, the composition of r with the epimorphism Z2[u, u

−1, b]→ Z2

sending both b and u to 1 is the zero map. Of course, CP∞ violates the finitedimensional hypothesis in Theorem 41.2.

Proof of Theorem 41.2. The proof is by induction on the dimension of X ,which starts trivially with X = ∅ (dimension −1). The induction step reduces toproving that, if the theorem is true for X , it is then true for Z = X ∪ C where Cis a finite family of G-cells. We consider the following commutative diagram.

hk−1G (X) //

rX≈

hkG(Z, X) //

rZ,X

hkG(Z) //

rZ

hkG(X) //

rX≈

hk+1G (Z, X)

rZ,X

hk−1

G (XG) // hkG(ZG, XG) // hk

G(ZG) // hkG(XG) // hk+1

G (ZG, XG)

The two lines are exact sequences, obtained by tensoring with Z2[u, u−1] the exact

sequence of (Z,X) for H∗G (as in in the proof of Lemma 27.17, we use that a directsum of exact sequences is exact and that, over a field, tensoring with a vector spacepreserves exactness). If rX is an isomorphism by induction hypothesis, it is enough,using the five lemma, to prove that rZ,X is an isomorphism. Note that C is adisjoint union of free G-cells Cf and of isotropic G-cells Gi. By excision, one hasthe following commutative diagram

h∗G(Z,X)≈ //

rZ,X

h∗G(C,BdC)≈ // h∗G(Cf ,BdCf)× h∗G(Ci,BdCi)

r

h∗G(ZG, XG)

≈ // h∗G(Ci,BdCi)

where r(a, b) = b. It is then enough to prove that h∗G(Cf ,BdCf) = 0. But thisfollows from the exact sequence of (Cf ,BdCf) for h∗G and, since Cf and BdCf arefree G-space, from Example 41.3.

We are now leading toward the Smith inequalities. Let us extend our groundring Z2[u, u

−1] to the fraction field Z2(u) of Z2[u] (this is just a field of character-istic 2, the grading is lost). For a space X , the total Betti number b(X) of X isdefined by

b(X) =

∞∑

i=0

dimH∗(X) ∈ N ∪ ∞ .

Lemma 41.8. Let X be a finite dimensional G-complex with b(X) <∞. Then,as a vector space over Z2(u),

dim Z2(u)⊗Z2[u] H∗G(X) ≤ b(X)

with equality if and only if X is equivariantly formal.


Proof. From the transfer exact sequence (39.12), we extract the followingexact sequence

Hk−1G (X)

u−−→ HkG(X)

ρ−→ Hk(X) −→ HkG(X)

u−−→ Hk+1G (X) .

We deduce that HkG(X) is generated by u ·Hk−1

G (X) and a number of elements ≤dimHk(X), which proves the first assertion. Moreover, dim Z2(u)⊗Z2[u] H

∗G(X) =

b(X) if and only if ρ : H∗G(X) → H∗(X) is surjective, that is X is equivariantlyformal.

Proposition 41.9. Let X be a finite dimensional G-complex with b(X) <∞.Then

(41.10) b(XG) ≤ b(X)


Proof.

b(X) ≥ dim Z2(u)⊗Z2[u] H∗G(X) by Lemma 41.8

= dim Z2(u)⊗Z2[u] H∗G(XG) by Theorem 41.2

= b(XG) ,

the last equality coming from Lemma 41.8, since XG is equivariantly formal. FromLemma 41.8 again, the above inequality is an equality if and only if X is equivar-iantly formal.

Formula (41.10) is an example of Smith inequalities, a development of the workof P. Smith started in 1938 [174]. The following corollary is a classical result in thetheory.

Corollary 41.11. Let X be a finite dimensional G-complex. Then,

(1) If H∗(X) ≈ H∗(pt), then H∗(XG) ≈ H∗(pt).(2) If X has the cohomology of a sphere, so does XG.

Proof. If X has the cohomology of a point, it is equivariantly formal and,by Proposition 41.9, b(XG) = 1 which proves (1). For Point (2), Proposition 41.9implies that b(XG) ≤ 2. Statement (2) is true if b(XG) = 2 or if b(XG) = 0 (since∅ = S−1). It remains to prove that b(XG) = 1 is impossible if H∗(X) ≈ H∗(Sn).

If b(XG) = 1, thenX is not equivariantly formal. Using Exact sequence (39.12),this implies that H∗G(X) ≈ H∗(RPn). As in Example 41.3, we deduce thath∗G(X) = 0, contradicting Theorem 41.2 (h∗G(XG) = Z2[u, u

−1] if b(XG) = 1).

Here is another consequence of Theorem 41.2.

Proposition 41.12. Let X be finite dimensional G-complex. Then, the fol-lowing statements are equivalent.

(1) X is equivariantly formal.(2) r : H∗G(X)→ H∗G(XG) is injective.

Proof. If Y is an equivariantly formal G-complex, then H∗G(Y ) is a free Z2[u]-module by Proposition 39.14 and thus j : H∗G(Y ) → Z2[u, u

−1] ⊗Z2[u] H∗G(Y ) =

h∗G(Y ) is injective. Conversely, if Y is not equivariantly formal, then there exists0 6= a ∈ H∗G(Y ) such that ua = 0 (using (3) of Proposition 39.14) and thus j(a) = 0.


Therefore, a G-complex is equivariantly formal if and only if j is injective. In thefollowing commutative diagram

H∗G(X)r //

j

H∗G(XG)jG

h∗G(X)

r

≈// h∗G(XG)

jG is then injective since XG is equivariantly formal and the bottom arrow is anisomorphism by Theorem 41.2. This shows that the top arrow r is injective if andonly if j is injective, proving that (1) ⇔ (2).

We shall now prove a localization theorem analogous to Theorem 41.2 for S1-spaces. Since we are working with Z2-cohomology, an important role is played bythe subgroup ±1 = S0 of S1. We also need the notion of a Γ-CW-complex for Γa topological group. If Γ0 is a closed subgroup of Γ, the Γ-space Γ/Γ0×Dn is calleda Γ-cell of dimension n (of type Γ0), with boundary Γ/Γ0×Sn−1 (the group Γ actson the left on Γ/Γ0 and trivially on Dn). One can attach a Γ-cell to a Γ-space Yvia a G-equivariant map ϕ : Γ/Γ0×Sn−1 → Y . A Γ-CW-structure on a Γ-space Xis a filtration

(41.13) ∅ = X−1 ⊂ X0 ⊂ X1 ⊂ · · · ⊂ X =⋃

n∈N

Xn

by Γ-subspaces, such that, for each n, the spaceXn (the n-skeleton) is Γ-homeomorphicto a Γ-space obtained from Xn−1 by attachment of a family Γ-cells of dimension n(of various type). A Γ-space endowed with a Γ-CW-structure is a Γ-CW-complex(or just a Γ-complex).The topology of X is the weak topology with respect to thefiltration (41.13).

If X is a Γ-complex, then X/Γ admits a CW-structure so that the projectionX → X/Γ is cellular. For Γ = G of order 2, the above definition is easily madeequivalent to that of p. 217 (compare also [35, pp. 101–102]). If Γ is a compact Liegroup acting smoothly on a smooth manifold X , then X admits a Γ-CW-structure(see [105]).

The Milnor classifying space BS1 for principal S1-bundles is homotopy equiv-alent to CP∞. Then, by Proposition 35.7, H∗S1(pt) = Z2[v] with v of degree 2. Fora pair (X,Y ) of S1-spaces, we thus define

h∗S1(X,Y ) = Z2[v, v−1]⊗Z2[v] H

∗S1(X,Y ) ,

with the notation h∗S1(X) = h∗S1(X, ∅). As in (41.1), h∗S1(X,Y ) is a Z-gradedZ2[v, v

−1]-algebra.

Theorem 41.14. Let X be a finite dimensional S1-complex such that XS1

=

XS0

. Then, the inclusion XS1 ⊂ X induces an isomorphism

h∗S1(X)≈−→ h∗S1(XS1

)

of Z-graded Z2[v, v−1]-algebras.

The hypothesis XS1

= XS0

is necessary in the above localization theoremTheorem 41.14. For example, let X = S1 with S1-action g · z = g2 z. Then

XS1 ≈ BS0 ≈ RP∞ by Example 40.4, so h∗S1(X) ≈ Z2[v, v−1, u] while XS1

= ∅.


Proof. The proof follows the plan of that of Theorem 41.2, by induction onthe skeleton of X , starting trivially with the (−1)-skeleton which is the empty set.The induction step reduces to proving that, if the theorem is true for X , it is thentrue for Z = X ∪ C where C is a family of S1-cells. As for Theorem 41.2, thiseventually reduces to proving that h∗G(C,BdC) = 0 when C is not an isotropy cell.

As XS1

= XS0

, the isotropy group Γ of C is then a finite group of odd order. Thepair (C,BdC) is of the form (S1/Γ×Dn, S1/Γ×Sn−1) and, as seen in Example 40.4,CS1 ≈ BΓ and (BdC)S1 ≈ BΓ × Sn−1. By Lemma 40.2, H∗S1(C) = H∗(pt) andH∗S1(BdC) = H∗(Sn−1). In particular, the multiplication by u is is the zero mapand thus h∗S1(C) = h∗S1(BdC) = 0. From the exact sequence of (C,BdC) for h∗G, itfollows that h∗G(C,BdC) = 0.

The Smith theory for S1-complexes with XS1

= XS0

is very similar to that ofS0-spaces. Let Z2(v) be the fraction field of Z2[v].

Lemma 41.15. Let X be a finite dimensional S1-complex with b(X) < ∞ and

XS1

= XS0

. Then, as a vector space over Z2(v),

dim Z2(v)⊗Z2[v] H∗S1(X) ≤ b(X)


Proof. The proof is the same as that of Lemma 41.8. The transfer exactsequence is replaced by the Gysin exact sequence of the S1-bundle X ×S∞ → XS1

which, as indicated in (7) p. 230, is induced from the universal bundle by p : XS1 →BS1 ≈ CP∞. Therefore, this Gysin sequence looks like

Hk−1S1 (X)

v−−→ Hk+1S1 (X)

ρ−→ Hk+1(X) −→ HkS1(X)

v−−→ Hk+2S1 (X)

and permits us the same arguments as for Lemma 41.8.

The proofs of 41.16–41.19 below are then the same as those of of 41.9–41.12,replacing Theorem 41.2 by Theorem 41.14.

Proposition 41.16. Let X be a finite dimensional S1-complex with b(X) <∞and XS1

= XS0

. Then

(41.17) b(XS1

) ≤ b(X)


Corollary 41.18. Let X be a finite dimensional S1-complex with XS1

= XS0

.Then,

(1) If H∗(X) ≈ H∗(pt), then H∗(XS1

) ≈ H∗(pt).(2) If X has the cohomology of a sphere, so does XS1

.

Proposition 41.19. Let X be finite dimensional S1-complex with XS1

= XS0

.Then, the following statements are equivalent.

(1) X is equivariantly formal.

(2) r : H∗S1(X)→ H∗S1(XS1

) is injective.


42. Equivariant cross products and Kunneth theorems

Let Γ1 and Γ2 be two topological groups; we set Γ12 = Γ1 × Γ2. Let X be aΓ1-space and Y be a Γ2-space. Then X × Y is a Γ12-space by the product action(γ1, γ2) · (x, y) = (γ1x, γ2y). The projections P1 : X × Y → X and P2 : X × Y → Yare equivariant with respect to the projection homomorphisms Γ12 → Γi. Passingto the Borel construction gives a map

(42.1) (X × Y )Γ12

P−→ XΓ1 × YΓ2

The map P is a homotopy equivalence, being induced by the homotopy equivalence

(42.2) P : EΓ12 × (X × Y )→ (EΓ1 ×X)× (EΓ2 × Y )

given by

P((ti(ai, bi), (x, y)

)=

((tiai, x), (tibi, y)

),

where (ti) ∈ ∆∞, (ai, bi) ∈ Γ12 and (x, y) ∈ X × Y . The case X = Y = pt providesa homotopy equivalence P0 : B(Γ12)→ BΓ1 ×BΓ2 and a commutative diagram

(42.3)

(X × Y )Γ12

P // XΓ1 × YΓ2

B(Γ12)

P0 // BΓ1 ×BΓ2

.

The cross product H∗(BΓ1) ⊗H∗(BΓ2)×−→ H∗(BΓ1 × BΓ2) post-composed with

H∗P0 gives a ring homomorphism

h : H∗Γ1(pt)⊗H∗Γ2

(pt) −→ H∗Γ12(pt) .

Note that, if BΓ1 or BΓ2 is of finite cohomology type, the Kunneth theorem impliesthat h is an isomorphism. The homotopy equivalence (42.1) together with (42.3)and the Kunneth theorem gives the following lemma.

Lemma 42.4. The composed map

×Γ12 : H∗Γ1(X)⊗H∗Γ2

(Y )× // H∗(XΓ1 × YΓ2)

H∗P

≈// H∗Γ12

(X × Y ) .

is an homomorphism of algebras. The (H∗Γ1(pt) ⊗ H∗Γ2

(pt))-module structure onH∗Γ1

(X)⊗H∗Γ2(Y ) and the H∗Γ12

(pt)-module structure on H∗Γ12(X×Y ) are preserved

via h. If YΓ2 is of finite cohomology type, then ×Γ12is an isomorphism.

Example 42.5. Let Γ1 = Γ2 = G = ±1. We let Γ1 act on the linearsphere X = Sm0 with XΓ1 = ω1

±, and Γ2 act on Y = Sn0 with Y Γ2 = ω2± (see

Example 39.30). Set H∗Γ1(pt) = Z2[u1] and H∗Γ2

(pt) = Z2[u2] (ui of degree 1). Asseen in Example 39.30, H∗Γ1

(X) and H∗Γ2(Y ) admit the following presentations

H∗Γ1(X) ≈ Z2[u1, A1, B1]

/(A1 +B1 + um1 , A

21 + um1 A1)

and

H∗Γ2(Y ) ≈ Z2[u2, A2, B2]

/(A2 +B2 + un2 , A

22 + un2A2) ,

where A1, B1 are of degree m and A2, B2 are of degree n. To shorten the formulae,we also denote by A1 the element A1 ×Γ12

1 ∈ H∗Γ12(X × Y ), by A2 the element

1×Γ12A2 ∈ H∗Γ12

(X × Y ), etc. By Lemma 42.4. we thus get the presentation

(42.6) H∗Γ12(X × Y ) ≈ Z2[u1, u2, A1, B1, A2, B2]

/I ,

42. EQUIVARIANT CROSS PRODUCTS AND KUNNETH THEOREMS 243


A1 +B1 + um1 , A21 + um1 A1 , A2 +B2 + un2 and A2

2 + un2A2 .

One can of course eliminate the Bi’s and get the shorter presentation

H∗Γ12(X × Y ) ≈ Z2[u1, u2, A1, A2]

/(A2

1 + um1 A1, A22 + un2A2) .

The commutative diagram

(42.7)

H∗Γ1(X)⊗H∗Γ2

(Y )×Γ12

≈//

rX⊗rY

H∗Γ12(X × Y )

r

H∗Γ1(XΓ1)⊗H∗Γ2

(Y Γ2)×Γ12

≈// H∗Γ12

((X × Y )Γ12 )

permits us to compute the image under r of the various classes of H∗Γ12(X × Y ).

Set

H∗Γ1(XΓ1) = Z2[u1]ω

1− ⊕ Z2[u1]ω

1+ and H∗Γ2

(Y Γ2) = Z2[u2]ω2− ⊕ Z2[u2]ω

2+ .

Denote the four points of (X × Y )Γ12 = XΓ1 × Y Γ2 by ω−− = (ω1−, ω

2−), ω−+ =

(ω1−, ω

2+), etc. With the notation R = Z2[u1, u2], one has

(42.8) H∗Γ12((X × Y )Γ12) ≈ Rω−− ⊕Rω+− ⊕Rω−+ ⊕Rω++

One has

r(A1) = rX(A1)×Γ12rY (1) = um1 ω

1+ ×Γ12

(1ω2− + 1ω2

+) = um1 ω+− + um1 ω++ .

Hence, the coordinates of r(A1) using (42.8) are (0, um1 , 0, um1 ). Similar computa-

tions provide the following table.

x coord. of r(x) in (42.8)

1 1 1 1 1

ui ui ui ui uiA1 0 um1 0 um1A2 0 0 un2 un2A1A2 0 0 0 um1 u

n2

B1 um1 0 um1 0

B2 un2 un2 0 0

B1B2 um1 un2 0 0 0

We now concentrate our interest to the case where Γ1 = Γ2 = Γ and seeX × Y as a Γ-space using the diagonal homomorphism ∆: Γ → Γ × Γ. We get ahomomorphism ∆∗ : H∗Γ×Γ(X × Y )→ H∗Γ(X × Y ). The composed map(42.9)

H∗Γ(X)⊗H∗Γ(Y )

×Γ

22×Γ×Γ // H∗(XΓ × YΓ)

H∗P

≈// H∗Γ×Γ(X × Y ) ∆∗ // H∗Γ(X × Y )


is called the equivariant cross product. For X = Y = pt, one has the followingcommutative diagram

(42.10)

H∗Γ(pt)⊗H∗Γ(pt)×Γ //

≈

H∗Γ(pt)

≈

H∗(BΓ)⊗H∗(BΓ) // H∗(BΓ)

.

Indeed, one has

(42.11)BΓ

B∆ //

∆BΓ

55B(Γ× Γ)

P

≈// BΓ×BΓ

and H∗∆BΓ(a× b) = a b by (27.6).

The equivariant cross product ×Γ will be useful in § 47 but one may wish toget some Kunneth theorem. As this is not even the case for X = Y = pt, someadaptation is needed. Lemma 42.4 together with diagram (42.10) implies that

(42.12) (w · a)×Γ b = a×Γ (w · b) = w · (a×Γ b) .

for all a ∈ H∗Γ(X), b ∈ H∗Γ(Y ) and w ∈ H∗Γ(pt). Therefore, ×Γ descend to thestrong equivariant cross product

×Γ : H∗Γ(X)⊗H∗Γ(pt) H∗Γ(Y )→ H∗Γ(X × Y ) .

The tensor product H∗Γ(X)⊗H∗Γ(pt)H∗Γ(Y ) still carries an H∗Γ(pt)-action, defined by

w · (a ⊗ b) = (w · a) ⊗ b = a ⊗ (w · b). Lemma 42.4 together with (42.12) impliesthat ×Γ is a morphism of H∗Γ(pt)-algebras.

Theorem 42.13 (Equivariant Kunneth theorem). Let Γ be a topological groupsuch that BΓ0 is of finite cohomology type for any closed subgroup Γ0 of Γ. LetX and Y be Γ-spaces, where X is a finite dimensional Γ-CW-complex. Supposethat Y is of finite cohomology type and is equivariantly formal. Then, the strongequivariant ×Γ cross product is an isomorphism of H∗Γ(pt)-algebras.

Proof. As ×Γ is a morphism of H∗Γ(pt)-algebra, it suffices to prove that itis a GrV-isomorphism. We follow the idea of the proof of the ordinary Kunneththeorem 27.15, fixing the Γ-space Y and comparing the “equivariant cohomologytheories”

h∗(X,A) = H∗Γ(X,A)⊗H∗Γ(pt) H∗Γ(Y ) and k∗(X,A) = H∗Γ(X × Y,A× Y )

defined for a Γ-pair (X,A). The definition of the strong equivariant cross productextends to pairs and we get a morphism of H∗Γ(pt)-algebras

×Γ : h∗(X,A)→ k∗(X,A)

One gets a commutative diagram

(42.14)

h∗(X)

×Γ

// h∗(A)

×Γ

δ∗ // h∗+1(X,A)

×Γ

// h∗+1(X)

×Γ

// h∗+1(A)

×Γ

h∗(X) // k∗(A)

δ∗ // k∗+1(X,A) // k∗+1(X) // k∗+1(A)


where the lines are exact. That the square diagram with the δ∗’s commutes comesfrom the definition of ×Γ, using the commutativity of Diagram (27.21).

The theorem is proven by induction on the dimension ofX . IfX is 0-dimensional,it is a disjoint union of homogeneous Γ-spaces. As the disjoint union axiom holds forour theories, the induction starts by proving the theorem for X = Γ/Γ0, where Γ0

is a closed subgroup of Γ. As Y is Γ-equivariantly formal, it is also Γ0-equivariantlyformal and one has

h∗(X) = H∗Γ(Γ/Γ0)⊗H∗Γ(pt) H∗Γ(Y )

≈ H∗(BΓ0)⊗H∗(BΓ)

(H∗(BΓ)⊗H∗(Y )

)

≈ H∗(BΓ0)⊗H∗(Y )

≈ H∗Γ0(Y ) .

On the other hand, consider the map α : EΓ × (Γ× Y )→ EΓ × Y given by

α(z, (γ, y)) = (zγ, γ−1y) .

It satisfies α(z, (δγ, δy)) = α(zδ, (γ, y)) and α(z, (γγ0, y)) = (zγγ0, γ−10 γ−1y); it

thus descends to a map

α : EΓ ×Γ (Γ/Γ0 × Y )≈−→ EΓ×Γ0 Y

which is a homeomorphism: its inverse is induced by the map β(z, y) = (z, ([e], y)),where e ∈ Γ is the unit element. Hence, k∗(X) is also isomorphic, as an H∗Γ(pt)-algebra, to H∗Γ0

(Y ). It remains to show that ×Γ is a GrV-isomorphism. As Yand BΓ0 are both of finite cohomology type, the graded vector space HΓ0(Y ) ≈H∗(BΓ0) ⊗ H∗(Y ) is finite dimensional in each degree. Therefore, it suffices toprove that ×Γ is surjective.

If Z is a Γ-space, we denote by i : Z → ZΓ the inclusion i(z) = [(1e, 0, ...), z](it induces the forgetful homomorphism H∗i = ρ : H∗Γ(Z) → H∗(Z)). One has acommutative diagram

Γ/Γ0 × Y oo s

i

Y

i

EΓ ×Γ (Γ/Γ0 × Y ) oo β

EΓ×Γ0 Y

where s is the slice inclusion s(y) = ([e], y]). We thus get a commutative diagram

H∗(Γ/Γ0)⊗H∗(Y )× // H∗(X × Y )

H∗s // H∗(Y )

H∗Γ(Γ/Γ0)⊗Γ H∗(Y )Γ

ρ⊗ρ

OO

×Γ // H∗Γ(X × Y )

ρ

OO

H∗β

≈// H∗Γ0

(Y )

ρ

OO

Let B be a GrV-basis of H∗(Y ). Let σ : Hk(Y ) → HkΓ(Y ) be a section of ρ.

For b ∈ B, one has

H∗s(ρ(1)× ρ(σ(b)) = H∗s(1×Γ b) = b ,

the last equality coming from Lemma 28.4. Therefore, H∗s× (ρ⊗ρ) is surjectiveand the formula

σ(a) = H∗β(1 ×Γ σ(a))


defines a section σ : H∗(Y )→ H∗Γ0(Y ) of ρ. The Leray-Hirsch theorem then implies

that H∗Γ0(Y ) is generated, as a H∗Γ(pt)-module, by σ(B). Hence, ×Γ is surjective.

The induction step reduces to proving that, if the theorem is true for A, it isthen true for X = A ∪C where C is a family of Γ-cells. By the five lemma in Dia-gram (42.14), it suffices to prove that ×Γ : h∗(X,A)→ k∗(X,A) is an isomorphism.By excision and the disjoint union axiom one can restrict ourselves to the case of apair (X0, A0) = Γ/Γ0×(Dn, Sn−1) (a Γ-cell). By the five lemma in Diagram (42.14)for the pair (X0, A0), it suffices to prove the theorem for X0 and for A0. The formeris covered by the 0-dimensional case (sinceX0 is Γ-homotopy equivalent to Γ/Γ0)and the latter is (n−1)-dimensional, thus covered by the induction hypothesis.

Remark 42.15. If Γ = e, Theorem 42.13 reduces to the ordinary Kunneththeorem 27.15. Therefore, the hypotheses that Y is of finite cohomology type isessential. Theorem 42.13 is also wrong if Y is not equivariantly formal. For example,set Γ = ±1, X = S1 and Y = S2, with the antipodal involution. These are freeΓ-spaces and, by Lemma 39.5,

H∗Γ(X) ≈ H∗(S1/± 1) ≈ Z2[u]/(u2) and H∗Γ(Y ) ≈ H∗(S2/± 1) ≈ Z2[u]/(u

3) .

Moreover,H∗Γ(pt) = H∗(RP∞) ≈ Z2[u] and, using Lemma 39.5 again together withProposition 24.21, the Z[u]-morphisms H∗Γ(pt)→ H∗Γ(X) and H∗Γ(pt)→ H∗Γ(Y ) aresurjective. Therefore,

H∗Γ(X)⊗H∗Γ(pt) H∗Γ(Y ) ≈ Z2[u]/(u

2)×Z2[u] Z2[u]/(u3) ≈ Z2[u]/(u

2) .

In particular, H∗Γ(X)⊗H∗Γ(pt) H∗Γ(Y ) vanishes in degree 3, while H3

Γ(X × Y ) =

Z2. Indeed H∗Γ(X ×Y ) ≈ H∗((X ×Y )/± 1) and (X ×Y )/± 1 is a closed manifoldof dimension 3.

The hypothesis that BΓ0 is of finite cohomology type is fulfilled if Γ0 is acompact Lie group. Note that it is only used in the proof for the stabilizers ofpoints of X . For other kind of equivariant Kunneth theorems (see [172]).

Example 42.16. Consider the diagonal action of the group G = ±1 on thethe product of linear spheres Sm0 ×Sn0 . Set H∗G(pt) = Z2[u], with u of degree 1. ByExample 42.5 and Theorem 42.13, one has

H∗G(Sm0 × Sn0 ) ≈ Z2[u,A1, B1, A2, B2]/I ,


A1 +B1 + um , A21 + umA1 , A2 +B2 + un and A2

2 + unA2 .

Using the notations of Example 42.5 for the fixed points, one has

(42.17) H∗G((Sm0 × Sn0 )G) ≈ Z2[u]ω−− ⊕ Z2[u]ω+− ⊕ Z2[u]ω−+ ⊕ Z2[u]ω++


and one has the following table for r : H∗G(Sm0 × Sn0 )→ H∗G((Sm0 × Sn0 )G)

x coord. of r(x) in (42.17)

1 1 1 1 1

u u u u u

A1 0 um 0 um

A2 0 0 un un

A1A2 0 0 0 um+n

B1 um 0 um 0

B2 un un 0 0

B1B2 um+n 0 0 0

For a generalization of this example, see Proposition 62.9.

We now define the equivariant reduced cross product, related to the equivariantcohomology of a smash product. Let X be a Γ1-space and Y be a Γ2-space, pointedby x ∈ XΓ1 and y ∈ Y Γ2 . Then, X ∨ Y is a Γ12-invariant subspace of X × Y .Consider the space

XΓ1∨YΓ2 = (XΓ1 × yΓ2) ∪ (xΓ1 × YΓ2) ⊂ XΓ1 × YΓ2 .

If the pairs (X, x) and (Y, y) are equivariant well cofibrant pairs, we saythat (X,x) and (Y, y) are equivariantly well pointed.

Lemma 42.18. Let (X,x) be an equivariantly well pointed Γ1-space and (Y, y) bean equivariantly well pointed Γ2-space. Then, the map P : (X×Y )Γ12 → XΓ1×YΓ2

of (42.1) sends (X ∨ Y )Γ12 onto XΓ1∨YΓ2 and induces an isomorphism

H∗P : H∗(XΓ1 ∨YΓ2)≈−→ H∗Γ12

(X ∨ Y )

Proof. That P ((X ∨ Y )Γ12) = XΓ1∨YΓ2 follows directly from the definitionof P , using (42.2). This gives a commutative diagram(42.19)

xΓ1 × yΓ2//

xΓ1 × YΓ2

(x × y)Γ12

P

h.e.

hhQQQQQQQQQQQ

//

(x × Y )Γ12

P

h.e.

66nnnnnnnnnn

(X × y)Γ12

P

h.e.vvmmmmmmmmmmm

// (X ∨ Y )Γ12

P

((PPPPPPPPPP

XΓ1 × yΓ2// XΓ1 ∨YΓ2

where the unlabelled arrows are inclusions and h.e. means “homotopy equiva-lence”. Our hypotheses and Lemma 40.22 imply that pairs like ((X×y)Γ12, (x×y)Γ12), etc, are good. Hence, the hypotheses of Proposition 12.83 to get Mayer-Vietoris sequences are fulfilled. We thus get a morphism from the Mayer-Vietorissequence for the outer square of (42.19) to that of the inner square, and the propo-sition follows from the five-lemma.


Remark 42.20. The map P : (X ∨ Y )Γ12 → XΓ1∨YΓ2 of Lemma 42.18 is ac-tually a weak homotopy equivalence, since the squares in (42.19) are homotopyco-Cartesian diagrams (see [36, Prop. 5.3.3]).

As X ∨ Y is a Γ12-invariant subspace of X × Y , the wedge product X ∧ Yinherits a Γ12-action.

Lemma 42.21. Let (X,x) be an equivariantly well pointed Γ1-space and (Y, y)be an equivariantly well pointed Γ2-space. Then, there is a natural isomorphism

H∗(XΓ1 × YΓ2 , XΓ1∨YΓ2)≈−→ H∗Γ12

(X ∧ Y )) .

Proof. By Lemma 42.18, the map P produces a morphism from the co-homology exact sequence of the pair (XΓ1 × YΓ2 , XΓ1 ∨YΓ2) to that of the pair((X × Y )Γ12 , (X ∨ Y )Γ12

). By Lemma 42.18 again and the fact that the map P

of (42.1) is a homotopy equivalence, the five lemma implies that

H∗P : H∗(XΓ1 × YΓ2 , XΓ1∨YΓ2)→ H∗((X × Y )Γ12 , (X ∨ Y )Γ12

)

is an isomorphism. By Lemma 40.23, the pair (X × Y,X ∨ Y ) is Γ12-equivariantlywell cofibrant. As X ∧ Y ) is not empty, Corollary 40.27 provides a natural isomor-

phism between H∗((X × Y )Γ12 , (X ∨ Y )Γ12

)and H∗Γ12

(X ∧ Y )).

Using the isomorphism of Lemma 42.21 as well as those of (40.21). one con-structs the following commutative diagram

(42.22)

H∗(XΓ1 , xΓ1)⊗H∗(YΓ2 , yΓ2)

×

≈ // H∗Γ1(X)⊗ H∗Γ2

(Y )

×Γ12

H∗(XΓ1 × YΓ2 , XΓ1∨YΓ2)≈ // H∗Γ12

(X ∧ Y )

which defines the equivariant reduced cross product ×Γ12 . The relative cross product(left vertical arrow) is indeed defined as in (27.7), since, as (Y, y) is equivariantlywell pointed, the couple (YΓ2 , yΓ2) is a good pair by Lemma 40.22.

In the case where Γ1 = Γ2 = Γ, one can see X ∧ Y as a Γ-space via thediagonal homomorphism ∆: Γ→ Γ×Γ. Composing ×Γ12 with ∆∗ : H∗Γ×Γ(X∧Y )→H∗Γ(X ∧ Y ), we get the equivariant reduced cross product

(42.23) H∗Γ(X)⊗ H∗Γ(Y )×Γ−−→ H∗Γ(X ∧ Y ) .

Lemma 42.24. Let (X,x) and (Y, y) be equivariantly well pointed Γ-spaces.Then,

(1) there is an equivariant reduced cross product

H∗Γ(X)⊗ H∗Γ(Y )×Γ−−→ H∗Γ(X ∧ Y )

which is a bilinear map.(2) the following diagram

(42.25)

H∗Γ(X)⊗ H∗Γ(Y )

ρ⊗ρ

×Γ // H∗Γ(X ∧ Y )

ρ

H∗(X)⊗ H∗(Y )× // H∗(X ∧ Y )

43. EQUIVARIANT BUNDLES AND EULER CLASSES 249

is commutative, where ρ is the forgetful homomorphism.(3) the hypotheses on X, Y are inherited by XΓ, Y Γ and there is a commu-

tative diagram(42.26)


r⊗r

×Γ // H∗Γ(X ∧ Y )

r

H∗Γ(XΓ)⊗ H∗Γ(Y Γ)OO≈

×Γ // H∗Γ(XΓ ∧ Y Γ)OO≈

[H∗(XΓ)⊗H∗(BΓ)]⊗ [H∗(Y Γ)⊗H∗(BΓ)]×⊗ // H∗(XΓ ∧ Y Γ)⊗H∗(BΓ)

Proof. The equivariant reduced cross product of (1) is obtained by post-

composing ×Γ12 of (42.22) (with Γ1 = Γ2 = Γ) with ∆∗ : H∗Γ×Γ(X ∧ Y )→ H∗Γ(X ∧Y ).

Let α : Γ′ → Γ is a continuous homomorphism. Then (X,x) and (Y, y) areΓ′-equivariantly well cofibrant. Our constructions are natural enough so that thereis a commutative diagram

(42.27)


α∗⊗α∗

×Γ // H∗Γ(X ∧ Y )

α∗

H∗Γ′(X)⊗ H∗Γ′(Y )×Γ′ // H∗Γ′(X ∧ Y )

.

For Γ′ = I, the homomorphism α∗ coincides with the forgetful homomorphism ρ(see 40.12), which proves (2).

To prove (3), we note that the upper square of (42.25) commutes by obviousnaturality of the equivariant reduced cross product with respect to equivariantmaps. The commutativity of the lower square is obtained using the considerationsof (42.10) and (42.11).

Example 42.28. Let (Z, z) be a well pointed space, considered with the trivial

action of G = I, τ. Then, H∗G(Z) ≈ H∗(Z)[u] and the bottom square in (42.26)becomes

H∗G(Z)⊗ H∗G(Z)×G //

OO≈

H∗G(Z ∧ Z)OO≈

H∗(Z)[u]⊗ H∗(Z)[u]×[u] // H∗(Z)[u]

,

where, for a, b ∈ H∗(Z), ×[u] is defined by

aum×[u] bun = (a× b)um+n .

43. Equivariant bundles and Euler classes

Although we are mostly interested in equivariant vector bundles, passing throughequivariant principal bundles is easier and more powerful. Let A be a topologicalgroup. A principal A-bundle ζ over over a space Y consists of a continuous map


p : P → Y , a continuous right action of A on P such that p(uα) = p(u) for allu ∈ P and all α ∈ A; in addition, the following local triviality should hold: for eachx ∈ X there is a neighbourhood U of x and a homeomorphism ψ : U ×A→ p−1(U)such that pψ(x, α) = x and ψ(x, αβ) = ψ(x, α)β. In consequence, A acts freelyon P and transitively on each fiber. Also, p is a surjective open map, descending to

a homeomorphism P/A≈−→ X (use [42, § I Chapter VI]). Two principal A-bundles

ζ = (Pp−→ X) and ζ = (P

p−→ X) are isomorphic if there exists an A-equivarianthomeomorphism h : P → P such that ph = p.

Let Γ be a topological group and let X be a (left) Γ-space. An A-principal

bundle ζ : Pp−→ X is called a Γ-equivariant principal A-bundle if it is given a left

action Γ × P → P commuting with the free right action of A and such that theprojection p is Γ-equivariant (a more general setting is considered in e.g. [123, 35,

136]). Two Γ-equivariant principal A-bundles ζ = (Pp−→ X) and ζ′ = (P

p−→ X)are isomorphic if there exists an (Γ,A)-equivariant homeomorphism h : P → P suchthat ph = p.

Example 43.1. Let p : P → pt be a Γ-equivariant principal A-bundle over apoint. The A-action on P is free and transitive. Hence, choosing a point s ∈ Pprovides a continuous map µ : Γ → A by the equation γs = sµ(γ). For γ, γ′ ∈ Γ,one has

sµ(γγ′) = (γγ′)s = γ(γ′s) = (γs)µ(γ′) = (sµ(γ))µ(γ′) = s(µ(γ)µ(γ′)) ,

which proves that µ is a homomorphism. Another point s ∈ P is of the form s = sαfor some α ∈ A. The map µ obtained from s is related to µ by

sαµ(γ) = sµ(γ) = γs = γsα = sµ(γ)α

and hence µ(γ) = α−1µ(γ)α. If p : P → pt is another Γ-equivariant principal A-bundle and if h : P → P is a (Γ, A)-equivariant homeomorphism, then γh(s) =h(s)µ(γ). This provides a map from the isomorphism classes of Γ-equivariant prin-cipal A-bundles over a point and the set hom(Γ, A)/A of the conjugation classes ofcontinuous homomorphisms from Γ to A. This map is a bijection. A homomor-phism µ : Γ→ A is realized by the bundle A→ pt with the Γ-action γ · α = µ(γ)α(hence, if e is the unit element of A, one has indeed γ · e = eµ(γ)). This proves thesurjectivity. The proof of the injectivity is left to the reader.

Let ζ : Pp−→ X be a Γ-equivariant principal A-bundle. Being Γ-equivariant, the

map p induces a map pΓ : PΓ → XΓ. Let i : X → XΓ be an inclusion as in (40.12).

Lemma 43.2. The map pΓ : PΓ → XΓ is a principal A-bundle, denoted by ζΓ.Moreover, ζ is isomorphic to the induced principal A-bundle i∗ζΓ.

Example 43.3. Let ξ be a Γ-equivariant principal A-bundle over a point, cor-responding to [µ] ∈ hom(Γ, A)/A (see Example 43.1). It is then isomorphic toA→ pt with the Γ-action γ ·α = µ(γ)α. Then ξΓ is the principal A-bundle over BΓinduced by the map Bµ : BΓ→ BA. Indeed, the map f : EΓ×ΓA→ EA given byf([(tiγi), α]) = (tiµ(γi)α) is A-equivariant and covers the map Bµ.

Before proving Lemma 43.2, let us recall the standard local cross-sections forthe Milnor construction of the universal Γ-bundle p : EΓ → BΓ. For i ∈ N, let(EΓ)i = (tjγj) ∈ EΓ | ti 6= 0 and let (BΓ)i = p((EΓ)i). There is a cross-section si of p over (BΓ)i sending b ∈ (BΓ)i to the unique element in (tjΓj) ∈


p−1(b) with γi = 1. If Z is a Γ-space, the map ψi : (BΓ)i × Z≈−→ (EΓ)i ×Γ Z

given by ψi(b, u) = [si(b), u] is a homeomorphism: its inverse is induced by thecorrespondence [(tjγj), u] 7→ (p(tjγj), γiz).

Proof of Lemma 43.2. The right A-action on PΓ = EΓ ×Γ P is defined by[z, u]α = [z, uα]. For i ∈ N, one has the commutative diagram

(BΓ)i × Pψi

≈//

id×p

(EΓ)i ×Γ P

pΓ

(BΓ)i ×Xψi

≈// (EΓ)i ×Γ X

.

The upper homeomorphism is A-equivariant and id× p is a principal A-bundle. As(EΓ)i×ΓXi∈N is an open covering of XΓ, the map pΓ admits local trivializationsof a principal A-bundle.

We have proven that pΓ is a principal A-bundle. If z ∈ EΓ, let iz : X → XΓ

be the inclusion defined in (40.12). Then, iz : P → PΓ is an A-equivariant mapcovering iz, inducing an isomorphism of principal A-bundles ζ ≈ i∗zζΓ.

Remark 43.4. When A is abelian, the last assertion of Lemma 43.2 may bestrengthened: a principal A-bundle over a Γ-space X admits a structure of a Γ-equivariant bundle if and only if it is induced from a principal A-bundle over XΓ

(see [124]).

The construction ζ 7→ ζΓ enjoys some functorialities. Let µ : Γ′ → Γ be acontinuous homomorphism between topological groups. Let X ′ be a Γ′-space, Xa Γ-space and let f : X ′ → X be a continuous map which is Γ′-equivariant withrespect to µ. Recall from (40.6) that f induces a map fΓ′,Γ : XΓ′ → XΓ. If ζ isa Γ-equivariant principal A-bundle over X , then f∗ζ is a Γ′-equivariant principalA-bundle over X ′.

Lemma 43.5. (f∗ζ)Γ′ ≈ f∗Γ′,ΓζΓ.

Proof. The map f is covered by a Γ′-equivariant map of principal A-bundlef : P (f∗ζ) → P , where P (f∗ζ) denotes the total space of f∗ζ. By functoriality ofthe Borel construction (see 40.5), there is a commutative diagram

P (f∗ζ)Γ′

fΓ′,Γ // PΓ

oo = // P (ζΓ)

X ′Γ′

fΓ′,Γ // XΓoo = // XΓ

Thanks to the description of the A-actions (see the the proof of Lemma 43.2), the

map fΓ′,Γ A-equivariant. Hence, fΓ′,Γ factor through an isomorphism (f∗ζ)Γ′ ≈f∗Γ′,ΓζΓ.

For another functoriality of ζΓ let ϕ : A → A′ be a continuous homomorphismbetween topological groups. It makes A′ a left A-space (α ·α′ = ϕ(α)α′). If ζ : P →X is a Γ-equivariant A-principal bundle, P ×A A′ is, in an obvious way, the total


space of a Γ-equivariant A′-principal bundle ϕ∗ζ. The tautological homeomorphismEΓ×Γ (P ×AA′) ≈ (EΓ×Γ P )×AA′ gives an isomorphism of A′-principal bundles

(43.6) (ϕ∗ζ)Γ ≈ ϕ∗ζΓ .

By a Γ-equivariant K-vector bundle ξ over X (K = R or C), we mean a Γ-equivariant map p : E = E(ξ)→ X which is a K-vector bundle, such that, for eachγ ∈ Γ and each x ∈ X , the map y 7→ γy is a K-linear map from p−1(x) to p−1(γx).The tangent bundle to a smooth Γ-manifold is an example for K = R.

It is convenient here to see a K-vector bundle ξ of rank r as associated to aprincipal GL(Kr)-bundle, the bundle Fra(ξ) of frames of ξ. Its total space is

Fra(ξ) = ν : Kr → E(ξ) | ν is a K-linear isomorphism onto some fiber of ξ .with the map pFra : Fra(ξ) → X given by pFra(ν) = pν(0). The right GL(Kr)-action on Fra(ξ) is by precomposition (we use the same notation for the bundleFra(ξ) and for its total space). The evaluation map sending [ν, t] ∈ Fra(ξ)×GL(Kr)

Kr to ν(t) ∈ E(ξ) defines an isomorphism of K-vector bundles

(43.7) Fra(ξ)×GL(Kr) Kr ≈−→ E(ξ) .

For more details and developments, see 51.12. If ξ is a Γ-equivariant K-vectorbundle, then Γ acts on Fra(ξ) by (γν)(t) = γ ·ν(t). Hence, Fra(ξ) is a Γ-equivariantprincipal GL(Kr)-bundle and (43.7) is an isomorphism of Γ-equivariant K-vectorbundles. The The tautological homeomorphism

EΓ×Γ (Fra(ξ)×GL(Kr) Kr) ≈ (EΓ×Γ Fra(ξ))×GL(Kr) Kr

implies that

(43.8) E(ξ)Γ ≈ Fra(ξ)Γ ×GL(Kr) Kr .

Using Lemma 43.2, this proves the following lemma.

Lemma 43.9. Let ξ = (p : E(ξ) → X) be a Γ-equivariant K-vector bundle ofrank r. Then, the map pΓ : E(ξ)Γ → XΓ is a K-vector bundle of rank r, denoted byξΓ. Moreover, ξ is isomorphic to the induced vector bundle i∗ξΓ.

Let µ : Γ′ → Γ and f : X ′ → X as for Lemma 43.5. If ξ is a Γ-equivariant K-vector bundle of rank r over X , then f∗ξ is a Γ′-equivariant K-vector bundle overX ′ of the same rank. One has an isomorphism of Γ′-equivariant GL(Kr)-principalbundles f∗Γ′,ΓFra(ξ) ≈ Fra(f∗ξ). Therefore, Lemma 43.5 gives an isomorphism ofK-vector bundles

(43.10) (f∗ξ)Γ′ ≈ f∗Γ′,ΓξΓ .The correspondence ξ 7→ ξΓ commutes with some operations on vector bundles,

like the Whitney sum or the tensor product. We first define these operations in thecategory of Γ-equivariant K-vector bundles. Let ξ (respectively ξ′) be two two suchbundles over a Γ-spaceX , of ranks r (respectively r′). Set F = Fra(ξ), F ′ = Fra(ξ′),

G = GL(Kr) and G′ = GL(Kr′). The diagonal inclusion ∆X : X → X ×X is Γ-equivariant with respect to the diagonal homomorphism ∆Γ : Γ → Γ × Γ. Hence,∆∗X(F × F ′) is a Γ-equivariant principal (G × G′)-bundle. The linear (G × G′)-action on Kr ⊕ Kr′ given by (R,R′) · (v, v′) = (Rv,R′v′) defines a continuous

homomorphism ϕ⊕ : G×G′ → GL(Kr⊕Kr′). This permits us to define the Whitneysum

ξ ⊕ ξ′ = ϕ⊕∗ ∆∗X(F × F ′)×GL(Kr⊕Kr′ ) (Kr ⊕Kr′)


as a Γ-equivariant K-vector space. The Γ-equivariant tensor product is definedaccordingly

(43.11) ξ ⊗ ξ′ = ϕ⊗∗ ∆∗X(F × F ′)×GL(Kr⊗Kr′ ) (Kr ⊗Kr′) ,

using the homomorphism ϕ⊗ : G×G′ → GL(Kr⊗Kr′) induced by the unique linear

action of G×G on Kr ⊗Kr′ satisfying (R,R′) · (v ⊗ v′) = (Rv ⊗R′v′).Lemma 43.12. (ξ ⊕ ξ′)Γ ≈ ξΓ ⊕ ξ′Γ and (ξ ⊗ ξ′)Γ ≈ ξΓ ⊗ ξ′Γ.Proof. One has

(ξ ⊕ ξ′)Γ =[ϕ⊕∗ ∆∗X(F × F ′)×GL(Kr⊕Kr′ ) (Kr ⊕Kr′)

]Γ

≈[ϕ⊕∗ ∆∗X(F × F ′)

]Γ×GL(Kr⊕Kr′ ) (Kr ⊕Kr′) by (43.8)

≈ ϕ⊕∗[∆∗X(F × F ′)

]Γ×GL(Kr⊕Kr′ ) (Kr ⊕Kr′) by (43.6)

whileξΓ ⊕ ξ′Γ = ϕ⊕∗ ∆∗XΓ

(FΓ × F ′Γ)×GL(Kr⊕Kr′ ) (Kr ⊕Kr′) .

Therefore, it is enough to construct an isomorphism of principal (G×G′)-bundles

(43.13) ∆∗XΓ(FΓ × F ′Γ) ≈

[∆∗X(F × F ′)

]Γ.

This will prove the lemma for Whitney sum, and also for the tensor product, usingϕ⊗ instead of ϕ⊕.

As ∆X : X → X ×X is Γ-equivariant with respect to the diagonal homomor-phism ∆Γ : Γ→ Γ× Γ, it induces a map Φ = (∆X)Γ,Γ×Γ : XΓ → (X ×X)Γ×Γ. ByLemma 43.5, one has

(43.14)[∆∗X(F × F ′)

]Γ≈ Φ∗(F × F ′)Γ .

For Γ-space Z and Z ′, a natural homotopy equivalence P : (Z × Z ′)Γ ≃−→ ZΓ × Z ′Γwas constructed in (42.1). For Z = F and Z ′ = F ′, we thus get a homotopy

equivalence P : (F ×F ′)Γ ≃−→ FΓ×F ′Γ which is (G×G′)-equivariant. The followingdiagram.

Φ∗(F × F ′)Γ

Φ // (F × F ′)Γ

P

≃// FΓ × F ′Γ

XΓ

Φ // (X ×X)Γ×ΓP

≃// XΓ ×XΓ

is commutative, thus each square is a morphism of principal (G × G′)-bundles.By the definition of P in § 42, one has P Φ = ∆XΓ . Therefore, Φ∗(F × F ′)Γ ≈∆∗XΓ

(FΓ×F ′Γ). This together with (43.14) gives the required isomorphism of (43.13).

Let ξ be a Γ-vector bundle of rank r over X . The equivariant Euler class eΓ(ξ)is the Euler class of ξΓ:

eΓ(ξ) = e(ξΓ) ∈ HrΓ(X) .

Example 43.15. Let χ : Γ→ GL(V ) be a representation of Γ on a vector spaceV of dimension r. This makes V a Γ-space. One can also see V as a vector bundlewith basis a point. This gives a Γ-vector bundle χ of rank r over a point and thena vector bundle χΓ = (VΓ → BΓ) of rank r over BΓ. Its equivariant Euler classeΓ(χ) is an element of Hr

Γ(pt) = Hr(BΓ). Its vanishing is related to the existenceof a non-zero fixed vector in V , as seen in the following lemma.


Lemma 43.16. If V Γ 6= 0, then eΓ(χ) = 0.

Proof. A non-zero fixed vector 0 6= v ∈ V Γ determines a nowhere-zero sectionof p : VΓ → BΓ (see (5) p. 230). This implies that eΓ(χ) = 0 by Lemma 28.61.

We now give a few recipes to compute an equivariant Euler class. A Γ-equivariant vector bundle p : E → X is called rigid if the Γ-action on X is trivial.

Lemma 43.17. Let ξ = (p : E → X) be a rigid Γ-vector bundle of rank r. Letχ : Γ → GL(Ex) be the representation of Γ on the fiber Ex over x ∈ X. Suppose

that Hk(X) = 0 for k < r. Then, the following equation

eΓ(ξ) = 1× e(ξ) + eΓ(χ)× 1

holds in HrΓ(X) = Hr(BΓ×X).

Proof. The inclusion j : x → X satisfies j∗ξ = χ. It induces jΓ : BΓ =xΓ → XΓ satisfying j∗ΓξΓ = χΓ. Hence, H∗jΓ(eΓ(ξ)) = eΓ(χ). By construction ofξΓ, one has i∗ξΓ = ξ, where i : X → XΓ denotes the inclusion. Hence, H∗i(eΓ(ξ)) =e(ξ). Using the homeomorphism XΓ ≈ BΓ × X , the maps jΓ and i are sliceinclusions. The lemma then follows from Corollary 28.5.

Let χ : Γ → O(1) be a continuous homomorphism, permitting Γ to act on R.This gives a Γ-line bundle χ over a point (see Example 43.15). Its equivariantEuler class lives in H1

Γ(pt) = H1(BΓ). As χ is continuous, it factors through thehomomorphism π0χ : π0(Γ) → O(1) ≈ Z2. As EΓ is contractible, the homotopyexact sequence of Γ → EΓ → BΓ identifies π0(Γ) with π1(BΓ). One thus gets(using Lemma 24.1) the isomorphism(43.18)

homcont(Γ, O(1)) ≈ hom(π0(Γ),Z2) ≈ hom(π1(BΓ),Z2) ≈ H1(BΓ) = H1Γ(pt) .

Lemma 43.19. Under the isomorphism of (43.18), one has eΓ(χ) = χ.

Proof. Note that EΓ×Γ O(1)→ BΓ is a 2-fold covering and that

E(χ) = EΓ×Γ R = (EΓ×Γ O(1))×O(1) R .

Then, EΓ ×Γ O(1) → BΓ is the sphere bundle S(χΓ) for the Euclidean structureon χΓ given by the standard Euclidean structure on R. By Proposition 28.58, theEuler class eΓ(χ) coincides with the characteristic class w(S(χΓ)) of the two-coveringS(χΓ)→ BΓ. But EΓ×Γ O(1) ≈ B kerχ and thus

π1(S(χΓ)) = π1(B kerχ) = π0(kerχ) = kerπ0χ = kerπ1Bχ .

Therefore, S(χΓ) → BΓ is the 2-covering with fundamental group to kerπ1Bχ ⊂π1(BΓ). Diagram (24.5) then implies that w(S(χΓ)) = χ.

A discrete group Γ is a 2-torus if it is finitely generated and if every elementhas order 2. It follows that Γ is isomorphic to ±1m, the integer m being calledthe rank of Γ. As seen in § 42,

(43.20) BΓ ≃ B(±1m) ≃ (B±1)m ≃ (RP∞)m .

Hence, H∗Γ(pt) is isomorphic to a polynomial algebra

(43.21) H∗Γ(pt) ≈ H∗((RP∞)m) ≈ H∗(RP∞)⊗· · ·⊗H∗(RP∞) ≈ Z2[u1, . . . , um] ,

where degree(ui) = 1. Under the identifications of (43.18), ui ∈ H1Γ(pt) corre-sponds to the homomorphism ±1n → ±1 which is the projection onto the ithfactor.


Example 43.22. Let Γ be the 2-torus formed by the diagonal matrices ofO(n). Then H∗(BΓ) ≈ Z2[u1, . . . , un], where ui ∈ H1(BΓ) corresponds to thehomomorphism πi : Γ → ±1 given by the i-th diagonal entry. The inclusionχ : Γ→ O(n) provides a Γ-equivariant vector bundle χ of rank n over a point. Notethat χ is a direct sum of 1-dimensional representations πi. Using Lemmas 43.12and 43.19, we get that

w(χ) =

n∏

i=1

(1+ui) .

Lemma 43.23. Let χ : Γ→ GL(V ) be a representation of a 2-torus Γ on a finitedimensional vector space V . Then the following two conditions are equivalent.

(1) V Γ = 0.(2) eΓ(χ) 6= 0.

Proof. That (2) implies (1) is given by Lemma 43.16. To prove the converse,we use the fact that χ is diagonalizable, with eigenvalues ±1: indeed, this is truefor a linear involution (see Example 39.26) and, if a, b ∈ GL(V ) commute, then bpreserves the eigenspaces of a. Thus, V = V1 ⊕ · · ·Vr and Γ acts on Vj through ahomomorphism χj : Γ→ ±1 = O(1). Hence, χ is the Whitney sum χ1⊕· · ·⊕ χr.Therefore,

eΓ(χ) = e(χΓ)

= e((χ1)Γ ⊕ · · · ⊕ (χr)Γ) by Lemma 43.12

= e((χ1)Γ) · · · e((χr)Γ) by Proposition 28.62

= χ1 · · · χr by Lemma 43.19.

The condition V Γ = 0 implies that none of the χj vanishes. As H∗(BΓ) is apolynomial algebra, this implies that eΓ(χ) 6= 0.

Proposition 43.24. Let ξ = (Ep−→ X) be a rigid Γ-vector bundle of rank r,

where Γ is a 2-torus. Suppose that EΓ consists only of the image of the zero section.Then, the cup-product with the equivariant Euler class

H∗Γ(X)−eΓ(ξ)−−−−−−→ H∗+rΓ (X)

is injective.

Proof. Without loss of generality, we may suppose that X is path-connectedand non-empty. Let x ∈ X . Consider the slice inclusion s : BΓ → BΓ × X withimage BΓ × x. Then, H∗Γs(eΓ(ξ)) = eΓ(ξx), where ξx = (Ex → x) is therestriction of ξ over the point x. As X is path-connected, Lemma 28.4 implies thatthe component of eΓ(ξ) ∈ Hr(BΓ⊗X) in Hr(BΓ)×H0(X) is equal to eΓ(ξx)× 1(as BΓ is of finite cohomology type, we identify Hr(BΓ×X) with H∗(BΓ)⊗H∗(X)by the Kunneth theorem).

Now let 0 6= a ∈ Hk(BΓ ×X). We isolate its minimal component amin by theformula

a = amin +A

with

0 6= amin ∈ Hk−p(BΓ)⊗Hp(X) and A ∈⊕

q>p

Hk−q(BΓ)⊗Hq(X) .


Then

a eΓ(ξ) = amin (eΓ(ξx)⊗ 1) +A′

with

amin (eΓ(ξx)⊗ 1) ∈ Hk−p+r(BΓ)⊗Hp(X) and A′ ∈M

q>p

Hk−q+r(BΓ)⊗Hq(X) .

Therefore it suffices to prove that amin (eΓ(ξx) ⊗ 1) 6= 0. The condition onEΓ implies that EΓ

x = 0 and thus, by Lemma 43.23, eΓ(ξx) 6= 0. As H∗(BΓ)is a polynomial algebra, this implies that a eΓ(ξ) 6= 0. Let B be a basis ofHk−p(BΓ) and C be a basis of Hp(X). Then, b ⊗ c | (b, c) ∈ B × C is a basis ofHk−p(BΓ)⊗Hp(X). As H∗(BΓ) is a polynomial algebra, the family b eΓ(ξx) |b ∈ B is free in Hk−p+r(BΓ). Hence, if

0 6= amin =∑

(b,c)∈B×C

λbc (b ⊗ c) (λbc ∈ Z2) ,

then

amin (eΓ(ξx)⊗ 1) =∑

(b,c)∈B×C

λbc((b eΓ(ξx))⊗ c

)6= 0 .

Statements 43.19, 43.23 and 43.24 have analogues, replacing O(1) by SO(2)and 2-tori by tori. Let Γ be a topological group and let χ : Γ → SO(2) be acontinuous homomorphism, making Γ to act on R2. This gives a Γ-vector bundleχ of rank 2 over a point (see Example 43.15). Its equivariant Euler class lives inH2

Γ(pt) = H2(BΓ). As SO(2) ≈ S1, one has BSO(2) ≈ CP∞ (see Example 40.1).Define

(43.25) κ(χ) = H∗Bχ(ı) ∈ H2Γ(pt) = H2(BΓ) ,

where ı is the non-zero element of H2(BSO(2)) = Z2.

Lemma 43.26. eΓ(χ) = κ(χ).

Proof. The homomorphism χ : Γ→ SO(2) makes SO(2) a Γ-space. The mapEχ : EΓ → ESO(2) descends to continuous maps EΓ ×Γ SO(2) → ESO(2) andthere is a commutative diagram

(43.27)

EΓ×Γ SO(2) //

ESO(2) oo ≈ //

S∞

BΓ

Bχ // // BSO(2) oo ≈ // CP∞

Note that EΓ×Γ SO(2)→ BΓ is an SO(2)-principal bundle and that

E(χ) = EΓ×Γ R2 = (EΓ×Γ SO(2))×SO(2) R2 .

As SO(2) ≈ S1, EΓ×ΓSO(2) is the sphere bundle S(χΓ) for the Euclidean structureon χΓ given by the standard Euclidean structure on R2. Diagram (43.27) impliesthat S(χΓ) is induced by Bχ from the Hopf bundle S∞ → CP∞, whose Euler classin ı ∈ H2(CP∞) (see Proposition 35.7). Hence,

eΓ(χ) = e(S(χΓ)) = H∗Bχ(ı) = κ(χ) .


A torus Γ is a Lie group isomorphic to (S1)m, the integer m being called therank of Γ. For instance, SO(2) is a torus of rank 1. As seen in § 42,

(43.28) BΓ ≃ B((S1)m) ≃ (BS1)m ≃ (CP∞)m .

Hence, H∗Γ(pt) is isomorphic to a polynomial algebra

(43.29) H∗Γ(pt) ≈ H∗((CP∞)m) ≈ H∗(CP∞)⊗ · · ·⊗H∗(CP∞) ≈ Z2[v1, . . . , vm] ,

where degree(vi) = 2. One has vi = κ(χi) where χi : (S1)n → S1 ≈ SO(2) isthe projection onto the ith factor. If Γ is a torus, its associated 2-torus Γ2 is thesubgroup of elements of order 2 in Γ.

Lemma 43.30. Let Γ be a torus and Γ2 be its associated 2-torus. Let χ : Γ →GL(V ) be a representation of Γ on a finite dimensional vector space V . Then thefollowing two conditions are equivalent.

(1) V Γ2 = 0.(2) eΓ(χ) 6= 0.

Moreover, if (1) or (2) holds true, then dimV is even.

Proof. As Γ is a torus, χ(Γ) is contained in a maximal torus T of GL(V ).Those are all conjugate (see [20, § IV.1]). If dim V = 2s+1, there is an isomorphismV ≈ R2⊕· · ·⊕R2⊕R intertwining T with SO(2)×· · ·SO(2)×1 (see [20, Chapter IV,Theorem 3.4]). This contradicts the condition V Γ 6= 0. We can then suppose thatdimV = 2s, in which case there is an isomorphism V ≈ R2 ⊕ · · · ⊕ R2 conjugatingT with SO(2)× · · ·SO(2). The homomorphism χ takes the form χ = (χ1, . . . , χs)where χj : Γ → SO(2). Hence χ = χ1 ⊕ · · · ⊕ χs and, using Lemma 43.12 andProposition 28.62,

eΓ(χ) = eΓ(χ1) · · · eΓ(χs) = κ(χ1) · · · κ(χs) ,

the last equality coming from Lemma 43.26. Since H∗(BΓ) is a polynomial algebra,the condition eΓ(χ) 6= 0 is equivalent to κ(χj) 6= 0 for all j. The condition V Γ =

0 = V Γ2 is equivalent to V Γj = 0 = V Γ2

j for all j, where Vj is the 2-dimensional

vector space corresponding to the jth factor R2 in the decomposition of V .We are thus reduced to the case dimV = 2 and χ : Γ→ SO(2). We start with

preliminaries. Choose isomorphisms Γ ≈ (S1)m, SO(2) ≈ S1 and S1 ≈ R/Z. Weget a commutative diagram

(43.31)

Zm // //

π1χ

Rm // //

χ

(S1)m

χ

Z // // R // // S1

where the vertical arrows are homomorphisms. Therefore, χ(x1, . . . , xm) =∑

i bixiwith bi ∈ Z and

(43.32) χ(γ1, . . . , γm) = γb11 · · ·γbmm .

We deduce that

(43.33) V Γ = 0 ⇐⇒ π1χ non-trivial ⇐⇒ χ surjective⇐⇒ bj 6= 0 ∀ j .If χ is surjective, one gets a fiber bundle kerχ→ Γ→ S1 and, using its homotopyexact sequence and (43.32), we get

(43.34) V Γ = 0 =⇒ cokerπ1χ ≈ π0(kerχ) ≈ Z/gcd(b1, . . . , bm)Z .


Since 0 ∈ V Γ ⊂ V Γ2 , the condition V Γ2 = 0 implies that V Γ = 0. Hence,using (43.31)–(43.34),

(43.35)

V Γ2 = 0 ⇐⇒ Γ2 ⊂ kerχ

⇐⇒ 12Z ⊂ kerπ1χ

⇐⇒ 2 | gcd(b1, . . . , bm)

⇐⇒ hom(π0(kerχ); Z2) = 0

.

As in the proof of Lemma 43.26, we consider S = EΓ ×Γ S1, which is the total

space for the sphere bundle S(χ). One has a commutative diagram

Γ

χ // EΓ

// BΓ

=

S1 // S // BΓ

whose rows are fiber bundles. Passing to the homotopy exact sequences, we get acommutative diagram

π2(BΓ)

=

≈ // π1(Γ)

π1χ

// 0

π2(BΓ)

≈ // π1(S1) // π1(S) // 0

whose rows are exact sequences. Hence,

(43.36) π1(S) ≈ cokerπ1χ

Now, the Gysin sequence for S → BΓ gives

(43.37) H1(BΓ)︸︷︷︸0

→ H1(S)→ H0(BΓ)︸︷︷︸Z2

eΓ(χ)−−−−−→ H2(BΓ) .

By Lemma 43.16, one knows that eΓ(χ) 6= 0 implies V Γ = 0. Therefore, us-ing (43.34)–(43.37),

(43.38)

eΓ(χ) 6= 0 ⇐⇒ H1(S) = 0

⇐⇒ hom(π1(S); Z2) = 0

⇐⇒ hom(cokerπ1χ; Z2) = 0

⇐⇒ hom(π0(kerχ); Z2) = 0

⇐⇒ V Γ2 = 0

.

Proposition 43.39. Let Γ be a torus and Γ2 be its associated 2-torus. Letξ = (p : E → X) be a rigid Γ-vector bundle of rank r. Suppose that EΓ2 consistsonly of the image of the zero section. Then r is even and the cup-product with theequivariant Euler class

H∗Γ(X)−eΓ(ξ)−−−−−−→ H∗+rΓ (X)

is injective.

Proof. The proof is the same as that of Proposition 43.24, using Lemma 43.30instead of Lemma 43.23.

44. EQUIVARIANT MORSE-BOTT THEORY 259

44. Equivariant Morse-Bott Theory

Let f : M → R be a smooth function defined on a smooth manifold M . A pointx ∈ M is critical for f if df(x) = 0. Let Crit f ⊂ M be the subspace of criticalpoints for f . Then, f(Crit f) ⊂ R is the set of critical values of f . We say that fis Morse-Bott if the following two conditions holds

• Crit f is a disjoint union of submanifolds. A connected component ofCrit f is called a critical manifold of f .• the kernel of the Hessian Hx at a critical point x equals the tangent space

to the critical manifold N containing x.

This definition coincides with that of a Morse function when Crit f is a discrete set.See e.g. [148, 146, 93, 17, 12] for presentations of Morse and Morse-Bott theory.The index of x ∈ Crit f is the number of negative eigenvalues of Hx. This numberis constant over a critical manifold and thus defines a function ind: π0(Crit f)→ N.Also, the normal bundle νN to a critical manifold N decomposes into a Whitneysum νN ≈ ν−N⊕ν+N of the negative and positive normal bundles, i.e. the bundlesspanned at each x ∈ N respectively by the negative and positive eigenspaces of Hx.Note that rank ν−(N) = indN .

A (continuous) map g : X → Y is called proper if the pre-image of any compactset is compact. For instance, if X is compact, then any map g is proper. Letf : M → R be a proper Morse-Bott function and let a < b be two regular values.Define Ma,b = f−1([a, b]), a compact manifold whose boundary is the union ofMa = f−1(a) and Mb = f−1(b). Denote by fa,b the restriction of f to Ma,b. TheMorse-Bott polynomial Mt(Ma,b) is defined by

Mt(fa,b) =∑

N∈π0(Crit fa,b)

t indNPt(N)

(the sum is finite since, as f is proper, Crit fa,b is compact).

Proposition 44.1 (Morse-Bott inequalities). There is a polynomial Rt, withpositive coefficients, such that

(44.2) Mt(fa,b) = Pt(Ma,b,Ma) + (1 + t)Rt .

Equation (44.2) implies that the coefficients of Mt(fa,b) are greater or equalto those of Pt(Ma,b,Ma) (whence the name of Morse-Bott inequalities). For theequivalence of (44.2) with other classical and more subtle forms of the Morse-Bottinequalities (see [12, § 3.4]).

Proof. The map fa,b has a finite number of critical values, all in the interiorof [a, b]. Let a = a0 < a1 < · · · < ar = b be regular values such that [ai, ai + 1]contains a single critical value. We shall prove by induction on i that (44.2) holdstrue for fa,ai

. The induction starts trivially for i = 0, with the three terms of (44.2)being zero.

As [ai, ai+1] contains a single critical level, there is a homotopy equivalence

(44.3) Mai,ai+1 ≃Mai∪Si

Di

where (Di, Si) is the disjoint union over N ∈ π0(Crit fai,ai+1) of the pairs formedby the disk and sphere bundles of ν−(N) (see [17, pp. 339–344]). By excision and


the Thom isomorphism,

(44.4) H∗(Mai,ai+1 ,Ma) ≈ H∗(Di, Si) ≈∏

N∈π0(Crit fai,ai+1)

H∗−indN (N) .

Therefore,

(44.5) Mt(fai,ai+1) = Pt(Mai,ai+1,Mai) .

On the other hand, Corollary 12.44 applies to the triple (Ma,ai+1 ,Ma,ai,Ma) gives

the equality

(44.6) Pt(Ma,ai+1 ,Ma) + (1 + t)Qt = Pt(Ma,ai+1 ,Ma,ai) + Pt(Ma,ai

,Ma) ,

for some Qt ∈ N[t]. By excision and (44.5), one gets

(44.7) Pt(Ma,ai+1 ,Ma,ai) = Pt(Mai,ai+1,Mai

) = Mt(fai,ai+1) .

Thus, (44.6) and (44.7) provide the induction step.

A proper Morse-Bott function f : M → R is called perfect if for any two regularvalues a < b, Equation (44.2) reduces to

(44.8) Mt(fa,b) = Pt(Ma,b,Ma) .

The easiest occurrence of perfectness is the following lacunary principle.

Lemma 44.9. Suppose that no consecutive powers of f occur in Mt(f). Then,f is perfect.

Proof. Suppose that Rt 6= 0 in (44.2). Then, two successive powers of t occurin (1 + t)Rr. The same happens then in Mt(fa,b), and then in Mt(f).

Other simple criteria for perfectness are given by the following three results.For a regular value x of f : M → R, set Wx = f−1(−∞, x].

Lemma 44.10. Let f : M → R be a proper Morse-Bott function. Then, thefollowing two conditions are equivalent.

(1) f is perfect.(2) For any regular values a < b < c of f , the cohomology exact sequence of

the triple (Wc,Wb,Wa) cuts into a global short exact sequence

0→ H∗(Wc,Wb)→ H∗(Wc,Wa)→ H∗(Wb,Wa)→ 0 .

Proof. Suppose that f is perfect. Then, by excision,

Pt(Wc,Wa) = Pt(Ma,c,Ma) = Mt(fa,c)

and analogously for Pt(Wb,Wa) and Pt(Wc,Wb). As

Mt(fc,a) = Mt(fc,b) + Mt(fb,a) ,

one has

Pt(Wc,Wa) = Pt(Wc,Wb) + Pt(Wb,Wa) .

By Corollary 12.44 and its proof, this implies that H∗(Wb,Wa) → H∗(Wc,Wb) issurjective, whence (2).

Conversely, suppose that (2) holds true. For two regular values a < c, we provethat Pt(Ma,c,Ma) = Pt(Wc,Wa) = Mt(fa,c), by induction on the number na,c ofcritical values in the segment [a, c]. This is trivial for na,c = 0, since then Ma,c

is then diffeomorphic to Ma × [a, c] (see [93, Chapter 6, Theorem 2.2]). When


na,c = 1, one uses (44.5). For the induction step, when na,c ≥ 2, choose a regularvalue b ∈ (a, c) such that na,b = na,c − 1. Then,

Pt(Wc,Wa) = Pt(Wc,Wb) + Pt(Wb,Wa) by (2)

= Mt(fc,b) + Mt(fb,a) by induction hypothesis

= Mt(fa,c) ,

which proves the induction step.

Lemma 44.11. Let f : M → R be a proper Morse-Bott function. Then if forany two regular values a < b, one has

(44.12) dimH∗(Ma,b,Ma) ≤ dimH∗(Crit fa,b)

and f is perfect if and only if (44.12) is an equality.

Proof. The evaluation of (44.2) at t = 1 implies (44.12) and the equality isequivalent to Rt = 0.

In the case where M is a closed manifold, one has the following result.

Proposition 44.13. Let f : M → R be a Morse-Bott function, where M is aclosed manifold. Then f is perfect if and only if

(44.14) dimH∗(M) = dimH∗(Crit f) .

Proof. Equation (44.12) implies (44.14) when f(M) ⊂ (a, b). Conversely, leta < b be two regular values of f . Let a′ < a and b′ > b such that f(M) ⊂ (a′, b′).Using Corollary 12.46 and excision, we get

(44.15)dimH∗(M) = dimH∗(Ma′,b′ ,Ma′)

≤ dimH∗(Ma′,b,M′a) + dimH∗(Ma′,b′ ,Ma′,b)

= dimH∗(Ma′,b,M′a) + dimH∗(Mb,b′ ,Mb) .

Doing the same for dimH∗(Ma′,b,M′a) and using (44.2) gives

(44.16)dimH∗(M) ≤ dimH∗(Ma′,a,M

′a) + dimH∗(Ma,b,Ma) + dimH∗(Mb,b′ ,Mb)

≤ dimH∗(Crit fa′,a) + dimH∗(Crit fa,b) + dimH∗(Crit fb,b′)= dimH∗(Crit f) .

Now, if (44.14) holds true, then all the inequalities occurring in (44.15) and (44.15)are equalities, including dimH∗(Ma,b,Ma) = dimH∗(Crit fa,b).

Theorem 44.17. Let M be a smooth Γ-manifold, where Γ is a 2-torus. Letf : M → R be a proper Γ-invariant Morse-Bott function which is bounded below.Suppose that Crit f = MΓ. Then

(1) f is perfect.(2) M is Γ-equivariantly formal.(3) the restriction morphism H∗Γ(M)→ H∗Γ(MΓ) is injective.

Remark 44.18. When Γ = ±1, Theorem 44.17 follows from Smith theory.Indeed, for any regular values a < b of f ,

dimCrit fa,b = dimH∗(MΓa,b) by hypothesis

≤ dimH∗(Ma,b) by Proposition 41.9

≤ dimCrit fa,b by Lemma 44.11.


Therefore, the above inequalities are equalities and f is perfect by Lemma 44.11and equivariantly formal by Proposition 41.9. Point (3) then follows from Proposi-tion 41.12.

Remark 44.19. Under the hypotheses of Theorem 44.17, when Γ = ±1 andf is a Morse function, M. Farber and D. Schutz have proven that each integralhomology group Hi(M ; Z) is free abelian with rank equal to the number of criti-cal points of index i [58, Theorem 4]. By the universal coefficients theorem [80,Theorem 3.2], such a function is perfect.

Proof of Theorem 44.17. For a regular value x of f , setWx = f−1(−∞, x].We first prove that

(44.20) H∗Γ(Wx)→ H∗Γ(WΓx ) is injective for all regular value x.

This is proven by induction on the number nx of critical values in the interval(−∞, x], following the argument of [194, proof of Proposition 2.1]. If nx = 0, thenWx = ∅ and H∗Γ(Wx) = 0, which, as f is bounded below, starts the induction.

Suppose that n ≥ 1 and that (44.20) holds true when nx < n. Let y be a regularvalue of f with ny = n. Choose z < y such that nz = n− 1 (this is possible sincethe set of critical values is discrete Morse-Bott function is discrete).. As in (44.3),one has a homotopy equivalence

Mz,y ≃My ∪S D ,

where (D,S) is the disjoint union over N ∈ π0(Crit fz,x) of the pairs formed by thedisk and sphere bundles of ν−(N). Using (44.4) and the proof of Proposition 28.54,we get the commutative diagram(44.21)

H∗Γ(Wy ,Wz) oo ≈excision

H∗Γ(Mz,y,Mz) oo ≈excision

H∗Γ(D,S) oo ≈Thom

∏N H

∗−indNΓ (N)

−(eΓ(ν−(N))

)

H∗Γ(Wy) // H∗Γ(Mz,y) // H∗Γ(MΓz,y)

≈ // ∏N H

∗Γ(N)

where N runs over π0(Crit fz,y) = π0(MΓz,y). As MΓ = Crit f , the linear Γ-action

of ν−(N) has fixed point set consisting only of the image of the zero section. ByLemma 43.24, the right vertical arrow of (44.21) is injective. Thus, we deducefrom (44.21) that H∗Γ(Wy,Wz)→ H∗Γ(Wy) is injective. This cuts the Γ-equivariantcohomology sequence of (Wy,Wz) into short exact sequences. The same cuttingoccurs for the pair (WΓ

y ,WΓz ) using Proposition 12.35, and one has a commutative

diagram

(44.22)

0 // H∗Γ(Wy ,Wz) //

rz,y

H∗Γ(Wy) //

ry

H∗Γ(Wz) //

rz

0

0 // H∗Γ(MΓz,y) // H∗Γ(WΓ

y ) // H∗Γ(WΓz ) // 0

where the vertical arrows are induced by the inclusions. The left vertical arrow isinjective by (44.21). Since nz = n − 1, the right one is injective by induction hy-pothesis. By diagram-chasing, we deduce that the middle vertical arrow is injective,which proves (44.20).


Warning: As WΓy is the disjoint union of MΓ

z,y and WΓz , one has H∗Γ(WΓ

y ) ≈H∗Γ(MΓ

z,y) ⊕ H∗Γ(WΓz ). Consider the image Im ry of ry under under this decom-

position. The above arguments imply that Im rz,y × 0 ⊂ Im ry. But, in general0× Im rz 6⊂ Im ry (see Example 44.27 below).

We deduce Point (3) from (44.20). Indeed as M =⋃xWx Corollary 12.22

provides a commutative diagram

(44.23)

H∗Γ(M)

≈ // lim←−

x

H∗Γ(Wx)

H∗Γ(MΓ)≈ // lim

←−x

H∗Γ(WΓx )

.

As the right vertical arrow is injective by (44.20), so is the left one.For Point (2), we first prove that ρx : H∗Γ(Wx) → H∗(Wx) is surjective for

all regular value x. This is also done by induction on nx, starting trivially whennx = 0. For the induction step, consider as above two regular values z < y suchthat ny = nz + 1. The cohomology exact sequences of the pair (Wy ,Wz) give thecommutative diagram(44.24)

0 // HkΓ(Wy ,Wz) //

ρy,z

HkΓ(Wy) //

ρy

HkΓ(Wz) //

ρz

0

Hk−1(Wz) // Hk(Wy ,Wz) // Hk(Wy) // Hk(Wz) // Hk+1(Wy ,Wz)

,

the top sequence being cut as seen above. Similarly to (44.21), we get a commutativediagram(44.25)

H∗Γ(Wy,Wz) oo ≈excision

ρy,z

H∗Γ(Mz,y,Mz) oo ≈excision

H∗Γ(D,S) oo ≈Thom

∏N H

∗−indNΓ (N)

ρCrit

H∗(Wy,Wz) oo ≈

excisionH∗(Mz,y,Mz) oo ≈

excisionH∗(D,S) oo ≈

Thom

∏N H

∗−indNΓ (N)

.

Since Crit f ⊂ MΓ, the map ρCrit is surjective and so is ρy,z. If ρz is surjectiveby induction hypothesis, a diagram-chasing argument proves that ρy is surjective.Now, recall that, for a Γ-space X , ρ : : H∗Γ(X)→ H∗(X) is equal to H∗i for somefiber inclusion i : X → XΓ. One can thus consider the Kronecker dual ρ∗ : H∗(X)→H∗(XΓ). One has a commutative diagram

(44.26)

lim−→

x

H∗(Wx)

lim−→

ρx,∗

≈ // H∗(M)

ρ∗

lim−→

x

H∗((Wx)Γ)≈ // H∗(MΓ)

.


As ρx is surjective, ρx,∗ is injective and thus lim−→

ρx,∗ is injective. By Diagram (44.26)

and Kronecker duality, ρ : H∗Γ(M) → H∗(M) is surjective and M is equivariantlyformal.

Let us finally prove Point (1). Consider two regular values z < y such thatny = nz + 1. As in (44.24) the vertical maps are surjective, the cohomology exactsequence of (Wy,Wz) cuts into a global short exact sequence. By the proof that (2)implies (1) in Lemma 44.10, this implies that f is perfect.

Example 44.27. Consider the action of Γ = ±1 on M = Sn ⊂ R×Rn givenby γ · (t, x) = (t, γ x), with fixed points p± = (±1, 0). Note that M is a sphere withlinear involution Sn0 in the sense of Example 39.26. The formula f(t, x) = t definesa Morse function M satisfying the hypotheses of Theorem 44.17. Taking z = 0 andy = 2 as regular values of f , one has Wy = M and Diagram (44.22) becomes

0 // H∗Γ(M,W0)j //

r+

H∗Γ(M) //

r

H∗Γ(W0) //

r−≈

0

0 // H∗Γ(p+) // H∗Γ(p+)⊕H∗Γ(p−) // H∗Γ(p−) // 0

.

Set H∗(BΓ) = Z2[u]. By Lemma 43.23, eΓ(ν−(p+)) = un. Together with Dia-gram (44.21), this shows that rj(U) = (un, 0), where

U ∈ HnΓ (M,W0) ≈ Hn

Γ (M0,2,M0) ≈ H0Γ(p+)

is the Thom class of ν−(p+). Let B = j(U). Using the diagram

H0Γ(p+)

Thom

≈//

≈

HnΓ (M,W0)

j //

HnΓ (M)

ρ

H0(p+)

Thom

≈// Hn(M,W0)

≈ // Hn(M)

one sees that ρ(B) is the generator of Hn(Sn) = Z2. Hence, ρ is surjective, asexpected by Theorem 44.17. By the Leray-Hirsch theorem, H∗Γ(M) is the free Z[u]-module generated by B. Now, r(B) = (un, 0) and r(u) = (u, u). As is r injective byTheorem 44.17, the relation B2 = unB holds true in H∗Γ(M). Given the dimensionof Hk

Γ(M), this establishes the GrA[u]-isomorphism

Z2[B, u]/(B2 + unB) ≈ H∗Γ(M)

Taking the image by ρ adds the relation u = 0 and we recover that H∗(Sn) ≈Z2[B]

/(B2). Note that (0, u) is not in the image of r, confirming the warning in the

proof of Theorem 44.17. But (0, un) = r(B+un) is in the image of r, correspondingto the generator A = B + un of H∗Γ(M) (see Example 39.30). Had we considered−f instead of f , the above discussion would have selected the generator A first.The relation A2 = unA also holds true and recall from Example 39.30 that H∗Γ(M)admits the presentation

Z2[u,A,B]/(A2 + unA,A+B + un) ≈ H∗Γ(M) .


Example 44.28. Let Γ be a 2-torus and let χ : Γ→ ±1 be a homomorphism,identified with χ ∈ H1(BΓ) under the bijection (43.18). Consider the Γ-action onM = S1 ⊂ R2 given by γ · (t, x) = (t, χ(γ)x), with fixed points p± = (±1, 0). Wecall M a χ-circle. As in Example 44.27, one sees that the image of

r : H∗Γ(M)→ H∗Γ(p±) ≈ H∗Γ(p−)⊕H∗Γ(p+) ≈ H∗(BΓ)⊕H∗(BΓ)

is the H∗(BΓ)-module generated by B = (χ, 0) and that H∗Γ(M) admits the pre-sentation

H∗Γ(M) ≈ H∗(BΓ)[B]/(B2 + χB) .

Moreover, the image of r is the set of classes (a, b) such that b − a is a multipleof χ.

Theorem 44.17 admits the following analogue for torus actions.

Theorem 44.29. Let Γ be a torus and Γ2 be its associated 2-torus. Let M bea smooth Γ-manifold. Let f : M → R be a proper Γ-invariant Morse-Bott functionwhich is bounded below. Suppose that Crit f = MΓ = MΓ2 . Then

(1) f has only critical manifolds of even index. In particular, if f is a Morsefunction, then M is of even dimension.

(2) f is perfect.(3) M is Γ-equivariantly formal.(4) the restriction morphism H∗Γ(M)→ H∗Γ(MΓ) is injective.

Proof. The proof is the same as that of Theorem 44.17. The hypothesisCrit f = MΓ implies that the negative normal bundles are Γ-vector bundles andthe hypothesis Crit f = MΓ2 permits us to use Proposition 43.39 instead of Propo-sition 43.24.

In Theorem 44.29, note that the perfectness of f is implied by (1). WhenΓ = S1, Points (3) and (4) follows from Smith theory, in the same way as inRemark 44.18.

Example 44.30. Let Γ be a torus with associated 2-torus Γ2. Let χ : Γ → S1

be a continuous homomorphism. Consider the Γ-action on M = S2 ⊂ R× C givenby γ · (t, x) = (t, χ(γ)x), with fixed points p± = (±1, 0). We call M a χ-sphere.Let us assume that the restriction of χ to the associated 2-torus Γ2 of Γ is nottrivial. This implies that MΓ = MΓ2 , so we can apply Theorem 44.29 and, as inExample 44.27, one sees that the image of

r : H∗Γ(M)→ H∗Γ(p±) ≈ H∗Γ(p−)⊕H∗Γ(p+) ≈ H∗(BΓ)⊕H∗(BΓ)

is theH∗(BΓ)-module generated by B = (κ(χ), 0), where κ(χ) ∈ H2(BT ) is definedin (43.25). Also, H∗Γ(M) admits the presentation

H∗Γ(M) ≈ H∗(BΓ)[B]/(B2 + κ(χ)B) .

Moreover, the image of r is the set of classes (a, b) such that b − a is a multipleof κ(χ).

A consequences of Theorem 44.17 an 44.29 are the surjectivity theorems a laKirwan (see Remark 44.39 below). For f : M → R is a continuous map, we set


M− = f−1((−∞, 0]), M+ = f−1([0 − ∞)) and M0 = M− ∩M+ = f−1(0). Theinclusions form a commutative diagram.

(44.31)

M0

i+ //

i−

i

""EEE

EEEE

M+

j+

M−

j− // M

.

Proposition 44.32. Let M be a closed smooth Γ-manifold, where Γ is a 2-torus. Let f : M → R be a Γ-invariant Morse-Bott function satisfying Crit f = MΓ.Suppose that 0 is a regular value of f . Then

H∗Γi : H∗Γ(M)→ H∗Γ(M0)

is surjective and its kernel is the ideal kerH∗Γj−+kerH∗Γj+, generated by kerH∗Γj−and kerH∗Γj+.

For applications of this proposition, see § 62.2.

Proof. We use the abbreviations i∗± = H∗Γi±, j∗± = H∗Γj±, etc. As M is com-pact, f is proper and bounded. By Theorem 44.17, the restriction homomorphismH∗Γ(M)→ H∗Γ(MΓ) = H∗Γ(Crit f) is injective. The following commutative diagram

H∗Γ(M)(j∗−,j

∗+)//

H∗Γ(M−)⊕H∗Γ(M+)

H∗Γ(MΓ)≈ // H∗Γ(MΓ

−)⊕H∗Γ(MΓ+)

shows that the Mayer-Vietoris sequence in equivariant cohomology for Diagram (44.31)splits into a global short exact sequence

(44.33) 0→ H∗Γ(M)(j∗−,j

∗+)−−−−−→ H∗Γ(M−)⊕H∗Γ(M+)

i∗−+i∗+−−−−→ H∗Γ(M0)→ 0 .

Suppose that x1 < x2 < x3 < . . . are regular values of f such that fxi,xi+1 hasonly one critical level (we use the notations of Theorem 44.17 and its proof). Then,by (44.22), H∗Γ(Wi+1) → H∗Γ(Wi) is surjective. As W0 = M− and M is compact,this argument shows that j∗− is surjective. By symmetry, replacing f by −f (usingagain that M is compact), one also has that j∗+ is surjective.

Let a ∈ H∗Γ(M0). Using (44.33), choose a± ∈ H∗Γ(M±) such that a = i∗−(a−)+i∗−(a+). As j∗± is surjective, there exist b± ∈ H∗Γ(M) with i∗±(b±) = a±. Then

i∗(b− + b+) = i∗(b−) + i∗(b+) = i∗−j∗−(b−) + i∗+j

∗+(b+) = i∗−(a−) + i∗+(a+) = a ,

which proves that i∗ = H∗Γi is surjective. As i∗ = i∗±j±, we have also proven thati∗± is surjective. Therefore, one has a commutative diagram

(44.34)

H∗Γ(M,M+) // //

≈

H∗Γ(M)j∗+ // //

j∗−i∗

&& &&LLLLLLLLH∗Γ(M+)

i∗+H∗Γ(M−,M0) // // H∗Γ(M−)

i∗− // // H∗Γ(M0)


where the horizontal and vertical sequences are exact and the left hand verticalarrow is an isomorphism by excision. Hence, ker i∗ = (j∗−)−1(ker i∗−), which yeldsto an exact sequence

0→ ker j∗− → ker i∗ → ker i∗− → 0 .

But Diagram (44.34) provides a section of ker i∗ → ker i∗−, whose image is equal toker j∗+. This proves the assertion on ker i∗ (which is actually GrV-isomorphic toker j∗− ⊕ ker j∗+).

Example 44.35. Consider the action of Γ = ±1 on M = RPn given by

γ · [x0, . . . , xn] = [x0, . . . , xn−1, γxn] .

The Morse-Bott function defined on M by f([x0, . . . , xn]) = 1 − 2x2n satisfies the

hypotheses of Theorem 44.17 and Proposition 44.32. Set Mc = f−1(c). The criticalsubmanifolds are M±1 and one has M− ≃M−1 = pt and M+ ≃M1 = RPn−1. Letu ∈ H1(BΓ) = Z2 be the generator.

The bundle projection p : MΓ → BΓ and its restriction pc to (Mc)Γ give ele-ments H∗pc(u) = uc ∈ H∗Γ(Mc) and H∗p(u) = v ∈ H∗Γ(M). One has

H∗Γ(M−) ≈ Z2[u−1] and H∗Γ(M+) ≈ H∗(BΓ×M1) ≈ Z2[b, u1]/(bn) ,

with the degree of b equal to 1. Consider the following commutative diagram

H0Γ(M1)

Thom

≈//

α

33H1Γ(M,M−) // H1

Γ(M)

r1

ρ // H1(M)

≈

H1Γ(M1)

ρ1 // H1(M1)

.

By Diagram (44.21), r1 α(1) = eΓ(ν−), the equivariant Euler class of the (negative)

normal bundle ν−(M1). By Lemma 43.17,

eΓ(ν−) = 1× e(ν−(M1)) + e(χ)× 1 ∈ H1(BΓ×M1) ,

where χ is the representation of Γ on a fiber of ν−(M1). Note that ν−(M1) = ν(M1)is the canonical line bundle over RPn−1 (since M = RPn is obtained by attachingan n-cell to M1 = RPn−1 by the Hopf map (two fold covering) Sn−1 → RPn−1).Then, by Proposition 28.58 and its proof, e(ν−(M1)) = b. As χ has no non-zerofixed vector, eΓ(χ) = u by Lemma 43.23. Therefore,

eΓ(ν−) = 1× b+ u× 1 = b+ u1 ,

the last formula making sense in the presentation H∗Γ(M1) ≈ Z2[b, u1]/(bn). As

ρ1(b+u1) = b, this proves that ρ is surjective, as already known by Theorem 44.17.Let a = α(1) + v. One also has ρ1r1(a) = b so, by the Leray-Hirsch theorem,H∗Γ(M) is the free Z2[v]-module generated by a, a2, . . . , an−1 and the Poincare seriesof MΓ is

(44.36) Pt(MΓ) = Pt(M) ·Pt(BΓ) =1− tn+1

(1− t)2 .

By Diagram (44.22), r−1 α(1) = 0 in H1Γ(M−) ≈ H1

Γ(M−1). Therefore, the homo-morphism r : H∗Γ(M) → H∗Γ(MΓ) ≈ H∗Γ(M−1) ⊕ H∗Γ(M1) satisfies r(a) = (u−1, b)and r(v) = (u−1, u1). Finally, we claim that there is a GrA-isomorphism

(44.37) Z2[v, a]/(an+1 + va)

≈−→ H∗Γ(M) .


Indeed, one already knows that H∗Γ(M) is GrA-generated by v and a, and, usingthe injective homomorphism r, one checks that the relation an+1 = va holds true.This gives the GrA-morphism of (44.37) which is surjective, and hence bijectivesince both sides of (44.37) have the same Poincare series, computed in (44.36).

Replacing Theorem 44.17 by Theorem 44.29 in the proof of Proposition 44.32gives the following result.

Proposition 44.38. Let M be a closed smooth Γ-manifold, where Γ is a toruswith associated 2-torus Γ2. Let f : M → R be a Γ-invariant Morse-Bott functionsatisfying Crit f = MΓ = MΓ2 . Suppose that 0 is a regular value of f . Then

H∗Γi : H∗Γ(M)→ H∗Γ(M0)

is surjective and its kernel is the ideal kerH∗Γj−+kerH∗Γj+, generated by kerH∗Γj−and kerH∗Γj+.

As an example, one can take the complex analogue of Example 44.35, i.e.Γ = S1 acting on M = CPn given by γ · [x0, . . . , xn] = [x0, . . . , xn−1, γxn] and theMorse-Bott function f([x0, . . . , xn]) = 1− 2x2

n. All the formulae of Example 44.35hold true, with the degrees of all the classes multiplied by 2.

Remark 44.39. For Γ = S1, the hypotheses of Proposition 44.38 are realizedwhen f is the moment map of a Hamiltonian circle action (see [11]). In this case, itfollows from F. Kirwan’s thesis [115, § 5] that H∗Γ(M ; Q)→ H∗Γ(M0; Q) is surjective(see e.g. [194, Theorem 2]). This justifies the terminology of surjectivity theorem ala Kirwan used above to introduce Proposition 44.32 and 44.38. For the assertionon kerHΓi in these propositions, compare [195, Theorem 2]; our proofs followedthe hint of [195, Remark 3.5].

CHAPTER 7

The Steenrod squares

45. Cohomology operations

This section contains some generalities on the (mod 2) cohomology operations,in order to present the Steenrod squares. We take a global approach which mayshed a new light with respect to existing texts on the subject.

A cohomology operation is a map

Q = Q(X,Y ) : H∗(X,Y )→ H∗(X,Y )

defined for any topological pair (X,Y ), satisfying the following two conditions:

(1) Q is functorial, i.e. if g : (X ′, Y ′)→ (X,Y ) is a continuous map of pairs,then

(45.1) H∗gQ = QH∗g .

(2) Q(X,Y ) =∑Q[i](X,Y ) where Q[i](X,Y ) is the restriction of Q(X,Y )

to Hi(X,Y ).

We may restrict the definition to some classes of pairs, like CW-pairs, etc. Forinstance, restricting to pairs (X, ∅) gives operations on absolute cohomology, since

H∗(X, ∅) ≈−→ H∗(X). Point (2) is a partial linearity (Q is not supposed to be linear)and permits us to define Q via its Q[i].

Examples of cohomology operations are given by Q = 0 or Q = id . A lesstrivial example is the cohomology operation Q such that by Q[n](a) = an for alln ∈ N, where an = a · · · a (n times). Cohomology operations may be addedand composed, giving rise to more examples.

Here below a few remarks about cohomology operations. They are used through-out this section, without always an explicit notice.

45.2. By Theorem 18.2, a topological pair has, in a functorial way, the samecohomology as a CW-pair. Hence, when studying cohomology operations, we do notlose generality by restricting to CW-pairs. For instance, a cohomology operationdefined for CW-pairs extends in a unique way to a cohomology operation definedfor all topological pairs.

45.3. Let (X,Y ) be a CW-pair with Y non-empty. The quotient map (X,Y )→(X/Y, [Y ]) induces an isomorphismH∗(X/Y, [Y ])

≈−→ H∗(X,Y ) (Proposition 12.71).Most questions on cohomology operations may thus be settled by considering theCW-pairs of type (X, ∅) and (X, pt). In particular, a cohomology operation definedfor these pairs extends to a unique cohomology operation. Moreover, a cohomologyoperation Q defined on absolute cohomology for CW-complexes extends to a unique

269

270 7. THE STEENROD SQUARES

cohomology operation for CW-pairs, using the following commutative diagram:

(45.4)

0 // H∗(X, pt) //

Q

H∗(X) //

Q

H∗(pt) //

Q

0

0 // H∗(X, pt) // H∗(X) // H∗(pt) // 0

,

where (X, pt) is a CW-pair.

45.5. Let (X,Y ) be a CW-pair. By Corollary 12.18, the family of inclusionsiA : A→ X for A ∈ π0(X) gives rise the following commutative diagram

H∗(X, ∅)

Q

(H∗iA)

≈//

∏A∈π0(X)H

∗(A, ∅)Q

Q

H∗(X, ∅)(H∗iA)

≈//

∏A∈π0(X)H

∗(A, ∅)

or, if pt ∈ A0 ∈ π0(X),

H∗(X, pt)

Q

(H∗iA)

≈// H∗(A0, pt)×

∏A∈π0(X)−A0

H∗(A, ∅)Q

Q

H∗(X, pt)(H∗iA)

≈// H∗(A0, pt)×

∏A∈π0(X)−A0

H∗(A, ∅)

In other words, a cohomology operation preserves the connected components. To-gether with 45.3, this permits us often to restrict, without loss of generality, acohomology operation to pairs (X,Y ) where X is path-connected.

Lemma 45.6. If Q be a cohomology operation, then Q(0) = 0.

Proof. The class 0 ∈ H∗(X,Y ) is in the image of H∗(X,X) → H∗(X,Y ).As, H∗(X,X) = 0, the lemma follows from functoriality.

An important property of cohomology operations is that it does not decreasedimensions.

Lemma 45.7. Let Q be a cohomology operation. Then, there is a functionN : N→ N, satisfying N(0) = 0 and N(m) ≥ m, such that

(45.8) Q(Hm(X,Y )) ⊂N(m)⊕

k=m

Hk(X,Y )

for all topological pairs (X,Y ).

Proof. By 45.2–45.5 above, it is enough to prove the lemma for CW-pairs(X,Y ) with X connected and Y = pt or ∅. As H0(X, pt) = 0, Q(H0(X, pt)) = 0by Lemma 45.6. The constant map X → pt induces an isomorphism H0(pt, ∅) →H0(X, ∅). As H>0(pt, ∅) = 0, the functoriality implies (45.8) for m = 0, withN(0) = 0.

45. COHOMOLOGY OPERATIONS 271

If m > 0, then Hm(X, pt)≈−→ Hm(X, ∅) ≈−→ Hm(X), so it is enough to

prove (45.8) in the absolute case. By functoriality, the following diagram is com-mutative.

Hm(X,Xm−1)

Q

// // Hm(X)

Q

H∗(X,Xm−1) // H∗(X)

As Hk(X,Xm−1) = 0 for k < m, this proves that the direct sum in (45.8) startsat k = m. Also, any class a ∈ Hm(X) is of the form a = H∗f(ι) for some mapf : X → Km. Thus, N(m) is the maximal degree of Q(ι) ∈ H∗(Km).

A cohomology operation Q restricts to a cohomology operation Q′ on absolutecohomology by Q′X = Q(X,∅). Not every absolute cohomology operation Q′ is suchrestriction because it may not satisfy Q′(0) = 0, contradicting Lemma 45.6. Asan example, we can, for X path-connected and non-empty, define Q′ by Q′(a) = 0for a ∈ Hm(X) (m 6= 1) and Q′(H1(X)) = 1 (the functoriality of Q′ coming fromLemma 5.7). In fact, we have the following lemma.

Lemma 45.9. Let Q′ be a cohomology operation defined on absolute cohomologyfor connected CW-complexes. Then, there exists a cohomology operation Q suchthat Q′X = Q(X,∅) if and only if Q′(0) = 0. The cohomology operation Q is unique.

Proof. The condition is necessary by Lemmas 45.6. For the converse, using45.2–45.5 above, it suffices to define Q(X,pt) for an path-connected non-empty CW-

complex X . On H0(X, pt) = 0, Q is defined by Q(0) = 0 (this is compulsoryby Lemma 45.6). The functoriality for the inclusion (X, ∅) → (X, pt) on H0 isguaranteed by the condition Q′(0) = 0. Let P : H∗(X)→ H0(X) be the projectiononto the component of degree 0. Let j : pt → X be an inclusion of a point in X .As Q′(0) = 0, the commutative diagram

H>0(X)Q′ //

H∗j

H∗(X)P //

H∗j

H0(X)

H∗j≈

H>0(pt)Q′=0 // H∗(pt)

P // H0(pt)

shows that Q′(H>0(X)) ⊂ H>0(X). Hence, the commutative diagram

H>0(X, pt)

Q

≈ // H>0(X)

Q′

H>0(X, pt)

≈ // H>0(X)

defines Q and shows its uniqueness.

The notion of cohomology operation makes sense for the reduced cohomology,with the same definition.

Lemma 45.10. A cohomology operation Q descends to a unique cohomologyoperation on reduced cohomology, also called Q.


Proof. Let p : X → pt be the unique map from X to a point. Consider thefollowing diagram

H∗(pt)

Q

H∗p // H∗(X)

Q

// // H∗(X)

Q

H∗(pt)H∗p // H∗(X) // // H∗(X)

where the line are exact. As Q is a cohomology operation, the left square is commu-tative, so there is a unique Q : H∗(X)→ H∗(X) so that the right square commutesand this construction is functorial. (Recall that, if X is path-connected and Y is

non-empty, then H∗(X,Y ) = H∗(X,Y )).

We now study the multiplicativity of a cohomology operation Q. Note that,by Lemma 22.19, the relative cup product H∗(X,Y ) ⊗ H∗(X,Y ) → H∗(X,Y ) isdefined for all topological pairs (X,Y ). Consider the following four statements.

(a) Q(a b) = Q(a) Q(b) for all a, b ∈ H∗(X) and all spaces X .(b) Q(a b) = Q(a) Q(b) for all a, b ∈ H∗(X,Y ) and all topological pairs

(X,Y ).(c) Q(a × b) = Q(a) × Q(b) for all a ∈ H∗(X1), b ∈ H∗(X2) and all spaces

X1 and X2.(d) Q(a × b) = Q(a) × Q(b) for all a ∈ H∗(X1), b ∈ H∗(X2) and all pointed

spaces X1 and X2.

Proposition 45.11. For a cohomology operation Q, Conditions (a), (b) and(c) are equivalent and (a) implies (d). If Q(1) = 1, then (d) implies (a).

Proof. Without loss of generality, we may suppose that the spaces X and Xi

are connected CW-complexes. Statement (b) is stronger than (a) since H∗(X) =H∗(X, ∅). To prove that (a) implies (b), it suffices to consider the case Y = pt,which is obvious.

Using the functoriality of Q, (a)⇒(c) follows from the definition of the crossproduct and (c)⇒(a) from the formula a b = ∆∗(a×b) (see Remark 27.5). That(c)⇒(d) is obvious, so (a)⇒(d). Now, (d) implies (c) for classes of positive degree.As (c)⇒(a), property (d) implies that Q(a b) = Q(a) Q(b) except possiblyfor a or b equal to 1. If, say a = 1, then

Q(1 b) = Q(b) = 1 Q(b) = Q(1) Q(b) ,

since Q(1) = 1. Thus, (d)⇒(a) if Q(1)) = 1.

Corollary 45.12. Let Q be a cohomology operation with Q(1) = 1. Then,(a) is equivalent to

(d’) the following diagram

Hm(Km)⊗ Hn(Kn)Q⊗Q

× // Hm+n(Km ∧ Kn)Q

H∗(Km)⊗ H∗(Kn)× // H∗(Km ∧ Kn)

is commutative for all positive integers m and n.

46. PROPERTIES OF THE STEENROD SQUARES 273

Proof. It is clear that (d)⇒(d’). The corollary will then follow from Propo-sition 45.11 if we prove that (d’)⇒(d).

It suffices to prove (d) for a ∈ Hm(X1) and b ∈ Hn(X2) where X1 and X2 areconnected CW-complexes, so m,n > 0. Then a = f∗a (ιm) and a = fb∗(ιn) for mapsfa : X1 → Km and fb : X2 → Kn. Condition (d’) says that Q(ιm × ιn) = Q(ιm) ×Q(ιn) and Q(a× b) = Q(a)×Q(b) from this special case, using the functoriality of× and of Q.

46. Properties of the Steenrod squares

One of the most remarkable feature of the mod 2-cohomology is the existenceof cohomology operations, introduced by N. Steenrod and H. Cartan in the late1940s (see, e.g. [38, pp. 510–523]), called the Steenrod squares Sqi : H∗(X,Y ) →H∗(X,Y ) (i ∈ N). For a ∈ Hm(X,Y ), one has Sqi(a) = 0 for i > m (see 2.a

in Theorem 46.2 below). Hence, the sum∑

i∈NSqi(a) has only a finite number of

non-zero terms and thus defines the total Steenrod square

(46.1) Sq: H∗(X,Y )→ H∗(X,Y ) , Sq(a) =∑

i∈N

Sqi(a) .

The main theorem is the following.

Theorem 46.2. There exists a cohomology operation Sq and Sqi as in (46.1),which enjoys the following properties:

(1) Sq is Z2-linear.

(2) if a ∈ Hn(X,Y ) then Sqi(a) ∈ Hn+i(X,Y ) and

(a) Sqi(a) = 0 for i < 0 and i > n.(b) Sq0(a) = a.(c) Sqn(a) = a a.

(3) Sq(a b) = Sq(a) Sq(b). This is equivalent to the formula

Sqk(a b) =∑

i+j=k

Sqi(a) Sqj(b) (Cartan’s formula).

(4) Σ∗Sq = SqΣ∗, where Σ∗ : H∗(X)≈−→ H∗+1(ΣX) is the suspension iso-

morphism of Proposition 12.76.

(5) The Adem relations:

SqiSqj =

[i/2]∑

k=0

(j−k−1i−2k

)Sqi+j−kSqk (0 < i < 2j) .

The Steenrod squares are characterised amongst cohomology operations bysome of these properties (see Proposition 49.18). Also, the Adem relations generate

all the polynomial relations amongst the compositions of Sqi’s which hold true forany space (see Corollary 49.17).

Example 46.3. Theorem 46.2 permits us to compute Sqi easily for the projec-tive spaces RPn, CPn and HPn. Indeed, one has the following results.


(a) Let a ∈ H1(X). Then (2) implies that Sq(a) = a + a2 (we write an forthe cup product of n copies of a). Then, (3) implies that

Sq(an) = (a+ a2)n = an (1 + a)n = an n∑

i=1

(ni

)ai .

Therefore

(46.4) Sqi(an) =(ni

)an+i .

(b) If a ∈ H2(X) satisfies Sq1(a) = 0, then Sq(a) = a+ a2 and, as in (a), onehas

(46.5) Sq2i(an) =(ni

)an+i and Sq2i+1(an) = 0 .

(c) If a ∈ H4(X) satisfies Sq(a) = a+ a2, then

(46.6) Sq4i+k(an) =

(ni

)an+i if k ≡ 0 mod 4.

0 otherwise.

Besides the trivial case of OP 2, there are no more such examples. Indeed, byCorollary 50.3 and Theorem 50.6 below, if a ∈ Hm(X) satisfies Sq(a) = a+a2 witha2 6= 0, then m = 1, 2, 4 or 8.

We finish by two more properties of the Steenrod squares. As Sq(a b) =Sq(a) Sq(b), Proposition 45.11 implies the following result.

Proposition 46.7. Let X1 and X2 be topological spaces (pointed for (b)).Then,

(1) Sq(a× b) = Sq(a)× Sq(b) for all a ∈ H∗(X1), b ∈ H∗(X2).

(2) Sq(a × b) = Sq(a) × Sq(b) for all a ∈ H∗(X1), b ∈ H∗(X2).

Proposition 46.8. Let (X,Y ) be a topological pair. Then

Sqδ∗ = δ∗Sq ,

where δ∗ is the connecting homomorphism δ∗ : H∗(Y )→ H∗+1(X,Y ).

Proof. By 45.2, we may suppose that (X,Y ) is a CW-pairs. Let Z = X ∪[−2, 1]× Y and Z ′ = X ∪ [−2,−1]× Y ∪ 1× Y . The projection p : [−2, 1]× Y →−2 × Y extends to a homotopy equivalence of pairs p : (Z, 1 × Y ) → (X,Y ).The following commutative diagram

H∗(1 × Y )

δ∗

oooo p∗

H∗(S0 × Y )

δ∗

H∗+1(Z, 1 × Y ) oo H∗+1(Z,Z ′) oo ≈

excision// H∗+1(D1, S0 × Y )

shows that it is enough to prove that Sqδ∗ = δ∗Sq for a CW-pair of the type(D1 × Y, S0 × Y ). In the proof of Proposition 28.71, we showed that

δ∗(a) = e×(H∗i+(a) +H∗i−(a)

),

47. CONSTRUCTION OF THE STEENROD SQUARES 275

where i± : ±1×F → S0×F denote the inclusions and 0 6= e ∈ H1(D1, S0) = Z2.One has Sq(e) = Sq0(e) = e. Using the linearity of Sq and Proposition 46.7, we get

Sqδ∗(a) = Sq(e×

(H∗i+(a) +H∗i−(a)

))

= Sq(e)× Sq(H∗i+(a) +H∗i−(a))

= e×(SqH∗i+(a) + SqH∗i−(a)

)

= e×(H∗i+Sq(a) +H∗i−Sq(a)

)

= δ∗Sq(a) .

Sections 47 and 48 contain the proof of Theorem 46.2. They are based on theideas of Steenrod (see [180, VII.1 and VIII.1]) using the equivariant cohomology.Other treatments of similar ideas are developed in [2, VI.7] and [80, Section 4.L].

47. Construction of the Steenrod squares

The involution τ on X ×X given by τ(x, y) = (y, x) makes X ×X a G-spacefor G = id, τ. We consider the cross-square map β : Hn(X) → H2n(X × X)defined by β(a) = a × a. Its image is obviously contained in H2n(X × X)G. ByLemma 39.17, the image of ρ : H∗G(X × X) → H∗(X × X) is also contained inH∗(X ×X)G. The same considerations are valid for the reduced cross-square map

β : Hn(X)→ H2n(X ∧X)G, defined for a space X which is well pointed by x ∈ X .

The maps β and β are not linear but they are functorial: if f : X ′ → X is acontinuous maps, then H∗(f × f)β = βH∗f and H∗(f ∧ f) β = βH∗f . UsingDiagram (28.18), the use of base points x ∈ X and (x, x) ∈ X ∧ X provide acommutative diagram

(47.1)

H∗(X ∧X) // // H∗(X ×X)

H∗(X) // //

β

OO

H∗(X)

β

OO

Lemma 47.2. Let X be a connected CW-complex, pointed by x ∈ X0. Then, thecross-square maps β and β admit liftings βG and βG so that the following diagram

H∗(X ∧X) // // H∗(X ×X)

H∗G(X ∧X)

ρ

ggOOOOOOOOO

// H∗G(X ×X)

ρ

ggOOOOOOOOOO

H∗(X) // //

β

OO

βG

77

H∗(X)

βOO

βG

77

is commutative. These liftings are functorial and satisfy βG(0) = 0 and βG(0) = 0.Such liftings are unique.


Proof. The maps β, ρ, β and ρ preserving the connected components, one maysuppose that X is connected. Lemma 47.2 is obvious when n = 0, giving βG(0) = 0

and βG(1) = 1. We can then assume n > 0. In this case, Hn(X) → Hn(X) is an

isomorphism, so it is enough to define βG.We first define βG when X = Kn = K(Z2, n) with its CW-structure given in

the proof of Proposition 19.1, whose 0-skeleton consists in a single point x. ByProposition 28.17, the G-space Kn ∧ Kn satisfies Hk(Kn ∧ Kn) = 0 for k < 2n.

Hence, Proposition 39.21 implies that ρ : H2nG (Kn × Kn) → H2n(Kn × Kn)G is an

isomorphism. We define βG = ρ−1 β.

Now, a cohomology class a ∈ Hn(X) is of the form a = H∗fa(ι) for a mapfa : X → Kn, well defined up to homotopy. We define

(47.3) βG(a) = H∗G(fa × fa) βG(ι) .

This definition makes βG functorial. Indeed, let g : Y → X be a continuous mapand a = H∗fa(ι) ∈ Hn(X). If b = H∗g(a), then b = H∗fb(ι) with fb = fag. By

definition of βG, one has

H∗G(g × g) βG(a) = H∗G(g × g)H∗G(fa × fa)βG(ι) = H∗G(fb × fb)βG(ι) = βG(b) .

The functoriality of βG follows from that of βG. The uniqueness of βG (then, thatof βG) is obvious, since there was no choice for X = Kn and Definition (47.3) iscompulsory by the required functoriality.

Remark 47.4. When Hi(X) = 0 for i < n, then Hj(X ∧ X) = 0 for j < 2nby Proposition 28.17. The homomorphism ρ : H2n(X ∧X)→ H2n(X ∧X)G is an

isomorphism which is natural. Then, by functoriality, the formula βG = ρ−1 β

holds true.

The map βG : Hn(X) → H2nG (X ×X) extends to βG : H∗(X) → H∗G(X × X)

by

(47.5) βG(∑

j∈N

aj) =∑

j∈N

βG(aj) , where aj ∈ Hj(X) .

The inclusion of (X×X)G intoX×X induces a GrA[u]-morphism r : H∗G(X×X)→H∗G((X × X)G). Observe that (X × X)G is the diagonal subspace (x, x) ofX × X , hence homeomorphic to X . We thus write r : H∗G(X × X) → H∗G(X),considering X as a G-space with trivial G-action. Using (39.7), we thus get aGrA[u]-isomorphism H∗G(X) ≈ H∗(X)[u]. We also consider the (non-graded) ringhomomorphism ev1 : H∗(X)[u]→ H∗(X) which extends the identity on H∗(X) bysending u to 1 (evaluation of a polynomial at 1).

Let X be a CW -complex. By definition, the Steenrod square Sq: H∗(X) →H∗(X) is the composition

(47.6) Sq = ev1rβG

making the following diagram commutative.

(47.7)

H∗G(X ×X)r // H∗G(X) oo ≈ // H∗(X)[u]

ev1

H∗(X)

βG

OO

Sq // H∗(X)

.


By (47.5) and Lemma 47.2, the map Sq is a cohomology operation, defined so faron absolute cohomology for connected CW-complexes. As βG(0) = 0, Lemma 45.9implies that this partial definition of Sq extends to a unique cohomology operationSq: H∗(X,Y )→ H∗(X,Y ) defined for all topological pairs (X,Y ).

For a ∈ Hn(X,Y ) let Sqi(a) be the component of Sq(a) in Hn+i(X,Y ). Again,defining Sqi : H∗(X,Y )→ H∗(X,Y ) by

(47.8) Sqi(∑

j∈N

aj) =∑

j∈N

Sqi(aj) , where aj ∈ Hj(X)

provides a family of cohomology operation Sqi. A priori, i ∈ Z but, by Lemma 45.7,Sqi = 0 if i < 0. It follows from these definitions that Sq =

∑i∈N

Sqi.

By Lemma 45.10, the Steenrod squares Sq and Sqi are also cohomology opera-tions on reduced cohomology. In this case, the following diagram is commutative.

(47.9)

H∗G(X ∧X)r // H∗G(X) oo ≈ // H∗(X)[u]

ev1

H∗(X)

βG

OO

Sq // H∗(X)

In the important case where Hj(X) = 0 for j < n (e.g. X = Kn), one can useRemark 47.4 to get the following commutative diagram

(47.10)

H2n(X ∧X)Gρ−1

≈// H2n

G (X ∧X)r // H∗G(X) oo ≈ // H∗(X)[u]

ev1

Hn(X)

β

OO

Sq //βG

66nnnnnnnnnnn

H∗(X)

We now prove Properties (1)-(4) of Theorem 46.2, for the absolute cohomologyH∗(X) of a connected CW-complex X .

Proof of (2). That Sqi sends Hn(X) to Hn+i(X) is by definition and wealready noticed that Sqi = 0 for i < 0. If a ∈ Hn(X), then rβG(a) ∈ H2n

G (X),

which implies that Sqi = 0 for i > n. Note that, from (47.6) and the definition ofthe Sqi, one has, for a ∈ Hn(X):

(47.11) rβG(a) =

n∑

i=0

Sqi(a)un−i .

Let us prove that Sqn(a) = a a. From the above equation, we deduce thatSqn(a) = ev0rβG(a). Using Diagrams (39.10) and (39.11), we get the followingdiagram

H∗(X)βG //

OO=

H∗G(X ×X)

ρ

r // H∗G(X) oo ≈ // H∗(X)[u]

ev0

H∗(X)

β // H∗(X ×X)∆∗ // H∗(X)

where ∆∗ is induced by the diagonal map ∆: X → X ×X . By (27.6) p. 132, onehas

Sqn(a) = ev0rβG(a) = ∆∗β(a) = ∆∗(a× a) = a a .


It remains to prove that Sq0(a) = a. By naturality and since Hn(Kn) = Z2, itsuffices to find, for each integer n, some space X with a class a ∈ Hn(X) such thatSq0(a) 6= 0. The space X will be the sphere Sn. Indeed, there is a homeomorphism

h : Sn ∧ Sn ≈−→ S2n (see Example 28.19). We leave as an exercise to the reader toconstruct such a homeomorphism h which conjugates the G-action on Sn ∧ Sn toa linear involution on S2n. Therefore, Sn ∧ Sn is G-homeomorphic to the sphereS2nn of Example 39.26. We now use Diagram (47.10). By the reduced Kunneth

theorem, β(a) = a, the generator of H2n(Sn ∧ Sn) = Z2. By Proposition 39.27,r ρ−1(a) = a un, whence Sq0(a) = a.

Linearity of Sq. We have to prove that for each n ∈ N, the map Sq: Hn(X)→H∗(X) is linear. By Point (2) already proven, the restriction of Sq toH0 is Sq0 = id,so we may assume that n ≥ 1. By functoriality of Sq, the following diagram

Hn(X/Xn−1)

Sq

// // Hn(X)

Sq

H∗(X/Xn−1) // H∗(X)

is commutative, where the horizontal maps are induced by the projection X →X/Xn−1. Therefore, it is enough to prove the linearity of Sq: Hn(X/Xn−1) →H∗(X/Xn−1). We may thus assume that Hk(X) = 0 for k < n and use Dia-

gram (47.10) to define Sq. It is then enough to show that r ρ−1 β : Hn(X) →

H2nG (X) is linear. One has

β(a+ b) = β(a) + β(b) + a × b+ b × a= β(a) + β(b) + a × b+ τ∗(a × b) .

Using Proposition 39.21, we get

σ β(a+ b) = β(a) + β(b) + tr∗(a × b) .As rtr∗ = 0 by Proposition 39.16, this prove that r ρ−1

β is linear.

From the already proven properties of Sq, we now deduce a structure resultabout H∗G(X×X) (Proposition 47.12 below) which will be used to prove the multi-plicativity of Sq. Let N : H∗(X×X)→ H∗(X×X) be the GrV-morphism definedby N(x× y) = x× y + y × x and let N be the image of N . Note that

kerN = H∗(X ×X)G = D ⊕Nwhere D is the subgroup generated by x × x | x ∈ H∗(X). By definition of thetransfer map tr∗ : H∗(X ×X)→ H∗G(X ×X) (see (39.12)), one has

ρtr(x× y) = x× y + τ∗(x× y) = N(x× y) .The correspondence (x × x,N(y × z)) 7→ βG(x) + tr∗(y × z) produces a sectionσ : D ⊕N → H∗G(X ×X) of ρ : H∗G(X ×X)→ H∗(X ×X)G. We identify D ⊕Nwith σ(D ⊕ N ) and thus see D ⊕ N as a subgroup of H∗G(X ×X). Let D be theZ[u]-module generated in H∗G(X×X) by D. The following result is due to Steenrod(unpublished, but compare [76, § 2]).

Proposition 47.12. With the above identifications, the following propertieshold true.


(a) The restriction of ρ to D ⊕ N coincides with the identity D ⊕ N id−→H∗(X ×X)G.

(b) N = Ann (u).

(c) D is a free Z2[u]-module with basis D−0. In particular, D is isomorphicto Z2[u]⊗D.

(d) As a Z2[u]-module, H∗G(X ×X) = D ⊕ N .

Proof. Point (a) is obvious from the identification via σ. Hence, H∗(X×X)G

is the image of ρ and one has the commutative diagram

0 // H∗G(X ×X)/(u) //

ρ

H∗(X ×X)tr∗ //

OO=

Ann (u) //

ρ

0

0 // H∗(X ×X)G // H∗(X ×X) // N // 0

where the lines are exact (the upper line is the transfer exact sequence (39.13)).By the techniques of the five lemma, we deduce that ρ : Ann (u) → N is an iso-morphism. As σ(N) is contained in the image Ann (u) of the transfer map, thisproves (b). Also, we get the isomorphism

(47.13) ρ : H∗G(X ×X)/(u)≈−→ H∗(X ×X)G .

Let B be Z2-basis H∗(X) formed by homogeneous classes. To prove (c), onehas to show that

(47.14)∑

a∈B0

βG(a)uk(a) 6= 0 .

for any non-empty finite subset B0 of B and any function k : B0 → N. Let B0min be

the subset formed by the elements in B0 which are of minimal degree. Then

ev1r( ∑

a∈B0

βG(a)uk(a))

=∑

a∈B0

Sq(a) =∑

a∈B0min

a+ terms of higer degrees 6= 0 .

To prove (d), let A = H∗G(X×X) and B = D ⊕N . If B 6= A, let a ∈ A−B beof minimal degree. By the above and Sequence (39.13), one has B/uB = A/uA ≈H∗(X ×X)G. Hence, there exists b ∈ B and c ∈ A such that a = b + uc. By theminimality hypotheses on a, one has c ∈ B and thus a ∈ B (contradiction).

Proof that Sq(a b) = Sq(a) Sq(b) (multiplicativity). One has

β(a b) = (a b)× (a b) = (a a)× (b b) = β(a) β(b) .

Hence,

βG(a b) = βG(a) βG(b) + x

with x ∈ kerρ = (u) (the last equality was established in (47.13)). We may supposethat a ∈ Hm(X) and b ∈ Hn(X). Let V be Z2-basis H<m+n(X) formed byhomogeneous classes. By Proposition 47.12, x =

∑v∈V0 βG(a)uk(v) for some finite

subset V0 of V , with k(v) > 0. Let V0min be the subset formed by the elements in

V0 which are of minimal degree. Then,

Sq(a b) = ev1rβG(a b) =∑

v∈V0min

a+ terms of higer degrees .


Since Sqi(a b) = 0 for i < 0, one has V0min = ∅ and therefore V0 = ∅. This

implies that βG(a b) = βG(a) βG(b) and thus Sq(a b) = Sq(a) Sq(b).

Proof that Σ∗Sq = SqΣ∗. By Lemma 28.20, using the reduced suspensionS1 ∧ X , the relation Σ∗Sq = SqΣ∗ is equivalent to the following equation inH∗(S1 ∧X):

(47.15) b × Sq(c) = Sq(b × c) ∀c ∈ Hn(X) ,

where b is the generator of H1(S1) = Z2. As noticed in Proposition 46.7, theformula Sq(b c) = Sq(b) Sq(c) already proven implies that Sq(b × c) =Sq(b)×Sq(c). Also, by (2) already established, Sq(b) = b. Hence, Equation (47.15)holds true.

The proof of Points (1)–(4) in Theorem 46.2 is now complete for pairs (X, ∅)with X a connected CW-complex. We check easily that the extension of Sq andSqi to topological pairs given by Lemma 45.9 satisfies the same properties.

48. The Adem relations

The Adem relations are relations amongst the compositions SqiSqj . They wereconjectured by Wu Wen Tsun around 1950 and first proved in 1952 by J. Ademin his thesis at Princeton University (summary in [3] and full proofs in [4]). Wepresent below a proof based on the idea of Steenrod [180, § ], using the equivariantcohomology for the symmetric group Sym4. The proof in [80, § 4.L] is anotheradaptation of the same idea. For different proofs, see [31] and Remark 49.16 below.

Let X be a topological space. Consider the map

Sq : H∗(X)βG−−→ H∗G(X ×X)

r−→ H∗G((X ×X)G) ≈ H∗(BG×X)

(recall that BG ≈ RP∞: see Example 40.1). The map Sq would sit diagonally

in (47.7), the diagram used to define the Steenrod squares. By § 47, the map Sq isfunctorial, Z2-linear and multiplicative.

Consider now the iterated map

Sq Sq : H∗(X)→ H∗(BG×BG×X) ≈ H∗(X)[u, v] ,

using the Kunneth theorem and that H∗(BG × BG) ≈ Z2[u, v] with u and v indegree 1.

Proposition 48.1. For any a ∈ H∗(X), the polynomial Sq Sq(a) is symmetricin the variables u and v.

Before proving Proposition 48.1, we do some preliminaries. Let Y be a K-space for a topological group K. The equivariant cross product ×K of § 42 givesrise to an equivariant cross square map βK : Hn

K(Y ) → H2nK (Y × Y )G, defined by

βK(a) = a×K a, where G = I, τ acting on Y × Y by exchanging the factors. A

map βK : HnK(Y )→ H2n

K (Y ∧Y )G is similarly defined, using the reduced equivariantcross product×K. The following is a generalization of Lemma 47.2.

Lemma 48.2. Let Y be a K-space, equivariantly well pointed by y ∈ Y K . Then,

48. THE ADEM RELATIONS 281

(1) The cross-square maps βK and βK admit liftings βKG

and βKG

so that thefollowing diagram

H2nK (Y ∧ Y )G // // H2n

K (Y × Y )G

H2nG×K(Y ∧ Y )

ρhhQQQQQQQQQQ

// H2nG×K(Y × Y )

ρhhQQQQQQQQQQQ

HnK(Y ) // //

βK

OO

βKG

66

HnK(Y )

βK

OO

βKG

66

is commutative, where ρ and ρ are induced by the homomorphism K →G ×K. When K is the trivial group, these lifting coincide with those ofLemma 47.2.

(2) The lifting βKG

is functorial in Y and G, i.e. if Y ′ is a K ′-space, f : Y →Y ′ is equivariant with respect to a continuous homomorphism ϕ : K → K ′,then the following diagram

H2nG×K′(Y

′ × Y ′) f∗×f∗// H2nG×K(Y × Y )

HnK′(Y

′)f∗ //

βK′

G

OO

HnK(Y )

βKG

OO

is commutative. The analogous property holds for βG.(3) Suppose that the K-action on Y is trivial. Then, the following diagram is

commutative

H2nG×K(Y × Y )

r // H2nG×K(Y )

≈ // H2n(B(G ×K)× Y )

HnK(Y ) oo ≈ //

βGK

OO

Hn(BK × Y )fSq // H2n(BG×BK × Y )

P ≈

OO


Proof. The lifting βKG

is defined using the lifting βG of Lemma 47.2 and thefollowing commutative diagram(48.3)

H2nG×K(Y × Y )

r //

ρ

rreeeeeeeeeeeeeeeeeeeeeeeeeeeeeeH2n

G×K(Y )

H2nK (Y × Y )G oo

=// H2n((Y × Y )K)G oo ρ

H2nG ((Y × Y )K)

r //

≈

OO

H2nG (YK)

≈

OO

H2n((Y × Y )K×K)G oo ρ

∆∗K

OO

H2nG ((Y × Y )K×K)

r //

∆∗K

OO

H2nG ((∆Y )∆K)

≈

OO

H2n(YK × YK)G oo ρ

P

OO

H2nG (YK × YK)

r //

P

OO

H2nG (∆(Y )K))

≈

OO

HnK(Y ) oo = //

βK

OO

Hn(YK)

β

OOβG

55jjjjjjjjjjjjjj fSq

22eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

(we do not need the last column for the definition of βKG

but it will be usedlater). The commutativity of the right rectangle is the definition of βK , using theformula (42.9) for the equivariant cross product. The top vertical isomorphismscome from the following fact: if Z is a (G ×K)-space, the homotopy equivalenceE(G × K) × Z → EG × EK × Z given by

((ti(gi, ki), z

)7→

((tigi, z), (tiki, z)

)

descends to a homotopy equivalence E(G × K)G×K × Z → EG ×G (EK ×K Z).The same homotopy equivalence is used to get the map

(48.4) ZK = EK ×K Z → EG×G (EK ×K Z) ≈ E(G×K)×G×K Z = ZG×K

giving rise (for Z = Y × Y ) to the homomorphism ρ. For βKG

, we use the followingcommutative diagram

(48.5)

H2nG∧K(Y ∧ Y )

ρ

qqccccccccccccccccccccccccccccccccccccccccccccccc

H2nK (Y ∧ Y )G oo ≈

(1)// H2n((Y × Y )K , (Y ∨ Y )K)G oo ρ

H2nG ((Y × Y )K , (Y ∨ Y )K)

≈(2)

OO

H2n((Y × Y )K×K , (Y ∨ Y )K×K)G oo ρ

∆∗K

OO

H2nG ((Y × Y )K×K , (Y ∨ Y )K×K)

∆∗K

OO

H2n(YK × YK , YK∨YK)G oo ρ

P

OO

H2nG (YK × YK , YK∨YK)

OO

HnK(Y ) oo ≈ //

βK

OO

Hn(YK , yK)

β

OO

βG

33ggggggggggggggggggg

As Y is equivariantly well pointed by y, the pair (YK , yK) is well cofibrant, sothe cross-square map β is defined. Also, the pair (Y ×Y, Y ∨Y ) is K-equivariantly


well cofibrant (the proof is the same as for Lemma 40.23). Hence, Identification(1) then comes from Corollary 40.27. The same argument gives Identification (2),

using (48.4) for Z = Y × Y and Z = Y ∨ Y . The lifting βG is defined using thefollowing diagram.

H∗K(YK ∧ YK)G // H2n

K (YK × YK , YK∨YK)G

H2nG (YK ∧ YK)

ρhhQQQQQQQQQQQ

// H2nG×K(YK × YK , YK∨YK)

ρjjVVVVVVVVVVVVVVV

Hn(YK)≈ //

β

OO

βG

66mmmmmmmmmmm

H∗(YK , yK)

β

OO

βG

44

With these definitions, Diagram (48.5) is mapped into Diagram (48.3), giving rise

to the diagram of Point (1) of our lemma. That βKG

and βKG

coincide with βG andβG when K is the trivial group follows from the definitions, as well as Point (2).

Point (3) comes from (48.3), where the occurrence of Sq is noticed.

Proof of Proposition 48.1. By naturality, it is enough to prove the propo-sition for X = Kn and a the generator of Hn(Kn) = Z2 (n ≥ 1). We consider X aspointed by x ∈ X0, using the CW-structure given in the proof of Proposition 19.1,whose 0-skeleton consists in the single point x.

Let the symmetric group Σ = Sym4 act on X∧4 = X ∧ X ∧ X ∧ X andX4 = X×X×X×X by permutation of the factors. This action may be restrictedto the the subgroup of Γ of Σ generated by s = (1, 2)(3, 4) and t = (1, 3)(2, 4).As for (47.1), the use the base points x ∈ X and (x, x, x, x) ∈ X ∧ X provide acommutative diagram

(48.6)

H∗(X∧4) // // H∗(X4)

H∗(X) // //

β β

OO

H∗(X)

ββ

OO.

By Proposition 28.17,Hk(X∧4) = 0 for k < 4n and H4n(X∧4)Σ = H4n(X∧4) = Z2.As in Lemma 47.2, using Proposition 40.28, we get liftings

(β β)Σ : Hn(X)→ H4nΣ (X∧4) and (ββ)Σ : Hn(X)→ H4n

Σ (X4)

of β β and ββ. Consider the composite map.

Φ: Hn(X)(ββ)Σ−−−−−→ H4n

Σ (X4)r−→ H∗Σ((X4)Σ) ≈ H∗(X)⊗H∗(BΣ) .

Let Γ be the subgroup of Γ generated by s = (1, 2)(3, 4) and t = (1, 3)(2, 4). As sand t commute, Γ is isomorphic to G×G. Let i : Γ→ Σ denote the inclusion. We


shall prove that the following diagram

(48.7)

H∗(X)⊗H∗(BΓ)

id⊗H∗Bi

H∗(X)⊗H∗(BΓ0)OO≈

Hn(X)

Φ

99

fSqfSq // H∗(X)[u, v]

.

is commutative. For the moment, let us show that this property implies Propo-sition 48.1. It is enough to prove that the image of H∗(BΣ) → Z2[u, v] consistsof symmetric polynomials. Under the isomorphism Z2[u, v] ≈ H∗(BΓ) the auto-morphism exchanging u and v corresponds to that induced by exchanging s andt. But in Σ, exchanging s and t is achieved by the conjugation by the transposi-tion (2, 4). Such an inner automorphism of Σ induces the identity on H∗(BΣ) byProposition 40.3.

It remains to prove that Diagram (48.7) is commutative. Let G1 and G2 bethe subgroups of Γ generated by s and t respectively, so Γ ≈ G1 × G2. Thecommutativity of Diagram (48.7) comes from that of the following diagram(48.8)

H4nΓ (X4)

≈

''NNNNNNNNN

r // H4nΓ (X)

≈

H4nG2×G1

(X4)r //

r×r

((QQQQQQQQQQ

II

H4nG2×G1

(X)OO=

H4nG2×G1

(X2)r //

III

H4nG2×G1

(X)

Hn(X)βG1 //

(ββ)Γ

OO

fSq

++WWWWWWWWWWWWWWWWWWWWWWWW

I

H2nG1

(X2)r //

βG1G2

OO

H2nG1

(X)OO≈

βG1G2

OO

H2n(BG1 ×X)fSq // H4n(BG2 ×BG1 ×X)

≈

OO

together with the obvious commutativity of the diagram

H4nΣ (X4)

j∗

&&MMMMMMMM

r // H4nΣ (X)

j∗

≈ // H4n(BΣ×X)

(Bj×id)∗

Hn(X)

β2Γ //

β2Σ

OO

H4nΓ (X4)

r // H4nΓ (X)

≈ // H4n(BΓ×X)

.

The commutativity of Diagram (48.8) comes from that of its subdiagram I, IIand III (the commutativity of the other diagrams is obvious). Diagrams II and IIIcommute because of Points (2) and (3) of Lemma 48.2. To verify the commutativity


of Diagram (I), we put it as the inner square of the following diagram.

H4nΓ (X∧4)

≈ //

&&MMMMMMMMM

(A)

(D)

H4nG2×G1

(X∧4)

wwnnnnnnnnnn

H4nΓ (X4)

≈ // H4nG2×G1

(X4)

Hn(X)βG1 //

(ββ)Γ

OO

H2nG1

(X2)

βG1G2

OO

Hn(X)βG1 //

(β β)Γ

OO

≈88qqqqqqqqq

H2nG1

(X∧2)

βG1G2

OO

ggPPPPPPPPPP

(C)

(B)

Diagrams (A), (B) and (C) commute because of Lemmas 47.2 and 48.2, and Dia-

gram (A) is obviously commutative. As Hn(X)→ Hn(X) is an isomorphism, theinner inner square will commute if the outer does. But the outer square commutesby default, since, using Proposition 40.28, all the groups are equal to Z2 and themaps are non trivial (for βG1 and βG1

G2, this is checked using the restrictions to the

trivial group and Point(2) of Lemma 48.2).

From Proposition 48.1, we deduce relations between SqiSqj called the Ademrelations.

Theorem 48.9 (Adem relations). Let X be a topological space. For each i, j ∈N with i < 2j, the following relation

SqiSqj =

[i/2]∑

k=0

(j − k − 1

i− 2k

)Sqi+j−kSqk (0 < i < 2j)

holds amongst the non-graded endomorphisms of H∗(X).

Examples 48.10. (1) When i = 1, the right hand member in the Adem relation

reduces to(j−11

)Sqj+1. Hence

Sq1Sqj =

Sqj+1 if j is even

0 if j is odd.

For instance, Sq1Sq1 = 0 and Sq1Sq2 = Sq3.

(2) In the limit case i = 2j−1 with j > 0, the binomial coefficient(j−k−1

2j−1−2k

)vanish

if k ≤ j − 1, since 2j − 1− 2k > j − k − 1. Then

Sq2j−1Sqj = 0 if j > 0 .

Proof of the Adem relations. Let a ∈ Hn(X). The homomorphism Sq: Hn(X)→H∗(RP∞ ×X)

≈−→ H∗(X)[u] satisfies

Sq(a) =n∑

µ=0

Sqµ(a)un−µ =∑

µ∈Z

Sqµ(a)un−µ .

The range extension µ ∈ Z is possible since the other summands vanish. We shalldo that repeatedly in the computations below and, without other specification, the


summations will be over the integers (with only a finitely many non-zero terms).This permits us to exchange the summation symbols.

Observe that Sq(u) = u2 + uv = u(u + v) thus, by multiplicativity of Sq, one

has Sq(uk) = Sq(u)k = uk(u+ v)k. Therefore

Sq Sq(a) =∑µ Sq(un−µ)Sq(Sqµ(a))

=∑µ u

n−µ(u+ v)n−µ∑λ SqλSqµ(a)vn+µ−λ

=∑µ,λ u

n−µvn+µ−λ(u+ v)n−µSqλSqµ(a)

=∑µ,λ,ν

(n−µν

)un−µ+νv2n−λ−νSqλSqµ(a)

Setting λ+ ν = i yelds

Sq Sq(a) =∑

ν,µ,i

(n−µν

)un−µ+νv2n−iSqi−νSqµ(a) .

Setting n− µ+ ν = 2n− q yelds

(48.11) Sq Sq(a) =∑ν,q,i

(2n−q−ν

ν

)u2n−qv2n−iSqi−νSqq+ν−n(a) .

By Proposition 48.1, for each i and q, the coefficient of u2n−qv2n−i in (48.11) mustbe equal to that of u2n−iv2n−q. This leads to the equation

∑ν

(2n−q−ν

ν

)Sqi−νSqq+ν−n(a) =

∑ν

(2n−i−ν

ν

)Sqq−νSqi+ν−n(a) .

In the left hand member, set j = q−n. In the right hand member, set k = i+ν−nand use the relation

(xy

)=

(x

x−y

). In both side, restrict the range of summation so

that the summands are not zero for obvious reasons. This gives

(48.12)

i∑

ν=0

(n−j−ν

ν

)Sqi−νSqj+ν(a) =

[i/2]∑

k=0

(n−ki−2k

)Sqj+i−kSqk(a) .

Now, i and j being fixed, suppose that n = 2r − 1 + j for r large. Then, byLemma 36.9, (

n−j−νν

)=

(2r−1−ν

ν

)= 0 if ν 6= 0

for, the dyadic expansion of 2r−1− ν has a zero where that of ν has a one. Hence,the left hand member of (48.12) reduces to the single term SqiSqj(a). Also,

(n−ki−2k

)=

(2r+j−k−1

i−2k

)=

(j−k−1i−2k

)if i < 2j

since the length of the dyadic expansions of i− 2k is not more than that of j − 1and adding 2r to j − k − 1 only puts a single 1 far to the left.

Thus, (48.12) proves the Adem relations classes of degree 2r−1+j with r large(2r > maxi, j). As Σ∗Sq = SqΣ∗, Equation (48.12) holds for a if and onlyif it holds for Σ∗(a). But the suspension isomorphism may be iterated on a classa ∈ Hn(X) till its degree becomes of the form 2r − 1 + j with r large. This provesthe Adem relations.

49. The Steenrod algebra

The Steenrod algebra A is the graded Z2-algebra generated by indeterminatesSqi (in degree i) and subject to the Adem relations and to Sq0 = 1. The propertiesof the Steenrod squares imply that the cohomology H∗(X) of a space X is anA-module.

49. THE STEENROD ALGEBRA 287

Lemma 49.1. As an algebra, A is generated by Sqn | n = 2r.Proof. Let m = 2r + s with s < 2r (r ≥ 1). As

(2r−1s

)≡ 1 mod 2, the Adem

relation

SqsSq2r

= Sqm +

[s/2]∑

k=1

(2r−k−1s−2k

)Sqm−kSqk (0 < i < 2j)

expresses Sqm as a sum of SqiSqj . If s > 0 then i, j < m, which permits us toprove the lemma by induction on m.

Here below a few examples of decompositions of Sqi according to Lemma 49.1and its proof:

(49.2)

Sq3 = Sq1Sq2

Sq5 = Sq1Sq4

Sq6 = Sq2Sq4 + Sq5Sq1 = Sq2Sq4 + Sq1Sq4Sq1

Sq7 = Sq1Sq6 = Sq1Sq2Sq4

.

For 0 6= a ∈ H1(RP∞), the formula Sq(a2n

) = a2n

+ a2n+1

shows that Sq2n

isnot a sum of SqiSqj with i, j < 2n. On the other hand, the Adem relations implythat

Sq2Sq2 = Sq3Sq1 = Sq1Sq2Sq1 .

Therefore, the Sq2r

do not generate A freely. In order to achieve that, we shall takeanother system of generators.

For a sequence I = (i1, . . . , ik) of positive integers, we set SqI = Sqi1 · · · Sqik ∈A. The degree of I is i1 + · · ·+ ik. The sequence I is called admissible if ij ≥ 2ij+1.

Let Admn be the (finite) set of admissible sequences of degree n. A monomial SqI

is called admissible if I is admissible.

Proposition 49.3. A is the polynomial algebra over the admissible monomials.

The family of admissible monomials is sometimes called the Cartan-Serre basisof A.

Before proving Proposition 49.3, we develop some preliminaries. Fix an inte-ger n and consider

(49.4) wn = x1 · · ·xn ∈ H∗((RP∞)n) ≈ Z2[x1, . . . , xn] .

As Sq(xi) = xi + x2i = xi(1 + xi), one has

Sq(wn) =

n∏

i=1

Sq(xi) = wn

n∏

i=1

(1 + xi) .

Hence,

(49.5) Sqk(wn) = wnσk

where σk is the k-th elementary symmetric polynomial:

(49.6)σ1 = x1 + · · ·+ xnσ2 = x1x2 + · · ·xn−1xnσk =

∑i1<···<ik

xi1 · · ·xik .For an integer p, we use the joker notation Lp for any polynomial in σ1, . . . , σp.

For instance, the equations Lp = Lp+q and Sqi(Lp) = Lp+i hold true but, as

Sqp(σp) = σ2p and Sqi(σp) = 0 for i > p, one has Sqi(Lp) = L2p−1 for all i.


Lemma 49.7. Let I = (i1, . . . , ir) be an admissible sequence. Then, for w = wnwith n ≥ i1, one has

SqI(w) = w(σi1Li1−1 + Li1−1) = wLi1 .

Proof. Only the first equation has to be proven. We proceed by induction onr. For r=1, the lemma follows from (49.5). Suppose that k ≥ 2.

Let I = (i2, . . . , ir). By induction hypothesis, Equation (49.5) and the Cartanformula, one has

SqI(w) = Sqi1SqI(w)

= Sqi1(wLi2 )

= Sqi1(w)Li2 +∑

j≥1 Sqi1−j(w)Sqj(Li2)

= wσi1Li2 + w∑j≥1 σi1−jL2i2−1

= w(σi1Li1−1 + Li1−1)

since i1 ≥ 2i2.

Proof of Proposition 49.3. The Adem relations imply that any monomialSqI is a sum of admissible ones. It remains to see that the admissible monomialsare linearly independent.

Suppose that∑

I∈I λISqI = 0 for I ⊂ Admn and λI ∈ Z2. We must prove thatλI = 0 for all I. We proceed by induction on the cardinality of I. Equation (49.5)for n ≥ i1 proves the assertion for I = (i1).

Let maxj I = max0, ij | (i1, . . . , ik) ∈ I and let w = wn as in (49.4)with n ≥ maxI. Set I = I0∪ I1 where I0 = (i1, . . . , ik) ∈ I | i1 = max1 I.By Lemma 49.7, one has

0 =∑

I∈I

λISqI(w) =∑

I∈I0

λISqI(w)+∑

I∈I1

λISqI(w) = w(σmax ILmaxI−1+LmaxI−1) .

Therefore,

(49.8)∑

I∈I0

λISqI(w) = 0 and∑

I∈I1

λISqI(w) = 0 .

(This uses a very easy case of the uniqueness of the expression of a symmetricpolynomial as a polynomial in the σi’s).

If the decomposition I = I0 ∪ I1 is non-trivial (i.e. I 6= I0), (49.8) permitsus to apply the induction hypothesis. Otherwise, we decompose I0 with respectto max2(I0): I0 = I00∪I01, where I0 = (i1, . . . , ik) ∈ I | i2 = max2 I0 and, asin (49.8), obtain that

(49.9)∑

I∈I00

λISqI(w) = 0 and∑

I∈I01

λISqI(w) = 0 .

If ♯ I > 1, iterating this process will once produce a non-trivial decomposition,enabling us to use the induction hypothesis.

The proof of Proposition 49.3 shows that the map A 7→ A(w) sends SqI | I ∈Admn into a free family of H∗((RP∞)n). This proves the following.

Proposition 49.10. Let 0 6= a ∈ H1(RP∞). The evaluation map A →H∗((RP∞)n) given by A 7→ A(a× · · · × a) (n times) is injective in degree ≤ n.

49. THE STEENROD ALGEBRA 289

We no turn our interest to the cohomology ring H∗(Km) of the Eilenberg-

McLane complex Km. It contains the classes SqI(ι) (0 6= ι ∈ Hm(Km) = Z2)and the admissible monomials play an important role. Define the excess e(I) of anadmissible sequence I = i1, . . . , ik by

e(I) = (i1 − 2i2) + (i2 − 2i3) + · · ·+ (ik−1 − 2ik) + ik = i1 − i2 − · · · − ik .The excess of an admissible monomial SqI is the excess of I. A famous theorem ofJ.P. Serre [171, § 2] is the following.

Theorem 49.11. H∗(Km) is the polynomial algebra generated by SqI(ι) for Iadmissible of excess < m.

The proof of this theorem uses spectral sequences and will not be given here.The condition e(I) < m is natural: SqI(ι) = 0 is e(I) > n since i1 = e(I) + i2 +· · ·+ ik > n+ i2 + · · ·+ ik. If e(I) = n, then

SqI(ι) = (Sqi2 · · · Sqik)2 = · · · = (Sqir · · ·Sqik)2r−1

where e(ir, . . . , ik) < n.

Example 49.12. (1) Only the empty sequence has excess 0. Then H∗(K1)is the polynomial algebra generated by ι ∈ H1(K1). This is not a surprise sinceK1 ≈ RP∞ by Proposition 19.6.

In order to formulate the other examples, observe that if I = (i1, . . . , ik) isadmissible, so is I+ = (2i1, i1, . . . , ik) and e(I+) = e(I). We denote by F(I) thefamily of admissible sequences obtained from I by iterating this construction.

(2) The family of admissible monomials with excess 1 is F(1). Thus H∗(K2) isa polynomial algebra with one generator degree 2i + 1, i ∈ N. Its Poincare series is

Pt(K2) =∏

i∈N

1

1− t2i+1.

For the Poincare series of Km, see Lemma 49.21 below.(3) The set of admissible monomials with excess 2 is the union of the families

F(2r + 1, 2r, . . . , 2, 1) for r ≥ 0.

Remark 49.13. The coefficient exact sequence 0→ Z2 → Z4 → Z2 → 0 givesrise to a Bockstein homomorphism β : H∗(X)→ H∗+1(X) (see [80, § 3.E]). As β isfunctorial and not trivial, one has β(ι) 6= 0 in Hn+1(Kn). But, by Theorem 49.11,the only non-trivial element in Hn+1(Kn) is Sq1(ι). By naturality of β and Sq1,this proves that β = Sq1. This argument illustrates the following corollary ofTheorem 49.11, saying that the actions of the Steenrod algebra on Hn(−) for alln ∈ N generate all the mod 2 cohomology operations.

Corollary 49.14. Let Q be a cohomology operation, and let Q[n] its restrictionto Hn(−). Then, there exists An ∈ A such that Q[n](x) = Anx for all x ∈ Hn(X)and all spaces X.

Proof. By functoriality, it suffices to prove the statement for X = K =K(Z2, n) and x = ι, the generator ofHn(K). ButQ[n](ι) ∈ H≥n(K) by Lemma 45.7

and H≥i(K) = A · ι by Theorem 49.11.

We now list other corollaries of Theorem 49.11. The following one comes fromProposition 49.10.


Corollary 49.15. Let 0 6= a ∈ H1(RP∞) and let y = a×· · ·×a ∈ Hn((RP∞)n).Let fy : (RP∞)n → Kn such that Hfw(ι) = y. Then, H∗f : Hi(Kn)→ Hi((RP∞)n)is injective for i ≤ 2n.

Remark 49.16. The proofs of both Theorem 49.11 and Proposition 49.10, andthen that of Corollary 49.15, do not use the Adem relations. Thus, one can useCorollary 49.15 to give an alternative proof of the Adem relations, as in [171, § 33],[151, pp. 29–31] or [26].

Corollary 49.17. The Adem relations are the only relations amongst theSqI ’s which hold true for all spaces.

Proof. A relation amongst the SqI ’s would be of the form P (SqI1 , . . . ,SqIr ) =0, where P is a Z2-polynomial in r variables. The Adem relations imply that anymonomial SqI is a sum of admissible ones. Therefore, there is a relation of the formP (SqJ1 , . . . ,SqJs) = 0, where P is a Z2-polynomial in s variables and J1, . . . , Jsare admissible sequences. Let m be the maximal excess of J1, . . . , Js. For ι thegenerator of Hm(K(Z2,m)), the equation P (SqJ1(ι), . . . ,SqJs(ι)) = 0 implies, by

Theorem 49.11, that P = 0. Hence, the original relation P (SqI1 , . . . ,SqIr ) = 0 wasa consequence of the Adem relations.

Another consequence of Theorem 49.11 is that the Steenrod squares are char-acterised by some of their properties listed in Theorem 46.2.

Proposition 49.18. Suppose that for each CW -complex X, there exists a mapP : H∗(X)→ H∗(X) satisfying the following properties.

(a) If g : Y → X is a continuous map, then H∗gP = P H∗g.(b) P (Hn(X)) ⊂ H≤2n(X).(c) If a ∈ H1(RP∞) then P (a) = a+ a a.(d) P (x y) = P (x) P (y) for all x, y ∈ H(X).

Then P = Sq.

Proof. Using (a) and (d) together with the definition of the cross product,we get

(d’) P (x× y) = P (x)× P (y) for all x ∈ H(X) and y ∈ H(Y ).

Let w = x1 · · ·xn ∈ H∗((RP∞)n) ≈ Z2[x1, . . . , xn]. Using (c) and (d’) we prove,as for (49.5), that P (w) = wσk. But, by Formula (49.5) again, this shows thatP (w) = Sq(w). Using (a), (b) and Corollary 49.15, we deduce that P = Sq onHn(Kn). By (a), this proves that P = Sq in general.

As a last application of Theorem 49.11, we compute the Poincare series of Km,following [171, § 17]. By Theorem 49.11, one has

Pt(Km) =

∞∏

r=0

1

1− tm+a(r),

wherea(r) = ♯ I | I admissible, e(I) < m and deg(I) = r .

To compute a(r), we note that an admissible sequence I = (i1, . . . , ik) is determinedby its excess components α1 = i1 − 2i2, . . . , αk−1 = ik−1 − 2ik, αk = ik. Therefore,

(49.19) a(r) = ♯ (α1, . . . , αk) |k∑

i=1

αi < m and

k∑

i=1

αi(2i − 1) = r .


Set α0 = m− 1−∑ki=1 αi. Then

(49.20) m+ r = 1 +

k∑

i=0

αi2i = 1 + 20 + · · ·+ 20

︸︷︷︸α0

+ · · ·+ 2k + · · ·+ 2k︸︷︷︸αk

.

Using that∑ki=0 αi = m − 1 and writing the power of 2 in (49.20) in decreasing

order h1 ≥ · · · ≥ hm−1, we get

m+a(r) = ♯ (h1, . . . , hm−1) ∈ Nr−1) | h1 ≥ · · · ≥ hm−1 and 2h1 + · · ·+2hr +1 = m+r .

This proves the following result of [171, § 17].

Lemma 49.21. The Poincare series of Km is

Pt(Km) =∏

h1≥···≥hm−1≥0

1

1− t2h1+···+2hm−1+1.

Let r < m. If I is an admissible sequence with deg(I) = r, the conditione(I) < m is automatic since e(I) ≤ deg(I). Using (49.19), we see that a(r) is equalto the number of partitions of r into integers of the form 2i − 1. Also, Hm+r(Km)

only contains classes of the form SqI(ı) (products like SqI(ı)SqJ(ı) have higherdegree). This proves the following result of [187, p. 37].

Lemma 49.22. If r < m, then dimHm+r(Km) is equal to the number of parti-tions of r into integers of the form 2i − 1.

50. Applications

Suspension of the Hopf maps. Recall that the non triviality of the cup-square map α(a) = a a is H∗(KP 2) for K = R,C,H or O implies that the Hopfmaps

h1,1 : S1 → S1 , h3,2 : S3 → S2 , h7,4 : S7 → S4 and h15,8 : S15 → S8

are not homotopic to a constant maps (see Corollary 35.12). This argument cannotbe applied to the suspensions of the Hopf maps Σkhp,q : Sp+k → Sq+k since the cupproduct in H>0(ΣkKP 2) vanish by dimensional reasons (also by Corollary 25.4).But, for instance in RP 2, α(a) = Sq1(a). As Σ∗Sq = SqΣ∗, one deduces thatSq1 is not trivial on ΣkRP 2 and therefore Σkh1,1 is not homotopic to a constantmap for all k ∈ N (though Hk+1Σkh1,1 vanishes on Hk+1(ΣkRP 2)). The sameargument applies for the other Hopf maps, so we get the following proposition.

Proposition 50.1. For all k ≥ 0, the k-th suspension of the Hopf maps

Σkh1,1 : Sk+1 → Sk+1 , Σkh3,2 : Sk+3 → Sk+2 ,

Σkh7,4 : Sk+7 → Sk+4 and Σkh15,8 : Sk+15 → Sk+8

are not homotopic to a constant maps.

Actually, for k ≥ 1, Σkh3,2 represents the generator of πk+3(Sk+2) ≈ Z2 (see

[193, Proposition 5.1]).


Restrictions on cup-squares. The action of the Steenrod algebra on the co-homology imposes strong restrictions for the existence of classes with non-vanishingcup-square. Let A<n be the subalgebra of A generated by Sqi | i < n.

Proposition 50.2. Let X be a topological space and let a ∈ Hm(X). If m isnot a power of 2, then a a ∈ A<m(a).

Proof. By Lemma 49.1, A is generated by Sqk | k = 2i, so Sqm ∈ A<m ifm is not a power of 2. As a a = Sqm(a), this proves the proposition.

Corollary 50.3. Let X be a topological space. Let a ∈ Hm(X) such that

a a 6= 0. Then, there exists k ≤ m with k = 2i such that Sqk(a) 6= 0.

As a consequence of Proposition 50.2, if a ∈ Hm(X) satisfies a a 6= 0 withm not a power of 2, there must be a non-zero group Hm+k(X) with 0 < k < m.This is not the case for the space X = Cf = D2m ∪f Sm used in § 37 to define theHopf invariant of f : S2m−1 → Sm. Therefore, Proposition 50.2 has the followingcorollary, a result due to Adem [3].

Corollary 50.4. Suppose that f : S2m−1 → Sm satisfies Hopf (f) = 1. Thenm = 2r.

When m is a power of 2, Proposition 50.2 is wrong, as seen with the projectivespaces KP 2. Using secondary cohomology operations, J.F. Adams proved deeperresults [1, Theorem 4.6.1 and §1.2] implying the following theorem.

Theorem 50.5. Let X be a topological space and let a ∈ Hm(X) with m =

2r+1. Suppose that Sq2i

(a) = 0 for i ≤ r. If r ≥ 3, then a a ∈ A<m ·H∗(X).

Combining this theorem with Corollary 50.4, Adams got his famous result [1,Theorem 1.1.1]:

Theorem 50.6. Continuous maps f : S2m−1 → Sm with Hopf invariant 1 existonly for m = 1, 2, 4, 8.

Remark 50.7. Recall that a non-singular map µ : Rm×Rm → Rm (see § 36.2)determines, using (36.5), a continuous multiplication µ : Sm−1 × Sm−1 → Sm−1,giving rise to a map fµ : S2m−1 → Sm with Hopf (fµ) = 1 (Proposition 37.3). Thus,by Theorem 50.6, non-singular maps Rm × Rm → Rm exit only for m = 1, 2, 4, 8.Note that Corollary 50.4 gives another proof of Proposition 36.8.

Somehow related to Theorem 50.6 are the results of E. Thomas on the cohomol-ogy of H-spaces (see e.g. [190]). They also heavily use Steenrod square techniques.

A variant of the Sullivan conjecture. If f : R → S is a continuous map,then H∗f is a morphism of algebras over the Steenrod algebra A. The followingresult, conjectured by H. Miller, was proven by J. Lannes [122, Theorem 0.4].

Theorem 50.8. Let Y be a simply connected space of finite cohomology type.Let B = (RP∞)n. Then the map

[B, Y ]→ homA(H∗(Y ), H∗(B))

is a bijection.

In particular, if Y is a finite dimensional CW -complex, then [B, Y ] is a single-ton. This is a weak version of the Sullivan conjecture (see Remark 35.5).

CHAPTER 8

Stiefel-Whitney classes

51. Trivializations and structures on vector bundles

We recall below some classical facts about vector bundles (defined on p. 157).Two vector bundles ξ = (p : E → X) and ξ′ = (p′ : E′ → X) over the samespace X are isomorphic if there exists a homeomorphism h : E → E′ such thatp′ h = p and such that the restriction of h to each fiber is linear. Two suchisomorphisms h0, h1 : E → E′ are isotopic if there exists a family of isomorphismsht : E → E′ depending continuously on t ∈ [0, 1] and joining h0 to h1. Unlessotherwise mentioned, the total space of a vector bundle ξ is denoted by E(ξ) andthe bundle projection by p.

Let ξ be a vector bundle of rank r over Y and let f : X → Y be a continuousmap. The induced vector bundle f∗ξ is the vector bundle of rank r over X definedby

E(f∗ξ) = (x, z) ∈ X × E(ξ) | f(x) = p(z) ,with the projection onto X as the bundle projection. The projection to E(ξ) gives

a map f : E(f∗ξ)→ E(ξ) which is a linear isomorphism on each fiber and such thatthe following diagram

(51.1)

E(f∗ξ)f //

prX

E(ξ)

p

X

f // Y

is commutative. Note that if f and g coincide over A ⊂ X , the induced bundlesf∗ξ and g∗ξ restrict to the same bundle over A. Two such maps are said homotopicrelative to A if there is a homotopy F : X × I → X between f and g such thatF (a, t) = f(a) = g(a) for all a ∈ A. The next proposition is proven in [177,Theorem 11.3].

Proposition 51.2. Let ξ be a vector bundle over a space B. Let f, g : X → Bbe two maps, where X is a paracompact space. Suppose that f and g coincide overA ⊂ X. If f and g are homotopic relative to A, then there exists an isomorphismbetween f∗ξ and g∗ξ which is the identity over A.

A trivialization of a vector bundle ξ over X is an isomorphism of ξ with theproduct bundle ηr = (prX : X×Rr → X). A vector bundle admitting a trivialization

is called trivial. Denote by T (ξ) the space of trivializations of ξ (endowed with the

compact-open topology). Then, T (ξ) = π0(T (ξ)) is the set of isotopy classes of

trivializations of ξ is π0(T (ξ)).By post-composition, the topological group Aut (ηr) of the automorphisms of

the product bundle ηr acts continuously on the left on T (ξ). As usual we denote by

293

294 8. STIEFEL-WHITNEY CLASSES

GL(r,R) the general linear group, i.e. the topological group of automorphisms ofRr. Alternatively, GL(r,R) is the group of invertible (r×r)-matrices with real coef-

ficients, topologized as an open set of Rr2

. Note that Aut (ηr) ≈ Map(X,GL(r,R)),so the group π0(Map(X,GL(r,R))) = [X,GL(r,R)] acts on T (ξ).

Lemma 51.3. Let ξ be a trivial vector bundle over a topological space X. Thenthe action

(51.4) [X,GL(r,R)]× T (ξ)→ T (ξ)

is simply transitive. In particular, given a trivialization T0 of ξ, the correspondence

λ 7→ λ · T0 induces a bijection [X,GL(r,R)]≈−→ T (ξ).

Proof. Let h, h′ ∈ T (ξ). Then h′ = (h′ h−1)h and h′ h−1 ∈ Aut (ηr). Thisshows that Aut (ηr) acts transitively on T (ξ), whence the action (51.4) is transitive.

On the other hand, if h1 = λ · h0 ∈ T (ξ) is isotopic to h0 via a homotopyht ∈ T (ξ), the formula λt = hth

−10 defines a continuous path λt ∈ Aut (ηr) joining

λ0 = id to λ1 = λ. This shows that [X,GL(r,R)] acts simply on T (ξ).

Remark 51.5. The bijection of Lemma 51.3 depends on the choice of [h0] ∈T (ξ). In general, there is no natural choice, except for the product bundle (like inthe next example).

Example 51.6. We see the projection S1 × C → S1 as the product vectorbundle ξ of rank 2 over the circle, identifying C with R2. By Lemma 51.3

T (ξ) ≈ [S1, GL(2,R)] ≈ [S1, O(2)] ≈ ±1 × Z

(by Gram-Schmidt orthonormalization, GL(2,R) has the homotopy type of O(2)which is homeomorphic to ±1×S1). The fiber linear map corresponding to (1, k)is given by (x, z) 7→ xkz and that corresponding to (−1, k) is given by (x, z) 7→ xkz.

Example 51.7. Let ξ be a vector bundle over Y and let f : X → Y be acontinuous map. If ξ is trivial, so is f∗ξ. More precisely, let h : E(ξ)→ Y ×Rr be a

trivialization of ξ. Write h under the form h(z) = (p(z), h(z)) where h : E(ξ)→ Rd

is a map whose restriction to each fiber is a linear isomorphism. Then the map

f∗h : E(f∗ξ) → X × Rr given by f∗h(x, z) = (x, h(z)) is a trivialization of f∗ξ.This process descends to a map

f∗ : T (ξ)→ T (f∗ξ)

which is equivariant for the actions given by (51.4), via the homomorphismf∗ : [Y,GL(r,R)] → [X,GL(r,R)] induced by λ 7→ λf . Indeed, for λ : Y →GL(r,R) and T ∈ T (ξ), the following formula holds true in T (f∗ξ).

(51.8) f∗(λT ) = (f λ) f∗T .

Proposition 51.9. Let ξ be a vector bundle over a space X. If X is paracom-pact and contractible, then ξ is trivial. Moreover, any trivialization of the restric-tion of ξ over a point x0 ∈ X extends to a trivialization of ξ which is unique up toisotopy.

Proof. Let ξ = (p : E → X) and let ξ0 = (p : E0 → x0) be the restrictionof ξ over the point x0. The vector bundle ξ is induced by the identity of X :ξ = id∗Xξ. As X is contractible, idX is homotopic to a constant map c onto x0

51. TRIVIALIZATIONS AND STRUCTURES ON VECTOR BUNDLES 295

and, by Proposition 51.2, ξ ≈ c∗ξ0. The contractibility of X also implies thatc∗ : [x0, GL(r,R)] → [X,GL(r,R)] is an isomorphism. Proposition 51.9 thenfollows from Lemma 51.3 and the considerations of Example 51.7.

We now consider a vector bundle ξ over a space X = Z ∪ϕ Y obtained byattaching a pair (Z,A) to a space Y using an attaching map ϕ : A→ Y (see p. 83).Let jY : Y → X and jZ : Z → X be the natural maps. Let ξY = j∗Y ξ, ξZ = j∗Zξand denote by ξA the restriction of ξZ over A.

Lemma 51.10. Let ξ be a vector bundle over X = Z∪ϕY as above. Suppose that

there exists trivializations hY ∈ T (ξY ) and hZ ∈ T (ξZ) such that the restrictionof hZ to ξA is isotopic to ϕ∗hY . Suppose that Z is paracompact and that the pair(Z,A) is cofibrant. Then, the bundle ξ is trivial.

Proof. Let hA ∈ T (ξA) be the restriction of hZ to ξA. An isotopy from hAto ϕ∗hY produces a trivialization h ∈ T (ξA× I) (where ξA× I is the vector bundlep× id : E(ξA)× I → A× I) restricting to hA on ξA×0 and to ϕ∗hY on ξA×1.The pair (Z,A) being cofibrant, there exists a retraction r : Z×I → Z×0∪A×Iof Z × I onto Z ×0∪A× I (see Lemma 12.62). Let ξ1 be the restriction of r∗ξZabove Z × 1 ≈ Z. As Z is paracompact and r is actually a strong deformationretraction (see Remark 12.68), ξ1 is isomorphic to ξZ relative to A by Lemma 51.2.Then, r∗h produces at level 1 a trivialization hZ of ξZ which is equal to ϕ∗hY onE(ξA). This condition implies that the formula

H(u) =

hZ(z, u) if p(u) = iZ(z)

hY (y, u) if p(u) = iY (y)

defines a trivialization H : E(ξ)→ X × Rr of ξ.

The above lemmas have analogues for principal bundles. Let A be a topologicalgroup. Recall that a A-principal bundle overX consists of a continuous map p : P →X , a continuous right action of A on P such that p(zα) = p(z) for all z ∈ P and allα ∈ A. In addition, the following local triviality should hold: for each x ∈ X thereis a neighbourhood U of x and a homeomorphism h : U × A → p−1(U) such thatph(x, a) = x and h(x, aα) = h(x, a)α. In consequence, A acts simply transitivelyon each fiber and p is a surjective open map, thus descending to a homeomorphism

P/A≈−→ X (use [42, § I Chapter VI]). An isomorphism of A-principal bundles from

p : P → X to p′ : P ′ → X is a A-equivariant homeomorphism h : P → P ′ suchthat p′ h = p. Two such isomorphisms h0, h2 : P → P ′ are isotopic if there exists afamily of isomorphisms ht : P → P ′ depending continuously on t ∈ [0, 1] and joiningh0 to h1. The notion of induced A-principal bundle works as for vector bundlesand Proposition 51.2 also follows from [177, Theorem 11.3].

A trivialization of an A-principal bundle is an isomorphism with the productbundle X × A → X . An A-principal bundle is trivial if and only if it admits asection (see, e.g. [177, I.8]). This gives a bijection between sections and trivializa-tions and between homotopy classes of sections and isotopy classes of trivialization.If σ1, σ2 : X → P are two sections of p, then σ2(x) = σ1(x)α(x) for a uniqueα : Map(X,A) whence the analogue of Lemma 51.3: The action of Map(X,A) onthe trivializations of P is simply transitive. Also, Lemma 51.9 holds true for prin-cipal bundles.


Let p : P → X be an A-principal bundle, p′ : P ′ → X be an A′-principal bundleand let ϕ : A→ A′ be a continuous homomorphism. A continuous map g : P → P ′

is called a ϕ-map if p′ g = p and g(zα) = g(z)ϕ(α) for all z ∈ P and α ∈ A.Let ρ : A → GL(r,R) be a representation (i.e. a continuous homomorphism)

of the topological group A. Let P → X be an A-principal bundle. Then theRr-associated bundle with total space

P ×(A,ρ) Rr = P × Rr/(zα, u) ∼ (z, ρ(α)u)

is a vector bundle of rank r over X . Most of the time, the representation ρ isimplicit and we just write P ×A Rr for P ×(A,ρ) Rr. A (A, ρ)-structure (or just A-structure) for a vector bundle ξ of rank r over X is an A-principal bundle P → Xtogether with a vector bundle isomorphism f : P ×ARr → E(ξ). Two A-structures(P, f) and (P ′, f ′) are

• strongly equivalent if there exists an isomorphism of A-principal bundlesh : P → P ′ such that f ′ (h× idRr) = f .• weakly equivalent if there exists an isomorphism of A-principal bundlesh : P → P ′ such that f ′ (h× idRr) is isotopic to f

Here below are a few examples.

51.11. A = 1, the trivial group. An 1-structure on a vector bundle ξ isjust a trivialization of ξ. Strong equivalence coincides here with equality. Weakequivalence classes of 1-structures correspond to isotopy classes of trivializations.

51.12. Each vector bundle ξ of rank r admits a GL(r,R)-structure which isunique up to strong equivalence. Indeed, consider the space of frames of ξ:

Fra(ξ) = ν : Rr → E(ξ) | ν is a linear isomorphism onto some fiber of ξ .with the map pFra : Fra(ξ) → X given by pFra(ν) = pν(0). (The image byν ∈ Fra(ξ) of the standard basis of Rr is a frame (basis) of ν(Rr), whence thename of space of frames). By precomposition, the topological group GL(r,R) actscontinuously and freely on the right upon Fra(ξ). The map pFra descends to acontinuous bijection Fra(ξ)/GL(r,R) → X . Using local trivializations, one checksthat this map is a homeomorphism. Also, a local trivialization of ξ gives rise to alocal section of pFra. Hence, pFra is a GL(r,R)-principal bundle, called the framedbundle of ξ.

Consider the evaluation map fFra : Fra(ξ) ×GL(r,R) Rr → E(ξ) sending [ν, t] toν(t). This map is a continuous bijection which is linear on each fiber and, usinglocal trivializations of ξ again, one checks that it is a homeomorphism. Hence,fFra is a GL(r,R)-structure on ξ. We claim that fFra is a universal structure inthe following sense. Each A-structure (for a representation ρ : A → GL(r,R))

f : P ×Rr → E(ξ) determines a ρ-map f : P → Fra(ξ). The map f sends z ∈ P tothe map f(z,−) ∈ Fra(ξ). We check that two A-structure (P, f) and (P ′, f ′) are

(a) strongly equivalent if there exists an isomorphism of A-principal bundles

h : P → P ′ such that f ′ h = f .(b) weakly equivalent if there exists an isomorphism of A-principal bundles

h : P → P ′ such that f ′ h is homotopic to f .

The case A = GL(r,R) give the uniqueness claimed above: any GL(r,R)-structureon ξ (for ρ = id) is strongly equivalent to fFra.

51. TRIVIALIZATIONS AND STRUCTURES ON VECTOR BUNDLES 297

51.13. GL+(r,R)-structures and orientations (see also § 52 below). Recall thatan orientation of a finite dimensional real vector space is an equivalence class of(ordered) basis, where two basis are equivalent if their change-of-basis matrix is inGL+(r,R), i.e. has positive determinant. An orientation of a vector bundle ξ is anorientation of each fiber which varies continuously, i.e. there are local trivializations

p−1(U)≈−→ U ×Rd whose restriction to each fiber is orientation-preserving (for the

standard orientation of Rd). A vector bundle admitting an orientation is calledorientable and the choice of an orientation makes it oriented.

If P is a GL+(r,R)-principal bundle, then P ×GL+(r,R) Rr is oriented, using the

standard orientation of Rr. Hence, a GL+(r,R)-structure (P, f) on a vector bundleξ makes it oriented. Conversely, an oriented vector bundle ξ admits a GL+(r,R)-structure: one just restricts the canonical (Fra(ξ), fFra) of 51.12 to Fra+(ξ), where

(51.14) Fra+(ξ) = ν ∈ Fra(ξ) | ν preserves the orientation .(for the standard orientation of Rd). As in 51.12, we check that (Fra+(ξ), fFra) isuniversal for the A-structures on ξ with a representation ρ : A→ GL+(r,R).

51.15. O(r)-structures. Consider the orthogonal group O(r) with its standard

representation O(r) → GL(r,R). Let f : P ×O(r) Rr≈−→ E(ξ) be a O(r)-structure

on ξ. Then, the standard inner product on Rr gives, via f , an Euclidean structureon ξ (see p. 157) for which f is an isometry on each fiber. On the other hand, anEuclidean bundle ξ of rank r admits an O(r)-structure: one restricts fFra to thesubbundle of Fra(ξ) formed by orthonormal frames:

Fra⊥(ξ) = ν : Rr → E(ξ) | ν is a linear isometry onto some fiber of ξ .As in (51.12), one shows that such an O(r)-structure compatible with a givenEuclidean structure on ξ is unique up to strong equivalence. This process providesa bijection between Euclidean structures on ξ and strong equivalences of O(r)-structures.

On the other hand, a vector bundle over a paracompact space admits Euclideanstructures which form a convex space. Let (ξ, et) be the vector bundle ξ endowedwith an Euclidean structure et depending continuously on t ∈ I. Then

Frae(ξ) = νt : Rr → E(ξ) | t ∈ I, νt is a linear isometry onto some fiber of (ξ, et)is the total space of an O(r)-principal bundle over X × I. Note that Frae(ξ) isthe union indexed by I of Fra⊥(ξ, et), the bundle of orthonormal frames for theEuclidean structure et. If X is paracompact, Frae(ξ)→ X × I is isomorphic to theprincipal bundle Fra⊥(ξ, e0) × I → X × I [103, Chapter 4, Theorem 9.8]. Hence,there is an isomorphism h : Fra⊥(ξ, e0)→ Fra⊥(ξ, e1) such that i1h is homotopic toi0 (where it : Fra⊥(ξ, et)→ Fra(ξ) denotes the inclusion). This proves the followingstatement: a vector bundle over a paracompact space admits an O(r)-structurewhich is unique up to weak equivalence.

51.16. The case of orthogonal representations. Let ρ : A → O(r) be an or-thogonal representation of a topological group A. Let f : P ×A Rr → E(ξ) bean A-structure on the vector bundle ξ (for the representation ρ). As ρ is orthog-onal, P ×A Rr inherits a natural Euclidean structure which is transported on ξvia f (see 51.15). As in 51.12, there is an O(r)-structure (PO, fO) on ξ such that

f = fO f for a ρ-map f : P → PO. If h : P → P ′ induces a strong equivalence

between the A-structures (P, f) and (P ′, f ′), then PO = P ′O and f ′ h = f . If h


only induces a weak equivalence, it descends to an isomorphism hO : PO → P ′Omaking the following diagram commutative

Ph //

f

P ′

f ′

PO

hO // P ′O

We can pull-back f ′ over PO via hO, getting another representative of the weakequivalence class of (P ′, f ′). This permits us to assume that PO = P ′O and hO = id;this also means that the Euclidean structure induced by f and f ′ coincide. Now,using the Gram-Schmidt orthonormalization process in each fiber of ξ, the isotopybetween f ′ (h× idRr ) and f may be deformed into an isotopy of isometries. This

implies that f and f ′ h are homotopic.These considerations drive us to the following point of view for an A-structure

on ξ, in the case of an orthogonal representation. One first fix an Euclidean struc-ture on ξ and consider only A-structures (P, f) on ξ for which f is an isometry. In

other words, an A-structure may be seen as a ρ-map f : P → Fra⊥(ξ) and strongor weak equivalences are described as in (a) and (b) of 51.12.

Note that, if ξ is an oriented Euclidean bundle, one can consider the orientedorthonormal frames

(51.17) Fra+⊥ (ξ) = Fra⊥(ξ) ∩ Fra+(ξ)

which is the total space of an SO(r)-principal bundle over X . Then (Fra+⊥ (ξ), fFra)

is an SO(r)-structure which is universal for the A-structures associated to a rep-resentation A → SO(r). The special case of the representation Spin(r) → SO(r)(spin structures) is treated in § 53 (compare [127, § II.1]).

The best known example of rep+⊥ (ξ) is for ξ = TSn, the tangent bundle to the

standard unit sphere Sn. One sees TSn as the space of couples (v, w) ∈ Rn+1×Rn+1

such that |v| = 1 and 〈v, w〉 = 0 (up to translation, w is tangent to v ∈ Sn). Thenthe map q : SO(n + 1) → Sn defined by q(A) = Ae1 (first column vector) is the

oriented frame bundle for TSn. The bundle isomorphism SO(n+1)≈−→ Fra+

⊥ (TSn)sends A to the map νA : Rn → TSn defined by

νA(t1, . . . , tn) =(Ae1,

n∑

i=1

tiAei+1

).

51.18. Complex vector bundles. By replacing R by C in the definition of avector bundle, we get the notion of a complex vector bundle. The notion of a A-structure is defined accordingly, using a complex representation ρ : A → GL(r,C).As in 51.12, a complex vector bundle ξ of rank r admits a GL(r,C)-structure whichis unique up to strong equivalence. Also, as in 51.15, using a Hermitian structureon ξ (those form a contractible space), ξ admits an U(r)structure which is uniqueup to weak equivalence.

51.19. Classifying spaces and structures. Let EGL(r,R) → BGL(r,R) be theuniversal bundle for the principal GL(r,R)-bundles and let ζ its associated vectorbundle. Recall that, for X paracompact, the correspondence [c] → c∗ζ provides abijection from [X,BGL(r,R)] and the set of isomorphism classes of vector bundles

52. THE CLASS w1 – ORIENTABILITY 299

over X of rank r (using (51.12)). We shall use the following consequence of thisresult.

Lemma 51.20. Let (X,A) be a cofibrant pair with X paracompact. Let ξ be avector bundle over X whose restriction over A is trivial. Then, there exists a vectorbundle ξ over X/A such that ξ ≈ π∗ξ, where π : X → X/A is the quotient map.

Proof. Let c : X → BGL(r,R) be a classifying map for ξ (r = rank of ξ),i.e. ξ ≈ c∗(ζ). In the definition of a cofibrant pair, A is supposed to be closed,so A is paracompact. The restriction of c to A is then null-homotopic. As (X,A)is cofibrant, there is a homotopy from c to c such that c|A is a constant map.Therefore, c descends to c : X/A→ BGL(r,R). Hence, ξ ≈ c∗ζ ≈ π∗(c∗ζ).

Structures on vector bundles and the classifying spaces are related as follows.Let ξ be a vector bundle of rank r over a paracompact space X . Fix a characteristicmap c : X → BGL(r,R) for Fra(ξ). A representation ρ : A→ GL(r,R) of A inducesa continuous map Bρ : BA→ BGL(r,R) which may be made a Serre fibration. Alifting of c is a continuous map c : X → BA such that Bρ c = c and two such liftingsc0 and c1 are homotopic if there exists a homotopy h : X × I → BA between c0and c1 such that Bρh(x, t) = c(x) for all (x, t) ∈ X × I. Then, the set of weakequivalence of (A, ρ)-structures on ξ is in bijection with the homotopy classes ofliftings of c. For details (see [22, § 4]).

52. The class w1 – Orientability

Let O(V ) be the set of the two orientations of a finite dimensional vector spaceV . Let ξ be a vector bundle of rank r over a topological space X . Using thecanonical GL(r,R)-structure (Fra(ξ), fFra) of 51.12, we define

O(ξ) = Fra(ξ)×GL(r,R) O(Rr) .

The projection O(ξ)→ X is a locally trivial bundle whose fiber over x ∈ X is, viafFra, in bijection with O(p−1(x)). In consequence, O(ξ)→ X is a two fold covering.An orientation of ξ (see 51.13) is clearly a continuous section of O(ξ)→ X .

The characteristic class w(O(ξ) → X) ∈ H1(X) (see § 24.2), is called the firstStiefel-Whitney class of ξ and is denoted by w1(ξ).

Proposition 52.1. A vector bundle ξ over a CW-complex X is orientable ifand only if w1(ξ) = 0 in H1(X). The set of orientations of an orientable bundle isin bijection with H0(X).

Proof. The 2-covering O(ξ) is trivial if and only if it admits a continuoussection, that is to say if and only if ξ is orientable. On the other hand, if X is aCW-complex, the 2-covering O(ξ) → X is trivial if and only if its characteristicclass vanish (see Lemma 24.11). There are two orientation for the restriction of ξover each connected component, whence the last assertion.

Remark 52.2. If η is a trivial vector bundle, then w1(ξ ⊕ η) = w1(ξ). Indeed,O(ξ ⊕ η) = O(ξ).

Proposition 52.3. Vector bundles over an 1-dimensional CW-complex areclassified by their rank and their first Stiefel-Whitney class.


Proof. Let ξ and ξ′ be two vector bundles over a CW-complex X . If ξ andξ′ are isomorphic, they have the same rank and the 2-coverings O(ξ) and O(ξ′) areisomorphic, which implies that w1(ξ) = w1(ξ

′).To prove the converse, note that, when X is 1-dimensional, any vector bundle

ζ over X is the Whitney sum of a line bundle λ with the trivial vector bundle [103,Chapter 8, Theorem 1.2]. By Remark 52.2, w1(ζ) = w1(λ). We are thus reducedto ξ and ξ′ being both of rank 1, in which case we use Proposition 52.4 below.

Let L(X) be the set of isomorphism classes of real lines bundles over a spaceX . The tensor product (see (43.11)) of two line bundles is again a line bundle. Thisprovides an operation ⊗ on L(X).

Proposition 52.4. Let X be a CW-complex. Then the first Stiefel-Whitneyclass provides an isomorphism

w1 : (L(X),⊗)≈−→ (H1(X),+) .

In particular, (L(X),⊗) is an elementary abelian 2-group.

Proof. Let ξ = (p : E → B) be a line bundle over X . Endow ξ with aEuclidean structure. The unit sphere bundle S(E) → X is then a 2-fold coveringwhich is clearly isomorphic to O(ξ). By (24.10), w(O(ξ)) determines O(ξ) and

then S(E). But S(E) determines ξ by the isomorphism S(E) ×O(1) R≈−→ E (an

O(1)-structure on ξ). Hence, w1 is injective. For the surjectivity, let a ∈ H1(X).

By (24.10), there is a 2-covering X → X with characteristic class a. We see it as an

O(1)-principal bundle. Then, X ×O(1) R → X is a Euclidean line bundle ξ whose

sphere bundle is X. By the above, w1(ξ) = a.It remains to show that w1(ξ⊗ξ′) = w1(ξ)+w1(ξ

′) for two line bundles ξ and ξ′

over X . We start with some preliminaries. The linear group GL(R) is isomorphicthe multiplicative group R× = R − 0. Set K = R× × R×. Using the R-vector

space isomorphism R⊗ R≈−→ R such that x⊗ y 7→ xy, the following diagram

GL(R)×GL(R)

≈

ϕ⊗ // GL(R⊗ R)

≈

Kϕ // R×

is commutative, where ϕ⊗ is the natural homomorphism (see p. 253) and ϕ is justthe standard product, which is a continuous homomorphism.

Let F = Fra(ξ), F ′ = Fra(ξ′) and F⊗ = Fra(ξ ⊗ ξ′) seen, using the above, asprincipal R×-bundles. Let ∆: X → X×X be the diagonal inclusion. The definitionof F⊗ in (43.11) becomes

F⊗ = ϕ⊗∗ ∆∗(F × F ′) ≈ ∆∗(ϕ⊗∗ (F × F ′)) .

52. THE CLASS w1 – ORIENTABILITY 301

Let β, β′ : X → BR× be characteristic maps for F and F ′. One has morphismsof R×-principal bundles, i.e. commutative diagrams

(52.5)

E(F⊗)

// E(F × F ′)×K R×

// (ER× × ER×)×K R×

X

∆ //(β,β′) 33

X ×X β×β′ // BR× ×BR×

where the maps on the top line are R×-equivariant.Recall from (42.3) that the two projections of K onto R× induce a homotopy

equivalence P : BK≃−→ BR××BR×. Let P ′ be a homotopy inverse of P . One has

more morphisms of R×-principal bundles:

(52.6)

(ER× × ER×)×K R× //

EK ×K BR×κ //

ER×

BR× ×BR×

P ′

≃// BK

Bϕ // BR×

where κ is defined by κ([(ti(gi, g′i), λ]) = (ti, gig

′iλ). Diagrams (52.5) and (52.6)

imply that BϕP ′ (β, β′) is a characteristic map for the R×-principal bundle F⊗.As the inclusion ±1 → R× is a homomorphism and a homotopy equivalence,

one has BR× ≃ B±1 ≃ RP∞ ≃ K(Z2, 1). The map BϕP ′ thus defines con-tinuous multiplication on K(Z2, 1). One checks that it is homotopy commutativeand admits a homotopy unit. Let u be the generator of H1(BR×) = Z2. ByProposition 28.85 and Lemma 28.89, one has

(52.7) H∗(BϕP ′)(u) = u× 1 + 1× u .Hence,

w1(ξ ⊗ ξ′) = H∗(β, β′)(u × 1 + 1× u) = H∗β(u) +H∗β′(u) = w1(ξ) + w1(ξ′) .

The above show that the correspondence [X → X ] 7→ [X×O(1) R→ X ] inducesa bijection between the set Cov2(X) of equivalence classes of 2-fold coverings of Xand L(X), with a commutative diagram

Cov2(X)≈ //

≈

w %%KKKKKKKKL(X)

≈

w1zzuuuuuuu

H1(X)

.

By Proposition 28.58, we get the following corollary.

Corollary 52.8. Let λ be a line bundle over a CW-complex X. Then w1(λ) ∈H1(X) coincides with the Euler class e(λ) of λ.

Example 52.9. The tautological line bundle over RPn. The 2-covering ζ =(Sn → RPn) (1 ≤ n ≤ ∞) is an O(1)-principal bundle. Its associated line bundleλ = (Sn ×O(1) R → RPn) satisfies w1(λ) = w(ζ) 6= 0 in H1(RPn) = Z2 (see the


proof of Proposition 24.21). Seeing RPn as the space of vector lines in Rn+1, λidentifies itself with the tautological line bundle over RPn, i.e.

E(λ) = (a, v) ∈ RPn × Rn+1 | v ∈ a , p(a, v) = a .

The identification Sn ×O(1) R≈−→ E(λ) is given by [z, t] 7→ (R z, tz).

Proposition 52.10. Let ξ be a vector bundle over a CW-complex X. Then,the following conditions are equivalent.

(1) The restriction of ξ over the 1-skeleton of X is trivial.(2) w1(ξ) = 0.

Proof. Let i : X1 → X denote the inclusion and let ξ1 ≈ i∗ξ be the restrictionof ξ overX1. By Lemma 24.11, w1(ξ1) = i∗(w1(ξ)) and w1(ξ1) = 0 if ξ1 is trivial. Asi∗ : H1(X)→ H∗(X1) is injective, this proves the implication (1)⇒ (2). Conversely,if w1(ξ) = 0, then w1(ξ1) = 0 and ξ1 is trivial by Proposition 52.3.

We finish this section by a describing a singular cocycle representing w1(ξ) interms of transporting orientations. Let c : I → X be a path and let α ∈ O(Ec(0))(where Ex is the fiber of ξ over x ∈ X). We see α as an element of the fiber ofFra(ξ) over c(0). The unique lifting c : I → Fra(ξ) of c with c(0) = α provides anorientation c∗α = c(1) ∈ O(Ec(1)). We say that c∗α is obtained from α by transportalong c.

Choose a set-theoretic section α : X → Fra(ξ), i.e. an assignation of an orienta-tion of Ex for each x ∈ X (which has not to vary continuously). We see a singularsimplex σ ∈ S1(X) as a path σ : I → X via the identification I ≈ ∆1 sending t to(t, 1− t). The cochain w1(ξ, α) defined by

〈w1(ξ, α), σ〉 =

1 if σ∗α(σ(0)) 6= α(σ(1))

0 otherwise.

The cochains w1(ξ, α) and w(Fra(ξ), α) of § 24.2 clearly coincide. Hence, by Propo-sition 24.12, w1(ξ, α) is a cocycle representing w1(ξ).

Let x0 ∈ X and α0 ∈ O(Ex0). We say that a loop c : I → X at x0 preservesthe orientation if c∗α0 = α0 (this condition does not depend on the choice of α0).Note that, if d is another loop at x0, then (dc)∗α0 = d∗(c∗α0). Also, c∗α0 dependsonly on the homotopy class of the loop c, by the homotopy lifting property of thecovering Fra(ξ)→ X . Hence the correspondence

c 7→

0 if c preserves the orientation

1 otherwise.

provides a homomorphism w : π1(X,x0)→ Z2 which corresponds to w1(ξ) ∈ H1(X)

via the isomorphism H1(X)≈−→ hom(π1(X,x0),Z2) of Lemma 24.1.

53. The class w2 – Spin structures

In this section, we define a cellular second Stiefel-Whitney class w2(ξ) ∈ H2(X),

when ξ is an orientable vector bundle over a regular CW-complex X (recall that H∗

denotes the cellular cohomology introduced in § 16). A more general w2(ξ) ∈ H2(X)(X any space) is considered in § 54, but the results below are used to establish therelationship between w2(ξ) and the existence of spin structures on ξ.

53. THE CLASS w2 – SPIN STRUCTURES 303

Let ξ be an orientable vector bundle of rank r ≥ 2 over a CW-complex X .Denote by ξi the restriction of ξ over the i-skeleton X i of X . By Propositions 52.1and 52.10, ξ1 is trivial. Fix an orientation of ξ. Choose a trivialization T1 of ξ1which is compatible with this orientation. The restriction T0 of T1 over X0 is thusuniquely determined up to isotopy.

Let e be a 2-cell of X with characteristic map ϕ : (D2, S1) → (X,X1). AsD2 is contractible, there is a unique (up to isotopy) trivialization Te of ϕ∗ξ overD2 which is compatible with the orientation (see Proposition 51.9). Thus, ϕ∗T1

and Te provides two trivialization of ϕ∗ξ restricted to the boundary S1 of D2. ByLemma 51.3, these two trivializations differ by the action of a map from S1 toGL(r,R), whose range is GL+(r,R) since ϕ∗T1 and Te are both compatible with agiven orientation. Let

(53.1) w2(e) ∈ [S1, GL+(r,R)] ≈ π1(GL+(r,R)) ≈ π1(SO(r))

be the homotopy class of this map. Here, the isomorphism [S1, GL+(r,R)] ≈π1(GL

+(r,R)) holds true since GL+(r,R) is a topological group and GL+(r,R) hasthe homotopy type of SO(r) by the Gram-Schmidt orthonormalization process. Ifr ≥ 3, then π1(SO(r)) = Z2. If r = 2 then π1(SO(r)) ≈ Z and, by convention, wetake w2(e) mod 2. The correspondence e 7→ w2(e) thus defines a cellular 2-cochain

w2 = w2(ξ, T1) ∈ C2(X).

Lemma 53.2. Suppose that X has no 3-cells or that X3 is regular complex.Then, the cochain w2(ξ, T1) is a cellular cocycle. Its cohomology class w2(ξ) ∈H2(X) depends only of the isomorphism class of ξ.

The class w2(ξ) ∈ H2(X) is called the second Stiefel-Whitney class of the vectorbundle ξ. If the rank of ξ is ≤ 1, we set by convention that w2(ξ) = 0.

Proof. We may assume that X is connected. Observe first that the cochainw2(ξ, T1) does not depend on the orientation of ξ, since the other choice would justchange all the orientations under consideration. Let us see how w2(ξ, T1) depends

on the isotopy class of the trivialization [T1] ∈ T (ξ1). Let T ′1 ∈ T (ξ1) be anothertrivialization compatible with the orientation. As seen above, this compatibilityimplies that the restriction T ′0 of T ′1 to ξ0 coincides with T0 up to isotopy. Such anisotopy may be realized in a neighbourhood of X0 in X1, so one may assume thatT0 = T ′0. By Lemma 51.3, there is a unique map a : X1 → GL+(r,R) such thatT ′1 = a ·T1. As T0 = T ′0, the restriction of a to X0 is constant to the identity of Rr.Hence, each 1-cell ε with characteristic map ϕε : D

1 → X1 gives rise to a homotopyclass

a(ε) = [aϕε] ∈ [(D1, S0), (GL+(r,R), id)] ≈ π1(GL+(r,R), id) ≈ π1(SO(r), id) = Z2

which does not depend on the choice of ϕε (since π1(SO(r), id) = Z2; again, ifr = 2, one takes by convention the value mod 2 of a ∈ π1(SO(2)) ≈ Z). The corre-

spondence ε 7→ a(ε) determines a cellular 1-cochain a ∈ C1(X). Let e be a 2-cellof X with characteristic map ϕ : (D2, S1)→ (X,X1). Using the cellular boundaryformula (16.3) and writing the abelian group π1(GL

+(r,R)) = Z2 additively, we getthat [aϕ] = δ(a). In the same way, using (51.8), one gets the following formula.

(53.3) w2(ξ, T′1) = w2(ξ, a T1) = w2(ξ, T1) + δ(a) .

We now prove that w2 is a cellular cocycle. If X has no 3-cells, there is nothingto prove. Let ǫ be a 3-cell of X which, as X3 is now supposed to be regular, can


be identified with a subcomplex (also called ǫ) of X . As ǫ is contractible, there isa unique (up to isotopy) trivialization T ǫ of ξ|ǫ, compatible with the orientation.

As above, one may assume that T ǫ coincides with T0 over ǫ0. A trivializationT ǫ1 ∈ T (ξ1) may be thus defined by

T ǫ1 (z) =

T ǫ(z) if p(z) ∈ ǫT1(z) otherwise.

By (53.3) and the construction of T ε1 , one has

(53.4) δ(w2(ξ, T1))(ǫ) = δ(w2(ξ, Tǫ1 ))(ǫ) = 0 .

Equation (53.4) shows that δ(w2(ξ, T1)))(ǫ) = 0 for all 3-cell ǫ, proving that

δ(w2(ξ, T1)) = 0. Also, formula (53.3) show that [w2(ξ, T1)] ∈ H2(X) does notdepend on the choice of T1 and thus depends only on ξ. Finally, if ξ′ isomorphic to

ξ via a homeomorphism h : E(ξ′)≈−→ E(ξ) over the identity of X , the trivialization

T ′1 = T1h satisfies w2(ξ′, T ′1) = w2(ξ, T1). Therefore, w2(ξ

′) = w2(ξ).

Lemma 53.5. Let ξ be an orientable vector bundle of rank r ≥ 2 over a CW-complex X satisfying the hypotheses of Lemma 53.2. If η is a trivial vector bundleover X, then w2(ξ ⊕ η) = w2(ξ).

Proof. Note that ξ ⊕ η is orientable since ξ is so, thus w2(ξ ⊕ η) is defined.

Let us represent w2(ξ) by a cocycle w2(ξ, T1) as above. If Tη : X × Rs≈−→ E(η)

is a fixed trivialization of η, one checks that w(ξ, T1 ⊕ Tη) = π1j w(ξ, T1), wherej : SO(r) → SO(r + s) is the inclusion. But π1j is an isomorphism if r ≥ 3 or thereduction mod 2 if r = 2, which proves the lemma in these cases.

If r = 1, then w2(ξ) = 0 by convention. Also, as ξ is orientable, it is trivial byProposition 52.4. Thus, ξ ⊕ η is trivial and w2(ξ ⊕ η) = 0 by the proof that (2)implies (3) in the next proposition (this part of the proof is valid for any r).

Example 53.6. Let X = S2, with its cellular decomposition with one 0-celland one 2-cell: X = D2 ∪S1 pt. Let α : S1 → GL+(r,R), representing [α] ∈π1(GL

+(r,R)). Then

E(ξ[α]) = D2 × Rr ∪α pt× Rr

is the total space of a vector bundle ξ[α] of rank r over X . This process gives a

bijection between π1(GL+(r,R)) and the isomorphism classes of vector bundles of

rank r over S2 (compare [177, § 18]). The two cochain associating to the 2-cell ofX the element [α] ∈ π1(GL

+(r,R)) = Z2 if r ≥ 3 (or its reduction mod 2 if r = 2)

represents w2(ξ[α]) ∈ H2(X) = Z2. Summing up, there are two vector bundles (up

to isomorphism) of rank ≥ 3 over S2: the trivial bundle η, satisfying w2(η) = 0,and the non-trivial bundle ξ, characterised by the property that w2(ξ) 6= 0. This isan example of Proposition 53.7 below.

Proposition 53.7. Let ξ be a vector bundle of rank r ≥ 3 over a CW-complexX. Suppose that X has no 3-cell or that X3 is a regular complex. Let ξi be therestriction of ξ over the X i. Then, the following conditions are equivalent.

(1) ξ3 is trivial.(2) ξ2 is trivial.(3) w1(ξ) = 0 and w2(ξ) = 0.

53. THE CLASS w2 – SPIN STRUCTURES 305

In the next section, Proposition 53.7 will be generalized, using the singularsecond Stiefel-whitney class (see Proposition 54.12).

Proof. By Proposition 52.10, w1(ξ) = 0 if ξ2 is trivial. Also, ξ is orientable byProposition 52.1, so w2(ξ) is defined. We can represent w2(ξ) by a cocycle w2(ξ, T1)

where T1 is the restriction of Γ ∈ T (ξ2). For each 2-cell e, we thus have T e = Γ|e,thus w2(ξ, T1) = 0, proving that w2(ξ) = 0. Thus, (2) implies (3).

Conversely, suppose that w1(ξ) = 0 and w2(ξ) = 0. Choose a trivialization T1

of ξ1 and let ϕ : Λ2 × (D2, S1)→ (X,X1) be a global characteristic map for the 2-

cells of X . Then, w2(ξ, T1) = δ(a) for a ∈ C1(X). As in the proof of Lemma 53.2,the cochain a may be used to modify (relative to ξ0) T1 into a trivialization T ′1such that w2(ξ, T

′1) = 0 (this uses that r ≥ 3). This means that, over Λ2 × S1,

the trivialization ϕ∗T ′1 coincides up to isotopy with the unique (up to isotopy)trivialization over Λ2×D2 compatible with the orientation. By Lemma 51.10, thisimplies that ξ2 is trivial.

We have so far proven that (2)⇔ (3). We now prove the equivalence (1)⇔ (2),which is true for any CW-complex. The implication (1)⇒ (2) is trivial. Conversely,let Γ ∈ T (ξ2). Let ϕ : Λ3 × (D3, S2)→ (X,X2) be a global characteristic map forthe 3-cells of X . As above, one compares the trivialization ϕ∗Γ over Λ3 × S2

with the unique (up to isotopy) trivialization over Λ3 × D3 which is compatiblewith the orientation. Their isotopy classes differ by the action of an element of[Λ3 × S2, GL+(r,R)]. But π2(SO(r)) = 0 (see, e.g. [177, 22.10]), thus

[S2, GL+(r,R)] ≈ [S2, SO(r)] ≈ π2(SO(r)) = 0 .

As above, using Lemma 51.10, this implies that ξ3 is trivial.

We now see the second Stiefel-Whitney class w2(ξ) as the obstruction to theexistence of the existence of a spin structure on ξ. This refers to the standardorthogonal representation ρ0 : Spin(r)→ SO(r) which is a 2-covering. As in 51.16,it is natural to fix an Euclidean structure and an orientation on ξ, in which case,

a spin structure is seen a ρ0-map f : P → Fra+⊥ (ξ), where P is a Spin(r)-principal

bundle. Note that f is a 2-covering whose restriction to each fiber is modelled by ρ0,and any such covering would be a spin structure. Alternative equivalent definitionsof spin structures are to be found in [147] or [127, § 2.1].

Strong and weak equivalences for spin structures may be expressed as in (a)and (b) in 51.12 but then, by the homotopy lifting property for coverings, these twonotions of equivalence coincide. Since π1(SO(r)) is cyclic, ρ0 is, up to equivalence,the only 2-covering of SO(r) which is non-trivial. These considerations prove thefollowing lemma.

Lemma 53.8. Let ξ be an oriented Euclidean bundle ξ of rank r over a CW-complex X. Then, there is a bijection between

• the strong (or weak) equivalence classes of spin structures on ξ.• the isomorphism classes of 2-coverings of Fra+

⊥ (ξ) whose restriction to thefibers of pFra is non-trivial.

In general, the above sets are empty. For the existence of a spin structure, onehas the following proposition (see Proposition 54.12 for a more general framework).

Proposition 53.9. Let ξ be a vector bundle of rank r ≥ 2 over a regularCW-complex X. Then, the following conditions are equivalent.


(a) ξ admits a spin structure.(b) w1(ξ) = 0 and w2(ξ) = 0.

Moreover, if (a) or (b) holds true, then the set of strong (or weak) equivalenceclasses of spin structures on ξ is in bijection with H1(X).

Proof. Let (P, f) be a spin structure on ξ and let P2 → X2 be the restrictionof P onto the 2-skeleton of X . As π1(Spin(r)) = 0 (if r ≥ 3), the principal bundleP2 → X2 admits a section by [177, Corollary 34.4] and is therefore trivial. Thisimplies that ξ2 is trivial and (b) holds by Proposition 53.7.

Conversely, suppose that w1(ξ) and w2(ξ) vanish. Fix an orientation and anEuclidean structure on ξ (thus inducing an orientation and an Euclidean structureon ξk). Suppose first that r ≥ 3. By Proposition 53.7, ξ2 is trivial and thus admits

a spin structure f2 : P2 → Fra+⊥ (ξ2). Consider the commutative diagram

(53.10)

π2(X2) //

π1(SO(r))i∗ //

≈

π1(Fra+⊥ (ξ2)) //

π1(X2) //

≈

π0(SO(r))

≈

π2(X) // π1(SO(r))

i∗ // π1(Fra+⊥ (ξ)) // π1(X) // π0(SO(r))

where the rows are the homotopy exact sequences of the bundles Fra+⊥ (ζ) for ζ = ξ2

or ξ. By Lemma 53.8, the set of strong equivalence classes of spin structures on ζis in bijection with the set

E(ζ) = κ ∈ hom(π1(Fra+⊥ (ζ)),Z2) | κi∗ 6= 0 .

Diagram (53.10) implies that π1(Fra+⊥ (ξ2))→ π1(Fra+

⊥ (ξ)) is an isomorphism (five-lemma) and that E(ξ) = E(ξ2). Hence, the spin structure on ξ2 extends to ξ.

In the case r = 2, we apply the above argument to ξ ⊕ η where η is a trivialbundle of rank 1. Note that w1(ξ ⊕ η) and w2(ξ ⊕ η) vanish by Remark 52.2 andLemma 53.5. Hence, ξ ⊕ η admits a spin structure. Taking the sum with a fixedframe of η gives a (SO(2) → SO(3))-map Fra+

⊥ (ξ) → Fra+⊥ (ξ ⊕ η) and thus a

commutative diagram(53.11)

π2(X) //

≈

π1(SO(2))i∗ //

π1(Fra+⊥ (ξ)) //

π1(X) //

≈

π0(SO(2))

≈

π2(X) // π1(SO(3))

i∗ // π1(Fra+⊥ (ξ ⊕ η)) // π1(X) // π0(SO(3))

As E(ξ ⊕ η) 6= ∅, Diagram (53.11) shows that E(ξ) 6= ∅ and hence ξ admits aSpin(2)-structure.

Finally, note that, in (53.10), the last horizontal map vanish (π0(SO(r)) →π0(Fra+

⊥ (ξ)) is injective). Also, hom(π1(Fra+⊥ (ζ)),Z2) is an Abelian group. Hence,

if E(ξ) 6= ∅, it is in bijection with

κ ∈ hom(π1(Fra+⊥ (ζ)),Z2) | κi∗ = 0 ≈ hom(π1(X),Z2) ≈ H1(X) ,

the last isomorphism being established in Lemma 24.1.

54. DEFINITION AND PROPERTIES OF THE STIEFEL-WHITNEY CLASSES 307

54. Definition and properties of the Stiefel-Whitney classes

Let ξ = (p : E → X) be a vector bundle of rank r over a paracompact spaceX . In Proposition 28.59 was established the Thom isomorphism

Φ∗ : Hk(X)≈−→ Hk+r(E,E0) ,

where E0 ⊂ E is the complement of the zero section. Using the Steenrod squaringSq: H∗(E,E0) → H∗(E,E0), the (total) Stiefel-Whitney class w(ξ) ∈ H∗(X) of ξis defined by

(54.1) w(ξ) = Φ−1SqΦ(1) ,

where 1 ∈ H0(X) is the unit class. The component of w(ξ) in Hi(X) is denotedby wi(ξ) and is called the i-th Stiefel-Whitney class of ξ.

Equation (54.1) is one of the multiple definitions of the Stiefel-Whitney classand it is due to Thom [186, § II and III]. For a history of the Stiefel-Whitneyclasses (see [150, p. 38] and [38, Chapter IV, § 1]). The main properties of theStiefel-Whitney class are given in the following proposition.

Theorem 54.2. Let ξ = (p : E → X) be a vector bundle of rank r over aparacompact space X.

(1) w(ξ) = 1 + w1(ξ) · · ·+ wr(ξ). In particular, wi(ξ) = 0 if i > r.(2) If f : Y → X be a continuous map, then w(f∗ξ) = H∗f(w(ξ)). In partic-

ular, if ξ is isomorphic to ξ′, then w(ξ) = w(ξ′).(3) If ξ is trivial, then w(ξ) = 1.(4) Let ξ′ be a vector bundle over a paracompact space X ′. Then,

(54.3) w(ξ × ξ′) = w(ξ) × w(ξ′) ∈ H∗(X ×X ′) .

If X ′ = X, then

(54.4) w(ξ ⊕ ξ′) = w(ξ) w(ξ′) ∈ H∗(X) .

(5) If η is a trivial vector bundle over X, then w(ξ) = w(ξ ⊕ η).(6) wr(ξ) is the Euler class of ξ.

Proof. Recall that Φ(1) is the Thom class U(ξ) ∈ Hr(E,E0); thus, (54.1) isequivalent to

(54.5) w(ξ) = Φ−1Sq(U(ξ)) .

Now, Theorem 54.2 comes from the properties of Sq established in Theorem 46.2.Since Sq0 = id, one has w0(ξ) = φ−1

Sq0(U(ξ)) = 1. As Sqi(U(ξ)) = 0 for i > r,this proves (1). The naturality (2) comes from the naturality of all the ingredientsof (54.5): the Thom class is natural (Lemma 28.50), and so is Φ, and Sq is alsonatural, being a cohomology operation. Now, (3) is a consequence of (2) since atrivial bundle is induced by a map to a point.

To prove (4), one has

(54.6)w(ξ × ξ′) = φ−1

Sq(U(ξ × ξ′))= φ−1

Sq(U(ξ)× U(ξ′)

)using (28.64)

= φ−1(Sq(U(ξ))× Sq(U(ξ′)) by (3) of Theorem 46.2


On the other hand, if a ∈ H∗(X) and a′ ∈ H∗(X ′), one has

(54.7)

φ(a× a′) = H∗(p× p′)(a× a′) U(ξ × ξ′)= [H∗p(a)×H∗p′(a′)] [U(ξ)× U(ξ′)]= [H∗p(a) U(ξ)]× [H∗p(a′) U(ξ′)]= Φ(a)× Φ(a′) .

Thus, (54.6) together with (54.7) proves (54.3). If X ′ = X , then ξ⊕ξ′ = ∆∗(ξ×ξ′)where ∆: X → X ×X is the diagonal inclusion. Therefore, (54.4) comes from (2)already proven, (54.3) and Remark 27.5:

w(ξ ⊕ ξ′) = H∗∆(w(ξ × ξ′)) = H∗∆(w(ξ) × (ξ′)) = w(ξ) w(ξ′) .

Property (5) is a consequence of (3) and (4). Finally, (6) follows from

wr(ξ) = Φ−1Sqr(U(ξ)) = Φ−1(U(ξ) U(ξ)) = e(ξ) ,

the last equality coming from (28.52).

Remark 54.8. Versions of Properties (1), (2), (54.4) and (6) uniquely charac-terise the total Stiefel-Whitney class. See Proposition 56.5, [150, Theorem 7.3] or[103, Chapter 16,§ 5]. This is the philosophy of the axiomatic presentation of theStiefel-Whitney class (see [150]), inspired by that of the Chern classes introducedby Hirzebruch [94, p. 58].

Remark 54.9. As the Steenrod squares are used for Definition (54.1), the Ademrelations provide constraints amongst the Stiefel-Whitney classes. For instance,the relation Sq2i+1 = Sq1Sq2i (see Example 48.10) implies that w2i+1(ξ) = 0 ifw2i(ξ) = 0. Also, if w2k(ξ) = 0 for k = 1 . . . , r, then, by Lemma 49.1, wj(ξ) = 0for 0 < j < 2r+1.

We now discuss the relationship with the classes w1 and w2 defined in § 52and 53.

Proposition 54.10. Let ξ be a vector bundle over a CW-complex X. Then, thefirst Stiefel-Whitney class w1(ξ) ∈ H1(X) defined above coincides with that definedin § 52. In particular, w1(ξ) = 0 if and only if ξ is orientable.

Proof. Both definitions enjoy naturality for induced bundles. We can thenrestrict ourselves to X being 1-dimensional, since H1(X) → H1(X1) is injective.In this case, ξ ≈ λ ⊕ η where λ is a line bundle and η a trivial vector bundle (seee.g. [103, Chapter 8, Theorem 1.2]). By Remark 52.2 and (5) of Theorem 54.2, weare reduced to the case of a line bundle. Then, both definitions coincide with theEuler class by Corollary 52.8 and Point (6) of Theorem 54.2.

A similar result holds for the second Stiefel-Whitney class.

Proposition 54.11. Let ξ be an orientable vector bundle over a CW-complexX. Suppose that X has no 3-cells or that X3 is a regular complex. Then, the secondStiefel-Whitney class w2(ξ) ∈ H2(X) defined above coincides with the cellular one

w2(ξ) ∈ H2(X) defined in § 53.

Proof. Recall that the condition on X (and the orientability of ξ) was nec-essary for us to define w2(ξ). The coincidence between w2(ξ) ∈ H2(X) and

w2(ξ) ∈ H2(X) holds under the identification of H2(X) and H2(X) as the same

55. REAL FLAG MANIFOLDS 309

subgroup of H2(X2) (see (16.8)). The class w2 is natural by Point (2) of Theo-rem 54.2 and, by construction, w2 is natural for the restriction to a subcomplex.We can thus suppose that X = X2 and that X is connected.

As ξ is orientable, its restriction over X1 is trivial. By Lemma 51.20, ξ ≈ p∗ξ,where p : X → X = X/X1. Again, w2(ξ) = H∗p(w2(ξ)) and, by construction of

w2, w2(ξ) = H∗p(w2(ξ)). We can thus suppose that X is a bouquet of 2-sphere, oreven that X = S2 with its minimal cell decomposition.

If η is a trivial bundle, both equations w2(ξ⊕η) = w2(ξ) and w2(ξ⊕η) = w2(ξ)hold true, by Point (5) of Theorem 54.2 and Lemma 53.5. We can thus suppose thatξ has rank ≥ 3. As seen in Remark 53.6, there is only non-trivial such bundle overS2, characterized by w2(ξ) 6= 0. Let γC be the tautogical bundle over CP 1 ≈ S2.By Proposition 35.13, one has

0 6= e(γC) = w2(γC) = w2(γC ⊕ η) .which fisinshes the proof of our proposition. Incidently, we have proven that γC isstably non-trivial.

Proposition 54.11 permits us to generalize the framework of Propositions 53.7and 53.9.

Proposition 54.12. Let ξ be a vector bundle of rank r ≥ 3 over a CW-complexX. Then, the following conditions are equivalent.

(1) w1(ξ) = 0 and w2(ξ) = 0.(2) the restriction ξ3 of ξ over X3 is trivial.

Proof. By Theorem 54.2, (1) implies w1(ξ2) = 0 and w2(ξ2) = 0. As ξ2 hasno 3-cells, w2(ξ2) is defined and, by Proposition 54.11, w2(ξ2) = 0. By Proposi-tion 53.7, ξ2 is trivial which, as seen in the proof of Proposition 53.7, implies thatξ3 is trivial. Thus, (1) implies (2). To prove that (2) implies (1), let j : X3 → Xdenote the inclusion. Then j∗(wi(ξ)) = wi(ξ3)) = 0 for and j∗ : Hk(X)→ Hk(X3)is injective for k ≤ 3. (We have also proven that (1) implies w3(ξ) = 0, but this isalready known by Remark 54.9).

Proposition 54.13. Let ξ be a vector bundle of rank r ≥ 2 over a CW-complexX. Then, the following conditions are equivalent.

(1) w1(ξ) = 0 and w2(ξ) = 0.(2) ξ admits a spin structure

Moreover, if (2) holds true, then the set of strong (or weak) equivalence classes ofspin structures on ξ is in bijection with H1(X).

Proof. Suppose first that r ≥ 3. If ξ admits a spin structure, then ξ2 istrivial (see the proof of Proposition 53.9), which implies (1) by Proposition 54.12.Conversely, if (1) holds true, then ξ2 is trivial by Proposition 54.12 and thus ξ2admits a spin-structure. That this structure extends to ξ is established as in theproof of Proposition 53.9. For the case r = 2 as well as for the last assertion of theproposition, the proofs are the same as those for for Proposition 53.9.

55. Real flag manifolds

Most of the results of this section come from [14], but we do not use spectral se-quences. The Leray-Hirsch theorem 28.26 for locally trivial bundles, together with


some perfect Morse theory, is sufficient for our needs. We shall deal with homoge-neous spaces of the form Γ/Γ0, where Γ is a Lie group and Γ0 a compact subgroup(therefore, a Lie subgroup). Then Γ/Γ0 inherits a smooth manifold structure [35,Chapter 1, Proposition 5.3]. More generally, [19, Chapter II, Theorem 5.8] impliesthe following lemma.

Lemma 55.1. Let Γ be a Lie group and H ⊂ G be compact subgroups of Γ.Then, the quotient map Γ/H → Γ/G is a smooth locally trivial fiber bundle withfiber G/H. If H = 1, then the quotient map Γ → Γ/G is a smooth G-principalbundle.

55.1. Definitions and Morse theory. Let n1, . . . nr be positive integers andlet n = n1 + n2 + · · ·nr. By the flag manifold Fl(n1, . . . , nr), we mean any smoothmanifold diffeomorphic to the homogenuous space

(55.2) Fl(n1, . . . , nr) ≈ O(n)/O(n1)×O(n2)× · · · ×O(nr) .

Here below are some examples.

(1) Nested subspaces. Fl(n1, . . . , nr) is the set of nested vector subspaces V1 ⊂· · · ⊂ Vr ⊂ Rn with dimVi =

∑ij=1 nj .

(2) Mutually orthogonal subspaces. Fl(n1, . . . , nr) is the set of r-tuples (W1, . . . ,Wr)of vector subspaces Rn which are mutually orthogonal and satisfy dimWi =ni. The correspondence from this definition to Definition (1) associates to(W1, . . . ,Wr) the nested family Vi where Vi is the vector space generatedby W1 ∪ · · · ∪Wi.

(3) Isospectral symmetric matrices. Let λ1 > · · · > λr be real numbers. Con-sider the manifold SM(n) of all symmetric real (n×n)-matrices, on whichO(n) acts by conjugation. Then Fl(n1, . . . , nr) occurs as the orbit of thediagonal matrix having entries λi with multiplicity ni.

(55.3) Fl(n1, . . . , nr) =R dia

(λ1, . . . , λ1︸︷︷︸

n1

, · · · , λr, . . . , λr︸︷︷︸nr

)R−1 | R ∈ O(n)

.

In other words, Fl(n1, . . . , nr) is here the space of symmetric real (n×n)-matrices with characteristic polynomial equal to

∑ri=1(x− λi)ni . Indeed,

elementary linear algebra teaches us that two matrices in SM(n) are in thesame O(n)-orbit if and only if they have the same characteristic polyno-mial. The correspondence from this definition to Definition (2) associates,to a matrix M , its eigenspaces for the various eigenvalues.

Concrete definition (3) will be our working definition for Fl(n1, . . . , nr) throughoutthis section. Special classes of flag manifolds are given by the Grassmanians

Gr(k; Rn) = Fl(k, n− k) ≈ O(n)/O(k) ×O(n− k)

of k-planes in Rn. This is a closed manifold of dimension

dim Gr(k; Rn) = dimO(n)− dimO(k) − dimO(n− k) = k(n− k) .For example, Gr(1; Rn) ≈ RPn−1, of dimension n− 1. Using Definition (3) above,our “concrete Grassmannian” will be

(55.4) Gr(k; Rn) =R dia

(1, . . . , 1︸︷︷︸

k

, 0, . . . , 0︸︷︷︸n−k

)R−1 | R ∈ O(n)

.


In other words, Gr(k; Rn) is the space of orthogonal projectors on Rn of rank k.Another interesting flag manifold is the complete flag manifold

Fl(1, . . . , 1) ≈ O(n)/O(1) × · · · ×O(1)

with dimFl(1, . . . , 1) = dimO(n) = n(n−1)2 .

We now define real functions on the flag manifolds by restriction of the weightedtrace on f : SM(n)→ R defined by

f(M) =

n∑

j=1

j Mjj

where Mij denotes the (i, j)-entry of M .

Proposition 55.5. Let Fl(n1, . . . , nr) ⊂ SM(n) be the flag manifold as pre-sented in (55.3). Then, the restriction f : Fl(n1, . . . , nr) → R of the weightedtrace is a perfect Morse function whose critical points are the diagonal matricesin Fl(n1, . . . , nr). The index of the critical point dia(x1, . . . , xn) is the number ofpairs (i, j) with i < j and xi < xj .

For a general discussion about such Morse functions on flag manifolds, see [12,Chapter 8].

Example 55.6. For Gr(2; R5) = Fl(2, 3), we get the the following (52) = 10critical points, with their index and value by f .

critical point index value

dia(1, 1, 0, 0, 0) 0 3

dia(1, 0, 1, 0, 0) 1 4

dia(1, 0, 0, 1, 0), dia(0, 1, 1, 0, 0) 2 5

dia(1, 0, 0, 0, 1), dia(0, 1, 0, 1, 0) 3 6

dia(0, 0, 1, 1, 0), dia(0, 1, 0, 0, 1) 4 7

dia(0, 0, 1, 0, 1) 5 8

dia(0, 0, 0, 1, 1) 6 9

Remark 55.7. The function f : Gr(k; Rn)→ R given by

f(M) = −k(k + 1)

2+ f(M)

is a Morse function which is self-indexed, i.e. f(M) = j if M is a critical point ofindex j.

Proof of Proposition 55.5. We introduce precise notations which will beused later. For 1 ≤ i < j ≤ n, let rij : M2(C)→Mn(C) defined by requiring thatthe entries of rij(N) are those of the identity matrix In, except for

rij(N)ii = N11 , rij(N)ij = N12 , r

ij(N)ji = N21 , rij(N)jj = N22 .

The restriction of rij to SO(2) gives an injective homomorphism rij : SO(2) →SO(n) whose image is formed by the matrices

Rijt = rij(

cos t − sin tsin t cos t

)(t ∈ R) .


The action of Rijt on Fl(n1, . . . , nr) ⊂ SM(n) by conjugation produces a flow and

thus a vector field V ij on Fl(n1, . . . , nr), whose value V ijM at M ∈ Fl(n1, . . . , nr) is

V ijM = ddt(R

ijt MRij−t)|t=0 (we identify TMFl(n1, . . . , nr) as a subspace of SM(n)).

A direct computation gives that

(55.8)

(Rijt MRij−t)ii = Mii cos2 t−Mij sin 2t+Mjj sin2 t

(Rijt MRij−t)jj = Mii sin2 t+Mij sin 2t+Mjj cos2 t

(Rijt MRij−t)ij = Mij cos 2t+ (Mii −Mjj) sin t cos t .

Moreover,

(55.9)

(Rijt MRij−t)ik = Mik cos t−Mjk sin t if i 6= k 6= j

(Rijt MRij−t)kj = Mki sin t+Mkj cos t if i 6= k 6= j

(Rijt MRij−t)kl = Mkl if i 6= k and j 6= l .

Let gij(t) = f(Rijt MRij−t). The first derivative gij(t) satisfies

gij(t) = (j − i)(Mii −Mjj) sin 2t+ 2(j − i)Mij cos 2t .

Hence,

(55.10) V ijM f = gij(0) = 2(j − i)Mij ,

which proves that only the diagonal matrices in Fl(n1, . . . , nr) may be critical pointsof the weighted trace.

Suppose that ∆ ∈ Fl(n1, . . . , nr) is a diagonal matrix. Let

J∆ = (i, j) | 1 ≤ i < j ≤ n and ∆ii 6= ∆jj .and let V∆ = V ij∆ | (i, j) ∈ J∆ ⊂ T∆Fl(n1, . . . , nr). By (55.8) and (55.9),ddt(R

ijt ∆Rij−t)(0) has only non-zero term away from the diagonal, namely d

dt(Rijt ∆Rij−t)ij(0) =

∆ii −∆jj . Hence, vectors of V∆ are linearly independent. But

♯J∆ =n(n− 1)

2−

rX

k=1

nk(nk − 1)

2= dim O(n)−

rX

k=1

dim O(nk) = dim Fl(n1, . . . , nr) .

Therefore, V∆ is a basis of T∆Fl(n1, . . . , nr). Using (55.10), this proves that thediagonal matrices in Fl(n1, . . . , nr) are exactly the critical points of the weightedtrace. The matrix of the Hessian form Hf on T∆Fl(n1, . . . , nr) is

Hf(V kl∆ , V ij∆ ) = V kl∆ (V ijf)

= V kl∆

(M 7→ 2(j − i)Mij

)by (55.10)

= 2(j − i)[ddt(R

klt ∆Rkl−t)|t=0

]ij.

Using (55.8) and (55.9), we see that the matrix of Hf in the basis V∆ is diagonal,with diagonal term

Hf(V ij∆ , V ij∆ ) = 2(j − i)(∆ii −∆jj) .

As (i, j) ∈ J∆, this proves that f is a Morse function as well as the assertion onthe Morse index of ∆.

It remains to prove that f is perfect. Let Γ be the subgroup of O(n) formed bythe diagonal matrices (with coefficients ±1). The O(n)-action on SM(n) by conju-gation may be restricted to Γ and f is Γ-invariant. Moreover, the diagonal matrices


in Fl(n1, . . . , nr) are exactly the fixed points of the Γ-action. The perfectness of fthen follows from Theorem 44.17.

Here below a first consequence of Proposition 55.5.

Corollary 55.11.

dimH∗(Fl(n1, . . . , nr)) =n!

n1! · · ·nr!.

In particular,

dimH∗(Fl(k, n− k)) = Gr(k; Rn) = (nk ) and dimH∗(Fl(1, · · · , 1)) = n! .

Proof. By Proposition 55.5, the weighted trace f : Fl(n1, . . . , nr) → R is aperfect Morse function. Hence, by Proposition 44.13, dimH∗(Fl(n1, . . . , nr)) =♯Crit f . But Crit f consists of the diagonal matrices in Fl(n1, . . . , nr), which are allconjugate to

dia(λ1, . . . , λ1︸︷︷︸

n1

, · · · , λr, . . . , λr︸︷︷︸nr

)

by a permutation matrix. Hence, Crit f is an orbit of the symmetric group Symn,with isotropy group Symn1

× · · · × Symnr , whence the formulae.

Remark 55.12. The critical points of f in Proposition 55.5 are related to theSchubert cells (see § 55.3).

Consider the inclusion SM(n) ⊂ SM(n + 1) with image the matrices withvanishing last row and column. Seeing Gr(k; Rn) ⊂ SM(n) as in (55.4), this givesan inclusion Gr(k; Rn) ⊂ Gr(k; Rn+1).

Lemma 55.13. The homomorphism Hj(Gr(k; Rn+1))→ Hj(Gr(k; Rn)) inducedby the inclusion is surjective for all j and is an isomorphism for j ≤ n− k.

Proof. Let us use the Morse function f : Gr(k; Rn+1) → R of Remark 55.7and let f ′ be its restriction to Gr(k; Rn). Then, f ′ and f are self-indexed andCritf ′ ⊂ Critf ⊂ N. For m ∈ N, let Wm = f−1((∞,m + 1/2]) and W ′m =(f ′)−1((∞,m + 1/2]). For the first assertion, we prove, by induction on m thatH∗(Wm)→ H∗(W ′m) is surjective for all m ∈ N. The induction starts with m = 0,since W0 ≃W ′0 ≃ pt. The induction step involves the cohomology exact sequences

(55.14)

0 // H∗(Wm,Wm−1) //

i∗m,m−1

H∗(Wm) //

i∗m

H∗(Wm−1) //

i∗m−1

0

0 // H∗(W ′m,W′m−1) // H∗(W ′m) // H∗(W ′m−1) // 0

obtained by Lemma 44.10, since f and f ′ are perfect by Proposition 55.5. FromProposition 55.5 again and its proof, the critcal points of f ′ have the negativenormal directions in Wm−1 or in W ′m−1. Hence, using excision, the Morse lemmaand Thom isomorphisms, we get the commutative diagram

H∗(Wm,Wm−1)≈ //

i∗m,m−1

∏C∈Critf∩f−1(m)H

∗−m(C)

proj

H∗(W ′m,W′m−1)

≈ //∏C∈Critf′∩f−1(m)H

∗−m(C)


which proves that i∗m,m−1 is onto. If i∗m−1 is surjective by induction hypothesis, weget that i∗m is surjective by diagram-chasing argument.

Note that the point D ∈ Critf − Critf ′ of lowest index is

D = dia(1, . . . , 1, 0, . . . , 0, 1) ∈ SM(n+ 1)

satisfies f(D) = index(D) = n− k + 1 (the number of zeros in D). Hence, Critf ∩Wn−k = Critf ′ ∩ W ′n−k. The same induction arguement as above shows that

Hj(Gr(k; Rn+1))→ Hj(Gr(k; Rn)) is an isomorphism for j ≤ n− k.

55.2. Cohomology rings. The cohomology ring of a flag manifold V willbe generated by the Stiefel-Whitney classes of some tautological bundles over V .Consider a flag manifold Fl(n1, . . . , nr), with n = n1 + · · · + nr. Consider thefollowing closed subgroups of O(n).

Bi = O(n1)× · · · × 1 × · · · ×O(nr) ⊂ O(n1)× · · · ×O(nr) ⊂ O(n) ,

where 1 sits at the i-th place. Then

Pi = O(n)/Bi →→ O(n)/O(n1)× · · · ×O(nr) = Fl(n1, . . . , nr)

is an O(ni)-principal bundle over Fl(n1, . . . , nr). Indeed, if K is a compact sub-group of a Lie group G, then G → G/K is a principal K-bundle (see, e.g. [11,Theorem 2.1.1, Chapter I]). Let ξi be the vector bundle of rank ni associated toPi, i.e. E(ξi) = Pi ×O(ni) Rni . The vector bundle ξi is called the i-th-tautologicalvector bundle over Fl(n1, . . . , nr). Being associated to an O(ni)-principal bundle, ξiis endowed with an Euclidean structure and its space of orthogonal frames Fra⊥(ξi)is equal to Pi.

In the mutually orthogonal subspaces description (presentation (2), p. 310) ofFl(n1, . . . , nr), we see that

E(ξi) = (W1 . . . ,Wr, v) ∈ Fl(n1, . . . , nr)× Rn | v ∈ Wi .Note that ξ1 ⊕ · · · ⊕ ξr is trivial. Indeed,

E(ξ1⊕ · · · ⊕ ξr) = ((W1, . . . ,Wr), (v1, . . . , vr)) ∈ Fl(n1, . . . , nr)× (Rn)r | vi ∈ Wiand the correspondence

(55.15) ((W1, . . . ,Wr), (v1, . . . , vr)) 7→ v1 + · · ·+ vr

restricts to a linear isomorphism on each fiber. Such a map thus provides a trivial-ization of ξ1 ⊕ · · · ⊕ ξr.

If one sees Fl(n1, . . . , nr) as the space of matrices M ∈ SM(n) with character-istic polynomial equal to

∑ri=1(x− λi)ni (presentation (3), p. 310), then

(55.16) E(ξi) = (M, v) ∈ Fl(n1, . . . , nr)× Rn |Mv = λiv .The vector bundle ξ1 over Fl(k, n − k) = Gr(k; Rn) is called the tautological

vector bundle over the Grassmannian Gr(k; Rn); it is of rank k and is denoted by ζ,ζk or ζk,n. The space of Fra⊥(ζk) is the Stiefel manifold Stief(k,Rn) of orthonormalk-frames in Rn.

The inclusion Rn ≈ Rn × 0 → Rn+1 induces an inclusion Gr(k; Rn) →Gr(k; Rn+1) and we may consider the inductive limit

Gr(k; R∞) = limn

Gr(k; Rn)

which is a CW-space. The tautological vector bundle ζk is also defined overGr(k; R∞) and induces that over Gr(k; Rn) by the inclusion Gr(k; Rn) → Gr(k; R∞).


It is classical that πi(Stief(k,Rn)) = 0 for i < n − k (see [177, Theorem 25.6]),thus Stief(k,R∞) = Fra(ζk) is contractible. Hence, the O(k)-principal bundleStief(k,R∞) → Gr(k; R∞) is a universal O(k)-principal bundle (see [177, § 19.4])and thus homotopy equivalent to the Milnor universal bundle EO(k) → BO(k).In particular, Gr(k; R∞) has the homotopy type of BO(k). As a consequence, anyvector bundle of rank k over a paracompact space X is induced from ζk by a mapX → Gr(k; R∞) (for a direct proof of that (see [150, Theorem 5.6]).

Theorem 55.17. The cohomology ring of BO(k) is GrA-isomorphic to thepolynomial ring

H∗(BO(k)) ≈ H∗(Gr(k; R∞)) ≈ Z2[w1, . . . , wk]

generated by the Stiefel-Whitney classes wi = wi(ζk) of the tautological bundle ζk.

Proof. Slightly more formally, we consider the polynomial ring Z2[w1, . . . , wk]with formal variables wi of degree i. The correspondence wi 7→ wi(ζk) provides aGrA-morphism ψ : Z2[w1, . . . , wk] → H∗(BO(k)) which we must show that it isbijective.

For the injectivity, we consider the tautological line bundle γ over RP∞ andits n-times product γn over (RP∞)n. As seen above, ζk is universal so γn isinduced by a map f : (RP∞)n → BO(n). Recall from Proposition 24.21 thatH∗(RP∞) = Z2[a] with a of degree 1 and, by Theorem 54.2, w(γ) = 1 + a. By

the Kunneth theorem, there is a GrA-isomorphism Z2[a1, . . . , an]≈−→ H∗((RP∞)n)

and, by Theorem 54.2, w(γn) = (1+a1) · · · (1+an). AsH∗f(w(ζk)) = w(γn),there is a commutative diagram

Z2[w1, . . . , wk]

φ

ψ // H∗(BO(k))

H∗f

Z2[a1, . . . , an]≈ // H∗((RP∞)n))

withφ(w(ζk)) = (1 + a1) · · · (1 + an) = 1 + σ1 + · · ·σn ,

where σi = σi(a1, . . . , an) is the i-th elementary symmetric polynomial in the vari-ables aj (see (49.6)). Thus, φ(wi) = σi. Now, if 0 6= A ∈ Z2[w1, . . . , wk] satisfiesψ(A) = 0, then φ(A) = 0 would be a non-trivial polynomial relation between theσi’s. But the elementary symmetric polynomials are algebraicly independent (seee.g. [120]). Thus, ψ is injective.

For d ∈ N, let

Bd = (d1, . . . , dk) ∈ Nk |k∑

j=1

j dj = d .

The correspondence (d1, . . . , dk) 7→ wd11 · · ·wdk

k is a bijection from Bd onto a basis

of the vector subspace Z2[w1, . . . , wk][d] formed by the elements in Z2[w1, . . . , wk]

which are of degree d. On the other hand, consider Gr(k; Rn) ⊂ SM(n) as in (55.4),with n large. Let Critdf ⊂ Gr(k; Rn) be the set of critical points of index d for theweighted trace. Then the cocorrespondence

(d1, . . . , dk) 7→ dia(0, . . . , 0︸︷︷︸dk

, 1, 0, . . . , 0︸︷︷︸dk−1

, 1, . . . , 0, . . . , 0︸︷︷︸d1

, 1, 0, . . . , 0)


provides a bijection Bd ≈−→ Critdf . As f is a perect Morse function by Proposi-tion 55.5, one has

♯Bd = ♯Critdf = dimHd(Gr(k; Rn)) = dimHd(BO(k)) ,

the last equality coming from Lemma 55.13 when n is large enough. Therefore,

dim Z2[w1, . . . , wk][d] = dimHd(BO(k)) .

As ψ is injective, it is then bijective.

Define

Qr(t) =1

1− tr = 1 + tr + t2r + · · · ∈ Z[[t]] ,

which is the Poincare series of Z2[x] if x is of degree r. Here below a direct conse-quence of Theorem 55.17.

Corollary 55.18. The Poincare series of BO(k) is

Pt(BO(k)) = Q1(t) · · ·Qk(t) .

Also, Theorem 55.17 together with Lemma 55.13 gives the following corollary.

Corollary 55.19. The cohomology ring H∗(Gr(k; Rn)) is generated, as a ring,by the Stiefel-Whitney classes w1(ζk), . . . , wk(ζk) of the tautological bundle ζk.

Theorem 55.17 permits us to compute the cohomology of BSO(k). The latter

also has a tautological bundle ζk = (ESO(k) ×SO(k) Rk → BSO(k)) which isorientable.

Corollary 55.20. The cohomology ring of BSO(k) is GrA-isomorphic to thepolynomial ring

H∗(BO(k)) ≈ H∗(Gr(k; R∞)) ≈ Z2[w2, . . . , wk]

generated by the Stiefel-Whitney classes wi = wi(ζk) of the tautological bundle ζk.

Proof. Let i : SO(k) → O(k) denote the inclusion. By Example 40.4, themap Bi : BSO(k)→ BO(k) is homotopy equivalent to a two fold covering, which isnon-trivial since BSO(k) is connected. By Lemma 24.11, its its characteristic classw(Bi) ∈ H1(BO(k)) is not trivial. By Theorem 55.17, the only non-zero elementin H1(BO(k)) is w1(ξk), so w(Bi) = w1(ξk).

By Theorem 55.17 and the transfer exact sequence (Proposition 24.17), thering homomorphism H∗Bi : H∗(BO(k)) → H∗(BSO(k)) is surjective with kernel

the ideal generated by w1(ζk). As Bi is covered by a bundle map from ζk to ζk,

one has H∗Bi(wi(ζk) = wi(ζk). The corollary follows.

Remark 55.21. In contrast with the simplicity of H∗(BSO(k)), the coho-mology ring H∗(BSpin(k)) is complicated and its computation requires spectralsequences (see [164]). The stable case BSpin = limk BSpin(k) is slightly simpler(see [189]).

We are now in position to give a GrA-presentation of H∗(Fl(n1, . . . , nr)). Let

(55.22) w(ξj) = 1 + w1(ξj) + · · ·+ wnj(ξj) ∈ H∗(Fl(n1, . . . , nr))

be the Stiefel-Whitney class of the tautological vector bundle ξj . As seen in (55.15),ξ1 ⊕ · · · ⊕ ξr is trivial. By Theorem 54.2, the equation

(55.23) w(ξ1) · · · w(ξr) = 1


holds true. Hence, the homogenuous components ofw(ξ1) · · · w(ξr) in positivedegrees vanish, giving rise to n equations.

Theorem 55.24. The cohomology algebra H∗(Fl(n1, . . . , nr)) is GrA-isomorphicto the quotient of the polynomial ring

Z2[wi(ξj)] , 1 ≤ i ≤ rj , j = 1, . . . , r

by the ideal generated by the homogeneous components of w(ξ1) · · ·w(ξr) in positivedegrees.

Proof. We first prove that H∗(Fl(n1, . . . , nr)) is, as a ring, generated by theStiefel-Whitney classes wi(ξj) (1 ≤ i ≤ rj , j = 1, . . . , r). This is done by inductionon r (note that r ≥ 2 in order to the definition of Fl(n1, . . . , nr) to make sense).For r = 2, as Fl(n1, n2) = Gr(n1; Rn1+n2), the result comes from Corollary 55.19.For the induction step, let us define a map π : Fl(n1, . . . , nr) → Fl(n − nr, nr) byπ(W1 . . . ,Wr) = (W1⊕· · ·⊕Wr−1,Wr) (using the mutually orthogonal definition (2)of the flag manifolds). By Lemma 55.1, this gives a locally trivial bundle

Fl(n1, . . . , nr−1)ι−→ Fl(n1, . . . , nr)

π−→ Fl(n− nr, nr) .

By induction hypothesis,H∗(Fl(n1, . . . , nr−1)) is generated, as a ring, by the Stiefel-Whitney classes of its tautological bundles, say wi(ξj). Note that these bundles areinduced by the tautological bundles (called ξj) over Fl(n1, . . . , nr): ξj = ι∗ξj . Then,H∗ι is surjective and wi(ξj) 7→ wi(ξj)) is a cohomology extension of the fiber (seep. 145). On the other hand,

Fl(n− nr, nr) ≈ Gr(n− r; Rn) ≈ Gr(nr; Rn) ,

the last isomorphism sending an (n − nr)-dimensional subspace of Rn to its or-thogonal complement. By Corollary 55.19, H∗(Gr(nr; Rn)) is GrA-generated byw1(ζnr

), . . . , wnr(ζnr

) and H∗π(wi(ζnr)) = wi(ξr). By the Leray-Hirsch theo-

rem 28.26, H∗(Fl(n1, . . . , nr)) is then GrA-generated by wi(ξj) (1 ≤ i ≤ nj andj = 1 . . . r).

Let Γ = O(n1)× · · · ×O(nr) ⊂ O(n) and consider the commutative diagram

(55.25)

O(n) //

EO(n) //

BO(n)

=

O(n)/Γ // EO(n)/Γ // BO(n)

where the top line is the O(n)-universal bundle. Hence, the bottom line is a locallytrivial bundle with fiber equal to O(n)/Γ = Fl(n1, . . . , nr); as EO(n) is contractible,there are homotopy equivalences

EO(n)/Γ ≃ BΓ ≃ BO(n1)× · · · ×BO(nr) ,


the last homotopy equivalence coming from (42.3). Hence, Diagram (55.25) maybe rewritten in the following way.

(55.26)

Γ= //

Γ

O(n) //

EO(n) //

BO(n)

=

Fl(n1, . . . , nr)β // BΓ

Bα // BO(n)

where α denotes the inclusion of Γ in O(n). The left column is a Γ-principalbundle which is a Γ-structure on ξ = ξ1 ⊕ · · · ⊕ ξr. The central column is theΓ-principal bundle associated to the vector bundle ζ = ζn1 × · · · × ζnr

over BΓ ≃BO(n1)× · · · ×BO(nr). Thus, the map β is a classifying map for the Γ-structureon ξ: it lifts the map Bαβ, which is classifying for ξ as a vector bundle (thatBαβ is null-homotopic is coherent with the triviality of ξ, seen in (55.15)). Hence

β∗ζnj≈ ξj , β∗ζ ≈ ξ

and thus

H∗β(wi(ζnj)) = wi(ξj) , H

∗β(wi(ζ)) = wi(ξ) .

As H∗(Fl(n1, . . . , nr)) is GrA-generated by by the classes wi(ξj), H∗β is surjec-

tive and one may apply the Leray-Hirsch theorem 28.26 and its corollaries. ByTheorem 55.17 and the Kunneth theorem, there is a GrA-isomorphism

Z2[wi(ζnj)]≈−→ H∗(BΓ) , (1 ≤ i ≤ rj , j = 1, . . . , r) .

On the other hand, H∗(BO(n)) ≈ Z2[wi(ζn)] and H∗Bα(wi(ζn)) = wi(ζ). ByCorollary 28.31, H∗(Fl(n1, . . . , nr)) is then GrA-isomorphic to the quotient ofH∗(BΓ) by the ideal generated by wi(ζ) (i > 0). Hence, one has the followingcommutative diagram

Z2[wi(ζj)]‹`

wi(ζ), i > 0´

≈ **TTTTTTTTTTTTT

≈ // Z2[wi(ξj)]‹`

wi(ξ), i > 0´

ttjjjjjjjjjjjjj

H∗(Fl(n1, . . . , nr))

which proves Theorem 55.24.

Corollary 55.27. The Poincare polynomial of Fl(n1, . . . , nr) is given by theformula

Pt(Fl(n1, . . . , nr)) =

∏rj=1[Q1(t) · · ·Qnj

(t)]

Q1(t) · · ·Qn(t).

In particular,

Pt(Gr(k; Rn)) = Pt(Fl(k, n− k)) =Q1(t) · · ·Qk(t)

Qn−k+1(t) · · ·Qn(t)and

Pt(Fl(1, . . . , 1)) =Q1(t)

n

Q1(t) · · ·Qn(t)=

(1− t)(1 − t2) · · · (1− tn)(1 − t)n .


Remark 55.28. The above formulae, evaluated at t = 1 using L’Hospital’srule, give dimH∗(Fl(n1, . . . , nr)), etc, giving again the formulae of Corollary 55.11.

Proof of Corollary 55.27. We have seen in the proof of Theorem 55.24that

Fl(n1, . . . , nr)β−→ BΓ

Bα−−→ BO(n)

is a locally trivial bundle satisfying the hypotheses of the Leray-Hirsch theorem.We know by Corollary 55.18 that Pt(BO(k)) = Q1(t) · · ·Qk(t). By the Kunnethformula, we get that

Pt(BΓ) = Pt(BO(n1)× · · · ×BO(nr)) =

r∏

j=1

[Q1(t) · · ·Qnj(t)] .

The first formula then comes from Corollary 28.32. The other formulae are conse-quences of the first one.

We now give some illustrations of Theorem 55.24.

Example 55.29. Consider the case of the complete flag manifold Fl(1, . . . , 1).Theorem 55.24 says that H∗(Fl(1, . . . , 1)) is generated by xi = w1(ξi) for i =1 . . . , n. In this generating system, wi(ξ1 ⊕ · · · ⊕ ξn) = σi, the ith elementarysymmetric polynomial in the variables xi. Hence, by Theorem 55.24,

H∗(Fl(1, . . . , 1)) ≈ Z2[x1, . . . , xn]/(σ1, . . . , σn

).

Example 55.30. Consider the case of the Grassmannian

Gr(k; Rn) = Fl(k, n− k) .

Set ζ = ξ1, with Stiefel-Whitney class w(ζ) = w = 1 +w1 +w2 + · · · , and ζ⊥ = ξ2,with w(ζ⊥) = w = 1 + w1 + w2 + · · · . Note that the fiber of the vector bundleζ⊥ over P ∈ Gr(k; Rn) is the set of vectors in Rn which are orthogonal to P .Equation (55.23) becomes

(55.31) w w = 1 ,

which is equivalent to the following system of equations:

(55.32) wi =k∑

r=1

wrwi−r (i = 1, . . . , n− k) and wi = 0 if i > n− k .

This system has the following unique solution.

Lemma 55.33. With the convention wi = wi = 0 for i < 0, the followingequation


wr =

∣∣∣∣∣∣∣∣∣∣∣

w1 1w2

...wr−1 1wr wr−1 · · · w2 w1

∣∣∣∣∣∣∣∣∣∣∣

= det(wi+1−j

)1≤i,j≤r

.

0

holds true in Hr(Gr(k; Rn)). The symmetric formula wr = det(wi+1−j

)1≤i,j≤r

holds true as well. These formulae are both equivalent to Equation (55.31).

Proof. The first equation is proved by induction on r, starting, for r = 1,with w1 = w1 (this also gives the uniqueness of the solution). The induction stepis achieved by expanding the determinant with respect to the first column: the(s, 1)-th minor is equal to wr−s by induction hypothesis and the result followsfrom (55.32). The symmetric equation follows from the symmetry in wi and wiof (55.32) (coming from the symmetry of (55.31)).

Here below two special case of Example 55.30.

Example 55.34. Consider the case of Gr(1; Rn) = Fl(1, n− 1) ≈ RPn−1. Therelation w w = 1 gives rise to the system of equations

w1 + w1 = 0wi + wi−1w1 = 0 (i = 2, . . . , n− 1)wn−1w1 = 0

from which we deduce the usual presentation H∗(RPn−1) ≈ Z2[w1]/(wn1 ).

Example 55.35. In the case Gr(2; R4) = Fl(2, 2), the relation w w = 1gives rise to four equations

(55.36)

w1 = w1

w2 = w21 + w2

w31 = 0

w2w21 + w2

2 = 0 .

and Theorem 55.24 says that H∗(Gr(2; R4)) is generated by w1, w2, w1 and w2,subject to Relations (55.36). The first two equations imply that H∗(Gr(2; R4))is generated by w1 and w2, as known since Corollary 55.19. We check that anadditive basis of H∗(Gr(2; R4)) is given by 1, w1, w2, w

21 , w2w1 and w2w

21 = w2

2 .The Poincare polynomial of Gr(2; R4) is given by Corollary 55.27:

Pt(Gr(2; R4)) =Q1(t)Q2(t)

Q3(t)Q4(t)=

(1− t3)(1− t4)(1− t)(1− t2) = 1 + t+ 2t2 + t3 + t4 .

For any bundle of ξ rank k over a space X the dual (or normal) Stiefel-Whitneyclasss wr(ξ) are defined by the equation of Lemma 55.33. Set w(ξ) = 1+w1(ξ)+· · ·for the total dual Stiefel-Whitney class. Equations (55.31) and (55.32) are satisfied.If there exists a vector bundle η over X such that ξ⊕η is trivial, then w(η) = w(ξ).Thus, if η is of rank r, then wi(ξ) = 0 for i > r. The same condition is necessaryfor ξ being induced from the tautological bundle ζ by a map f : X → Gr(k; Rk+r).

For example, let M be a smooth manifold of dimension k which admits animmersion β : M → Rk+r . Let x ∈ M . By identifying Tβ(x)R

k+r with Rk+r the


k-vector space Txβ(TM) becomes an element of Gr(k; Rk+r). This produces a map

β : M → Gr(k; Rk+r) and TM = β∗ζ. We thus get the following result.

Proposition 55.37. If a smooth manifold M of dimension k admits an im-mersion into Rk+r, then wi(TM) = 0 for i > r.

For improvements of Proposition 55.37 concerning also smooth embeddings,see Proposition 58.36 and Corollary 58.46. Usually, Proposition 55.37 does not givethe smallest integer r for which M immerses into Rk+r . This is however the casein the following example, taken from [150, Theorem 4.8].

Proposition 55.38. For k = 2j (j ≥ 1), the projective space RP k immersesinto RN if and only if N ≥ 2k − 1.

Proof. That a manifold of dimension k ≥ 2 immerses into R2k−1 is a classicaltheorem of H. Whitney [206]. Conversly, we shall see in Proposition 58.16 that

w(TRP 2j

) = (1 + a)2j+1 = 1 + a+ a2j

,

where 0 6= a ∈ H1(RP 2j

) = Z2. Hence,

w(TRP 2j

) = 1 + a+ a2 + · · ·+ a2j−1 ,

which, using Proposition 55.37 implies that RP 2j

does not immerses into R2j+1−2.

55.3. Schubert cells and Stiefel-Whitney classes. Let f : M → R be aMorse function on a manifold M . It is classical that M has the homotopy typeof a CW-complex whose r-cells are in bijection with the critical points of indexr of f (see, e.g. [12, Theorem 3.28]). For the weighted trace f (or f) definedon M = Gr(k; Rn) in Proposition 55.5 (or Remark 55.7), a very explicit suchCW-structure is given, using the Schubert cells (there are generalizations for flagmanifolds). Inspired by works of H. Schubert on enumerative geometry in theXIXth century (see e.g. [166]), Schubert cells were introduced in 1934 (for complexGrassmannians) by Ch. Ehresmann [45] ([46] for the real Grassmannians). See[38, pp. 224–25] for a history. We restrict ourselves here to a very elementary pointof view, Schubert calculus being a huge subject in algebraic geometry.

Recall that Crit f are diagonal matrices in SM(n). We write dia(λ1, . . . , λn) =dia(λ), where λ = λ1 · · ·λn is a binary word of length n. Let [nk ] be the set orsuch words with

∑λi = k (they are (nk ) in number). The correspondence λ 7→ λ0

identifies [nk ] with a subset of [n+1k ], permitting us to define [∞k ] as the direct limit

of [nk ].Let F = (F1 ⊂ · · · ⊂ Fn) be a complete flag in Rn (adding the convention that

F0 = 0). For λ ∈ [nk ], the Schubert cell CFλ with respect to F is defined by

CFλ = P ∈ Gr(k; Rn) | dim(P ∩ Fi) =i∑

j=1

λj ⊂ Gr(k; Rn) .

(This convention is close to that of [117], except for the binary words being writtenin the reverse order, so it works for n =∞, in the spirit of [150, § 6]). The followingfacts may be proven.


(1) The Schubert cells CFλ | λ ∈ [nk ] are the open cells of a CW-structureXF on Gr(k; Rn) (see [150, § 6]). The dimension of CFλ is

d(λ) = index (dia(λ)) = f(dia(λ)) = −k(k + 1)

2+

∑

i≥1

λi .

By Proposition 55.5, the the cellular chains have then the same Poincarepolynomial as the homology. Therefore, XF is a perfect CW-structure.

(2) The closure CFλ , called the Schubert variety, satisfies

CFλ = P ∈ Gr(k; Rn) | dim(P ∩ Fi) ≥i∑

j=1

λj ⊂ Gr(k; Rn)

and is a subcomplex of XF (see e.g. [45, § 10]). As XF is perfect, so is CFλand thus CFλ defines a homology class

[λ] = [CFλ ] ∈ Hd(λ)(Gr(k; Rn)) (n ≤ ∞)

which does not depend on F since, by Proposition 55.5, the complete flagmanifold is path-connected. It corresponds, under the isomorphism (16.13)between cellular and singular homology, to the cellular homology class forXF indexed by λ. It follows that the S = [λ] ∈ H∗(Gr(k; Rn)) | λ ∈ [nk ]is a basis of H∗(Gr(k; Rn)) (n ≤ ∞).

(3) Let P ∈ Gr(k; Rn). Using a basis of Rn compatible with the flag F , letMP be the matrix of a linear epimorphism Rn → Rn−k with kernel P .The condition P ∈ CFλ is equivalent to the vanishing of various minorsof MP . Therefore, P ∈ CFλ is a compact real algebraic variety. This isanother proof of the existence of the class [λ], since such a variety carriesa fundamental class (see [188, p. 67] or [15, Theorem 3.7 and § 3.8]).

(4) Suppose that F is the standard flag (Fi = Ri×0). Then f(CFλ ) = [0, d(λ)]and CFλ ∩ f−1(d(λ)) = dia(λ). Recall from Proposition 55.5 that d(λ) isequal to the number of pairs (i, j) with 1 ≤ i < j ≤ n such that λi < λj .

For such a pair (i, j), let Rijt be the one-parameter subgroup of SO(n)

considered in the proof of Proposition 55.5. Then, the curve Rijt dia(λ)Rij−tis contained in CFλ for t ∈ R and stays in CFλ when |t| < π/2. By the proofof Proposition 55.5, these curves generate the negative normal bundle forf at dia(λ).

Example 55.39. Consider the case of Gr(1; Rn+1) ≈ RPn. For F the standardflag in Rn+1, the Schubert cells give the standard CW-structure on RPn, the cellCFλ for λ = 0r10n−r being of dimension r. The Schubert variety CFλ is equal toRP r (a rare case where it is a smooth manifold).

Note that H∗α([λ]) = [λ0] where α : Gr(k; Rn) → Gr(k; Rn+1) is induced bythe inclusion Rn ≈ Rn ⊕ 0 → Rn ⊕R. We often identify [λ] with [λ0]. In this way,for instance, [100101] determines a class in H5(Gr(k; Rn)) for n ≥ 6.

Let S♯ = [λ]♯ | λ ∈ [nk ] (n ≤ ∞) be the additive basis of H∗(Gr(k; Rn)) whichis dual for the Kronecker pairing to the basis S (see (2) above): the class [λ]♯ isdefined by

〈[λ]♯, [µ]〉 = δλµ,

where δλµ is the Kronecker symbol. The basis S♯ was studied in [29, 30]. Because ofintersection theory, a more widely used additive basis for H∗(Gr(k; Rn)) (defined


only or n < ∞) is SPD, is formed by the Poincare duals [λ]PD for all λ ∈ [nk ].Though some intersection theory in used in the proof of Proposition 55.47 below,we shall not use SPD. We just note the following result.

Lemma 55.40. For any k ≤ n <∞, the two sets S♯ and SPD in H∗(Gr(k; Rn))are equal.

Proof. Let F be the standard flag (Fi = Ri × 0) and let F− be the anti-standard one (F−i = 0 × Ri). For λ ∈ [nk ], define λ− ∈ [nk ] by λ−i = λn+1−i.

The cycles CFλ and CF−

λ− are of complementary dimensions and, by (4) above,they intersect transversally in a single point (the k-plane generated by λiei fori = 1, 2, . . . , n). In the same way, if µ ∈ [nk ] satisfies d(λ) = d(µ) but µ 6= λ, then

CFλ ∩ CF−

µ− = ∅. Analogoulsy to Proposition 33.15, one has

(55.41) [µ−]PD [λ]PD = [CFλ ∩ CF−

λ− ]

(see Remark 55.42 below). This implies that

〈[µ−]PD [λ]PD, [Gr(k; Rn)]〉 = δµλ

But, using (26.18),

〈[µ−]PD [λ]PD, [Gr(k; Rn)]〉 = 〈[µ−]PD, [λ]PD [Gr(k; Rn)]〉= 〈[µ−]PD, [λ]〉 .

This proves that [λ]♯ = [λ−]PD (or [λ]PD = [λ−]♯).

Remark 55.42. In the above proof, (55.41) is not a consequence of Proposi-

tion 33.15, which would require that CFλ and CF−

λ− are submanifolds of Gr(k; Rn). In

this simple situation, one could use the Morse function f to isolate the intersectionpoint around the critical level d(λ) and deal with an intersection of submanifolds(with boundaries). For more general situation (see the proof of Proposition 55.47below), we must rely on the intersection theory for real algebraic varieties (see, e.g.[15, (1.12) and § 5]).

In addition to the above ambient inclusion α : Gr(k; Rn) → Gr(k; Rn+1) wealso consider the fattening inclusion β : Gr(k; Rn)→ Gr(k + 1; Rn+1) sending P toR⊕P ⊂ R⊕Rn. Then H∗j([λ]) = [1λ] for all λ ∈ [kn]. This drives us to decomposea word λ ∈ [nk ] into its prefix, stem and suffix, delimited by the first 0 and the last1 of λ:

λ = 111111111︸︷︷︸prefix

00101101︸︷︷︸stem

0000︸︷︷︸suffix

.

Given n and k, a word λ ∈ [nk ] (and then a class [λ] ∈ H∗(Gr(k; Rn)) or [λ]♯ ∈H∗(Gr(k; Rn))) is determined by its stem. For example, 0101 is, for k = 4 andn = 9 is the stem of the unique class [11010100]♯ ∈ H3(Gr(4; R8)). The stemof 1 ∈ H0(Gr(k; Rn)) is just 0. Here below a first use of the prefix-stem-suffixdecomposition.

Proposition 55.43. Let n, k and i be integers with 0 ≤ i ≤ k. Then, fork+1 ≤ n ≤ ∞, the Stiefel-Whitney class wi = wi(ζk) is the class in Hi(Gr(k; Rn))with stem 01i. For example, w3(ζ4) = [1011100]♯ ∈ H3(Gr(4; R7)).


Proof. The proposition is true if i = 0, since w0(ζk) = 1. Let us as-sume that i ≥ 1. We first prove that wi(ζk) = [01i]♯ in Hi(Gr(i; R∞)). RecallH∗(Gr(i; R∞)) ≈ Z2[w1, . . . , wi] (where wj = wj(ζi)) so

K∗ =(kerH∗β : H∗(Gr(i; R∞))→ H∗(Gr(i− 1; R∞))

)

is the ideal generated by wi. Hence, Ki is one-dimensional generated by wi. AsH∗β([λ]) = [1λ], one has

(55.44) H∗β([µ]♯) =

[λ]♯ if µ = 1λ

0 otherwise.

Hence 0 6= [01i]♯ ∈ Ki. Therefore, wi(ζk) = [01i]♯. Proposition 55.43 followsfrom the above particular case since H∗α(wi(ζk)) = wi(ζk) and H∗β(wi(ζk)) =wi(ζk−1).

Let λ, µ ∈ [nk ]. As S is a basis for H∗(Gr(k; Rn)) and S♯ is the Kronecker dualbasis for H∗(Gr(k; Rn)), we can write

[λ]♯ [µ]♯ =∑

ν∈[nk]

Γνλµ [ν]♯

where

Γνλµ = 〈[λ]♯ [µ]♯, [ν]〉 ∈ Z2 .

Computing the ”structure constants” Γνλµ is a version of the Schubert calculus

(mod 2). The usual Schubert calculus deals with the structure constants Cνλµ for

the basis SPD, defined by

[λ]PD [µ]PD =∑

ν∈[nk]

Cνλµ [ν]PD .

By Lemma 55.40 and its proof, one has Γνλµ = Cν−

λ−µ− . Again, Schubert calculus

was initiated by Ch. Ehresmann in [45, 46] and further developed in e.g. [29, 30,73, 64]. For a more recent as well as an equivariant version, see [117]. Note thatΓνλµ = 0 unless d(λ) + d(µ) = d(ν).

A binary word λ ∈ [nk ] is determined by its Schubert symbol, i.e. the k-tuple ofintegers indicating the positions of the 1’s in λ. For instance, the Schubert symbolof 0100101 is (2, 5, 7). We use the Schubert symbol of λ for all the cohomologyclasses [λ0j ]♯ (λ and λ0 having the same symbol). For the reverse correspondence,we decorate the Schubert symbol by a flat sign . Example:

[01001010r]♯ = (2, 5, 7) in H7(Gr(3; R7+r))

(2, 5, 7) = [01001010r] in H7(Gr(3; R7+r))

Our notation for Schubert symbols are that of [150], close to the original oneof [45]. Other conventions are used in e.g. [30, 73].

Remark 55.45. Fix the integers k ≤ n and let a = (a1, . . . , ak) be a k-tuple of integers. In order for a to be a Schubert symbol determining a classin H∗(Gr(k; Rn)), it should satisfy

(55.46) 1 ≤ a1 < a2 < · · · < an ≤ n .By convention, if this is not the case, we decide that a represents the class 0.


A Schubert cell CFλ will be also labelled by the Schubert symbol of λ: CFλ = CFaif a = [λ]♯. For the Poincare duality (see Lemma 55.40), we set

a− = [λ−]♯ = [λ]PD .

If a = (a1, . . . , ak) then a− = (n+ 1− ak, . . . , n+ 1− a1). The definition of Γνλµ isalso transposed for Schubert symbols:

a b =∑

c

Γcab c

where the sum runs over all Schubert symbols c and

Γcab = 〈a b, c〉 ∈ Z2 .

The following proposition and its proof is a variant, in our language, of ReductionFormula I of [73, p. 202].

Proposition 55.47 (Reduction formula). Let k ≥ 2 be an integer. Let r, sand t be positive integers ≤ k satisfying t = r + s − 1. Let a, b and c be Schubertk-symbols. Then

Γcab =

0 if ct < ar + bs − 1

Γcab

if ct = ar + bs − 1,

where a, b and c are the Schubert (k − 1)-symbols

a = (a1, . . . , ar−1, ar+1 − 1, . . . , ak − 1)

b = (b1, . . . , bs−1, bs+1 − 1, . . . , bk − 1)

c = (c1, . . . , ct−1, ct+1 − 1, . . . , ck − 1)

Example 55.48. Let us use the formula for s = 1 and suppose that bs = 1.Thus b = [µ] = [1µ] with µ ∈ [n−1

k−1 ]. The condition t = r + s− 1 reduces to t = r

and ct = ar + bs − 1 becomes cr = ar. Writing it in terms of a = [λ] and c = [ν]this means that if λr = νr = 1 for some r, one can remove λr from λ and νr fromν and replace µ by µ. For instance,

Γ0110110101,11010 = Γ0110

1010,1010 s = 1 and r = t = 5

= Γ010100,010 s = 1 and r = t = 3

= 1 since [100]♯ = 1 in H0(Gr(1; R3)).

Proof of Proposition 55.47. Let F , F ′ and F ′′ be three complete flags inRn. If chosen generically, then CFx , CF

′

y and CF′′

z are pairwise transverse for any

Schubert symbols x, y and z. Therefore, if d(a) + d(b) = d(c), Ca− ∩ Cb− ∩ Cc is0-dimensional and

(55.49)

Γcab = 〈a b, c〉= 〈a b, (c)PD [Gr(k; Rn)]〉= 〈a b (c)PD, [Gr(k; Rn)]〉= ♯(Ca− ∩ Cb− ∩ Cc) mod 2 ,

the last equality coming from the intersection theory analogous to Proposition 33.15but for algebraic cycles (see Remark 55.42).


Let P ∈ Gr(k; Rn). If P ∈ Ca− ∩ Cb− ∩ Cc then

dim(P ∩ Fn+1−ar) ≥ k + 1− r

dim(P ∩ F ′n+1−bs) ≥ k + 1− s

dim(P ∩ F ′′ct) ≥ t .

Therefore, the condition t = r + s− 1 implies that

dim(P ∩ Fn+1−ar∩ F ′n+1−bs

∩ F ′′ct) ≥ 1 .

On the other hand, as F , F ′ and F ′′ are transverse flags,

dim(Fn+1−ar∩ F ′n+1−bs

∩ F ′′ct) = t− r − s+ 2 .

Thus, Γcab = 0 if ct < ar + bs − 1. If ct = ar + bs− 1, then Fn+1−ar∩F ′n+1−bs

∩F ′′ct

is a line L, which must be contained in any P ∈ Ca− ∩ Cb− ∩ Cc. Let L⊥ be theorthogonal complement of L and let π : Rn → L⊥ be the orthogonal projection.For 1 ≤ i ≤ n− 1, define

Fi =

π(Fi) if i ≤ n− arπ(Fi+1) if i ≥ n+ 1− ar .

As L ⊂ Fn+1−arbut L 6⊂ Fn−ar

, the sequence of vector spaces Fi constitutes acomplete flag F in for L⊥. Define F ′ accordingly and F ′′ by

F ′′i =

π(F ′′i ) if i ≤ ctπ(F ′′i+1) if i ≥ ct + 1 .

Then, F , F ′ and F ′′ are transverse flags and, by linear algebra, one checks that

P = π(P )⊕ L ∈ Ca− ∩ Cb− ∩ Cc ⇐⇒ π(P ) ∈ C(a)− ∩ C(b)− ∩ Cc .Hence,

♯(CFa− ∩ CF′

b− ∩ CF′′

c ) = ♯(CF(a)− ∩ CF′

(b)− ∩ CF′′

c )

which, using (55.49), proves that Γcab = Γcab

.

Corollary 55.50. Let Let a, b and c be Schubert k-symbols. Then Γcab = 0unless ci ≥ maxai + b1 − 1, bi + a1 − 1 for all 1 ≤ i ≤ k. In particular, Γcab = 0unless ci ≥ maxai, bi for all 1 ≤ i ≤ k.

Proof. If cr < ar + b1 − 1 for some integer r, then Γcab = 0 by the reductionformula for s = 1. As Γcab = Γcba, this proves the corollary.

We now compute, for a Schubert symbol a, the expression of wi a in thebasis S♯. For J ⊂ 1, 2, . . . , k, we define a map a 7→ aJ from Nk to itself by

aJi =

ai + 1 if i ∈ Jai if i /∈ J .

Proposition 55.51. Let a be a Schubert k-symbol. The following equation

(55.52) wi a =∑

J⊂1,2,...,k♯J=i

aJ

holds in H∗(Gr(k; Rn)) (with the convention of Remark 55.45).

As in the right side of (55.52), we use the convention of Remark 55.45, Propo-sition 55.51 holds true for any n and any i (wi = 0 if i ≥ k).


Example 55.53.

w2 (1, 3, 4, 6) =

(2, 3, 5, 6) + (2, 3, 4, 7) + (1, 3, 5, 7) in H6(Gr(4; Rn)) for n ≥ 7.

(2, 3, 5, 6) in H6(Gr(4; R6)).

Proof. It suffices to prove the proposition for n =∞. We identify wi with itSchubert symbol which, by Lemma 55.43, is

wi = (1, 2, . . . , k − i, k − i+ 1, . . . , k + 1) .

Then, the notation Γca,wiis meaningful. Let c be a Schubert k-symbol such that

Γca,wi6= 0. Then d(c) = d(a) + i and, as cj ≥ aj by Lemma 55.50, there is

K ⊂ 1, 2, . . . , k with ♯K = k − i such that cj = aj for j ∈ K. By iterating thereduction formula for s = 1 with the indices in K, we get that Γca,wi

= Γca,wi, where

a and c are Schubert i-symbols and wi = (2, 3, . . . , i+1). By Lemma 55.50, we havecj ≥ aj + 1 for all 1 ≤ j ≤ i. This implies that c = aJ for J = 1, 2, . . . , k −K.

Conversely, let J ⊂ 1, 2, . . . , k with |J | = i. We have to prove that Γ(J) =

ΓaJ

a,wi= 1 if aJ is a Schubert symbol for Gr(k; R∞). By repeating the reduction

formula for s = 1 with all the indices not in J , we get that

Γ(J) = ΓaJ

a,wi

where a is a Schubert i-symbol, J = 1, 2, . . . , i and wi = (2, 3, . . . , i + 1). Byiterating again the reduction formula for s = 1 with the indices i, i− 1, etc, till 2,we get that

Γ(J) = Γ(a1+1)(a1),(2)

.

This coefficient is equal to 1, as (u) w1 = (u+1) in H∗(Gr(1; R∞)) ≈ H∗(RP∞).

For λ ∈ [nk ], let λ⊥ ∈ [nn−k] be obtained from λ by exchanging 0’s and 1’s and

reverse the order: 100101⊥ = 010110; in formula:

(55.54) λ⊥j = 0 ⇐⇒ λn+1−j = 1 .

Note that d(λ⊥) = d(λ). This formal operation is related to the homeomorphismh : Gr(k; Rn)→ Gr(n−k; Rn) sending k-plane P to its orthogonal complement P⊥.

Lemma 55.55. H∗h([λ]♯) = [λ⊥]♯.

Proof. Let F = (F1 ⊂ · · · ⊂ Fn) be a complete flag in Rn and let F− be thedual flag, defined by F−i = F⊥n−i (we add the convention that F0 = 0 = F−0 ). To

establish Lemma 55.55, we shall prove that h(C(F )λ ) = C

(F−)

λ⊥.

Let P ∈ Gr(k; Rn). Write P = (P ∩ Fi)⊕Qi. Then

P⊥ ∩ F−n−i = v ∈ F−n−i | 〈v,Qi〉 = 0 .

Hence

(55.56) codimF−n−i(P⊥ ∩ F−n−i) = dimQi = codimP (P ∩ Fi) .


Suppose that P ∈ C(F )λ for λ ∈ [nk ]. Then, P⊥ ∈ C(F−)

µ µ ∈ [nn−k]. We must prove

that µ = λ⊥, that is to say (λi = 0 ⇐⇒ µn+1−i = 1). But, using (55.56)

λi = 0 ⇐⇒ dim(P ∩ Fi) = dim(P ∩ Fi−1)

⇐⇒ codimF−n−i(P⊥ ∩ F−n−i) = codimF−n+1−i

(P⊥ ∩ F−n+1−i)

⇐⇒ dim(P⊥ ∩ F−n+1−i) = dim(P⊥ ∩ F−n−i) + 1

⇐⇒ µn+1−i = 1 .

Let w = w(ζ⊥k ) = 1 + w1 + · · · wn−k be the total Stiefel-Whitney class of thetautological (n− k)-vector bundle over Gr(k; Rn) (see Example 55.30).

Proposition 55.57. Suppose that n ≥ i+ k − 1. Then wi ∈ Hi(Gr(k; Rn) isthe class of stem 0i1. Its Schubert symbol is (1, 2, . . . , k − 1, k + i).

Example: w5 = [11100001]♯ = (1, 2, 3, 8) in H5(Gr(4; Rn) for n ≥ 8.

Proof. The homeomorphism h : Gr(k; Rn) → Gr(n − k; Rn) is covered bythe tautological bundle map ζ⊥k → ζn−k. Hence, h∗ζn−k = ζ⊥k and thus wi =H∗h(wi(ζn−k)). Therefore, wi = (wi)

⊥ by Lemma 55.55. As stem (λ⊥) = stem (λ)⊥,Proposition 55.57 follows from Proposition 55.43.

We now give the expression of wi (a). As w1 = w1, we can use For-mula (55.52) for i = 1: for a Schubert k-symbol a, one has

(55.58) w1 a = w1 a =∑

b

b

where the sum runs over all the Schubert k-symbols b such that

aj ≤ bj ≤ aj + 1 and

k∑

j=1

(bj − aj) = 1 .

Example:

w21 = w1 (1, 2, . . . , k − 1, k + 1)

= (1, 2, . . . , k − 2, k, k + 1) + (1, 2, . . . , k − 1, k + 2)

= w2 + w2 .

Formula (55.58) admits the following generalization, called the Pieri formula, whichis a sort of a dual of Proposition 55.51.

Proposition 55.59 (Pieri’s formula). Let a be a Schubert k-symbol. The fol-lowing equation

(55.60) wi a =∑

b

b

holds in H∗(Gr(k; Rn)), where the sum runs over all the Schubert k-symbols b suchthat

(55.61) aj ≤ bj < aj+1 and

k∑

j=1

(bj − aj) = i

(with the convention of Remark 55.45).

56. SPLITTING PRINCIPLES 329

Proof. For Schubert k-symbols a, b, the proposition says that

(55.62) 〈wi a, b〉 = 1⇐⇒ (a, b) satisfies (55.61) .

(Note that the implication ⇒ follows from Lemma 55.50 and from wi being ofdegree i). Rewriting (55.62) with λ, µ ∈ [nk ] gives that 〈wi [λ]♯, [µ]〉 = 1 if andonly if λ and µ satisfy the following pair of conditions

(i) λ = A110r1A210r2 · · ·As10rsAs+1, with∑s

j=1 rj = i, and

(ii) µ = A10r11A20

r21 · · ·Ars0rs1As+1.

(Intuitively: a certain quantity of 1’s are shifted by one position to the right oftotal amount shifting being i). The pair of conditions (i) and (ii) is equivalent tothe following one

(i)⊥ λ⊥ = A⊥s+11rs0A⊥s 1rs−10 · · ·A⊥2 1r10A⊥1 with

∑sj=1 rj = i, and

(ii)⊥ µ⊥ = A⊥s+101rs A⊥s 01rs−1 · · ·A⊥2 01r1.

Recall from Lemma 55.55 and the proof of Proposition 55.57 that the home-omorphism h : Gr(k; Rn) → Gr(n − k; Rn) satisfies H∗([ν]) = [ν⊥] and wi =H∗(wi(ζn−k)). Therefore

(55.63) 〈wi [λ]♯, [µ]〉 = 1⇐⇒ 〈wi(ζn−k) [λ⊥]♯, [µ⊥]〉 = 1 .

By Proposition 55.51, the right hand equality in (55.63) is equivalent to the pair ofconditions (i)⊥ and (ii)⊥, which proves Proposition 55.59.

We finish this subsection by mentioning the Giambelli’s formula, which expressa cohomology class given by a Schubert symbol as a polynomial in the wi’s. TheGiambelli and the generalized Pieri formulae together provides a procedure forcomputing the structure constants Γcab.

Proposition 55.64 (Giambelli’s formula).

(a1, . . . , ak) = det(wai−j

)1≤i,j≤k

.

with the convention that wu = 0 if u < 0.

For wr = (1, 2, . . . , k − r, k − r + 1, . . . , k + 1), Proposition 55.64 reproves thesecond formula of Lemma 55.33.

Proof. By induction on k, starting trivially if k = 1. The lengthy inductionstep, using the Pieri formula, may be translated in our language from [73, pp. 204–205] (see also [30, p. 366]).

56. Splitting principles

Let α : Γ→ O(n) denotes the inclusion of the diagonal subgroup of O(n)

Γ ≈ O(1)× · · · ×O(1) ⊂ O(n) .

This induces an inclusion Bα : BΓ → BO(n) between the classifying spaces. Thesymmetric group Symn acts on O(1) × · · · × O(1) by permutting the factors, andthen on BΓ. As in § 55, ζn denotes the tautological vector bundle on BO(n) ≃Gr(n; R∞). It is the vector bundle associated to the universal O(n)-bunle EO(n)→BO(n).

Theorem 56.1. The GrA-morphism H∗Bα : H∗(BO(n))→ H∗(BΓ) is injec-tive and its image is H∗(BΓ)Symn . The induced vector bundle Bα∗ζn splits into aWhitney sum of line bundles.


Proof. We have seen in (55.25) that the homotopy equivalence

BΓ ≃ EO(n)/O(1)× · · · ×O(1)

makes Bα homotopy equivalent to the locally trivial bundle

(56.2) Fl(n1, . . . , nr)β−→ BΓ

Bα−−→ BO(n) .

We have also established in the proof of Theorem 55.24 that H∗β is surjective.Hence, by Corollary 28.31, H∗Bα is injective. Also, using (42.3) and that O(1) ≈±1. one has a homotopy equivalence

BΓψ

≃// BO(1)× · · · ×BO(1) ≃ RP∞ × · · · × RP∞

and thus a GrA-isomorphism

ψ∗ : Z2[x1, · · · , xn] ≈−→ H∗(BΓ)

where xi has degree 1. By Theorem 55.17,

H∗(BO(n)) ≈ Z2[w1(ζn), . . . , wn(ζn)] .

Note that Bα is covered by a morphism of principal bundles

EO(1)× · · · ×EO(1)Eα //

EO(n)

BO(1)× · · · ×BO(1)

Bα // BO(n) .

One has a similar diagram for the associated vector bundles γ = (EO(1)×O(1) R→BO(1)) (corresponding to the tautological line bundle over RP∞) and ζn. Thisimplies that Bα∗ζn ≈ γ × · · · × γ. As w(γ × · · · × γ) =

∏ni=1(1 + xi), one has

H∗Bα(wi(ζn)) = wi(γ × · · · × γ) = σi ,

where σi is i-th elementary symmetric polynomial in the variables xj . The secondassertion of Theorem 56.1 follows, since the elementary symmetric polynomialsGrA-generate Z2[x1, · · · , xn]Symn ≈ H∗(BΓ)Symn .

Finally, the homotopy equivalence ψ is of the form ψ = (ψ1, . . . , ψn), withψi : BΓ→ BO(1). In other words, ψ coincides with the composition

BΓ∆−→ BΓ× · · · ×BΓ

ψ1×···×ψn−−−−−−−→ BO(1) × · · · ×BO(1) ,

where ∆ is the diagonal map. Hence,

Bα∗ζn ≈ ψ∗(γ × · · · × γ) = ∆∗(ψ∗1γ × · · · × ψ∗nγ) = ψ∗1γ ⊕ · · · ⊕ ψ∗nγ ,which shows that Bα∗ζn is isomorphic to a Whitney sum of line bundles.

Theorem 56.1 may be generalized as follows. Consider the inclusion homomor-phism

αn1,...,nr: O(n1)× · · · ×O(nr)→ O(n)

sending (A1, . . . , Ar) to the diagonal-block matrix with blocksA1, . . . , Ar. Using thehomotopy equivalence BO(n1)×· · ·×BO(nr) ≃ B(O(n1)×· · ·×O(nr)) (see (42.3)),the homomorphism αn1,...,nr

induces a continuous map

Bαn1,...,nr: BO(n1)× · · · ×BO(nr)→ BO(n) .

Theorem 56.3. The map Bαn1,...,nrsatisfies the following properties.

56. SPLITTING PRINCIPLES 331

(1) The GrA-morphism

H∗Bαn1,...,nr: H∗(BO(n))→ H∗(BO(n1)× · · · ×BO(nr))

is injective.(2) H∗Bαn1,...,nr

(wi) = wi(ζn1 × · · · × ζnr) for each i ≥ 0. In particular, the

image of H∗Bαn1,...,nris generated by wi(ζn1 × · · · × ζnr

) (i ≥ 0).(3) The induced vector bundle Bα∗n1,...,nr

ζn splits into a Whitney sum of vectorbundles of ranks n1, . . . , nr.

Proof. Using the inclusion factorization

O(1)n≈ // O(1)n1 × · · · ×O(1)nr // O(n1)× · · · ×O(nr)

α // O(n) ,

where α = αn1,...,nr, the injectivity of H∗Bα comes from that of H∗Bα1,...,1, es-

tablished in Theorem 56.1. As Bα is covered by a morphism of principal bundles

EO(n1)× · · · ×EO(nr)α //

EO(n)

BO(n1)× · · · ×BO(nr)

Bα // BO(n)

,

one deduces (2) and (3) as in the proof of Theorem 56.1.

Proposition 56.4. Let ξ be a vector bundle over a paracompact space X.Then, there is a map f : Xξ → X such that

(1) H∗f is injective.(2) f∗ξ splits into a Whitney sum of line bundles.

Proposition 56.4 is called the splitting principle. For ξ = ζn over BO(n),Theorem 56.1 says that on can take BO(n)ζn

and f = Bα.

Proof. As X is paracompact, ξ admits an Euclidean structure and there isa classifying map ϕ : X → BO(n) for ξ, i.e. ξ ≈ ϕ∗ζn. Consider the pull-backdiagram

Xξf //

ϕ

X

ϕ

BΓ

Bα // BO(n)

,

where Bα is defined as in (56.2). As Bα is a locally trivial bundle with fiberFl(n1, . . . , nr), so is f (this is the Fl(n1, . . . , nr)-bundle associated to Fra⊥ξ). Wesaw in the proof of Theorem 56.1 that H∗(BΓ)→ H∗(Fl(n1, . . . , nr)) is surjective.Then, so is H∗(Xξ) → H∗(Fl(n1, . . . , nr)). Hence, by Corollary 28.31, H∗f isinjective. Now,

f∗ξ = f∗ϕ∗ζn = ϕ∗Bα∗ζn .

As, by Theorem 56.1, Bα∗ζn is a Whitney sum of line bundles, so does f∗ξ.

One consequence of the splitting principle is the uniqueness of the Stiefel-Whitney classes (compare [150, Theorem 7.3] or [103, Chapter 16,§ 5]).

Proposition 56.5. Suppose that w is a correspondence associating, to eachvector bundle ξ over a paracompact space X, a class w(ξ) ∈ H∗(X), such that


(1) if f : Y → X be a continuous map, then w(f∗ξ) = H∗f(w(ξ)).(2) w(ξ ⊕ ξ′) = w(ξ) w(ξ′).(3) if γ is the tautological line bundle over RP∞, then w(γ) = 1 + a, where

0 6= a ∈ H1(RP∞).

Then w = w, the total Stiefel-Whitney class.

Proof. Condition (2) implies that

(2.bis) w(ξ1 ⊕ · · · ⊕ ξn) = w(ξ1) · · · w(ξn).

As in the proof of Theorem 56.1, Conditions (1), (2.bis) and (3) imply that themap Bα : BΓ→ BO(n) satisfies

H∗Bα(w(ζn)) = (1 + xi)n ∈ H∗(BΓ) ≈ Z2[x1, · · · , xn] .

Thus, still by the proof of Theorem 56.1, H∗Bα(w(ζn)) = H∗Bα(w(ζn)). AsH∗Bα is injective, this implies that w(ζn) = w(ζn). The bundle ζn being universal,Condition (1) implies that w(ξ) = w(ξ) for any vector bundle ξ over a paracompactspace X .

Another consequence of the splitting principle is the action of the Steenrodalgebra on the Stiefel-Whitney classes. The following proposition was proved byWu Wen Tsun [209].

Proposition 56.6. Let ξ be a vector bundle over a paracompact space X. Then

(56.7) Sqiwj(ξ) =∑

0≤k≤i

(j−i+k−1

k

)wi−k(ξ)wj+k(ξ) .

Example 56.8. Setting wi = wi(ξ), we get

Sq1wj = w1wj + (j − 1)wj+1

Sq2wj = w2wj + (j − 2)w1wj+1 +(j−12

)wj+2

Sq3wj = w3wj + (j − 3)w2wj+1 +(j−22

)w1wj+2 +

(j−33

)wj+3 .

Proof of Proposition 56.6. By naturality of w and Sq, it suffices to prove (56.7)for ξ = ζn, the tautological vector bundle onBO(n). By Theorem 56.1 and its proof,

H∗Bα(wj(ζn)) = σj ∈ Z2[x1, · · · , xn]where xj corresponds to the non-trivial element in H1(RP∞), and σj is the j-th el-ementary symmetric polynomial in the variables xr. As H∗Bα is injective by Theo-rem 56.1, Formula (56.7) reduces to the computation of Sqiσj in Z2[x1, · · · , xn]Symn ,using that Sq(xr) = xr + x2

r . This technical computation may be found in full de-tails in [14, Theorem 7.1]. The reader may, as an exercise, prove the special casesof Example 56.8.

The splitting principle also gives the following result about the Stiefel-Whitneyclasses of a tensor product (for a more general formula, see [150, p. 87]).

Lemma 56.9. Let η and ξ be vector bundles over a paracompact space X. Sup-pose that ξ is of rank r and that η is a line bundle. Then

(56.10) w(η ⊗ ξ) =r∑

k=0

(1 + w1(η))kwr−k(ξ) .

57. COMPLEX FLAG MANIFOLDS 333

Proof. Set u = w1(η). Suppose first that ξ splits into a Whitney sum of r linebundles ξj , of Stiefel-Whitney class 1 + vj . Then, letting σk = (v1, . . . , vr) denotethe kth elementary symmetric polynomial, one has

w(η ⊗ ξ) = w(⊕rj=1(η ⊗ ξj))=

∏rj=1 w(η ⊗ ξj) by (54.4)

=∏rj=1(1 + u+ vj) by Proposition 52.4

=∑r

k=0(1 + u)kσr−k(v1, . . . , vr).

Since σr−k(v1, . . . , vr) = wr−k(ξ), we have shown (56.10) when ξ splits into aWhitney sum of r line bundles. If this is not the case, Formula (56.10) still holdstrue by the splitting principle of Proposition 56.4.

57. Complex flag manifolds

The plan of this section follows that of § 55. We shall indicate the slight changesto get from the real flag manifolds to the complex ones, without repeating all theproofs.

Let n1, . . . nr be positive integers and let n = n1 + n2 + · · ·nr. By the complexflag manifold FlC(n1, . . . , nr), we mean any smooth manifold diffeomorphic to thehomogeneous space

(57.1) FlC(n1, . . . , nr) ≈ U(n)/U(n1)× U(n2)× · · · × U(nr) .

The most usual concrete occurrence of complex flag manifolds are as below.

(1) Nested subspaces. FlC(n1, . . . , nr) is the set of nested complex vector sub-

spaces V1 ⊂ · · · ⊂ Vr ⊂ Cn with dimC Vi =∑ij=1 nj .

(2) Mutually orthogonal subspaces. FlC(n1, . . . , nr) is the set of r-tuples (W1, . . . ,Wr)of complex vector subspaces Cn which are mutually orthogonal (for thestandard Hermitian product on Cn) and satisfy dimWi = ni. The corre-spondence from this definition to Definition (1) associates to (W1, . . . ,Wr)the nested family Vi where Vi is the complex vector space generated byW1 ∪ · · · ∪Wi.

(3) Isospectral Hermitian matrices. Let λ1 > · · · > λr be real numbers. Con-sider the manifold HM(n) of all Hermitian (n × n)-matrices, on whichU(n) acts by conjugation. Then FlC(n1, . . . , nr) occurs as the orbit of thediagonal matrix having entries λi with multiplicity ni.

(57.2) FlC(n1, . . . , nr) =R dia

(λ1, . . . , λ1︸︷︷︸

n1

, · · · , λr, . . . , λr︸︷︷︸nr

)R−1 | R ∈ U(n)

.

In other words, FlC(n1, . . . , nr) is here the space of Hermitian (n × n)-matrices with characteristic polynomial equal to

∑ri=1(x− λi)ni . Indeed,

two matrices in HM(n) are in the same U(n)-orbit if and only if theyhave the same characteristic polynomial. The correspondence from thisdefinition to Definition (2) associates, to a matrix M , its eigenspaces forthe various eigenvalues.

Concrete definition (3) is our working definition for FlC(n1, . . . , nr) throughout thissection. Special classes of flag manifolds are given by the complex Grassmanians

Gr(k; Cn) = FlC(k, n− k) ≈ U(n)/U(k)× U(n− k)


of complex k-planes in Cn. This is a closed manifold of dimension

dimGr(k; Cn) = dimU(n)− dimU(k)− dimU(n− k) = 2k(n− k) .For example, Gr(1; Cn) ≈ CPn−1, of dimension 2(n− 1).

Using Definition (3) above, our “concrete Grassmannian” will be

(57.3) Gr(k; Cn) =R dia

(1, . . . , 1︸︷︷︸

k

, 0, . . . , 0︸︷︷︸n−k

)R−1 | R ∈ U(n)

.

As, in the real case, we define the complete complex flag manifold

FlC(1, . . . , 1) ≈ U(n)/U(1)× · · · × U(1)

with dimFlC(1, . . . , 1) = dimU(n)− n = n2 − n = n(n− 1).As in § 55, we define real functions on the flag manifolds by restriction of the

weighted trace on f : HM(n)→ R defined by

f(M) =

n∑

j=1

j Mjj

where Mij denotes the (i, j)-entry of M .

Proposition 57.4. Let FlC(n1, . . . , nr) ⊂ HM(n) be the complex flag manifoldas presented in (55.3). Then, the restriction f : FlC(n1, . . . , nr)→ R of the weightedtrace is a perfect Morse function whose critical points are the diagonal matrices inFlC(n1, . . . , nr). The index of the critical point dia(x1, . . . , xn) is twice the numberof pairs (i, j) with i < j and xi < xj . In consequence, dimFlC(n1, . . . , nr) =2 dimFl(n1, . . . , nr) and

(57.5) Pt(FlC(n1, . . . , nr)) = Pt2(Fl(n1, . . . , nr)) .

Recall that dimFl(n1, . . . , nr) was computed in Corollary 55.11 and that thePoincare polynomial Pt(Fl(n1, . . . , nr)) was described in Corollary 55.27. Equal-ity (57.5) implies the following corollary.

Corollary 57.6. The cohomology groups of FlC(n1, . . . , nr) vanish in odddegrees.

Remark 57.7. The manifold FlC(n1, . . . , nr) ⊂ HM(n) admits an U(n)-invariantsymplectic form, induced from the non-degenerate symmetric form (X,Y ) 7→ trace (XY )on HM(n) (see [11, Chapter II, Example 1.4]). The weighted trace is the momentmap for the Hamiltonian circle action given by the conjugation by dia(eit, e2it, . . . , enit).The involution τ given on FlC(n1, . . . , nr) by the complex conjugation is anti-symplectic and anti-commutes with the circle action. Its fixed point set is Fl(n1, . . . , nr).Note that f is τ -invariant and the critical point of f or f |Fl(n1, . . . , nr) are the same.This, together with (57.5), is a particular case of a theorem of Duistermaat [43](see also Remark 57.14).

Proof of Proposition 57.4. We use the injective homomorphism rij : SU(2)→U(n), introduced in the proof of Proposition 57.4, whose image contains the matri-ces

Rijt = rij(

cos t − sin tsin t cos t

)and Rijt = rij

(cos t

√−1 sin t√

−1 sin t cos t

)(t ∈ R) .


Suppose that ∆ ∈ FlC(n1, . . . , nr) is a diagonal matrix. Then, a basis ofT∆FlC(n1, . . . , nr) is represented by the curves

∆ij(t) = Rijt ∆Rij−t and ∆ij(t) = Rijt ∆Rij−t .

As in the proof of Proposition 55.5, this shows that the critical points of f areexactly the diagonal matrices and computes the indices.

As the critical points are all of even index, the function f is a perfect Morsefunction by Lemma 44.9. One can also proceed as in the proof of Proposition 55.5,using that f is invariant for the action of the diagonal subgroup T of U(n), whichis the torus U(1)× · · · × U(1), and use Theorem 44.29.

As in § 55, consider the inclusionHM(n) ⊂ HM(n+1) with image the matriceswith vanishing last row and column. Seeing Gr(k; Cn) ⊂ HM(n) as in (55.4), thisgives an inclusion Gr(k; Cn) ⊂ Gr(k; Cn+1). The proof of the following lemma isthe same as that of Lemma 55.13.

Lemma 57.8. The homomorphism Hj(Gr(k; Cn+1))→ Hj(Gr(k; Cn)) inducedby the inclusion is surjective for all j and is an isomorphism for j ≤ 2(n− k).

Tautological bundles. Consider a complex flag manifold FlC(n1, . . . , nr), with n =n1 + · · ·+ nr and the following closed subgroups of U(n).

Bi = U(n1)× · · · × 1 × · · · × U(nr) ⊂ U(n1)× · · · × U(nr) ⊂ U(n) .

Then

Pi = U(n)/Bi →→ U(n)/U(n1)× · · · × U(nr) = FlC(n1, . . . , nr)

is an U(ni)-principal bundle (see p. 314) over FlC(n1, . . . , nr). Its associated com-plex vector bundle of rank ni, i.e. E(ξi) = Pi ×U(ni) Cni , is called i-th-tautologicalvector bundle over FlC(n1, . . . , nr). Being associated to an U(ni)-principal bundle,ξi is endowed with an Hermitian structure and its space of orthonormal framesFra⊥(ξi) is equal to Pi. In the mutually orthogonal subspaces presentation (2) ofFlC(n1, . . . , nr), we see that

E(ξi) = (W1 . . . ,Wr, v) ∈ FlC(n1, . . . , nr)× Cn | v ∈ Wi .Note that ξ1 ⊕ · · · ⊕ ξr is trivial (see § 55, p. 314).

The complex vector bundle ξ1 over FlC(k, n−k) = Gr(k; Cn) is called the tauto-logical vector bundle over the complex Grassmannian Gr(k; Cn); it is of (complex)rank k and is denoted by ζ or ζk. The space of Fra⊥(ζk) is the complex Stiefelmanifold Stief(k,Cn) of orthonormal k-frames in Cn.

The inclusion Cn ≈ Cn × 0 → Cn+1 induces an inclusion Gr(k; Cn) →Gr(k; Cn+1) and we may consider the inductive limit

Gr(k; C∞) = limn

Gr(k; Cn)

which is a CW-space. The tautological vector bundle ζk is also defined overGr(k; C∞) and induces that over Gr(k; Cn) by the inclusion Gr(k; Cn) → Gr(k; C∞).It is classical that πi(Stief(k,Cn)) = 0 for i < 2(n − k) + 1 (see [177, 25.7]),thus Stief(k,C∞) = Fra(ζk) is contractible. Hence, the U(k)-principal bundleStief(k,C∞) → Gr(k; C∞) is a universal U(k)-principal bundle (see [177, § 19.4])and thus homotopy equivalent to the Milnor universal bundle EU(k) → BU(k).In particular, Gr(k; C∞) has the homotopy type of BU(k). As a consequence, any


complex vector bundle of rank k over a paracompact space X is induced from ζkby a map X → Gr(k; C∞).

To emphasise the analogy with § 55, we introduce the total Chern classes c(ξ) ∈H2∗(X) of a complex vector bundle ξ of rank k over a space X by

c(ξ) =

k∑

j=1

w2j(ξR) ,

where ξR is the vector bundle ξ seen as a real vector bundle of (real) rank 2k. Thecomponent of c(ξ) in H2j(X) is

(57.9) cj(ξ) = w2j(ξR) ∈ H2j(X)

is called the i-th Chern class of ξ.

Theorem 57.10. The cohomology ring of BU(k) is GrA-isomorphic to thepolynomial ring

H∗(BU(k)) = H∗(Gr(k; C∞)) ≈ Z2[c1, . . . , ck]

generated by the Chern classes ci = ci(ζk) of the tautological bundle ζk.

Remark 57.11. Our Chern classes cj(ξ) are the reduction mod 2 of the integralChern classes (see [150, § 14] or [103, Chapter 16]). That the restriction mod 2of cj(ξ) coincides with w2j(ξR) (whence our definition (57.9)) is proven in [177,Theorem 41.8]. Note that, by Theorem 57.10, w2j+1(ξR) = 0.

Proof of Theorem 57.10. It is the same as that of Theorem 55.17, usingProposition 57.4. To see that the Chern classes are algebraically independent, weuse the tautological complex line bundle γ over CP∞ and its n-times product γn

over (CP∞)n.

Theorem 57.10 together with Lemma 57.8 gives the following corollary.

Corollary 57.12. The cohomology ring H∗(Gr(k; Cn)) is generated, as a ring,by the Chern classes c1(ζk), . . . , ck(ζk) of the tautological bundle ζk.

Let c(ξj) = 1 + c1(ξj) + · · ·+ cnj(ξj) ∈ H∗(FlC(n1, . . . , nr)) be the Chern class

of the tautological vector bundle ξj . The following theorem is proven in the sameway as Theorem 55.24. Actually, replacing Z2[ci(ξj)] by Z[ci(ξj)], the statement istrue for the integral cohomology (as we wrote a minus sign in the last expression).

Theorem 57.13. The cohomology algebra H∗(FlC(n1, . . . , nr)) is GrA-isomorphicto the quotient of the polynomial ring

Z2[ci(ξj)] , 1 ≤ i ≤ rj , j = 1, . . . , r

by the ideal generated by the homogeneous components of 1− c(ξ1) · · · c(ξr).

Remark 57.14. By Theorems 57.13 and 55.24, the the correspondence ci(ξj) 7→wi(ξj) provides a ring isomorphism

H2∗(FlC(n1, . . . , nr))≈−→ H∗(Fl(n1, . . . , nr)) .

Actually, FlC(n1, . . . , nr) with the complex conjugation is a conjugation space (see§ 61). Given Remark 57.7, this is established in [85, Theorem 8.3].


We have seen in Proposition 52.4 that the first Stiefel-Whitney class classifiesthe real lines bundles. The full analogue for complex line bundles requires coho-mology with Z-coefficients: the first integral Chern class provides an isomorphism

(LC(X),⊗)≈−→ H2(X ; Z), where LC(X) be the set of isomorphism classes of com-

plex lines bundles over a CW-complex X (see [94, pp. 62–63]). But, staying withinthe mod 2-cohomology, one can prove the following result.

Proposition 57.15. Let ξ and ξ′ be two complex line bundles over a CW-complex X. Then c1(ξ ⊗ ξ′) = c1(ξ) + c1(ξ

′).

Proof. The argument follows the end of the proof of Proposition 52.4. Onehas to replace R× by C× and K by KC = C× × C×. The only thing to prove isthat the composed map

BC× ×BC×P ′

≃// BKC

Bϕ // BC×

corresponding to that of of Diagram (52.6) satisfies

(57.16) H∗(BϕP ′)(v) = v × 1 + 1× v ,where v is the generator of H2(BC×) = H2(BU(1)) = H2(CP∞). The complexconjugation of C× induces an involution τ onBC× corresponding to the conjugationon CP∞, with fixed point RP∞ = BR×. The map BϕP ′ is τ -equivariant. Hence,Equation (57.16) follows from (52.7), using that the inclusion j : RP∞ → CP∞

satisfies H∗j(v) = u2 (see Proposition 35.14).

Finally, the splitting principle results of § 56 have their correspondents forcomplex bundles. One uses the inclusion of the diagonal subgroup of U(n)

Γ ≈ U(1)× · · · × U(1) ⊂ U(n) .

The following result is proven in the same way as for Theorem 56.1.

Theorem 57.17. The GrA-morphism H∗Bα : H∗(BU(n)) → H∗(BΓ) is in-jective and its image is H∗(BΓ)Symn . The complex vector bundle Bα∗ζn inducedfrom the universal bundle ζn splits into a Whitney sum of complex line bundles.

As for Theorem 56.3, Theorem 57.17 generalizes in the following way for theinclusion

αn1,...,nr: U(n1)× · · · × U(nr)→ U(n) .

Theorem 57.18. The map Bαn1,...,nrsatisfies the following properties.

(1) The GrA-morphism

H∗Bαn1,...,nr: H∗(BU(n))→ H∗(BU(n1)× · · · ×BU(nr))

is injective.(2) H∗Bαn1,...,nr

(ci) = ci(ζn1 × · · · × ζnr) for each i ≥ 0. In particular, the

image of H∗Bαn1,...,nris generated by ci(ζn1 × · · · × ζnr

) (i ≥ 0).(3) The induced complex vector bundle Bα∗n1,...,nr

ζn splits into a Whitney sumof complex vector bundles of ranks n1, . . . , nr.

As in § 56, we deduce from Theorem 57.17 the following proposition (splittingprinciple for complex bundles).


Proposition 57.19. Let ξ be a complex vector bundle over a paracompact spaceX. Then, there is a map f : Xξ → X such that

(1) H∗f is injective.(2) f∗ξ splits into a Whitney sum of complex line bundles.

As in Proposition 56.5, we get an axiomatic characterisation of Chern classes.

Proposition 57.20. Suppose that c is a correspondence associating, to eachcomplex vector bundle ξ over a paracompact space X, a class c(ξ) ∈ H2∗(X), suchthat

(1) if f : Y → X be a continuous map, then c(f∗ξ) = H∗f(c(ξ)).(2) c(ξ ⊕ ξ′) = c(ξ) c(ξ′).(3) if γ is the tautological complex line bundle over CP∞, then c(γ) = 1 + a,

where 0 6= a ∈ H2(CP∞).

Then c = c, the total Chern class.

Thanks to our definition of Chern classes via the Stiefel-Whitney classes, thefollowing proposition is a direct consequence of Proposition 56.6.

Proposition 57.21. Let ξ be a complex vector bundle over a paracompact spaceX. Then

(57.22) Sq2icj(ξ) =∑

0≤k≤i

(j−i+k−1

k

)ci−k(ξ) cj+k(ξ) .

Remark 57.23. As in § 55.3, the Schubert calculus may be developed forcomplex Grassmanians. The degrees of (co)homology classes are doubled. TheStiefel-Whitney classes wi are replaced by the Chern classes ci. The Stiefel-Whitneyclasses wi corresponds, in the literature, to the Segre classes.

We finish this section with the complex analogue of Lemma 56.9.

Lemma 57.24. Let η and ξ be complex vector bundles over a paracompact spaceX. Suppose that ξ is of rank r and that η is a line bundle. Then

(57.25) c(η ⊗ ξ) =

r∑

k=0

(1 + c1(η))kcr−k(ξ) .

Proof. The proof is the same as that of Lemma 56.9. The use of Proposi-tion 52.4 has to be replaced by that of Proposition 57.15.

58. The Wu formula

58.1. Wu’s classes and formula. Let Q be a closed manifold of dimensionn. The map

Hn−k(Q)Sqk

−−→ Hn(Q)〈−,[Q]〉−−−−−→ Z2

is a linear form on Hn−k(Q). By Poincare duality (see Theorem 32.18), there is

a unique class vk(Q) ∈ Hk(Q) such that 〈Sqk(a), [Q]〉 = 〈vk(Q) a, [Q]〉 for alla ∈ Hn−k(Q). In other words,

(58.1) Sqk(a) = vk(Q) a

for all a ∈ Hn−k(Q). The left hand side of (58.1) vanishing if k > n − k, one hasvk(Q) = 0 if k > n/2. The class vi(Q) is the i-th Wu class of Q (for Wu classes in

58. THE WU FORMULA 339

a more general setting, see [121, § 3]). Note that v0(Q) = 1. The total Wu classv(Q) is defined by

v(Q) = 1 + v1(Q) + · · ·+ v[n/2](Q) ∈ H∗(Q) .

As an exemple the next lemma shows the role of vk(Q) when n = 2q. Let Vbe a Z2-vector space and let B : V × V → V be a bilinear form. A symplectic basisof V for B is a basis a1, . . . , ak, b1 . . . , bk of V such that B(ai, aj) = B(bi, bj) = 0and B(ai, bj) = B(bj , aj) = δij . By convention, the empty basis for V = 0 is alsosymplectic.

Lemma 58.2. Let Q be a closed smooth manifold of dimension 2k such that itsWu class vk(Q) vanishes. Then the bilinear form B : Hk(Q)×Hk(Q)→ Z2 givenby B(x, y) 7→ 〈x y, [Q]〉 admits a symplectic basis.

Note that the lemma implies that Hk(Q) has even dimension and thus, byPoincare duality, χ(Q) is even. This can be also deduced from Corollary 33.20 and

Theorem 54.2 since, by the Wu formula (see below), w2k(TQ) = Sqk(vk(Q)) = 0.

Proof. By definition of the Wu class, vk(Q) = 0 is equivalent to B being alter-nate, i.e. B(x, x) = 0 for all x ∈ Hk(Q). By Theorem 32.18, B is son-degenerate.We are thus reduced to prove the following classical claim: on a Z2-vector spaceV of finite dimension, a non-degenerate bilinear form B which is alternate ad-mits a symplectic basis. As B is non-degenerate, there exists a1, b1 ∈ V such thatB(a1, b1) = 1. Hence B(b1, a1) = 1 since alternate implies symmetric in character-istic 2. One has an exact sequence

0→ A→ Vφ−→ Z2 ⊕ Z2 → 0 ,

where φ is the linear map φ(v) = (B(a1, v), B(v, b1)) and A = kerφ. As B is non-degenerate, so is its restriciton to A × A. This permits us to prove the claim byinduction on the dimV .

The Wu’s formula below relates the Wu class of Q to the Stiefel-Whitney classw(TQ) of the tangent bundle TQ of Q (often called the Stiefel-Whitney class of Q).

Theorem 58.3 (Wu’s formula). For any smooth closed manifold Q, one has

w(TQ) = Sq(v(Q)) .

The Wu formula was proved by Wu Wen Tsun in 1950 [208] by direct com-putations in H∗(Q×Q). We follow below the proof of Milnor-Stasheff [150, The-orem 11.14] (for a proof using equivariant cohomology, see Remark 58.34). Thecomputations are lightened by the the use of the slant product

(58.4) H∗(X × Y )/−→ H ∗ (X)

(Hk(X × Y )⊗Hm(Y )

/−→ Hk−m(X))

which is defined as follows. Consider the map H∗(X)⊗H∗(Y )⊗H∗(Y )→ H∗(X)defined by the correspondence a⊗ b⊗β 7→ 〈b, β〉a, using the Kronecker pairing 〈 , 〉.As H∗(X × Y ) ≈ H∗(X)⊗H∗(Y ) by the Kunneth theorem (we assume that Y isof finite cohomology type), this gives the bilinear map (58.4). The slant product ischaracterized by the equation

(a× b)/β = 〈b, β〉 a


for all a ∈ H∗(X ×X) and β ∈ H∗(Y ). It is also a morphism of H∗(X)-module,i.e.

(58.5) [(u× 1) a]/β = u (a/β)

for all u ∈ H∗(X), a ∈ H∗(X ×X) and β ∈ H∗(Y ).

Proof of Wu’s formula. Consider Q as the diagonal submanifold of M =Q×Q, with normal bundle ν = ν(Q,M). By Lemma 33.21, TQ is isomorphic to ν.

A Riemannian metric provides a smooth bundle pair (D(ν), S(ν)) with fiber(Dr, Sr−1) and there is a diffeomorphism from D(ν) to a closed tubular neighbour-hood W of Q in M . One has the following diagram

Q∆ //

i

AAA

AAAA

M==j

||||

|||

Wπ

YY

where π is the bundle projection and the other maps are inclusions. By excision,

H∗(M,M −Q) oo j∗

≈H∗(W,BdW ) ≈ H∗(D(ν), S(ν)) .

Hence, the Thom class of ν may be seen as an element U ∈ Hq(W,BdW ) satisfyingU = j∗(U ′) for a unique U ′ ∈ Hq(M,M − Q). Let U ′′ ∈ Hq(M) be the image ofU ′ under the restriction homomorphisms Hq(M,M −Q)→ Hq(M).

By definition of the Stiefel-Whitney class w = w(TQ) = w(ν), one has

π∗(w) U = SqU .

One has ∆∗(1×w) = 1 w = w, whence j∗(1×w) = π∗(w). Hence, the equation(1× w) U ′ = SqU ′ holds true in H∗(M,M −Q) which, in H∗(M), implies

(58.6) (1× w) U ′′ = SqU ′′ .

Without loss of generality, we may assume thatQ is connected. LetA = a1, a2, . . . and B = b1, b2, . . . of H∗(Q) be an additive bases of H∗(Q) which are Poincaredual, i.e. 〈ai bj , [Q]〉 = δij . We suppose that a0 = 1. By Lemma 33.5 and (33.2),one has

U ′′ =∑

i

ai × bi

and therefore

U ′′/[Q] = (1× b0)/[Q] = 1 .

Applying this together with Equations (58.6) and (58.5) gives

(58.7) SqU ′′/[Q] = [(1× w) U ′′]/[Q] = w (U ′′/[Q]) = w .

We now express the Wu class v = v(M) in the A-basis: v =∑

i λiai. Then,〈v bj , [Q]〉 = λj , which implies that

(58.8) v =∑

i

〈v bi, [Q]〉 ai =∑

i

〈Sq bi, [Q]〉ai, .


Hence, using (58.7), we get

Sq v =∑i〈Sq bi, [Q]〉Sq ai

=∑i

(Sq ai × Sq bi

)/[Q]

= SqU ′′/[Q]

= w .

The remaining of this subsection is devoted to general corollaries of Wu’s for-mula. The first one says that the Stiefel-Whitney class w(TQ) depends only on thehomology type of Q.

Corollary 58.9. Let f : Q′ → Q be continuous map between smooth closedmanifolds Q and Q′ of the same dimension. Suppose that H∗f : H∗(Q)→ H∗(Q′)is surjective. Then H∗f(w(TQ)) = w(TQ′).

Proof. By Kronecker duality, H∗f is injective and thus π0f is injective. Theconnected components of Q out of the image of f play no role, so one may assumethat π0f is a bijection. This implies that H∗f([Q′]) = [Q].

Let v′ = H∗f(v(Q)). For b ∈ H∗(Q), one has

〈v(Q) b, [Q]〉 = 〈v(Q) b,H∗f [Q′]〉= 〈H∗f(v(Q) b), [Q′]〉= 〈v′ H∗f(b), [Q′]〉 .

On the other hand

〈v(Q) b, [Q]〉 = 〈Sq b,H∗f [Q′]〉= 〈Sq(H∗f(b)), [Q′]〉= 〈v(Q′) H∗f(b), [Q′]〉 .

Therefore, the equality

〈v′ H∗f(b), [Q′]〉 = 〈v(Q′) H∗f(b), [Q′]〉is valid for all b ∈ H∗(Q). As H∗f is surjective, Theorem 32.18, this implies thatv′ = V (Q′), so H∗f(v(Q)) = v(Q′). By Wu’s formula,

w(TQ′) = Sq v(Q′) = SqH∗f(v(Q)) = H∗f(Sq v(Q)) = H∗f(w(TQ))) .

Corollary 58.10. Let f : Q → Q be continuous map of degree one. ThenH∗f(w(TQ)) = w(TQ′).

Proof. By Proposition 31.8, H∗f is surjective and then H∗f is injective byKronecker duality. As H∗(Q) is a finite dimensional vector space, this implies thatH∗f (and H∗f) are bijective. The results then follows from Corollary 58.9.

58.2. Orientability and spin structures. A smooth manifold is orientableif its tangent bundle is orientable. The following corollary generalizes Proposi-tion 23.5.

Corollary 58.11. Let Q be a smooth closed n-dimensional manifold. ThenQ is orientable if and only if Sq1 : Hn−1(Q)→ Hn(Q) vanishes.


Proof. By Proposition 54.10, Q is orientable if and only if w1(TQ) = 0 which,by Wu’s formula, is equivalent to v1(Q) = 0. By the definition of v1(Q), its vanish-ing is equivalent to Sq1 : Hn−1(Q)→ Hn(Q) being zero.

The same argument, using Proposition 54.13, implies the following result.

Corollary 58.12. Let Q be a smooth closed n-dimensional manifold which isorientable. Then TQ admits a spin structure if and only if Sq2 : Hn−2(Q)→ Hn(Q)vanishes.

Example 58.13. A closed manifold M such that H∗(M) is GrA-isomorphicto H∗(RPn) is orientable if and only if n is odd. Indeed, let 0 6= a ∈ H1(M) = Z2.By the Cartan formula, Sq1(an−1) = 0 if and only if n is odd. As Hn−1(M) isgenerated by an−1, the assertion follows from Corollary 58.11. A similar argument,using Corollary 58.12, proves that a closed manifold M such that H∗(M) is GrA-isomorphic to H∗(CPn) admits a spin structure if and only if n is odd.

In the particular case n = 4, Corollary 58.12 gives the following result.

Corollary 58.14. Let Q be a smooth connected closed 4-dimensional manifoldwhich is orientable. Then,

(1) a a = w2(TQ) a for all a ∈ H2(Q).(2) TQ admits a spin structure if and only if the cup-square map H2(Q) →

H4(Q) vanishes.(3) w4(TQ) = w2(TQ) w2(TQ).

Point (b) is the analogue of Proposition 23.5 for surfaces. In particular, TCP 2

does not admit a spin structure.

Proof. If w1(TQ) = 0, then w2(TQ) = v2(Q) by Wu’s formula. Hence,a a = Sq2(a) = v2(Q) a = w2(TQ) a for all a ∈ H2(Q), which proves (a).Point (b) thus follows from Corollary 58.12. For Point (c), we use Wu’s formulaagain:

w4(TQ) = Sq2(v2(Q)) = Sq2(w2(TQ)) = w2(TQ) w2(TQ) .

Corollary 58.15. Let Q be a smooth closed manifold of dimension n ≤ 7. IfTQ admits a spin structure, then w(TQ) = 1.

Proof. The proposition is obvious for n ≤ 2. Otherwise, by Proposition 54.12,the existence of a spin structure implies that the restriction of TQ over the 3-skeleton of a triangulation of Q is trivial. Hence w3(TM) also vanishes which, byWu’s formula, implies that vi(M) vanishes for i ≤ 3. As n ≤ 7, this implies thatv(Q) = 1 and thus w(TQ) = Sq v(Q) = 1.

An interesting example is given by the projective spaces.

Proposition 58.16. Let 0 6= a ∈ H1(RPn). The Stiefel-Whitney class of thetangent space of RPn is

w(TRPn) = (1 + a)n+1

and the Wu class of RPn is

v(RPn) =

[n/2]∑

i=0

(n−ii

)ai .


Here below a few examples.

n v(RPn) w(TRPn)

2 1 + a 1 + a+ a2

3 1 14 1 + a+ a2 1 + a+ a5

5 1 + a2 1 + a2 + a4

6 1 + a+ a3 1 + a+ a2 + a3 + a4 + a5 + a6

7 1 1

Remark 58.17. The formulae of Proposition 58.16 imply the following.

(1) RPn is orientable if and only if n is odd (this is not a surprise!). Moregenerally, w2i+1(TRP 2k+1) = 0.

(2) TRPn admits a spin structure if and only if n ≡ 3 mod 4. In this case,there are two spin structures, since H1(RPn) = Z2. For a discussionabout the two structures for RP 3 ≈ SO(3), see [127, Example 2.5, p. 87].

(3) w(TRPn) = 1 if and only if n = 2k − 1. But TRPn is trivial if and onlyif n = 1, 3, 7 by Adams Theorem [1, p. 21].

Proof of Proposition 58.16. The two formulae will be proved separately.Checking Wu’s formula is left as an exercise. By (46.4),

vi(RPn) an−i = Sqian−i =

(n−ii

)an =

(n−ii

)ai an−i .

which proves that vi(RPn) =(n−ii

)ai. This proves the formula for the Wu class.

As for the Stiefel-Whitney class, the idea is the following. Recall that RPn =Gr(1; Rn+1) = Fl(1, n). Write γ = ξ1 and γ⊥ = ξ2 for the tautological bundles.Then,

(58.18) TRPn ≈ hom(γ, γ⊥)

(see [150, Lemma 4.4] for a proof). But γ ⊕ γ⊥ is the trivial bundle ηn+1 of rankn+ 1. Adding to both side of (58.18) the bundle hom(γ, γ) ≈ η1, we get

TRPn ⊕ η1 ≈ hom(γ, ηn+1) .

The latter is the Whitney sum of (n + 1)-copies of γ∗ = hom(γ, η1). But γ∗ ≈ γ,using an Euclidean metric on γ. For details (see [150, Theorem 4.5]). Hence, theformula for w(TRPn) follows from (54.4).

Remark 58.19. The argument of the proof of Proposition 58.16 essentiallyworks for computing the Chern class of the tangent bundle to TCPn (which is acomplex vector bundle). The slight difference is that γ∗ = hom(γ, η1) is not, ascomplex vector bundle, isomorphic to γ but to the conjugate bundle γ (the complexstructure on each fiber is the conjugate of that of γ (see [150, pp. 169–170]). But thisdoes not alter our Chern classes which are defined mod 2: ci(TCPn) = w2i(TCPn).Thus

c(TCPn) = (1 + a)n+1 and v(CPn) =

[n/2]∑

i=0

(n−ii

)ai

where 0 6= a ∈ H2(CPn). The first formula holds as well for the integral Chern classwith a suitable choice of a generator of H2(CPn; Z) (see [150, Theorem 14.10]).


58.3. Applications to 3-manifolds. Wu’s formula has two important con-sequences for closed 3-dimensional manifolds. The first one is the following.

Proposition 58.20. Let Q be a smooth closed 3-dimensional manifold which isorientable. Then TQ is a trivial vector bundle (in other words: Q is parallelizable).

Proof. For any smooth closed manifold, one has w1(TQ) = v1(Q) by Wu’sformula. Thus, v1(Q) = 0 if Q is orientable. In dimension 3, this implies thatv(Q) = 1 and, by Wu’s formula again, w(TQ) = 1. The result then follows fromProposition 54.12.

The second application is Postnikov’s characterization of the cohomology ringof a closed connected 3-dimensional manifold [161]. Let M be such a manifold.Consider the symmetric trilinear form πM : H1(M)×H1(M)×H1(M)→ Z2 definedby

πM(a, b, c) = 〈a b c, [M ]〉 .The first observation is that πM determines the ring structure of H∗(M).

Lemma 58.21. Let M and M be two closed connected 3-dimensional manifolds.Suppose that there exists an isomorphism h1 : H1(M)→ H1(M) such that

πM(h1(a), h1(b), h1(c)) = πM (a, b, c) .

Then, h1 extends to a GrA-isomorphism h∗ : H∗(M)→ H∗(M).

Proof. LetA = a1, . . . , am be a Z2-basis ofH1(M). The set A = a1, . . . , amwhere ai = h1(ai) is then a Z2-basis of H1(M). Let B = b1, . . . , bm andB = b1, . . . , bm be the bases of H2(M) and H2(M) which are Poincare dualto A and A, i.e. the equations

(58.22) 〈ai bj , [M ]〉 = δij and 〈ai bj , [M ]〉 = δij

are satisfied for all i, j. Let h2 : H2(M) → H2(M) be the isomorphism such thath2(bi) = bi and let h3 be the unique isomorphism from H3(M) to H3(M). Thisproduces a GrV-isomorphism h∗ : H∗(M)→ H∗(M). To prove that h∗ is a GrA-morphism, write ai aj =

∑ml=1 λ

lijbl and, using (58.22), note that

πM(ai, aj , ak) = 〈ai aj ak, [M ]〉 =m∑

l=1

λlij〈bl ak, [M ]〉 = λkij .

Therefore,

〈h2(ai aj) ak, [M ]〉 =∑m

l=1 πM (ai, aj , al)〈h2(bl) ak, [M ]〉=

∑ml=1 πM (ai, aj , al)〈bl ak, [M ]〉

= πM (ai, aj , ak)

and

〈h1(ai) h1(aj) ak, [M ]〉 = 〈h1(ai) h1(aj) h1(ak), [M ]〉= πM (h1(ai), h

1(aj), h1(ak)) .

Since πM (h1(ai), h1(aj), h

1(ak)) = πM(ai, aj, ak), this proves that h2(ai aj) =h1(ai) h1(aj). On the other hand, h3 formally satisfies 〈h3(u), [M ]〉 = 〈u, [M ]〉.Hence, the equality h1(ai) h2(bj) = h3(ai bj) follows from (58.22). We havethus established that h∗ is a GrA-morphism


The trilinear form πM is linked to the Wu class v(M) ∈ H1(M).

Lemma 58.23. Let M be a closed connected 3-dimensional manifold. Then, theWu class v = v1(M) satisfies

(58.24) πM(v, b, c) = πM(b, b, c) + πM(b, c, c)

for all b, c ∈ H1(M).

Proof. This comes from that

v1(M) (b c) = Sq1(b c) = Sq1(b) c + b Sq1(c) = b b c + b c c .

The following “realizability result” is due to Postnikov [161].

Proposition 58.25. Let (V, π) a symmetric trilinear form, with V a finitedimensional Z2-vector space. Let v ∈ V satisfying (58.24). Then, there exists aclosed connected 3-manifold M with an isometry (H1(M), πM) ≈ (V, π), sendingv1(M) onto v.

Proof (indications). The full proof may may be found in [161]. When Mis orientable, the form πM is alternate since the left hand side of (58.24) vanishes

(v = w1(TM) = 0). Hence, πM ∈∧3

H1(M). An alternate form π ∈ ∧3V may

be lifted to π ∈ ∧3V , where V is a free abelian group’ with V ⊗ Z2 ≈ V . In

[183], D. Sullivan constructed a closed connected orientable 3-manifold M with

(H1(M ; Z), πM ) ≈ (V , π), which thus proves Proposition 58.25 in the orientablecase.

58.4. The universal class for double points. The material of this sectionis essentially a rewriting in our language of results of A. Haefliger [76]. Let M be aclosed manifold of dimensionm. LetG = 1, τ acting onM×M by τ(x, y) = (y, x),with fixed point set (M ×M)G = ∆M , the diagonal submanifold of M ×M . The

diagonal inclusion δ : M → M ×M induces a diffeomorphism δ : M≈−→ ∆M . For

N > 1, SN ×G (M ×M) is a closed manifold containing RPN × ∆M as a closedsubmanifold of codimension m. Let PDG,N(M) = PD(RPN ×∆M ) ∈ Hm(SN ×G(M ×M)), the Poincare dual of RPN ×∆M (see § 33.1). If N is big enough,

HmG (M ×M) ≈ Hm(S∞ ×G (M ×M))→ Hm(SN ×G (M ×M))

is an isomorphism. Therefore, there is a unique class PDG(M) ∈ HmG (M × M)

whose image in Hm(SN ×G (M ×M)) is equal to PDG,N (∆M ).The class PDG(M) is called the universal class of double points for continuous

maps into M , terminology justified by Lemma 58.27 below. For a space X , denoteby j : X0 → (X ×X) the inclusion of X0 = (X ×X) −∆X into (X ×X). As Gacts freely on X0, the quotient space X∗ = X0/G, called the reduced symmetricsquare of X , has the homotopy type of X0

G by Lemma 39.5.

The diffeomorphism δ : M≈−→ ∆M is covered by a bundle isomorphism δ : TM

≈−→ν(M ×M,∆M ) (see Lemma 33.21), which intertwines τ with the antipodal invo-

lution on TM . Via δ, the sphere bundle T 1M becomes G-diffeomorphic with theboundary BdW of a G-invariant tubular neighbourhood of ∆M in M ×M . Thus,

(58.26) W ∗ ≈ (W −∆M )G ≈ (BdW )G ≈ (T 1M)G ≈ (T 1M)/G .


As j is G-equivariant, it induces H∗Gj : H∗G(X×X)→ H∗G(X0). Let f : Q→Mbe a continuous map. Consider the homomorphism

Ψ: HmG (M ×M)

H∗G(f×f)−−−−−−→ HmG (Q×Q)

H∗Gj−−−→ HmG (Q0)

≈−→ Hm(Q∗) .

We denote by Ψ : HmG (M×M)→ Hm(T 1Q/G) the post-composition of Ψ with the

homomorphism Hm(Q∗)→ Hm(W ∗) ≈ Hm(T 1N/G). Define

OfG = Ψ(PDG(M)) ∈ HmG (Q∗) and OfG = Ψ(PDG(M)) ∈ Hm

G ((T 1Q)/G) .

Lemma 58.27. Let f : Q →M be a continuous map between closed manifolds.Then

(1) if f is homotopic to an embedding, then OfG = 0.

(2) if f is homotopic to an immersion, then OfG = 0.

Proof. The classes OfG and OfG depend only on the homotopy class of f . If fis injective, then (f × f)(Q0) ⊂M0 and thus (f × f)G(Q0

G) ⊂M0G = (M ×M)G −

(∆M )G. By Lemma 33.5, PDG(M) has image zero in HmG (M0), which proves (1)

(this does not use that Q is a manifold). Suppose that f is an immersion, so f islocally injective. As Q is compact, there is a G-invariant tubular neighbourhood Wof ∆Q in Q×Q such that (f × f)(W 0) ⊂M0. We deduce (2) as above for (1).

In order to get applications of Lemma 58.27, we now express PDG,N(M) withinthe description of Hm

G (M ×M) given by Proposition 47.12, which can be rephrasedas follows. There is a GrA[u]-isomorphism from H∗G(M ×M) to (Z2[u]⊗D) ⊕ N ,where D is the Z2-vector space generated by x × x | x ∈ Hk(M), k ≥ 0 so thatρ : H∗G(M ×M) → H∗(M ×M)G sends the the elements of D ⊕ N (elements ofu-degree 0) isomorphically to H∗(M ×M)G. The Z2-vector space N is generatedby x× y + y × x | x, y ∈ H∗(M) and coincides with the ideal ann (u).

Proposition 58.28. Using the isomorphism HmG (M ×M) ≈ (Z2[u]⊗D) ⊕ N ,

we have

(1) PDG(M) ≡∑[m/2]k=0 um−2k(vk(M)× vk(M)) mod N , where vk(M) is the

k-th Wu class of M .

(2) ρ(PDG(M)) = PD(∆M ), the Poincare dual of ∆M in M ×M .

Example 58.29. Let M = RP 2. One has H∗(M) = Z2[a]/(a3) and v(M) =

1 + a. Then, PDG(M) ≡ u2 + a × a mod N and, according to Equation (33.2),ρ(PDG(M)) = PD(∆M ) = 1× a2 + a× a+ a2 × 1. Therefore,

PDG(RP 2) = u2 + a× a+N(1× a2) .

Proof. It is enough to prove (1) for PDG,N (M) with N big enough. Leti : Q→ P be the inclusion of a closed manifold Q into a compact manifold P . Forx ∈ Hj(P ), one has

(58.30) 〈x PD(Q), [P ]〉 = 〈x,PD(Q) [P ]〉 = 〈x,H∗i([Q])〉 = 〈H∗i(x), [Q]〉 .


We shall apply (58.30) to Q = RPN × M and P = SN ×G (M ×M) and x =uN−m+2i(a× a) ∈ HN+m(P ), where a ∈ Hm−i(M). One has(58.31)〈H∗i(x), [Q]〉 = 〈uN−m+2iH∗i(a× a), [Q]〉

= 〈uN−m+2i∑m−i

j=0 um−i−j Sqj(a), [Q]〉 by definition of Sq(a)

= 〈uNSqi(a), [Q]〉 only non-zero term

= 〈Sqi(a), [M ]〉 .

For y, z ∈ Hj(M), we have (y × y) + (z × z) ≡ (y + z) × (y + z) mod N anduN = 0. Therefore, for k > 0, uk PDG,N(Q) admits an expression of the form

uk PDG,N(Q) = uk∑[m/2]

j=0 um−2j(yj × yj) with yj ∈ Hj(M). Hence,

(58.32)

〈x PDG,N(Q), [P ]〉 = 〈uN−m+2i(a× a) ∑[m/2]j=0 um−2j(yj × yj), [P ]〉

= 〈uN (a× a) (yi × yi), [P ]〉= 〈uN(a yi)× (a yi), [P ]〉= 〈a yi, [M ]〉

Using (58.30), Formulae (58.31) and (58.32) imply that yi = vi(M) for all i =0 . . . , [m/2]. This proves (1).

For Point (2) we must prove that ρN(PDG,N) = PD(M) where ρN is inducedby the fiber inclusion M ×M → SN × (M ×M)→ P . But this map is transversalto RPN ×∆M . Point (2) thus comes from Proposition 33.10.

Proposition 58.28 enables us to compute the image of PDG(M) under the ho-momorphism

H∗G(M ×M)r // H∗G((M ×M)G)

≈ // H∗(M)[u]ev1 // H∗(M)

Corollary 58.33. ev1r(PDG(M)) = w(TM), the total Stiefel-Whitneyclass of the tangent bundle TM .

Proof.

ev1 r(PDG(M)) =∑[m/2]

k=0 ev1 r(vk(M)× vk(M)

)by Proposition 58.28

=∑[m/2]

k=0 Sq(vk(M)) by (47.7)

= Sq(v(M))

= w(TM) by the Wu formula.

Remark 58.34. The formula of Corollary 58.33 may be proven directly inthe following way. By Lemma 33.7 and the considerations before (58.26), one hasr(PDG(M)) = eG(TM), where G acts on TM by the antipodal action on eachfiber. This equivariant Euler class satisfies ev1(eG(TM)) = w(TM) (see (63.16) in§ 63), which proves Corollary 58.33. Moreover, using the proof of Corollary 58.33,with the last line removed, gives a new proof of the Wu formula.


Lemma 58.35. Let Q be a closed manifold of dimension p. There is a commu-tative diagram

0 // H∗−pG (∆Q)GysG //

H∗G δ≈

H∗G(Q×Q) //

r

H∗(Q∗)

// 0

0 // H∗−pG (Q)Φ // H∗G(Q) // H∗((T 1Q)/G) // 0

where the rows are exact sequences. Here, Φ(a) = a eG(TQ), where G acts onTQ via the antipodal involution.

Proof. The diagram comes from Proposition 33.13 applied, for N big, to thepair (SN ×G (Q × Q),RPN ×∆Q). The long diagram of Proposition 33.13 splitsinto the above diagram because Φ is injective (see Proposition 43.24). The bottomline is the Gysin sequence for the sphere bundle (T 1Q)G → QG.

The above results permit us to express an obstruction to embedding in termsof the dual Stiefel-Whitney classes. Let f : Q → M be a smooth map. Definewf = w(TQ) H∗f(w(TM)) ∈ H∗(Q) where w(TQ) is the dual Stiefel-Whitneyclasses of TQ (see p. 320).

Proposition 58.36. Let f : Qq →Mm be a smooth map between closed mani-folds.

(1) If f is homotopic to a smooth immersion, then wfk = 0 for k > m− q.(2) If f is homotopic to a smooth embedding, then

(wfm−q × 1) PD(∆Q) = H∗(f × f)(PD(∆M ))

in Hm(Q×Q).

Proof. We shall argue with the help of the diagram

(58.37)

Hm−pG (∆Q)

GysG

H∗Gδ

≈// Hm−p

G (Q)ev1 //

−eG(TQ)

H∗(Q)

−w(TQ)

HmG (Q×Q)

r // HmG (Q)

ev1 // H∗(Q)

The left square comes from Lemma 58.35 and is thus commutative. So is the rightsquare by Remark 58.34.

If f is homotopic to a smooth immersion, then OfG = 0 by Lemma 58.27. By

Lemma 58.35, this implies that there exists b ∈ Hm−qG (Q) such that the equations

ev1rH∗G(f × f)(PDG(M)) = ev1(b eG(TQ)) = ev1(b) w(TQ)

hold true in H∗(Q). But

ev1 rH∗G(f × f)(PDG(M)) = H∗f ev1r(PDG(M))

= H∗f(w(TM)) by Corollary 58.33.

Hence

(58.38) ev1(b) w(TQ) = H∗f(w(TM)) .


Since w(TQ) w(TQ) = 1, multiplying both sides of (58.38) by w(TQ) gives

(58.39) ev1(b) = H∗f(w(TM)) w(TQ) = wf .

As b is of degree m− q, Equation (58.39) implies that wfk = 0 for k > m − q. Wehave thus proven (1). Also, (58.39) is equivalent to

(58.40) b =

m−q∑

i=0

wfi um−q−i .

To prove Point (2), we use the diagram

(58.41)

Hm−pG (Q) oo H

∗Gδ

≈

ρ

Hm−pG (∆Q)

GysG //

ρ

HmG (Q×Q)

ρ

Hm−p(Q) oo H

∗ δ

≈Hm−p(∆Q)

Gys // Hm(Q×Q)

The left square is obviously commutative and the right square is so by Propo-sition 33.14, since the fibre inclusion Q × Q → SN ×G (Q × Q) is transversal

to RPN × ∆Q. If f is homotopic to a smooth embedding, then OfG = 0 byLemma 58.27. Also, (1) holds and, by the above and Diagram (58.37), one has

(58.42) ρH∗G(f × f)(PDG(M)) = ρGysG (H∗Gδ)−1(b)

By Point (2) of Proposition 58.28, one has(58.43)

ρH∗G(f × f)(PDG(M)) = H∗(f × f)ρ(PDG(M)) = H∗(f × f)(PD(∆M )) .

Let ı : ∆Q → Q×Q be the inclusion and pr2 : Q× Q → Q be the projection ontothe first factor. Then pr1ı δ = idQ. Hence, any a ∈ H∗(Q) satisfies

(58.44) a = H∗δH∗ıH∗pr1(a) = H∗δH∗ı(a× 1) .

Hence,(58.45)ρGysG (H

∗Gδ)−1(b) = Gys(H∗δ)−1

ρ (b) by commutativity of (58.41)

= Gys(H∗δ)−1(wfm−q) by (58.40)

= GysH∗ı(wfm−q × 1) by (58.44)

= (wfm−q × 1) PD(∆Q) by Lemma 33.11

Combining (58.42), (58.43) and (58.45) provides the proof of Point (2).

Corollary 58.46. Let Q be a closed manifold of dimension q.

(1) If Q may be immersed in Rm, then wk(TQ) = 0 for k > m− q.(2) If Q may be embedded in Rm, then wm−q(TQ) = 0.

Point (1) was already proven in Proposition 55.37.

Proof. Let f0 : Q → Rm be a smooth map. Composing with the inclusionRm → Sm = Rm ∪ ∞ gives a smooth map f : Q→ Sm, homotopic to a constantmap. Then wf = w(TQ) and H∗(f × f) = 0. Corollary 58.46 thus follows fromProposition 58.36.


Examples 58.47. 1. LetQ be a closed non-orientable surface. Then, w1(TQ) =w1(TQ) 6= 0. Therefore, Q cannot be embedded in R3. Note that M can be em-bedded in R4 by Whitney’s theorem and immersed in R3 using Boy’s surface.

2. Let Q be a closed 4-dimensional orientable manifold which is not spin (forinstance CP 2). Then, w2(TQ) = w2(TQ) 6= 0. Therefore, Q cannot be embed-ded in R6. Note that CP 2 embeds in R7. Indeed, CP 2 is diffeomorphic to thespace FlC(1, 2) of Hermitian (3 × 3)-matrices with characteristic polynomial equalto x2(x − 1) (see (3) on p. 333). The vector space of Hermitian (3 × 3)-matriceswith trace 1 is isomorphic to R8 and each radius intersects FlC(1, 2) at most once.This gives an embedding of CP 2 in S7 = R7 ∪ ∞ and thus in R7.

3. The quaternionic projective plane Q = HP 2 has Wu class v4(Q) 6= 0. Hence,w4(TQ) = w4(TQ) 6= 0. Therefore, HP 2 cannot be embedded in R12. In thesame way, the octonionic projective plane OP 2 of dimension 16 cannot be embed-ded in R24. Improving the technique explained in the previous example producesembeddings HP 2 → R13 and OP 2 → R25 (see [132, § 3]).

59. Thom’s theorems

This section is a survey of some results of Thom’s important paper [187], which,amongst other things, was the foundation of cobordism theory. Some proofs arealmost complete and others are just sketched.

59.1. Representing homology classes by manifolds.

Theorem 59.1. Let X be a topological space and α ∈ Hk(X). Then, thereexists a closed smooth manifold M of dimension k and a continuous map f : M → Xsuch that H∗f([M ]) = α.

This theorem is due to Thom [187, Theorem III.2]. The result is wrong forintegral cohomology (see [187, Theorem III.9]). This section is devoted to the proofof Theorem 59.1. We start with some preliminaries.

Let ξ be a vector bundle of rank r over a paracompact space Y . Let (D(ξ), S(ξ))be the pair of the disk and sphere bundles associated to ξ via an Euclidean structure.The Thom space T (ξ) of ξ is defined by

T (ξ) = D(ξ)/S(ξ) .

The homeomorphism class of T (ξ) does not depend on the choice of the Euclideanstructure (see p. 157). Also, S(ξ) has a collar neighborhood in D(ξ). Hence,by Lemma 12.64, the pair (D(ξ), S(ξ)) is well cofibrant. Using Proposition 12.71together with the Thom isomorphism theorem provides the following isomorphisms

Hk(B)≈−→ Hk+r(D(ξ), S(ξ))

≈−→ Hk+r(T (ξ)) .

We now specialize to ξ = ζr,N , the tautological vector bundle over the Grass-mannian Gr(r; RN ) (N ≤ ∞). (The Thom space T (ζr,∞) is also called MO(r)in the literature; it is the r-th space of the Thom spectrum MO). We get someinformation on the cohomology ring H∗(T (ζr,N )) using the Gysin exact sequenceof the sphere bundle q : S(ζr,N )→ Gr(r; RN ), whose Euler class is wr.(59.2)

Hi−r(Gr(r; RN ))wr−−−→ Hi(Gr(r;RN ))

H∗q−−−→ Hi(S(ζr,N ))→ Hi−r+1(Gr(r; RN ))

wr−−−→ · · ·

59. THOM’S THEOREMS 351

together with the exact Sequence of Corollary 12.74 for the pair (D(ζr,N ), S(ζr,N ))(using that D(ζr,N ∼= Gr(r; RN ))

(59.3) Hi(T (ζr,N))→ Hi(Gr(r; RN ))H∗q−−−→ Hi(S(ζr,N ))→ Hi+1(T (ζr,N)) .

For N = ∞, − wr is injective by Theorem 55.17. Together with Lemma 55.13,Sequences (59.2) and (59.3) gives the following lemma.

Lemma 59.4. (1) The GrA-momorphism

H∗p : H∗(T (ζr,∞))→ H∗(Gr(r; R∞))

is injective and its image in positive degrees is the ideal generated by wr.In particular, H∗p(U) = wr.

(2) The GrA-momorphism Hi(T (ζr,∞))→ Hi(T (ζr,N )) generated by the in-clusion is bijective for N ≥ N − r − 1.

(Note that the equation H∗p(U) = wr is coherent with (28.51)). Consider thefollowing diagram

(59.5)

Gr(r; R∞) oo g

fwr

&&MMMMMMMMMM

p

(RP∞)r

fa×···×a

T (ζr,∞)

fU // Kr

in which the following notations are used. If X is a CW-complex and y ∈ Hr(X),then fy : X → Kr = K(Z2, r) denotes a map representing y. Then wr ∈ Hr(Gr(r; R∞))is the r-th Stiefel-Whitney class, U ∈ Hr(Tζr,∞) is the Thom class and 0 6= a ∈H1(RP∞). The map g classifies the r-th product of the tautological line bundle overRP∞ and p : Gr(r; R∞) ∼= D(ζr,∞)→ T (ζr,∞) is the quotient map. Diagram (59.5)is homotopy commutative. The commutativity of the lower triangle comes from thealready mentioned equation H∗p(U) = e(ζr,∞) = wr . For the the upper triangle(see e.g. the proof of Theorem 55.17).

Applying the cohomology functor to Diagram (59.5) provides the followingcommutative diagram.

(59.6)

H∗(Gr(r; R∞)) //H∗g //

hhH∗fwr

QQQQQQQQQQQOOH∗pOO

H∗((RP∞)r)OOH∗fa×···×a

H∗(T (ζr,∞)) oo H∗fU

H∗(Kr)

By Corollary 49.15, H∗fa×···×a : Hi(Kr)→ Hi((RP∞)r) is injective for i ≤ 2r.Therefore, H∗fU : Hi(Kr)→ Hi(T (ζr,∞)) is also injective for i ≤ 2r. As indicatedin Diagram (59.6), H∗p and H∗g are injective. The injectivity of H∗p was provenin Lemma 59.4 and that of H∗g was established in the proof of Theorem 55.17 orin Theorem 56.1 and its proof.

The map H∗fU is of course not surjective. By Lemma 59.4, dimHr+j(Tζr,∞)is the number of partitions of j while, for i < r, dimHr+j(Kr) is equal, byLemma 49.22, to the number of partitions of j into integers of the form 2i− 1. LetD(j) be the number of partitions of j into integers with none of them of the form


2i − 1. For each ω ∈ D(j) with j ≤ r, Thom constructs a class Xω ∈ Hr+j(Tζr,∞)represented by a map fω : Tζr,∞ → Kj . Together with fU , this gives a map

(59.7) F : Tζr,∞ → Y = Kr ×r∏

j=1

K♯D(j)j = Kr ×Kr+2 × · · · .

Thom proves that H∗F is an isomorphism up to degree 2r. Some analogous resultis proved for the cohomology with coefficients in a field of characteristic 6= 2 andboth Tζr,∞ and Y are simply connected. This enables Thom to prove the followingresult (see [187, pp.35 42]).

Lemma 59.8. If N ∈ N ∪ ∞ is big enough, there exists a map ψ from the2r-skeleton of Y to T (ζr,N ) such that the restrictions of ψF and F ψ to the(2r − 1)-skeleta of Y and Tζr,N respectively are homotopic to the identity.

As a corollary, we get the following result (see [187, Corollary II.12]).

Corollary 59.9. If N ∈ N ∪ ∞ is big enough, there exists a map ψ fromthe 2r-skeleton of Kr to Tζr,N such that H∗ψ(U) = ı, the fundamental class of Kr.

We are now ready to prove Theorem 59.1.

Proof of Theorem 59.1. By Theorem 18.11, there is a simplicial complexKX and a map φ : |KX | → X such that H∗φ is an isomorphism. By § 17, Thehomology of X is isomorphic to the simplicial homology of KX . By the definitionof the simplicial homology, there is a finite simplicial subcomplex K of KX , ofdimension k, such that α is in the image of Hk(|K|) → Hk(X). Now, there isa PL-embedding ψ : |K| → R2k+1 (see e.g. [175, Theorem 3.3.9]) and the theoryof smooth regular neighborhoods [92] produces a smooth compact codimension 0submanifold W of R2k+1 which is a regular neighborhood of ψ(|K|) for some C1-triangulation of R2k+1. In particular, W retracts by deformation on ψ(K). Bygeneral position, ψ is isotopic to an embedding ψ′ such that ψ′(|K|) avoids someregular neighborhood W ′ of ψ(|K|) contained in the interior of W . The closure ofW ′−W is homeomorphic toM×[0, 1], whereM = BdW (see [102, Corollary 2.16.2,p. 74]). We can thus construct a map ψ′′ : |K| →M such that the composite map

|K| ψ′′

−−→M →W → |K|is isotopic to id|K|. Hence α is in the image of H∗(M) → Hk(X). Therefore, it isenough to prove Theorem 59.1 when X is a closed manifold of dimension 2k.

Let a ∈ Hk(X) be the cohomology class which is Poincare dual to α. As asmooth manifold, X admits a C1-triangulation by a simplicial complex of dimension2k. There exists thus a continuous map fa : X → Kk representing the class a and,by cellular approximation, one may suppose that fa(X) is contained in the 2k-

skeleton of K. Let f = ψfa : X → T = T (ζk,N ), where ψ is a map as provided

by Corollary 59.9 for N large enough. Then fU f is homotopic to fa. Note thatT is a smooth manifold except at the point [S(ζk,N )]. Using standard techniques

of differential topology (see [93, § 2.2 and 3.2]), f is homotopic to f1 such that f1is a smooth map around f−1

1 (Gr(r; RN ))) which is transverse to Gr(r; RN ). ThenM = f−1

1 (Gr(r; RN ))) is a closed submanifold of codimension k in X with normalbundle ν = f∗1 ζk,N .


Let a′ ∈ Hk(X) be the Poincare dual of the homology class generated by[M ]. As in § 33.1, we consider the Thom class U(X,M) of ν as an element ofHk(X,X −M) and, if j : (X, ∅)→ (X,X −M) denotes the pair inclusion, one has

a′ = H∗j(U(X,M)) by Lemma 33.5

= H∗f1(U)

= H∗(ψfa)(U)

= H∗faH∗ψ(U)

= H∗fa(ı)

= a .

As Poincare duality is an isomorphism, this proves that [M ] = α.

Observe that, in the proof of Theorem 59.1, we have established the followingresult, due to Thom [187, Theorem II.1, p. 29].

Proposition 59.10. Let α ∈ Hk(X), where X is a closed smooth manifold ofdimension k + q > 2k. Let a = PD(α) ∈ Hq(X) be the Poincare dual of α. Then,the following statements are equivalent.

(1) There exists a closed submanifold M in X such that [M ] represents α.(2) There exists a continuous map F : X → T (ζq,∞) such that H∗F (U) = a.

59.2. Cobordism and Stiefel-Whitney numbers. Let M be a (smooth,possibly disconnected) manifold of dimension n. For a polynomial P ∈ Z2[X1, . . . , Xn],we set PM = P (w1(TM), . . . , wn(TM)) ∈ H∗(M). If M is closed, the numbermod 2

〈PM , [M ]〉 ∈ Z2

is called the Stiefel-Whitney number of M associated to P . We use the conventionthat 〈a, α〉 = 0 if a ∈ Hr(X) and α ∈ Hs(X) with r 6= s.

Two closed manifolds of the same dimension are called cobordant if their dis-joint union is the boundary of a compact manifold. One fundamental result ofThom [187, Theorema IV.3 and IV.10] is the following theorem, generalizing Corol-lary 32.15.

Theorem 59.11. Two closed manifolds of the same dimension are cobordantif and only if their Stiefel-Whitney numbers coincide.

Example 59.12. Let M be a closed 3-dimensional manifold. Its Wu class isv(M) = 1 + v1(M) = 1 + w1(TM). By Wu’s formula, w(TM) = Sq(v(M)) =1 + w1(TM) + w1(TM)2, so w2(TM) = w1(TM)2. The only possible non-zeroStiefel-Whitney number is then 〈w1(TM)3, [M ]〉. But

w1(TM)3 = w1(TM)2 v1(M) since w1(TM) = v1(M)

= Sq1(w1(TM)2) by definition of v1, since dim M = 3

= 0 by the Cartan formula.

Therefore, M is the boundary of a compact manifold. Note that, if M is orientable,the vanishing of its Stiefel-Whitney numbers follows from Proposition 58.20.

Example 59.13. The complex projective space CP 2 and the manifold RP 2 ×RP 2 have the same Stiefel-Whitney numbers by Proposition 58.16 and Remark 58.17.Therefore, they are cobordant.


Example 59.14. Let M be a closed orientable 4-dimensional manifold. Then,w1(TM) = 0 and w4(TM) = w2(TM)2 (see Corollary 58.14). Its only possiblenon-vanishing Stiefel-Whitney number is thus

〈w4(TM), [M ]〉 = 〈e(TM), [M ]〉 = χ(M) mod 2

(using Corollary 33.20). Therefore, M a boundary of a (possibly non-orientable)compact 5-manifold if and only if its Euler characteristic is even.

Proof of Theorem 59.11. Let M1 and M2 be two closed manifolds of thesame dimension and let M = M1∪M2. For any P ∈ Z2[X1, . . . , Xn], one has

〈PM , [M ]〉 = 〈PM1 , [M1]〉+ 〈PM2 , [M2]〉 .Hence, Theorem 59.11 is equivalent to the following statement: a closed manifoldM bounds if and only if its Stiefel-Whitney numbers vanishes.

Suppose that M = BdW for some compact manifold W . Then TM ⊕ η ≈TW|M where η is the trivial bundle of rank 1 over M . If j : M → W denotes theinclusion, one has

〈PM , [M ]〉 = 〈H∗j(PW ), [M ]〉 = 〈PW , H∗j([M ])〉 = 0 ,

since H∗j([M ]) = 0 (see Equation (32.11) and the end of the proof of Theo-rem 32.10).

For the converse, we shall prove that if a closed manifold M of dimension ndoes not bound, then at least one of its Stiefel-Whitney numbers is not zero. Letus embed M into Rn+r for r large, with normal bundle ν. Let f : M → Gr(r; R∞)be a map such that ν ≈ f∗ζr,∞. The map f induces a map Tf : Tν → Tζr,∞.A closed tubular neighbourhood N of M is diffeomorphic to D(ν). We considerRn+r ⊂ Sn+r. The projection N ≈ D(ν) → T (ν) extends to a continuous mapπ : Sn+r → Tν by sending the complement of N onto the point [S(ν)]. This gives

a map f = Tf π : Sn+r → Tζr,∞ (called the Pontryagin-Thom construction) Byan argument based on transversality, one can prove that, for r large enough, M

bounds if and only if f is homotopic to a constant map [187, Lemma IV.7 and itsproof].

Let us compose f with the map F : Tζr,∞ → Y of (59.7). By Lemma 59.8,

f is not homotopic to a constant map if and only if F f is not homotopic to a

constant map. As Y is a product of Eilenberg-McLane spaces, F f is not homo-

topic to a constant map if and only if H∗(F f) 6= 0. The latter implies thatH∗Tf : Hn+r(Tζr,∞) → Hn+r(Tν) does not vanish. Using the Thom isomor-phisms, this implies that H∗f : Hn(Gr(r; R∞)) → Hn(M) does not vanish. Thisimplies that there is a polynomial P in the Stiefel-Whitney classes of ν such that〈P , [M ]〉 6= 0. These classes wi are the normal Stiefel-Whitney classes of M and, byLemma 55.33, the Stiefel-Whitney classes wj = wj(TM) have polynomial expres-sions in the wi. Therefore, there is a polynomial P in wj such that 〈P, [M ]〉 6= 0,producing a non-zero Stiefel-Whitney number for M .

Corollary 59.15. Let M and M ′ be two closed smooth manifolds of the samedimension. Suppose that there exists a map f : M ′ → M such that H∗f is anisomorphism. Then, M and M ′ are cobordant.

As a consequence of Corollary 59.15, a Z2-homology sphere bounds.


Proof. As H∗f is an isomorphism, π0f is a bijection and then H∗f([M ′]) =[M ]. Let P ∈ Z2[X1, . . . , Xn]. By Corollary 58.9, H∗f(w(TM)) = w(TM ′) andthen H∗f(PM ′) = PM . Therefore,

〈PM ′ , [M ′]〉 = 〈PM ′ , H∗f([M ])〉= 〈H∗f(PM ′), [M ]〉= 〈PM , [M ]〉 .

Hence, M and M ′ have the same Stiefel-Whitney numbers. By Theorem 59.11,they are cobordant.

For closed manifolds of dimension n, “being cobordant” is an equivalence re-lation. The set of equivalence classes (cobordism classes) is denoted by Nn. Thedisjoint union endows Nn with an abelian group structure, actually a Z2-vectorspace structure since 2M = M ∪M is diffeomorphic to the boundary of M × [0, 1].The Cartesian product of manifolds makes N∗ =

⊕n Nn a Z2-algebra, called the

cobordism ring. A development of the results of this section and the previous one en-abled Thom to compute the cobordism ring N∗ [187, § IV]; the results are summedup in the the following theorem.

Theorem 59.16. (1) Nn is isomorphic to lim→k

πn+k(T (ζk,∞)).

(2) dimNn is the number of partitions of n into integers with none of themof the form 2i − 1.

(3) N∗ is GrA-isomorphic to a polynomial algebra Z2[X2, X4, X5, X6, X8, X9, . . . ]with one generator Xk for each integer k not of the form 2i − 1.

A representative for X2k is given by the cobordism class of RP 2k [187, p. 81].Odd dimensional generator of dimension 6= 2i − 1 were first constructed by Dold[39]. For details and proofs (see [187] or [182, Chapter VI]).

For example, N3 = 0, confirming Example 59.12. Another simple consequenceof Theorem 59.16 is the following corollary.

Corollary 59.17. Let M and N be closed manifolds which are not boundaries.Then M ×N is not a boundary.

CHAPTER 9

Miscellaneous applications and developments

This chapter, contains various applications and developments of the techniquesof mod2-(co)homology. Most of them are somehow original.

60. Actions with scattered or discrete fixed point sets

Let X be a finite dimensional G-complex (G = id, τ) with b(X) < ∞. BySmith theory (Proposition 41.9), we know that b(XG) ≤ b(X), which implies that♯(π0(X

G)) ≤ b(X). Inspired by the work of V. Puppe [162], we study in this sectionthe extremal case ♯(π0(X

G)) = b(X) (scattered fixed point set). Analogous resultsfor S1-actions are presented in the end of this section.

Proposition 60.1. Let X be a finite dimensional G-complex with b(X) <∞.

Suppose that ♯(π0(XG)) = b(X). Let a ∈ Hk

G(X). Then Sqi(a) = (ki )uia.

Proof. By Proposition 41.9, H>0(XG) = 0 and X is equivariantly formal.Therefore, XG has the cohomology of b(X) points and (XG)G ≈ BG × XG ishomotopy equivalent to a disjoint union of b(X) copies of RP∞. By (46.4), anyclass b ∈ Hk

G(XG) satisfies Sqi(b) = (ki )uib. As the restriction homomorphism

H∗G(X)→ H∗G(XG) is injective by Proposition 41.12, this proves the assertion.

As seen in the above proof, the G-space X of Proposition 60.1 is equivariantlyformal. Thus, ρ : H∗G(X)→ H∗(X) is surjective. As ker ρ = u ·H∗G(X) by (39.12),Proposition 60.1 has the following corollary (compare [162, Corollary 1]).

Corollary 60.2. Let X be a finite dimensional G-complex with b(X) < ∞.Suppose that ♯(π0(X

G)) = b(X). Then, any a ∈ H∗(X) satisfies Sq(a) = a (i.e.

Sqi = 0 for i > 0). In particular, a a = 0 if a ∈ H>0(X).

Let us restrict the above results to the case where X is a smooth closed G-manifold. Then, XG is a union of closed manifolds (see, e.g. [11, Corollary 2.2.2]).We have seen in the proof of Proposition 60.1 that each component of XG has thecohomology of a point. Hence XG is a discrete set of b(X) points (the smoothinvolution τ is called an m-involutions in [162]). Examples involve linear spheresSn0 ; if X1 and X2 are such G-manifold, so is X1×X2 with the diagonal involution; ifdimX1 = dimX2, the G-equivariant connected sun X1♯X2 around fixed points car-ries a m-involution. Thus, an orientable surface carries an m-involution. Also, if Xadmits a G-invariant Morse function, then τ is an m-involution by Theorem 44.17.Corollary 60.2 has the following consequence on the Stiefel-Whitney w(TX) formanifold X admitting an m-involution.

Corollary 60.3. Let X be a smooth closed G-manifold such that ♯(π0(XG)) =

b(X). Then w(TX) = 1.

357

358 9. MISCELLANEOUS APPLICATIONS AND DEVELOPMENTS

In consequence, a closed manifold X carrying an m-involution is orientable andadmits spin structures. Also, X is the boundary of a (possibly non-orientable)compact manifold by Thom’s theorem Theorem 59.11.

Proof. By Corollary 60.2, Sqi = 0 for i > 0. Hence, the Wu class V (X) isequal to 1. Therefore, using Wu’s formula 58.3, w(TX) = Sq(V (X)) = 1.

We now generalize to smooth closed G-manifolds with XG discrete (withoutasking that ♯XG = b(X)). This will lead us toward the link between closed G-manifold with discrete fixed point set and coding theory; such a link was initiatedin [162] and further developed in [119]. We start with the following lemma.

Lemma 60.4. Let X2k+1 be a smooth closed G-manifold such that XG is dis-crete. Then, ♯XG is even.

Proof. Let r = ♯XG. Let W = X − intD where D is a closed G-invarianttubular neighborhood of XG. Then W is a compact free G-manifold with boundaryV . The orbit space W = W/G is a compact manifold whose boundary V = V /Gis a disjoint union of r copies of RP 2k. By Proposition 32.13, the image B ofHk(W )→ Hk(V ) satisfies

2 dimB = dimHk(BdW ) = r

which shows that r is even.

Remark 60.5. If X is a finite dimensional G-CW-complex with b(X) <∞, it isknown that b(X) ≡ b(XG) mod 2 [8, Corollary 1.3.8]. If X is an odd dimensionalclosed manifold, then b(X) is even by Poincare duality. This provides another proofof Lemma 60.4

We use the notation of the proof of Lemma 60.4. Let 〈〈−,−〉〉 be the bilinearform on Hk(V ) given by 〈〈a, b〉〉 = 〈a b, [V ]〉. By Proposition 32.13 and its proof,one has 〈〈B,B〉〉 = 0 and r = 2 dimB. Labelling the r points of XG produces anisomorphism Hk(V ) ≈ Zr2 intertwining 〈〈−,−〉〉 with the standard bilinear form onZr2. Hence, in terms of coding theory (see, e.g. [44]), B is a binary self-dual linearcode in Zr2. Choosing another labelling for the points of XG changes B by anisometry of Zr2 obtained by coordinate permutations. The class of B modulo theseisometries thus provides an invariant of the G-action.

The self-dual code B has other descriptions. For instance, the diagram ofinclusions

Vi //

j

D

W // X

gives rise to the commutative diagram

(60.6)

HkG(X) //

=

HkG(W )⊕Hk

G(D) //

≈

HkG(V )

≈

HkG(X) // Hk(W )⊕Hk

G(XG) // Hk(V )

60. ACTIONS WITH SCATTERED OR DISCRETE FIXED POINT SETS 359

whose row are the Mayer-Vietoris exact sequences. The map V = VG → (XG)G is,on each component, homotopy equivalent to the inclusion RP 2k → RP∞. There-fore, the homomorphism Hk

G(XG) → Hk(V ) is an isomorphism. Hence, a chasingdiagram argument in (60.6) shows that B = image(Hk(W ) → Hk(V )) coincideswith

image(HkG(X)→ Hk

G(XG))

(pushed into Hk(V )). For other descriptions of B (see [162, § 2]).The following theorem is proved in [119, Theorem 3].

Theorem 60.7. Every binary self-dual linear code may be obtained from closedsmooth 3-dimensional G-manifold X with scattered fixed point set.

As in § 41, the above results have analogues for S1-actions. The proofs ofProposition 60.8 and Corollary 60.9 below are the same as for Proposition 60.1and Corollary 60.2, replacing Propositions 41.9 and 41.12 by Propositions 41.16and 41.19, etc. Recall that H∗S1(pt) ≈ Z2[v] with v of degree 2.

Proposition 60.8. Let X be a finite dimensional S1-complex with b(X) <∞and XS1

= XS0

. Suppose that ♯(π0(XG)) = b(X). Then Hodd

S1 (X) = 0 and, if

a ∈ H2kS1(X), then Sq2i(a) = (ki ) v

ia.

Corollary 60.9. Let X be a finite dimensional S1-complex with b(X) < ∞and XS1

= XS0

. Suppose that ♯(π0(XG)) = b(X). Then, Hodd(X) = 0 and any

a ∈ H∗(X) satisfies Sq(a) = a

Analogously to Corollary 60.3, one has the following result, with the sameproof.

Corollary 60.10. Let X be a smooth closed S1-manifold such that XS1

=

XS0

and ♯(π0(XG)) = b(X). Then, Hodd(X) = 0 and w(TX) = 1.

In particular, the manifold X of Corollary 60.10 is even-dimensional. Notethat this is necessary for an S1-action admitting an isolated fixed point (the action,restricted to an invariant sphere around the fixed point has discrete stabilizers, sothe sphere is odd-dimensional). The analogue of Lemma 60.4 is Lemma 60.11 below.To simplify, we restrict ourselves to semi-free actions (A Γ-action is called semi-free if the stabilizer of any point is either id or Γ). Incidentally, the hypothesis

XS1

= XS0

is not required.

Lemma 60.11. Let X be a smooth closed S1-manifold such that XS1

is discrete.Suppose that the action is semi-free. Then, ♯XG is even.

Proof. As seen above, X is even-dimensional, say dimX = 2k + 2. Letr = ♯XG. Let W = X−intD whereD is a closed S1-invariant tubular neighborhoodof XG. Then W is a compact free S1-manifold with boundary V . The orbitspace W = W/G is then a compact manifold of dimension 2k+ 1 whose boundary

V = V /G is a disjoint union of r copies of CP k. By Proposition 32.13, the imageB of Hk(W )→ Hk(V ) satisfies

2 dimB = dimHk(BdW ) = r

which shows that r is even.


As for the case of an involution, Lemma 60.11 permits us to associate, to a

closed smooth semi-free S1-manifold with XS1

discrete, the self-dual linear code

B ⊂ Hk(V ) ≈ Zr2. One can also see B as the image of HkS1(X) into Hk

S1(X)S1

.

61. Conjugation spaces

Introduced in [85], conjugation spaces are equivariantly formal G-spaces (G =id, τ) quite different from those with scattered fixed point sets studied in § 60:here, the cohomology ring of the fixed point set most resembles that of the totalspace. This similarity should be given by a “cohomology frame”, a notion which wenow define. We use the notations of § 39 for a G-space X , for example the forgetfulhomomorphism ρ : H∗G(X) → H∗(X) and the GrA[u]-morphism r : H∗G(X) →H∗G(XG) ≈ H∗(XG)[u] induced by the inclusion XG ⊂ X .

Let (X,Y ) be a G-pair. A cohomology frame or an H∗-frame for (X,Y ) is apair (κ, σ), where

(a) κ : H2∗(X,Y ) → H∗(XG, Y G) is an additive isomorphism dividing thedegrees in half; and

(b) σ : H2∗(X,Y ) → H2∗G (X,Y ) is an additive section of the natural homo-

morphism ρ : H∗G(X,Y )→ H∗(X,Y ),

satisfying, in H∗(XG) ≈ H∗(XG)[u], the conjugation equation

(61.1) rσ(a) = κ(a)um + ℓtm

for all a ∈ H2m(X,Y ) and all m ∈ N; in (61.1), ℓtm denotes any element inH∗(X,Y )[u] which is of degree less than n in the variable u. An involutionadmitting an H∗-frame is called a conjugation. An even cohomology pair (i.e.Hodd(X,Y ) = 0) together with a conjugation is called a conjugation pair. A G-space X is a conjugation space if the pair (X, ∅) is a conjugation pair. The existenceof the section σ implies that a conjugation space is equivariantly formal. Note thatthere are examples of G-spaces which admit pairs (κ, σ) satisfying (a) and (b) abovebut none of them satisfying the conjugation equation (see [61, Example 1]). Forsimplicity, we shall mostly restrict this survey to conjugation spaces; the corre-sponding statements for conjugation pairs may be found in [85].

Any space X such that H∗(X) = H∗(pt) = H∗(XG) is a conjugation space.For instance, a finite dimensional G-CW-complex X satisfying H∗(X) = H∗(pt)and XG 6= ∅ is a conjugation space by Corollary 41.11. Another easy example isthe G-sphere S2m

m of Example 39.26. Indeed, one has the following lemma.

Lemma 61.2. Let X be a finite dimensional G-CW-complex. Suppose thatH∗(X) ≈ H∗(S2n) and Hn(XG) 6= 0. Then X is a conjugation space.

Proof. By Corollary 41.11, H∗(XG) ≈ H∗(Sn). By Proposition 41.9, X isequivariantly formal and, by Proposition 41.12, r : H∗G(X) → H∗G(XG) is injec-tive. The proof of the existence of an H∗-frame then proceeds as in the proof ofCorollary 39.32.

An other important example is the complex projective space.

Example 61.3. Let a ∈ H2(CPm) and b ∈ H1(RPm) (m ≤ ∞). Then,the section σ : H∗(CPm) → H∗G(CPm) of Proposition 39.34, together with theisomorphism κ : H2∗(CPm)→ H∗(RPm) sending a to b makes an H∗-frame for the

61. CONJUGATION SPACES 361

complex conjugation on CPm. By Corollary 39.34, the conjugation equation takesthe form

(61.4) rσ(ak) = (κ(a)u + b2)k = κ(ak)uk + ℓtk .

The same treatment may be done for HPm or OP 2 (see Remark 39.38).

These examples are actually spherical conjugation complexes, i.e. G-spacesobtained from the empty set by countably many successive adjunctions of collectionsof conjugation cells. A conjugation cell (of dimension 2k) is a G-space which is G-

homeomorphic to the cone over S2k−1k−1 , i.e. to the closed disk of radius 1 in R2k,

equipped with a linear involution with exactly k eigenvalues equal to −1. At eachstep, the collection of conjugation cells consists of cells of the same dimension but,as in [72], the adjective “spherical” is a warning that these dimensions do not needto be increasing. For less standard examples of spherical conjugation complexes,see [85, 5.6, p. 22].

It is proven in [85, Proposition 5.2] that a spherical conjugation complex is aconjugation space. For example, complex flag manifold (with τ being the complexconjugation) are conjugation spaces because the Schubert cells (see § 55.3) areconjugation cells. This example generalizes to co-adjoint orbits of compact Liegroups for the Chevalley involution (see [85, § 8.3]) and more examples comingfrom Hamiltonian geometry (see [85, § 8.2 and 8.4] and [84]).

Other examples may be obtained from the previous ones since the class ofconjugation spaces is closed under many construction, such as

• direct products, with the diagonal G-action, when one of the factor is offinite cohomology type (see [85, Proposition 4.5]).• inductive limits (see [85, Proposition 4.6]).• if (X,Y, Z) is a G-triple so that (X,Y ) and (Y, Z) are conjugation pairs,

then (X,Z) is a conjugation pair. A direct proof using H∗-frames is givenin [85, Proposition 4.1]; a shorter proof using the conjugation criterion of[155, Theorem 2.3] is provided in [154, Proposition 2.2.1].• if F → E → B be G-equivariant bundle (with a compact Lie group as

structure group) such that F is a conjugation space and B is a sphericalconjugation complex, then E is a conjugation space (see [85, Proposi-tion 5.7]).

We now show the naturality of H∗-frames under G-equivariant maps, as provenin [85, Proposition 3.11]. Let X and Y be two conjugation spaces, with H∗-frames(κX, σX) and (κY , σY ). Let f : Y → X be a G-equivariant map. We denote byfG : Y G → XG the restriction of f to Y G.

Proposition 61.5 (Naturality ofH∗-frames). The equations H∗Gf σX = σY H∗f

and H∗fGκX = κY H∗f hold true.

Proof. Let a ∈ H2k(X). As σX and σY are sections of ρX : H∗G(X)→ H∗(X)and ρY : H∗G(Y )→ H∗(Y ) respectively, one has

(61.6) ρY H∗Gf σX(a) = H∗f ρX σX(a) = H∗f(a) = ρY σY H

∗f(a) .

This implies that

H∗Gf σX(a) ≡ σY H∗f(a) mod ker

(ρ2k

Y: H2k

G (Y )→ H2k(Y )).


The section σ produces an isomorphism H∗G(X) ≈ H∗(X)[u] and ker ρY is the idealgenerated by u (see Remark 39.15). As Hodd(Y ) = 0, we deduce that there existsdi ∈ Hi(Y ) such that

(61.7) H∗Gf σX(a) = σY H∗f(a) + σY (d2k−2)u

2 + · · ·+ σY (d0)u2k .

Let us apply rY to both sides of (61.7). For the left hand side, we get(61.8)rY H

∗Gf σX(a) = H∗Gf

GrX σX(a)

= H∗GfG(κX(a)uk + ℓtk) by the conjugation equation

= H∗GfG(κX(a))uk + ℓtk H∗

GfG being a GrA[u]-morphism.

But, using the right hand side of (61.7), we get

(61.9) rY H∗Gf σX(a) = κY (d0)u

2k + ℓt2k.

Comparing (61.8) with (61.9) and using that κY is injective implies that d0 = 0.Then, (61.9) may be replaced by

(61.10) rY H∗Gf σX(a) = κY (d2)u

2k−2 + ℓt2k−2.

Again, the comparison with (61.8) implies that d2 = 0. This process may becontinued, eventually giving that H∗Gf σX(a) = σY H

∗f(a). Applying rY to theright hand member of this equation gives

(61.11) rY σY H∗f(a) = κY H

∗f(a)uk + ℓtk

by the conjugation equation. Comparing the leading terms of (61.11) and (61.8)gives that H∗fGκX(a) = κY H

∗f(a).

Applying Proposition 61.5 to X = Y and f = id, we get the following corollary.

Corollary 61.12 (Uniqueness of H∗-frames). Let (κ, σ) and (κ′, σ′) be twoH∗-frames for the conjugation space X. Then (κ, σ) = (κ′, σ′).

We can thus speak about the H∗-frame of a conjugation space.

Proposition 61.13. Let (κ, σ) be the H∗-frame of a conjugation space X.Then κ and σ are multiplicative.

Proof. Let a ∈ H2m(X) and b ∈ H2n(X) One has a b = ρσ(a b) andρ(σ(a) σ(b)) = ρ(σ(a)) ρ(σ(b)) = a b. Hence, σ(a) σ(b) is congruentto σ(a b) modulo ker ρ. The same proof as for Proposition 61.5 then proves theproposition (details may be found in [85, Theorem 3.3]).

Much more difficult to prove, the following result was established in [61, The-orem 1.3].

Proposition 61.14. Let (κ, σ) be the H∗-frame of a conjugation space X.Then Sqiκ = κSq2i for all integer i.


Remark 61.15. It is not true in general that σSq = Sqσ. For example,consider the conjugation space CPm for 1 ≤ m ≤ ∞, with the notations of Exam-ple 61.3. Of course, Sq1(a) = 0 and then σSq1(a) = 0. On the other hand,

rSq1σ(a) = Sq1

rσ(a)

= Sq1((bu+ b2)

)by the conjugation equation (61.4)

= Sq1(b)u+ bSq1(u) by the Cartan formula

= b2u+ bu2

= r(σ(a)u) .

Since r is injective (see Lemma 61.16 below), this proves that

Sq1(σ(a)) = σ(a)u .

The following lemma is recopied with its short proof from [85, Lemma 3.8].

Lemma 61.16. Let X be a conjugation space. Then the restriction homomor-phism r : H∗G(X)→ H∗G(XG) is injective.

Proof. Suppose that r is not injective. Let 0 6= x = σ(y)uk+ℓtk ∈ H2n+kG (X)

be an element in ker r. The conjugation equation guarantees that k 6= 0. Wemay assume that k is minimal. By the conjugation equation again, we have 0 =r(x) = κ(y)un+k + ℓtn+k. Since κ is an isomorphism, we get y = 0, which is acontradiction.

TheH∗-frame of a conjugation space behaves well with respect to the character-istic classes of G-conjugate-equivariant bundles. A G-conjugate-equivariant bundle

over a G-space X (with an involution τ) is a complex vector bundle η = (Ep−→ X),

together with an involution τ on E such that p τ = τ p and τ is conjugate-linearon each fiber: τ (λx) = λτ (x) for all λ ∈ C and x ∈ E. This was called a “real

bundle” by Atiyah [10]. Note that ηG = (EGp−→ XG) is a real vector bundle and

rankR ηG = rankC η. The following result is proven in [85, Proposition 6.8]-

Proposition 61.17. Let η be a G-conjugate-equivariant bundle over a sphericalconjugation complex X. Then κ(c(η)) = w(ητ ).

A theory of (integral) equivariant Chern clases forG-conjugate-equivariant bun-dles over a conjugation space is developed in [157].

Another relationship between conjugation spaces and the Steenrod squares wasdiscovered by M. Franz and V. Puppe in [61]. It is illustrated by the case of CPm,with the notations of Example 61.3, where the conjugation equation (61.4) for CPm

may be written as follows.

(61.18) rσ(ak) = (κ(a)u + κ(a)2)k =

k∑

j=0

(kj )κ(a)k+juk−j =

k∑

j=0

Sqj(κ(ak))uk−j .

It was proven in [61, Theorem 1.1] that (61.18) holds true in general, leading tothe the following universal conjugation equation.


Theorem 61.19. Let X be a conjugation space, with H∗-frame (κ, σ). Then,for x ∈ H2k(X), one has

rσ(x) = (κ(a)u+κ(a)2)k =

k∑

j=0

Sqj(κ(x))uk−j .

Let r : H∗(X)→ H∗(XG) be the restriction homomorphism in non-equivariantcohomology. The following corollary was observed in [61, Corollary 1.2].

Corollary 61.20. For x ∈ H∗(X), one has r(x) = κ(x) κ(x).

Proof. Suppose that x ∈ H2k(X). Denote by ρG : H∗G(XG) → H∗(XG) theforgetful homomorphism for XG. Then

r(x) = rρσ(x) since ρσ = id

= ρGrσ(X) using (39.10)

= ρG(∑k

j=0 Sqj(κ(x))uk−j) by Theorem 61.19

= Sqk(κ(x) since ρG = evu=0, see (39.11)

= κ(x) κ(x)) .

Another consequence of Theorem 61.19 is the commutativity of the followingdiagram.

(61.21)

H∗(X)σ //

κ

H∗G(X)r // H∗G(XG) oo ≈ // H∗(XG)[u]

ev1

H∗(X)

Sq // H∗(XG)

Theorem 61.19 has also consequences on conjugation manifolds, i.e. a closedmanifold X with a smooth conjugation τ . Then, XG is a closed manifold (see, e.g.

[11, Corollary 2.2.2]) whose dimension, because of the isomorphism κ : H∗(X)≈−→

H∗(XG), is half of the dimension of X . By looking at the derivative of τ arounda fixed point, one checks that τ preserves the orientation if and only if dimX ≡0 mod4. For various properties of conjugation manifolds, see [78, § 2.7]), fromwhich we extract the following results (see also [157, Appendix A]).

Proposition 61.22. Let X be a smooth conjugation manifold of dimension 2n,with H∗-frame (κ, σ). Then κ preserves the Wu and Stiefel-Whitney classes:

κ(v(X)) = v(XG) and κ(w(TX)) = w(TXG) .

Proof. The Wu class v2i(X) is characterised by the equation

(61.23) v2i(X) a = Sq2i(a) for all a ∈ H2n−2i(X) .

Applying the ring isomorphism κ to (61.23) and using Proposition 61.14 gives

(61.24) κ(v2i(X)) κ(a) = Sqi(κ(a)) for all a ∈ H2n−2i(X) .

As κ is bijective, (61.24) implies that

κ(v2i(X)) b = Sqi(b) for all b ∈ Hn−i(XG) ,


which implies that κ(v2i(X)) = vi(XG), and, as Hodd(X) = 0, that κ(v(X)) =

v(XG). Using this and the Wu formula, one gets

κ(w(TX)) = κSq(v(X)) by the Wu formula

= Sqκ(v(X)) Proposition 61.14

= Sq(v(XG))) as already seen

= w(TXG) by the Wu formula.

In particular, X admits a spin structure if and only if XG is orientable. Also,the Stiefel-Whitney numbers of X all vanish if and only if those of XG do so. ByThom’s theorem 59.11, this gives the following

Corollary 61.25. Let X be a conjugation manifold. Then X bounds a com-pact manifold if and only if XG does so.

Two natural problems occur for conjugation manifolds.

(i) Given a closed connected smooth manifold Mn, does there exist a conju-gation 2n-manifold X with XG diffeomorphic to M ?

(ii) Classify, up to G-diffeomorphism, conjugation manifolds with a given fixedpoint set.

The circle is the fixed point set of a unique conjugation 2-manifold, namelyS2

1 ; the uniqueness may be proved using the Schoenflies theorem (compare [32,Theorem 4.1]).

For n = 2, recall that RP 2 is the fixed point set of the conjugation manifoldCP 2 and S1×S1 is that of S2×S2. The equivariant connected sum (around a fixedpoint) of conjugation manifolds being again a conjugation manifold (see [85, Propo-sition 4.7]), any closed surface is the fixed point set of some conjugation 4-manifold(of course, S2 = (S4

2)G). Answering Question (ii) is the main object of [78], usingthe following idea. For a smooth G-action on a manifold X with XG being ofcodimension 2, the quotient space X/G inherits a canonical smooth structure. IfX is a conjugation 4-manifold, then H∗(X/G) ≈ H∗(S4). Conversely, let (Y,Σ)be a manifold pair such that Y is a 4-dimensional Z2-homology sphere containingΣ as a codimension 2 closed submanifold. By Alexander duality (Theorem 32.20),one has H1(Y −Σ) = Z2. Thus, Y −Σ admits a unique non-trivial 2-fold covering(see § 24); the latter extends to a unique branched covering X → Y , with branchedlocus Σ, and X turns out to be a conjugation 4-manifold with XG = M . The finalstatement is thus the following ([78, Theorem A]).

Theorem 61.26. The correspondence X 7→ (X/G,XG) defines a bijection be-tween

(a) the orientation-preserving G-diffeomorphism classes of oriented connectedconjugation 4-manifolds, and

(b) the orientation-preserving G-diffeomorphism classes of smooth manifoldpairs (Y,Σ), where Y is an oriented 4-dimensional homology sphere andΣ is a closed connected surface embedded in M .

The conjugation sphere S42 corresponds to the trivial knot S2 ⊂ S4. Under

the bijection of Theorem A, any knot S2 → S4 corresponds to a conjugation 4-manifold X with XG ≈ S2. In general, X is not simply connected. On the other


hand, Gordon [67], [68] and Sumners [185] found infinitely many topologicallydistinct knots in S4 which are the fixed point set of smooth involutions. Theseexamples produce infinitely many topologically inequivalent smooth conjugationson S4 (see [78, Proposition 5.12]).

If X is a simply-connected conjugation 4-manifold, it is known that X/G is atleast homeomorphic to S4 (see [78, Proposition 5.3]). In addition, X is homeo-morphic (not necessarily equivariantly) to a connected sum of copies of S2 × S2,

CP 2, and CP2

(see [78, Proposition 2.17]). These are severe restrictions on asimply-connected closed smooth 4-manifold to carry a smooth conjugation.

M. Olbermann, in his thesis [154], was the first to address Question (i); heproved the following result (see [155, Theorem 1.2]).

Theorem 61.27. Any closed smooth orientable 3-manifold is diffeomorphic tothe fixed point-set of a conjugation 6-manifold.

The case of non-orientable 3-manifolds is not known. Any 3-dimensional Z2-homology sphere is the fixed point of infinitely many inequivalent conjugations onS6, [156]; this gives a partial answer to Question (ii) in this case.

Finally, as observed by W. Pitsch and J. Scherer, the answer to Question (i)is not always positive. For example, the octonionic projective plane OP 2, whichis a smooth closed 16-manifold (see Remark 35.11), is not the fixed point set ofany conjugation space. Indeed, H∗(OP 2) ≈ Z2[x]/(x

3) by Proposition 35.10 but,by Theorem 50.5, Z2[x]/(x

3) is not the cohomology ring of a topological space ifdegree (x) > 8.

62. Chain and polygon spaces

Chain and polygon spaces are examples of configuration spaces, a main conceptof classical mechanics. In recent decades, starting in [198, 81] (inspired by talksof W. Thurston on linkages [191]), new interests arose for polygon spaces, in con-nections with Hamiltonian geometry (see e.g. [116, 109, 86, 89]), mathematicalrobotics [139, 54] and statistical shape theory [88]. This section contains originalresults on the equivariant cohomology of chain spaces, giving new proofs for knownstatements about their ordinary cohomology.

We use the notations of [88, 83, 57], inspired by those of statistical shapetheory [110]. In order to make some formula more readable, we may write |J | forthe cardinality ♯J of a finite set J .

62.1. Definitions and basic properties. Let ℓ = (ℓ1, . . . , ℓn) ∈ Rn>0 and let

d be an integer. We define the subspace Cnd (ℓ) of (Sd−1)n−1 by

Cnd (ℓ) =z = (z1, . . . , zn−1) ∈ (Sd−1)n−1 |

n−1∑

i=1

ℓizi = ℓn e1,

where e1 = (1, 0, . . . , 0) is the first vector of the standard basis e1, . . . , ed of Rd. Anelement of Cnd (ℓ), called a chain, may be visualised as a configuration of (n − 1)successive segments in Rd, of length ℓ1, . . . , ℓn−1, joining the origin to ℓne1. Thevector ℓ is called the length vector. The chain space Cnd (ℓ) is contained in the big

62. CHAIN AND POLYGON SPACES 367

chain space BCnd (ℓ) defines as follows:

BCnd (ℓ) =z = (z1, . . . , zn−1) ∈ (Sd−1)n−1 | 〈

n−1∑

i=1

ℓizi, e1〉 = ℓn,

(successions of (n− 1) segments in Rd, of length ℓ1, . . . , ℓn−1, joining the origin tothe affine hyperplane with first coordinate ℓn).

The group O(d − 1), viewed as the subgroup of O(d) stabilizing the firstaxis, acts naturally (on the left) upon the pair (BCnd (ℓ), Cnd (ℓ)). The quotientCnd (ℓ)

/SO(d − 1) is the polygon space Nn

d , also defined as

Nnd (ℓ) = Nn

d (ℓ)/SO(d) ,

where

Nnd (ℓ) =

z ∈ (Sd−1)n

∣∣∣∣∣n∑

i=1

ℓizi = 0

the free polygon space (called “space of polygons” in [55, 63]).

The map from SO(d) × Cnd (ℓ) to Nnd (ℓ) given by

(A, (z1, . . . , zn−1)) 7→ (Az1, . . . , Azn−1,−Ae1)descends to an SO(d)-homeomorphism

(62.1) SO(d) ×SO(d−1) Cnd (ℓ)≈−→ Nn

d (ℓ) .

Recall that the map SO(d)→ Sd−1 given by A 7→ −Ae1 is the orthonormal orientedframe bundle for the tangent bundle to Sd−1 (see p. 298). Thus, by (62.1), we geta locally trivial bundle

(62.2) Cnd (ℓ)→ Nnd (ℓ)→ Sd−1 .

When d = 2 the space of chains Cn2 (ℓ) coincides with the polygon space Nn2 (ℓ).

The axial involution τ on Rd = R × Rd−1 given by τ(t, y) = (t,−y) inducesan involution, still called τ , on the pair (BCnd (ℓ), Cnd (ℓ)) and on (Sd−1)n−1. As τcommutes with the O(d − 1)-action on Cnd (ℓ), it descends to a G-action on Nn

d (ℓ).A bar above a G-space denotes its orbit space:

BCnd (ℓ) = BCnd (ℓ)/G , Cnd (ℓ) = Cnd (ℓ)/G , Nnd (ℓ) = Nn

d (ℓ)/G .

We shall compute the G-equivariant cohomology (G = id, τ) of BCnd (ℓ) andCnd (ℓ), as algebras over H∗G(pt) = Z2[u] (u of degree 1). This uses some G-invariantMorse theory on M = (Sd−1)n−1. We start with the robot arm map Fℓ : M → Rd =R× Rd−1 defined by

(62.3) Fℓ(z) =

n−1∑

i=1

ℓizi , z = (z1, . . . , zd−1) .

Consider Rd as the product R × Rd−1, which defines the projections p1 : Rd → Rand pd−1 : Rd → Rd−1. Define fℓ : M → R by

fℓ(z) = −p1(Fℓ(z)) = −n−1∑

i=1

ℓi〈zi, e1〉 .

Note that Fℓ is O(d)-equivariant and fℓ is O(d− 1)-invariant. For n = 2, it is clearthat fℓ is Morse function on Sd−1, with two critical points, namely e1 of index 0and −e1 of index d− 1. The following lemma follows easily.


Lemma 62.4. The function fℓ : (Sd−1)n−1 → R defined by

fℓ(z1, . . . , zn−1) = −n−1∑

i=1

ℓi〈zi, e1〉

is a G-invariant Morse function, with one critical point PJ for each J ⊂ 1, . . . , n−1, where PJ = (z1, . . . , zm−1) with zi equal to −e1 if i ∈ J and e1 otherwise (acollinear chain). The index of PJ is (d− 1) |J |.

A length vector ℓ ∈ Rn>0 is generic if Cn1 (ℓ) = ∅, that is to say there are nocollinear chains or polygons. In this section, we shall only deal with generic lengthvectors.

Corollary 62.5. If ℓ is a generic length vector, then BCnd (ℓ), Cnd (ℓ) and Nnd (ℓ)

are smooth closed manifolds of dimension

dimBCnd (ℓ) = dim Nnd (ℓ) = (n−1)(d−1)−1 and dim Cnd (ℓ) = (n−2)(d−1)−1 .

Proof. If ℓ is generic, then −ℓ is a regular value of fℓ. Indeed, if p2(Fℓ(z)) 6= 0,this follows from the O(d)-equivariance of Fℓ. If p2(Fℓ(z)) = 0, then, as ℓ is generic,z is not a critical point of Fℓ (these are the the collinear configurations zi = ±zj:see [81, Theorem 3.1]). Since BCnd (ℓ) = f−1

ℓ (−ℓn), this proves the assertion onBCnd (ℓ).

Define P : BCnd (ℓ) → Rd−1 by P (z) = pd−1(Fℓ(z)). As seen above, as ℓ isgeneric, P−1(0) contains no critical points of Fℓ. Therefore, P is transversal to 0and thus Cnd (ℓ) = P−1(0) is a closed submanifold of codimension d− 1 of BCnd (ℓ).

When ℓ is generic, the O(d − 1)-action on Cnd (ℓ)) is smooth. Hence, the bun-

dle (62.2) is a smooth bundle and the assertion on Nnd (ℓ) follows form (62.1). For

another proof of that Nnd (ℓ) is a manifold, see [55, Proposition 3.1]).

We now see how chain and polygon spaces are determined by some combina-torics of their length vector ℓ = (ℓ1, . . . , ℓn). A subset J of 1, . . . , n is calledℓ-short (or just short) if

∑i∈J ℓi <

∑i/∈J ℓi. The complement of a short subset is

called long. If ℓ is generic, subsets are either short or long. Short subsets form, withthe inclusion, a poset Sh(ℓ). Define Shn(ℓ) = J ∈ 1, . . . , n−1 | J∪n ∈ Sh(ℓ).

For J ⊂ 1, . . . , n, let HJ be the hyperplane (wall) of Rn defined by

HJ :=

(ℓ1, . . . , ℓn) ∈ Rn∣∣∣∑

i∈J

ℓi =∑

i/∈J

ℓi

.

The union H(Rn) of all these walls determines a set of open chambers in (R>0)n

whose union is the set of generic length vectors (a chamber is a connected componentof (R>0)

n − H(Rn)). We denote by Ch(ℓ) the chamber of a generic length vectorℓ. Note that Ch(ℓ) = Ch(ℓ′) if and only if Sh(ℓ) = Sh(ℓ′).

Let Symn be the group of bijections of 1, . . . , n; we see Symn−1 as thesubgroup of Symn formed by those bijections fixing n. If X is a set, the groupSymn acts on the Cartesian product Xn by

(x1, . . . , xn)σ = (xσ(1), . . . , xσ(n)) .

The notation emphasizes that this action is on the right: an element x ∈ Xn

is formally a map x : 1, . . . , n → X (xi = x(i)) and σ ∈ Symn acts by pre-composition, i.e. xσ = xσ. We shall use this action on various n-tuples, inparticular on length vectors.


Lemma 62.6. Let ℓ = (ℓ1, . . . , ℓn) and ℓ′ = (ℓ′1, . . . , ℓ′n) be two generic length

vectors. Then, the following conditions are equivalent

(1) Shn(ℓ) and Shn(ℓ′) are poset isomorphic.

(2) Sh(ℓ) and Sh(ℓ′) are poset isomorphic via a bijection σ ∈ Symn−1.(3) Ch(ℓ′) = Ch(ℓσ) for some σ ∈ Symn−1.

Moreover, if one of the above conditions is satisfied, there are O(d−1)-diffeomorphismsof manifolds pairs

h : (BCnd (ℓ), Cnd (ℓ))≈−→ (BCnd (ℓ′), Cnd (ℓ′))

andh : (BCnd (ℓ), Cnd (ℓ))

≈−→ (BCnd (ℓ′), Cnd (ℓ′)) .

Proof. Implications (1)⇐ (2)⇔ (3) are obvious. Let us prove that (1)⇒ (2).Let σ ∈ Symn−1 be the permutation giving the poset isomorphism Shn(ℓ) ≈Shn(ℓ

′). Replacing ℓ by ℓσ, we may assume that Shn(ℓ) = Shn(ℓ′). We now observe

that Shn(ℓ) determines Sh(ℓ). Indeed, let J ⊂ 1, . . . , n. Then, either n ∈ J orn ∈ J , and thus Shn(ℓ) tells us whether J ∈ Sh(ℓ) (or J ∈ Sh(ℓ)).

It remains to prove that (3) implies the existence of the O(d−1)-diffeomorphismsh and h. As the G action commutes with the O(d − 1)-action (G is naturallyin the center of O(d − 1)), it suffices to construct h, which will induce h. Ifσ ∈ Symn−1, then the correspondence z → zσ defines an O(d− 1)-diffeomorphism

(BCnd (ℓ), Cnd (ℓ))≈−→ (BCnd (ℓσ), Cnd (ℓσ)). Replacing ℓ by ℓσ, we may thus assume that

Ch(ℓ) = Ch(ℓ′) and σ = id.Consider the smooth map L : (Rd−0)n−1 → (R>0)

n given, for x = (x1, . . . , xn−1)by

L(x) =(|x1|, . . . , |xn−1|, 〈F (x), e1〉

),

where F (x) =∑n−1

i=1 xi. Observe that the map (Sd−1)n−1 → (Rd − 0)n−1 send-

ing (z1, . . . , zn−1) to (ℓ1z1, . . . , ℓn−1zn−1) induces a diffeomorphism γℓ : BCnd (ℓ)≈−→

L−1(ℓ) such that F γ = F , the robot arm map of (62.3). If ℓ is generic, then ℓis a regular value of L. Indeed, let x = (x1, . . . , xn−1) ∈ L−1(ℓ). For each i =1, . . . , n− 1, one can construct a path xi(t) = (xi1(t), . . . , x

in−1(t)) ∈ (Rd −0)n−1

with x(0) = x such that L(xi(t)) = (ℓi1(t), . . . , ℓin(t)) satisfies ℓij(t) = ℓj for j 6= i

and ℓii(t) = ℓi + αt with α 6= 0. For i = n, this follows from the proof of Corol-lary 62.5. Suppose that i ≤ n − 1. If x is not a lined configuration, then xi and∑j 6=i xj are linearly independent and generate a 2-dimensional plane Π, containing

F (x). There are rotations ρit and ρt of Π, depending smoothly on t, such that

ρit((1 + t)xi) + ρt(∑

j 6=i xj) = F (x). Hence, X i(t) may be defined as

xij(t) =

ρit((1 + t)xi) if j = i

ρt(xj) if j 6= i .

Finally, if x is a lined configuration, then F (x) and e1 are linearly independent(since ℓ is generic). They thus generate a 2-dimensional plane Ω. Let xi(t) definedby xii(t) = (1 + t)xi and xij(t) = xj when j 6= i. If t is small enough, there is

a unique rotation rt of Ω such that 〈rt(F (xi(t)), e1〉 = ℓn. We can thus definexij(t) = ρt(x

ij(t)).

As Ch(ℓ) is convex, it contains the segment [ℓ, ℓ′] consisting of only genericlength vectors. What has been done above shows that the map L is transversal to


[ℓ, ℓ′]. Therefore, X = L−1([ℓ, ℓ′]) is O(d)-cobordism between BCnd (ℓ) and BCnd (ℓ′).Let pd−1 : Rd → Rd−1 be the projection onto the last d − 1 coordinates. As in

the proof of Corollary 62.5, the map P : X → Rd−1 defined by P (x) = pd−1(F (x))is transversal to 0. Thus, Y = P−1(0) is a submanifold of X of codimensionn − 1 and the pair (X,Y ) is a cobordism of pairs between (BCnd (ℓ), Cnd (ℓ)) and(BCnd (ℓ′), Cnd (ℓ′)). The map L : X → [ℓ, ℓ′] has no critical point. The standardRiemannian metric on (Rd)n induces an O(d − 1)-invariant Riemannian metric on(X,Y ). Following the gradient lines of π for this metric provides the requiredO(d− 1)-equivariant diffeomorphism h.

For n ≤ 9, a list of all chambers (modulo the action of Symn) was obtained in[88]. Their numbers are

n 3 4 5 6 7 8 9

Nb of chambers 2 3 7 21 135 2470 175428

Geometric descriptions of several chain and polygon spaces for ℓ-generic areprovided in [83], as well as some general constructions. Among them, the operationof “adding a tiny edge”, which we now describe. Let ℓ = (ℓ2, . . . , ℓn) be a genericlength vector. If ε > 0 is small enough, the n-tuple ℓ+ := (δ, ℓ2, . . . , ℓn) is a genericlength vector for 0 < δ ≤ ε.

Lemma 62.7. There are O(d− 1)-equivariant diffeomorphisms

BΦ: BCnd (ℓ+)≈−→ Sd−1 × BCn−1

d (ℓ) and Φ: Cnd (ℓ+)≈−→ Sd−1 × Cn−1

d (ℓ) ,

where Sd−1×BCmd−1(ℓ) and Sd−1×Cmd−1(ℓ) are equipped with the diagonal O(d−1)-action.

Proof. The diffeomorphism Φ is constructed in [83, Proposition 2.1]. Theconstruction can be easily adapted to give BΦ.

62.2. Equivariant cohomology. Let M = (Sd−1)n−1. The G-invariantMorse function f = fℓ : M → R of Lemma 62.4 satisfies the hypotheses of Propo-sition 44.32, i.e. MG = Crit f . Therefore, M is G-equivariantly formal and therestriction morphism

(62.8) r : H∗G(M)→ H∗G(MG) ≈⊕

J

H∗G(PJ ) ≈⊕

J

Z2[uJ ]

is injective (this also follows from Lemma 41.8 and Proposition 41.12). The variablesuJ are of degree one and the Z2[u]-module structure on H∗G(MG) is given by theinclusion u 7→∑

J uJ .In the remaining of this section, whenever xi (i ∈ N) are formal variables in a

polynomial ring and J ⊂ N, we set xJ =∏j∈J xj . In particular, x∅ = 1.

Proposition 62.9. For n ≥ 2, there is a GrA[u]-isomorphism

(62.10) Z2[u,A1 . . . , An−1, B1, . . . , Bn−1]/I ≈−→ H∗G((Sd−1)n−1)

where the variables Ai and Bi are of degree d − 1 and I is the ideal generated bythe following families of relators


(a) Ai +Bi + ud−1 i = 1, . . . , n− 1(b) A2

i +Aiud−1 i = 1, . . . , n− 1

Moreover, using (62.8), one has for J ⊂ 1, 2, . . . , n− 1:

(62.11)

r(AJ ) =∑

J⊂K

u|J|(d−1)K

r(BJ ) =∑

J∩K=∅

u|J|(d−1)K

Proposition 62.9 generalizes Examples 44.27 and 42.16, with the slightly differ-ent notations of (62.8) for the equivariant cohomology of the fixed point set.

Proof. The proof proceeds by induction on n. It starts with n = 2 by usingExample 44.27. For the induction step, assume the proposition to be true forn = m−1. LetM = (Sd−1)m−2 = M×M0, where M = (Sd−1)m−2 andM0 = Sd−1.The induction hypothesis implies that

H∗G(M) ≈ Z2[u, A1 . . . , An−1, B1, . . . , Bn−1

/I

where I is the ideal generated by the families Ai + Bi + ud−1 and A2i + Aiu

d−1

(i = 1, . . . ,m − 2). The Z2[u]-module structure is obtained by identifying u withu. Also,

H∗G(M0) ≈ Z2[u0, A,B]/(A+B + ud−1

0 , A2 = ud−10 A)

and the Z2[u]-module structure is obtained by identifying u with u0. By Theo-rem 42.13, the strong equivariant cross product provides an isomorphism

×G : H∗Γ(X)⊗Z2[u] H∗Γ(Y )

≈−→ H∗Γ(X × Y ) .

Setting Ai = Ai ×G 1, Bi = Bi ×G 1 (i = 1, . . . ,m − 2), Am−1 = 1 ×G A andBm−1 = 1×G B gives the induction step for the isomorphism (62.10).

We now prove the induction step for (62.11). The fixed points of M are denoted

by PJ , indexed by J ⊂ 1, . . . ,m− 2. We denote the fixed point of M0 = Sd−1 ⊂R × Rd−1 by ωmin = (1, 0) and ωmax = (−1, 0) (corresponding to the extremaof the Morse function (t, x) 7→ −t). Set H∗G(MG) ≈ Z2[umin] ⊕ Z2[umax]. For

J ⊂ 1, . . . ,m− 2, then PJ = PJ × ωmin and PJ∪m−1 = PJ × ωmax. Hence, fori = 1, . . . , n− 2, one has, using the obvious notations, that

r(Ai) = r(Ai×G 1)

= r(Ai)×G r0(1) by naturality of ×G

=[ ∑

J⊂1,...,m−2i∈J

u|J|(d−1)J

]×G [1min + 1max] by induction hypothesis

=∑

Ji∈J

u|J|(d−1)J +

∑

J∪m−1i∈J

u|J|(d−1)J (J ⊂ 1, . . . ,m− 2)

=∑

J⊂1,...,m−1i∈J

u|J|(d−1)J .


As for i = m− 1, one has

r(Am−1) = r(1×G A)

= r(1)×G r0(A)

=[ ∑

J⊂1,...,m−2

1J]×G ud−1

max

=∑

J⊂1,...,m−2

ud−1J∪m−1

=∑

J⊂1,...,m−1m−1∈J

u|J|(d−1)J .

This proves (62.11) for r(Ai), i = 1, . . . ,m− 1. The formula for r(Bi) are deduced,using relators (a). The formulae for r(AJ ) and r(BJ ) follow since r is multiplicative.

We are now ready to compute the G-equivariant cohomology of BCnd (ℓ).

Theorem 62.12. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. There is aGrA[u]-isomorphism

Z2[u,A1 . . . , An−1, B1, . . . , Bn−1]/Iℓ ≈−→ H∗G(BCnd (ℓ))

where the variables Ai and Bi are of degree d − 1 and Iℓ is the ideal generated bythe following families of relators

(a) Ai +Bi + ud−1 i = 1, . . . , n− 1(b) A2

i +Aiud−1 i = 1, . . . , n− 1

(c) AJ J ⊂ 1, . . . , n− 1 and J ∪ n is long(d) BJ J ⊂ 1, . . . , n− 1 and J is long.

Proof. LetM = (Sd−1)n−1,M− = f−1((−∞,−ℓn]) andM+ = f−1([−ℓn,∞]),with the inclusions j± : M± → M (f = fℓ, the Morse function of Lemma 62.4).One has M− ∩M+ = B = BCnd (ℓ) = f−1(−ℓn). The G-invariant Morse functionf : M → R satisfies the hypotheses of Proposition 44.32. The latter implies thatthe morphism H∗G(M)→ H∗G(B) induced by the inclusion is surjective with kernelequal to kerH∗Gj−+ kerH∗Gj+. By Proposition 62.9, H∗G(M) is GrA[u]-isomorphicto Z2[u,A1 . . . , An−1, B1, . . . , Bn−1]

/I where I is the ideal generated by families

(a) and (b). We shall prove that kerH∗Gj− is the ideal generated by relators (c)and that kerH∗Gj+ is the ideal generated by relators (d).

The critical point PJ satisfies

f(PJ) =∑

i∈J

ℓi −∑

i/∈J

ℓi .

Therefore,

(62.13) PJ ∈M− ⇐⇒ f(PJ ) < −ℓn ⇐⇒ J ∪ n is short .


Therefore, one has a commutative diagram

(62.14)

H∗G(M)

H∗Gj−

// r // H∗G(MG)

H∗GjG−

≈ //⊕

J⊂1,...,n−1

Z2[uJ ]

pr−

H∗G(M−) // r− // H∗G(MG− )

≈ //

⊕

J⊂1,...,n−1J∪n short

Z2[uJ ]

That r and r− are injective follows from Theorem 44.17. Hence, for x ∈ H∗G(M),H∗j−(AJ ) = 0 if and only if pr−r(x) = 0. Since M is equivariantly formal (byTheorem 44.17 again), Theorem 62.12 implies that

H∗(M) ≈ H∗G(M)/(u) ≈ Z2[A1, . . . , An−1]/(A2i ) .

By the Leray-Hirsch theorem, H∗G(M)/(u) is then isomorphic to the free Z2[u]-module with basis Aj | J ⊂ 1, . . . , n− 1 (or Bj | J ⊂ 1, . . . , n− 1). Thus,x ∈ H∗G(M) may be uniquely written as x =

∑J⊂1,...,n−1 λJAJ , with λJ ∈ Z2[u].

Let J0 ⊂ 1, . . . , n− 1 minimal (for the inclusion) such that λJ0 6= 0. By (62.13),one has

r(x) = λJ0uJ0 mod⊕

J⊂1,...,n−1J 6=J0

Z2[uJ ] .

Hence, if x ∈ kerH∗Gj−, we deduce using Diagram (62.14) that J0 ∪ n is long.Therefore, λJ0AJ0 ∈ kerH∗Gj− and x + λJ0AJ0 ∈ kerH∗Gj−. Repeating the aboveargument with x+ λJ0 and so on proves that

x =∑

J⊂1,...,n−1

λJAJ ∈ kerH∗Gj− ⇐⇒ λJ = 0 whenever J ∪ n is short .

This proves that kerH∗Gj− is the Z2[u]-module generated by relators (c) (sinceAJAK = AJ∪K , this is an ideal).

In the same way, we prove that kerH∗Gj+ is the Z2[u]-module generated byrelators (d). Details are left to the reader.

Corollary 62.15. For a generic length vector ℓ = (ℓ1, . . . , ℓn), there is aGrA[u]-isomorphism

Z2[u,A1 . . . , An−1]/Iℓ ≈−→ H∗G(BCnd (ℓ))

where the variables Ai are of degree d − 1 and Iℓ is the ideal generated by thefollowing families of relators

(1) A2i +Aiu

d−1 i = 1, . . . , n− 1(2) AJ J ⊂ 1, . . . , n− 1 and J ∪ n is long

(3) ud−1∑

K⊂J

AK u(|J−K|−1)(d−1) J ⊂ 1, . . . , n− 1 and J is long

Note that, by (2), only the sets K ⊂ J with K ∪ n being short occur in thesum of Relators (3).


Proof. This presentation of H∗G(BCnd (ℓ)) is algebraically deduced from that ofTheorem 62.12. The generators Bi are eliminated using relators (a). Realtors (b)and (c) become respectively (1) and (2). Relators (d) become relators (3). Indeed,

BJ =∏i∈J Bi

=∏i∈J (Ai + ud−1) using (a)

=∑

K⊂J AK u|J−K|(d−1) plain extension

= ud−1∑

K⊂J

AK u(|J−K|−1)(d−1) as AJ = 0 since J is long.

Example 62.16. Elementary geometry easily shows that BCnd (ℓ) = ∅ if and onlyif n is long (see also Example 62.56 below). In Corollary 62.15, we see that if nis long, then relator (2) for J = ∅ implies that 1 ∈ Iℓ and thus H∗G(BCnd (ℓ)) = 0.Compare Example 62.54.

Example 62.17. Suppose that ℓ is generic and that ℓn = −α+∑n−1

i=1 ℓi, withα > 0 small enough such that J ∪ n is short only for J = ∅. Hence, relators (2)imply that Ai = 0 for i = 1, . . . , n− 1. The only subset J of 1, . . . , n− 1 which islong is 1, . . . , n− 1 itself. Thus, the family of relators (3) contains one element,in which the only non-zero term in the sum occurs for K = ∅. This relator has thusthe form u(n−1)(d−1) and we get

(62.18) H∗G(BCnd (ℓ)) ≈ Z2[u]/(u(n−1)(d−1)) .

Notice that, with our hypothesis, −ℓn is a regular value of f which is just above aminimum. Thus, BCnd (ℓ) = f−1(−ℓ) is G-diffeomorphic to the sphere S(n−1)(d−1)−1

endowed with the antipodal involution (the isotropy representation of G on thetangent space to M at the minimum P∅ of f). As the G-action on BCnd (ℓ) is free,one has

H∗G(BCnd (ℓ)) ≈ H∗(BCnd (ℓ)/G) ≈ H∗(RP (n−1)(d−1)−1)

which is coherent with (62.18).

Proposition 62.17 will help us to compute H∗G(Cnd (ℓ)), after introducing somepreliminary material. We use the robot arm map Fℓ : M = (Sd−1)n−1 → Rd =R × Rd−1 defined in (62.3). Let N = F−1

ℓ (R × 0). If ℓ′ = (ℓ1, . . . , ℓn−1) is itselfgeneric, then N is a closed submanifold of codimension d− 1 in M . Indeed, exceptat F−1

ℓ (0), the robot arm map is clearly transverse to R× 0 (use that F is SO(d)-equivariant). If ℓ′ is generic, then 0 is a regular value of Fℓ (see the proof ofCorollary 62.5). Hence, Fℓ is everywhere transversal to R× 0.

A slight change of e.g. ℓ1 (which does not change the G-diffeomorphism typeof the pair (BCnd (ℓ), Cnd (ℓ)) by Lemma 62.6) will make ℓ′ is generic. Hence, withoutloss of generality, one may assume that N is a closed G-invariant submanifold ofM . One has Cnd (ℓ) = N− ∩ N+ where N± = N ∩M± (notation of the proof ofProposition 62.15).

There is a G-equivariant map φt : M− →M− such that φ0 = id and φ1(M−) =N−. Indeed, for z ∈ M−, denote by Πz the 2-plane in Rd generated by e1 andFℓ(z). Define ρt(z) ∈ SO(d) to be the rotation of angle cos−1(t|Fℓ(z)|/|fℓ(z)|)on Πz, the identity on Π⊥z and such that 〈ρtFℓ(z), e1〉 ≥ fℓ(z). The retraction


by deformation φt is defined by φt(z1, . . . , zn−1) = (ρt(z1), . . . , ρt(zn−1)). Theexistence of φt implies that

(62.19) H∗G(M−) ≈ H∗G(N−) .

The restriction of fℓ,N : N → R of fℓ is also a G-invariant Morse function onM , with Crit fℓ,N = Crit fℓ (see [81, §3]; the index of a critical point P is differentfor fℓ,N and f when fℓ(P ) < 0).

Proposition 62.20. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector and leti : Cnd (ℓ) → BCnd (ℓ) be the inclusion. Then, H∗Gi : H

∗G(BCnd (ℓ)) → H∗G(Cnd (ℓ)) is

surjective, with kernel equal to Ann (ud−1), the annihilator of ud−1.

Proof. Let (B, C) = (BCnd (ℓ), Cnd (ℓ)). Consider the following commutativediagram

H∗G(M−) // //

H∗Gj≈

H∗G(B)

H∗Gi

H∗G(N−) // // H∗G(C)

where all the arrows are induced by the inclusions. The horizontal maps are indi-cated to be surjective: this follows from Proposition 44.32 since Crit f = MG = NG.That H∗Gj is an isomorphism was noticed in (62.19). Hence, H∗Gi is surjective.

Let B = B/G and C = C/G. As the G-action on (B, C) is free, the vertical mapsin the following diagram

CGiG //

BG

C i // B

are homotopy equivalences (see Lemma 39.5). As B and C are smooth closed man-ifolds, Proposition 33.8 implies that kerH∗Gi is the annihilator of the Poincare dual

PD(C) ∈ Hd−1(B) ≈ Hd−1G (B). It thus remains to show that PD(C) = ud−1.

Let pd−1 : Rd = R × Rd−1 → Rd−1 be the projection onto the second factor.The map ϕ : B → (Rd−1)n−1 − 0 defined by

ϕ(z1, . . . , zn−1) = (pd−1(z1), . . . , pd−1(zn−1))

is smooth, G-equivariant (for the involution x 7→ −x on (Rd−1)n−1) and satis-fies C = ϕ−1((Rd−1)n−2 − 0). It thus descends to a smooth map ϕ : B →RP (n−1)(d−1)−1, such that C = ϕ−1(RP (n−2)(d−1)−1). As in the proof of Corol-lary 62.5, one shows that ϕ is transversal to RP (n−2)(d−1)−1. By Proposition 33.10,one has

PD(C) = H∗ϕ(PD(RP (n−2)(d−1)−1)) = H∗ϕ(ud−1) = ud−1 .

The two occurrences of the letter u in the above formulae is a slight abuse oflanguage, permitted by the considerations of Lemma 39.5: the G-action under


consideration are all free and in the commutative diagram

H∗(RP (n−1)(d−1)−1)≈ //

H∗ϕ

H∗G((Rd−1)n−1 − 0)

H∗Gϕ

H∗(B)≈ // H∗G(B)

the generator of H1(RP (n−1)(d−1)−1) is sent to u.

We are ready to compute H∗G(Cnd (ℓ).

Theorem 62.21. For a generic length vector ℓ = (ℓ1 . . . , ℓn), there is a GrA[u]-isomorphism

Z2[u,A1 . . . , An−1]/Iℓ ≈−→ H∗G(Cnd (ℓ)) ≈ H∗(Cnd (ℓ))

where the variables Ai are of degree d − 1 and Iℓ is the ideal generated by thefollowing families of relators

(1) A2i +Aiu

d−1 i = 1, . . . , n− 1(2) AJ J ⊂ 1, . . . , n− 1 and J ∪ n is long

(3′)∑

K⊂J

AK u(|J−K|−1)(d−1) J ⊂ 1, . . . , n− 1 and J is long

Proof. We use the notations of the proof Theorem 62.12, withM = (Sd−1)n−1,etc. Recall from Proposition 62.9 that

H∗G(M) ≈ Z2[u,A1 . . . , An−1]/I1

where I1 is the ideal generated by relators (1). Denote by I2, I3′ and I3 theideals of H∗(M) generated by, respectively, relators (2), (3’) and relators (3) ofCorollary 62.15. It was shown in Theorem 62.12 and Corollary 62.15 that

(62.22) J = ker(H∗G(M)→ H∗G(BCnd (ℓ)) = I2 + I3 .In view of Proposition 62.20, we have to prove that the “quotient ideal”

J = x ∈ H∗G(M) | ud−1x ∈ J is equal to I2 + I3′ .

That I2 +I3′ ⊂ J is obvious. For the reverse inclusion, let x ∈ J . By (62.22),one has ud−1x = y2 + y3 for some y2 ∈ I2 and y3 ∈ I3. As I3 = ud−1I3′ , we canwrite y3 = ud−1y3′ with y3′ ∈ I3′ . Let z = y + y3′ . Then ud−1z ∈ I2. We shallprove that z ∈ I2.

As noticed in the proof of Theorem 62.12, H∗G(M) is the free Z[u]-modulegenerated by AJ (J ⊂ 1, . . . , n− 1). Thus, z admits a unique expression

z =∑

J⊂1,...,n−1

λJAJ ,

with λJ ∈ Z2[u]. Hence,

(62.23) ud−1z =∑

J⊂1,...,n−1

(ud−1λJ )AJ .


But, as ud−1z ∈ I2, one has

(62.24) ud−1z =∑

J⊂1,...,n−1J∪nlong

µJAJ .

As, H∗G(M) is the free Z[u]-module generated by the classes AJ , one deducesfrom (62.23) and (62.24) that λJ = 0 if J ∪ n is short. Thus, z ∈ I2.

The equality J = I2 + I3′ may be also obtained using a partial Groebnercalculus with respect to the variable u, as presented in [87, §6].

Remark 62.25. In the case d = 2, where Cn2 (ℓ) = Nn2 (ℓ), the presentation of

H∗(Cn2 (ℓ)) of Theorem 62.21 was obtained in [87, Corollary 9.2], using techniquesof toric manifolds.

Example 62.26. It is easy to see that Cnd (ℓ) = ∅ if and only k is long forsome k ∈ 1, . . . , n. If k = n then relator (2) for J = ∅ implies that 1 ∈ Iℓ andthus H∗G(Cnd (ℓ)) = 0. If k < n, it is relator (3’) for J = k which implies that1 ∈ Iℓ.

Example 62.27. Let ℓ = (1, 1, 1, ε), with ε < 1. The presentation ofH∗G(BC4d(ℓ))

given by Corollary 62.15 takes the form

H∗G(BCnd (ℓ)) ≈ Z2[u,A1, A2, A3]/Iℓ

with Iℓ being the ideal generated by A2i +Aiu

d−1 (i = 1, 2, 3), AJ for |J | = 2, andrelators (3) for J = 1, 2, 1, 3 and 2, 3, which are

ud−1(ud−1 +A1 +A2)

ud−1(ud−1 +A2 +A3)

ud−1(ud−1 +A1 +A3) .

The sum of these relators equals u2(d−1) which thus belongs to Iℓ. Relator (3) for

J = 1, 2, 3 does not bring new generators for Iℓ.The presentation of H∗G(C4

d(ℓ)) given by Theorem 62.21 is similar, with relators(3) replaced by relators (3’):

ud−1 +A1 +A2

ud−1 +A2 +A3

ud−1 +A1 +A3 .

The sum of these relators being equal to ud−1, we get that the three classes Ai ∈Hd−1G (BC4

d(ℓ)) are mapped to the same class A ∈ Hd−1G (C4

d(ℓ)). Therefore,

(62.28) H∗G(C4d(ℓ)) ≈ Z2[u,A]

/(ud−1, A2) .

Note that C4d(ℓ) is G-diffeomorphic to the unit tangent space T 1Sd−1 (by orthonor-

malizing (z1, z2)). Thus, (62.28) is a presentation of H∗(C4d(ℓ)) ≈ H∗((T 1Sd−1)/G).

In the presentations of H∗G(BCnd (ℓ)) and H∗G(Cnd (ℓ)) given in Corollary 62.15and Theorem 62.21, the integer d is only used to fix the degree of the variables Ai.Here is an application of that.


Lemma 62.29. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. For d ≥ 2, thereis an isomorphism of graded rings

ΨBCd : H∗G(BCn2 (ℓ))≈−→ H

∗(d−1)G (BCnd (ℓ)) and ΨCd : H∗G(Cn2 (ℓ))

≈−→ H∗(d−1)G (Cnd (ℓ))

which multiply the degrees by d− 1.

Proof. For an integer a ≥ 2, set Ma = (Sa−1)n−1 and BCa = BCna (ℓ). AsMa is equivariantly formal, Corollary 28.32 applied to the bundle Ma → (Ma)G →RP∞ implies that

Pt(H∗G(Ma)) = Pt(Ma) ·Pt(RP

∞) =(1 + ta−1)n−1

1− t .

Hence

Pt(H∗(d−1)G (Md)) =

(1 + td−1)n−1

1− td−1= Ptd−1(H∗G(M2))

which implies that

(62.30) dimHpG(M2) = dimH

p(d−1)G (Md)

for all p ∈ N.By eliminating the variables Bi in the presentation of H∗G(Ma) given in Propo-

sition 62.9, we get the presentation

H∗G(Ma) ≈ Z2[ua, Aa1 . . . , A

an−1]

/((Aai )

2 = ua−1a Aai

),

where Aai is of degree a− 1 and ua is of degree 1. Therefore, the correspondences

u2 7→ ud−1d and A2

i 7→ Aai define a homomorphism of graded rings Ψd : H∗G(M2)→H∗(d−1)G (Md), multiplying the degrees by d−1, which is clearly surjective. By (62.30),

Ψd is an isomorphism.By Corollary 62.15, H∗G(BCa) is the quotient of H∗G(Ma) by Ia2 + Ia3 , where

Iaj is the ideal generated by relators (j) of Corollary 62.15. As Ψd(I2j ) = Idj ∩

H∗(d−1)G (Md), the isomorphism Ψd descends to the required isomorphism ΨBCd . In

the same way, we construct ΨCd using Theorem 62.21.

62.3. Non-equivariant cohomology. The G-cohomology computations ofCorollary 62.15 and Theorem 62.21 give informations on the non-equivariant coho-mology of BCnd (ℓ) and Cnd (ℓ). We start with the big chain space.

Theorem 62.31. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. The Poincarepolynomial of BCnd (ℓ) is

(62.32) Pt(BCnd (ℓ)) =∑

J⊂1,...,n−1J∪nshort

t|J|(d−1) +∑

K⊂1,...,n−1K long

t|K|(d−1)−1 .

The proof of Theorem 62.31 makes use the simplicial complex Sh×n (ℓ) whosesimplexes are the non-empty subsets of Shn(ℓ).

Proof. Let BCd = BCnd (ℓ). By (39.13), one has a short exact sequence

(62.33) 0→ H∗G(BCd)/(u) ρ−→ H∗(BCd) tr∗−−→ Ann (u)→ 0 ,

whence

(62.34) Pt(BCd) = Pt(H∗G(BCd)/(u)) + Pt(Ann (u)) .


The presentation of H∗G(BCd) given in Corollary 62.15 implies that

(62.35) H∗G(BCd)/(u) ≈ Z2[A1, . . . , An−1]/J

where J is the ideal generated by the squares A2i of the variables and the monomials

AJ when J ∪ n is long. Therefore,

(62.36) H∗G(BCd)/(u) ≈ Λd−1(Sh×n (ℓ)) ,

the face exterior algebra of the simplicial complex Sh×n (ℓ) (see § 28.8). Then, byCorollary 28.83,(62.37)

Pt(H∗G(BCd)/(u)) = Pt(Λd−1(Sh×

n (ℓ))) =X

σ∈S(Sh×n (ℓ))

t(dim σ+1)(d−1) =X

J∈Shn(ℓ)

t|J|(d−1) .

Let us assume that d ≥ 3. The graded algebra H∗G(BCd) is concentrated indegrees ∗(d − 1). We claim that Ann (u) is concentrated in degrees ∗(d − 1) −1. Indeed, let us write H∗G(BCd) as the quotient H∗(M)/J as in the proof ofTheorem 62.21. A class 0 6= z ∈ Hp

G(BCd) is the image of z ∈ H∗G(M). As M isequivariantly formal, one has uz 6= 0. Hence, if z ∈ Ann (u), one has 0 6= uz ∈ J .As the ideal J is concentrated in degrees ∗(d− 1), we deduce that p = q(d− 1)− 1.Together with (62.33), this implies that

(62.38) H∗(d−1)(BCd) ≈ H∗G(BC)/(u) , H∗(d−1)−1(BCd) ≈ Ann (u)

and H∗(BCd) vanishes in other degrees. Since dimBCd = (n−1)(d−1)−1, Poincareduality gives the following formula

(62.39) Pt(BCd) =∑

J∈Shn(ℓ)

t|J|(d−1) +∑

J∈Shn(ℓ)

t(n−1−|J|)(d−1)−1 .

Using (62.33), we thus get, for d ≥ 3, that

(62.40) Pt(Ann (u)) =∑

J∈Shn(ℓ)

t((n−1−|J|)(d−1)−1)(d−1) =∑


t|K|(d−1)−1 ,

where the last equality is obtained by re-indexing the sum with K = 1, . . . , n −1 − J . It remains to prove that (62.40) is also valid when d = 2.

Let us fix some integer d ≥ 3. By Lemma 62.29 and its proof, there is an

isomorphism of graded rings ΨBCd : H∗(BC2) ≈−→ H∗(d−1)(BCd) such that

(62.41) ΨBCd(Ann (u;H∗(BC2)

)= Ann (ud−1;H∗(d−1)(BCd)) ,

where the second argument in Ann( ) specifies the ring in which the first argument isconsidered. As the relators of the presentation of H∗(BCd) given in Corollary 62.15are in degree ∗(d− 1), the correspondence x 7→ ud−2x provides, for every p ≥ 0, anisomorphism of Z2-vector spaces

Φd : Hp(d−1)(BCd) ≈−→ H(p+1)(d−1)−1(BCd) .We thus get an isomorphism of Z2-vector spaces

Φd : H∗(d−1)(BCd) ≈−→ H∗(d−1)−1(BCd)multiplying the degrees by d− 2 and satisfying

(62.42) Ψd

(Ann (ud−1;H∗(d−1)(BCd))

)= Ann (u;H∗(BCd)) .


From (62.41) and (62.42), we get

(62.43) td−2 Ptd−1

(Ann (u;H∗(BC2))

)= Pt(Ann (u;H∗(BCd))) .

The right hand of (62.43) being given by (62.40), we checks that

Pt(Ann (u;H∗(BC2))) =∑


t|K|−1

is the unique solution of Equation (62.43).We have thus proven that (62.40) is valid for all d ≥ 2. Together with (62.37)

and (62.34), this establishes the proposition.

Here below the counterpart of Theorem 62.31 for chain spaces. It requires thelength vector ℓ be dominated, i.e. satisfying ℓn ≥ ℓi for i ≤ n.

Theorem 62.44. Let ℓ be a generic length vector which is dominated. Then,the Poincare polynomials of Cnd (ℓ) is

(62.45) Pt(Cnd (ℓ)) =∑

J⊂1,...,n−1J∪nshort

t|J|(d−1) +∑


t(|K|−1)(d−1)−1.

Theorem 62.44 above reproves the computations of the Betti numbers of Cnd (ℓ)obtained by other methods in [58, Theorem 1] and [56, Theorem 2.1].

Proof. The proof is the same as that of Theorem 62.31, using the isomorphismΨCd of Lemma 62.29, instead of ΨBCd . The hypothesis that ℓ is dominated is used toobtain the analogue of Equation (62.36), namely

(62.46) H∗G(Cnd (ℓ))/(u) ≈ Λd−1(Sh×n (ℓ)) .

Indeed, let J ⊂ 1, . . . , n − 1 be a long subset and k ∈ J . As ℓ is dominated,the set (J − k) ∪ n is long. Therefore, the constant terms in relators (3’) ofTheorem 62.21 vanish and these relators are all multiples of ud−1. Equation (62.46)thus follows from Theorem 62.21.

Example 62.47. The length vector ℓ = (1, 1, . . . , 1) is dominated and is genericif n = 2r + 1. A subset J of 1, . . . , n is short if and only if |J | ≤ r. Hence, ford = 2, Equation (62.45) gives

Pt(C2r+12 (1, . . . , 1)) = Pt(N 2r+1

2 (1, . . . , 1)) =∑

k≤r−1

(n−1k

)tk +

∑

k≥r−1

(n−1k+2

)tk .

This formula was first proven in [108, Theorem C].

Remark 62.48. The hypothesis that ℓ is dominated is necessary (for any d)in Theorem 62.44, as shown by the example C = C4

d(ℓ) for ℓ = (1, 1, 1, ε) (seeExample 62.27). As C is diffeomorphic to the unit tangent space T 1Sd−1, one hasPt(C) = 1+ td−2 + td−1 + t2(d−1)−1, as seen in Example 33.22, while Theorem 62.44would give 1+3td−2+3td−1+t2(d−1)−1. What goes wrong is Formula (62.46). Usingthe presentation of H∗G(C) given in Theorem 62.21, one gets that H∗(d−1)(C) is the

quotient of Λd−1(Sh×n (ℓ)) by the constant terms of relators (3’) in Theorem 62.21,namely

∑j∈J AJ−j for all J ⊂ 1, . . . , n− 1 which are long.


Remark 62.49. Let C = Cnd (ℓ) with ℓ generic and dominated. As observed

in [63, Proposition A.2.4], H∗(C) is determined by H∗(d−1)(C) when d > 3, usingPoincare duality. Indeed, by Theorem 62.21 and Equation (62.46), Z = ZJ =ρ(AJ) | J ∈ Sh(ℓ) is a Z2-basis of H∗(d−1)(C) (X∅ = 1). The bilinear mapH∗(d−1)(C)×H∗(d−1)−1(C)→ Z2 given by

(x, y) 7→ 〈x y, [C]〉is non degenerate (see Theorem 32.18) and thus identifiesH∗(d−1)−1(C) withH∗(d−1)(C)♯.Let Y = YJ | J ∈ Sh(ℓ) be the Z2-basis of H∗(d−1)−1(C) which is dual to Z underthis identification. In particular, Y∅ = [C], the generator of H(n−2)(d−1)−1(C) = Z2

(we say that Y is the Poincare dual basis to the basis Z). One has then the followingmultiplication table.

(62.50) ZJ ZK =

ZJ∪K if J ∩K = ∅ and J ∪K ∈ Shn(ℓ)

0 otherwise,

(62.51) ZJ YK =

YJ−K if K ⊂ J0 otherwise

and

(62.52) YJ YK = 0 .

Indeed, (62.50) comes from the corresponding relation amongst the classes AJ .Formula (62.52) is true for dimensional reasons, since d > 3. For (62.51), note that,ZJ YK ∈ H(n−2−(|K|−|J|)(d−1)−1(C) and hence may be uniquely written as alinear combination

ZJ YK =∑

L∈L

λLYL ,

where L is the set L ∈ Shn(ℓ) with |L| = |K| − |J |. If I ∈ L, one has on one hand

〈ZI ∑

L∈L

λLYL, [C]〉 = λI

and on the other hand

〈ZI (ZJ YK), [C]〉 = 〈ZI∪J YK , [C]〉 =

1 if J ∩K = ∅ and I ∪ J = k

0 otherwise.

This shows that λL = 1 if and only if L = K − J .

We finish this subsection with some illustrations and applications of Theo-rems 62.31 and 62.44. The lopsidedness lops (ℓ) of a length vector ℓ = (ℓ1, . . . , ℓn)is defined by

lops (ℓ) = mink | ∃ J ⊂ 1, . . . , n− 1 with J long and |J | = k .The terminology is inspired by that of [89]. We note the equation

(62.53) dimSh×n (ℓ) = n− lops (ℓ)− 2 .

Example 62.54. For a generic length vector ℓ = (ℓ1, . . . , ℓn), the conditionlops (ℓ) = 0 is equivalent to n being long. By Theorem 62.31 this is equivalentto BCnd (ℓ) = ∅: otherwise J = ∅ produces a non-zero summand in the first sumof (62.32) (Compare Example 62.16). The chamber of ℓ is unique, represented bye.g. ℓ0 = (ε, . . . , ε, 1) with ε < 1/(n− 1).


Example 62.55. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector with lops (ℓ) =

1. From the second sum of (62.32), we see that this is equivalent to Hd−2(BCnd (ℓ)) 6=0 (the reduced cohomology is relevant for the cases d = 2, where it says that BCn2 (ℓ)is not connected). We check that Shn(ℓ) is poset isomorphic to Shn(ℓ

0) whereℓ0 = (ε, . . . , ε, 2, 1), with ε < 1/(n− 2). By Lemma 62.6, the chamber of ℓ is welldetermined modulo the action of Symn−1. The O(d − 1)-diffeomorphism type of

BCnd (ℓ0) may be easily described. It is clear that BC2d(2, 1) ≈ Sd−2. Therefore, by

Lemma 62.7, BCnd (ℓ0) ≈ (Sd−1)n−2×Sd−2. When d = 2, this is the only case whereBCn2 (ℓ) is not connected, as shown by Theorem 62.31. Note that Cnd (ℓ0) is emptyby Theorem 62.44, which is coherent with Proposition 62.20.

Example 62.56. Let ℓ = (ℓ1, . . . , ℓn) be a dominated generic length vectorwith lops (ℓ) = 2. We check that Shn(ℓ) is poset isomorphic to Shn(ℓ

0) where ℓ0 =(ε, . . . , ε, 1, 1, 1), with ε < 1/(n−3) (compare [83, Remark 2.4]. By Lemma 62.6, thechamber of ℓ is well determined modulo the action of Symn−1. As in the previousexample, we can describe the O(d − 1)-diffeomorphism type of BCnd (ℓ0). Suppose

first that n = 3. The Morse function f : (S(d−1)2 → [−3, 3] of Lemma 62.4 has nocritical point between its minimum and the level set f−1(−1) = BC3

d(1, 1, 1). By

the Morse Lemma, BC3d(1, 1, 1) is diffeomorphic to S2(d−1)−1. Using Lemma 62.7,

we deduce that BCnd (ℓ0) ≈ (Sd−1)n−3 × S2(d−1)−1. In the same way, one provesthat Cnd (ℓ0) ≈ (Sd−1)n−3 × Sd−2. Using Formula (62.45), we see that lops (ℓ) = 2if and only if Cn2 (ℓ) is not connected.

The following lemma uses nilpotency class nil introduced in § 25).

Lemma 62.57. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. Then

(a) If lops (ℓ) > 1, then lops (ℓ) = n− nilH>0(BCnd (ℓ)) + 1.

(b) Suppose that ℓ is dominated. If d > 2 or lops (ℓ) > 2, thenlops (ℓ) = n− nilH>0(Cnd (ℓ)) + 1.

Proof. Let BC = BCnd (ℓ). Suppose that lops (ℓ) = k ≥ 2. By Sequence (62.33),

the algebra H∗(BC) contains a copy of H∗G(BC) ≈ Λd−1(Sh×n (ℓ)). By (62.53),

dimSh×n (ℓ) = n− k− 2. Therefore, there exists x1, . . . , xn−k−1 ∈ Hd−1(BC) whosecup product v does not vanish in H(n−k−1)(d−1)(BC). By Poincare duality Theo-rem 32.18, there is w ∈ Hk(d−1)−1(BC) such that v w 6= 0 in H(n−1)(d−1)−1(BC).As k ≥ 2 the number k(d− 1)− 1 is strictly positive since d ≥ 2. Thus, v w is anon-vanishing cup product of length n−k. Such a length is the maximal possible, asseen using Sequence (62.33). Hence, nilH>0(BCnd (ℓ)) = n− k+ 1. This proves (a).

The proof of (b) is similar, using Theorem 62.44 and its proof instead of Theo-rem 62.31. As dim C = (n− 2)(d− 1)− 1, the class v is of degree (k− 1)(d− 1)− 1.The latter is strictly positive if d > 2 or k > 2.

Corollary 62.58. Let ℓ = (ℓ1, . . . , ℓn) and ℓ′ = (ℓ′1, . . . , ℓ′n) be two generic

length vectors. Suppose that, for some d ≥ 2, there exists a GrA-isomorphismH∗(BCnd (ℓ)) ≈ H∗(BCnd (ℓ′)). Then

lops (ℓ) = lops (ℓ′) .

If ℓ and ℓ′ are both dominated, then the above equality holds true if there exists aGrA-isomorphism H∗(Cnd (ℓ)) ≈ H∗(Cnd (ℓ′)).


Proof. For the big chain space BCnd ( ), this is follows from Lemma 62.57,except when d = 2 and lops (ℓ) ≤ 1, cases which are covered by Examples 62.54and 62.55. The argument for Cnd ( ) is quite similar. The case lops (ℓ) = 2 is coveredby Example 62.56. The case lops (ℓ) = 1 is not possible if ℓ is dominated, solops (ℓ) = 0 is equivalent to Cnd (ℓ) = ∅.

62.4. The inverse problem. By Lemma 62.6, the diffeomorphism type ofBCnd (ℓ) or Cnd (ℓ) is determined by the chamber Ch(ℓ) (up to the action of Symn−1).The inverse problem consists of recovering Ch(ℓ) by algebraic topology invariantsof BCnd (ℓ) (or Cnd (ℓ)). We start by the big chain space.

Proposition 62.59. Let ℓ = (ℓ1, . . . , ℓn) and ℓ′ = (ℓ′1, . . . , ℓ′n) be two generic

length vectors. Then, the following conditions are equivalent.

(1) Ch(ℓ′) = Ch(ℓσ) for some σ ∈ Symn− 1.

(2) BCnd (ℓ) and BCnd (ℓ′) are O(d − 1)-diffeomorphic.

(3) H∗G(BCnd (ℓ)) and H∗G(BCnd (ℓ′)) are GrA[u]-isomorphic.

Moreover, if d > 2 or n > 3, any condition (1)–(3) above is equivalent to

(4) H∗(BCnd (ℓ)) and H∗(BCnd (ℓ′)) are GrA-isomorphic.

Finally, if d > 2 or if lops (ℓ) 6= 2, then any condition (1)–(3) above is equiva-lent to

(5) H∗(BCnd (ℓ)) and H∗(BCnd (ℓ′)) are GrA-isomorphic.

That (5) implies (1) is not known in general if d = 2. That (4) implies (3) is

wrong if n = 3 and d = 2. Indeed, BC32(1, 3, 1) and BC32(1, 1, 1) are connected closed1-dimensional manifolds, thus both diffeomorphic to S1 but, by Corollary 62.15, onehas

H∗G(BC32(1, 3, 1)) ≈ Z2[u,A1]

/(A2

1, u) while H∗G(BC32(1, 1, 1)) ≈ Z2[u]

/(u2) .

Implications like (4) ⇒ (2) or (5) ⇒ (2) are in the spirit of Proposition 23.7:characterising a closed manifold (within some class) by algebraic topology tools.This was the historical goal of algebraic topology (see p. 169).

Proof. As ℓ and ℓ′ are generic, one has H∗(BCnd (ℓ)) ≈ H∗G(BCnd (ℓ)) and thesame for ℓ′. The following implications are then obvious, except (a) which wasestablished in Lemma 62.6.

(1)(a) +3 (2) +3

$BB

BBBB

BBBB

BB(3) +3 (4)

(5)

We shall now prove that (3) ⇒ (1), (4) ⇒ (3) and finally (5) ⇒ (1).

(3) ⇒ (1). A GrA[u]-isomorphism H∗G(BCnd (ℓ))≈−→ H∗G(BCnd (ℓ′)) descends to a

GrA-isomorphism : H∗G(BCnd (ℓ))/(u)≈−→ H∗G(BCnd (ℓ′))/(u). By (62.36), this implies

that Λd−1(Sh×n (ℓ)) and Λd−1(Sh×n (ℓ′)) are GrA-isomorphic. Using Lemma 28.80and Proposition 28.79, we deduce that the simplicial complexes Sh×n (ℓ) and Sh×n (ℓ′)are isomorphic. It follows that Shn(ℓ) and Shn(ℓ

′) are poset isomorphic. ByLemma 62.6, this implies (1).


(4) ⇒ (3). Let β : H∗(BCnd (ℓ))≈−→ H∗(BCnd (ℓ′)) is a GrA-isomorphism and let

β(u) = v. We must prove that v = u. This is obvious for d > 2 since H1(BCnd (ℓ′)) =Z2. If d = 2, Corollary 62.15 implies that

(62.60) xu = xv = x2

for all x ∈ H1(BCn2 (ℓ′)). By Corollary 62.15 again, H∗(BCn2 (ℓ′)) is generated in

degree 1, so (62.60) implies that (u + v)x = 0 for all x ∈ H∗(BCn2 (ℓ′)). By Corol-

lary 62.5, BCn2 (ℓ′) is a closed manifold of dimension > 1 (since n ≥ 4). We concludethat v = u by Poincare duality, using Theorem 32.18.

(5) ⇒ (1). Suppose first that d > 2. By (62.38) and (62.36), Condition (5) impliesthat Λd−1(Sh×n (ℓ)) and Λd−1(Sh×n (ℓ′)) are GrA-isomorphic. The argument is thenthe same as that for (3) ⇒ (1).

We now assume that d = 2. By Corollary 62.58, Condition (5) implies thatlops (ℓ) = lops (ℓ′). Let L = lops (ℓ) = lops (ℓ′). The cases L = 0, 1 were treated inExamples 62.54 and 62.55. Let us assume that L > 2. By Theorem 62.31 and itsproof, the subalgebra of H∗(BCn2 (ℓ)) (respectively: H∗(BCn2 (ℓ′))) generated by theelements of degree one is isomorphic to Λ1(Sh×n (ℓ)) (respectively: Λ1(Sh×n (ℓ′))). ByCondition (5), this implies that Λ1(Sh×n (ℓ)) and Λ1(Sh×n (ℓ′)) are GrA-isomorphicand the proof that Ch(ℓ′) = Ch(ℓσ) proceeds as that for (3) ⇒ (1).

Here is the analogue of Proposition 62.59 for the chain spaces.

Proposition 62.61. Let ℓ = (ℓ1, . . . , ℓn) and ℓ′ = (ℓ′1, . . . , ℓ′n) be two generic

length vectors. Suppose that ℓ and ℓ′ are dominated. Then, the following conditionsare equivalent.

(1) Ch(ℓ′) = Ch(ℓσ) for some σ ∈ Symn− 1.

(2) Cnd (ℓ) and Cnd (ℓ′) are O(d − 1)-diffeomorphic.

(3) H∗G(Cnd (ℓ)) and H∗G(Cnd (ℓ′)) are GrA[u]-isomorphic.

Moreover, if d > 2 or n > 4, then any condition (1)–(3) above is equivalent to

(4) H∗(Cnd (ℓ)) and H∗(Cnd (ℓ′)) are GrA-isomorphic.

Finally, if d > 2 or if lops (ℓ) 6= 3, then any condition (1)–(3) above is equiva-lent to

(5) H∗(Cnd (ℓ)) and H∗(Cnd (ℓ′)) are GrA-isomorphic.

Proof. The proof is the same as that of Proposition 62.59, except for thefollowing small differences. For (3) ⇒ (1), instead of (62.36), one uses Equa-tion (62.46), using that ℓ and ℓ′ are dominated. For (4) ⇒ (3), the hypothesisthat n > 4 guarantees that dim Cn2 ( ) > 1. For (5) ⇒ (1), one uses Theorem 62.44instead of Theorem 62.31.

Remark 62.62. In Proposition 62.61, implication (4)⇒ (1) is wrong for d = 2and n = 4: BC4

2(1, 1, 1, 2) and BC42(1, 2, 2, 2) are connected closed 1-dimensional

manifolds, thus both diffeomorphic to S1. Implication (5) ⇒ (1) is not known ingeneral if d = 2. It is however true if one uses the integral cohomology: this difficultresult, conjectured by K. Walker in 1985 [198] was proved by D. Schutz in 2010[167], after being established when lops (ℓ) 6= 3 in [56, Theorem 4] (length vectorswith lopsidedness > 3 are called normal in [56, 167]).

The hypothesis that ℓ is dominated in Proposition 62.61 is essential, as seen byProposition 62.63 and Lemma 62.64 below.


Proposition 62.63. Let ℓ be a generic length vector and let σ ∈ Symn. Ifd 6= 3, then H∗(Cnd (ℓ)) and H∗(Cnd (ℓσ)) are GrA–isomorphic.

Proposition 62.63 was first proved by V. Fromm in his thesis [63, Cor. 1.2.5].It is wrong if d = 3: for ε small, C4

3(ε, 1, 1, 1) is diffeomorphic to S2×S1 (see Exam-ple 62.56) while C4

3(1, 1, 1, ε) is diffeomorphic to T 1S2 ≈ RP 3 (see Example 62.27).We give below a proof of Proposition 62.63 based on an idea of D. Schutz, using

the following lemma.

Lemma 62.64. If d = 2, 4, 8, then Cnd (ℓ) is diffeomorphic to Cnd (ℓσ) for anyσ ∈ Symn.

The hypothesis d = 2, 4, 8 is essential in the above lemma. Indeed, for ε small,C4d(ε, 1, 1, 1) is diffeomorphic to Sd−1×Sd−2 (see Example 62.56) while C4

d(1.1.1, ε)is diffeomorphic to T 1Sd−1 (see Example 62.27). As d ≥ 2, these two spaces havethe same homotopy type only when d = 2, 4, 8 (see Example 33.22).

Proof. Identifying Rd with C, H or O, we get a smooth multiplication onSd−1 with e1 as unit element. Consider the smooth map π : Nn

d (ℓ) → Cnd (ℓ) givenby

π(z1 . . . , zn) = −z−1n (z1, . . . , zn−1) .

The embedding j : Cnd (ℓ)→ Nnd (ℓ) given by

j(z1, . . . , zn−1) = (z1, . . . , zn−1,−e1)is a section of π. Consider the composed map

Cnd (ℓ)j // Nn

d (ℓ)hσ

≈// Nn

d (ℓσ)π // Cnd (ℓσ)

where hσ(z) = zσ. Then, πhσ j is a diffeomorphism: a direct computation showsthat its inverse is π(hσ)−1

j.

Proof of Proposition 62.63. The case d = 2 is covered by Lemma 62.64,so we assume that d ≥ 4. As observe in Remark 62.48, one has a GrA-isomorphism

(62.65) H∗(d−1)(Cnd (ℓ) ≈ Ωd(ℓ)

where Ωd(ℓ) is the quotient of Z2[A1, . . . , An−1] (Ai of degree d − 1) by the idealgenerated by A2

i , AJ when J ∪ n is ℓ-long and∑j∈J AJ−j when J is ℓ-long.

By Lemma 62.64, there exists a ring isomorphism q4 : Ω4(ℓ)≈−→ Ω4(ℓ

σ). But, inthe definition of Ωd( ), the integer d is only used to fix the degree of the vari-ables Ai, and thus the grading of Ωd( ) (as non-graded rings, the rings Ωd(ℓ) areisomorphic for all d). Therefore, the isomorphism q4 defines a GrA-isomorphism

qd : Ωd(ℓ)≈−→ Ωd(ℓ

σ) for all d ≥ 2. Together with (62.65), this gives a GrA-

isomorphism qd : H∗(d−1)(Cnd (ℓ))≈−→ H∗(d−1)(Cnd (ℓσ)) when d ≥ 3.

Without loss of generality, we may assume that ℓ is dominated. Remark 62.49thus provides an additive basis Z ∪ Y of H∗(Cnd (ℓ)). The set Z ′ = ZJ = qd(ZJ) |J ∈ Sh(ℓ) is then a Z2-basis of H∗(d−1)(Cnd (ℓσ). Let Y ′ = Y ′J | J ∈ Sh(ℓ) be

the basis of H∗(d−1)−1(Cnd (ℓσ) which is Poincare dual to Z ′, as in Remark 62.49.Relations (62.50)–(62.52) hold true both for Z ∪ Y in H∗(Cnd (ℓ)) and for Z ′ ∪ Y ′in H∗(Cnd (ℓσ)). Therefore, qd extends to a GrA-isomorphism qd : H∗(Cnd (ℓ))

≈−→H∗(Cnd (ℓσ)) when d > 3.


62.5. Spatial polygon spaces and conjugation spaces. The integral co-homology ring of the spatial polygon space Nn

3 (ℓ) has been computed in [87, The-orem 6.4]. The result is as follows.

Theorem 62.66. Let ℓ = (ℓ1 . . . , ℓn) be a generic length vector. Then, there isa graded ring isomorphism

Z[v,A1 . . . , An−1]/Iℓ ≈−→ H∗(Nn

3 (ℓ); Z)

where the variables v and Ai are of degree 2 and Iℓ is the ideal generated by thefollowing families of relators

(1) A2i +Aiv i = 1, . . . , n− 1

(2) AJ J ⊂ 1, . . . , n− 1 and J ∪ n is long

(3′)∑

K⊂J

AK v|J−K|−1 J ⊂ 1, . . . , n− 1 and J is long

Theorem 62.66 says in particular that Hodd(Nn3 (ℓ); Z) = 0. The Bockstein

exact sequence for 0 → Z → Z → Z2 → 0 (see [175, Ch. 5, Sec. 2, Theorem 11])thus implies that H∗(Nn

3 (ℓ)) ≈ H∗(Nn3 (ℓ); Z) ⊗ Z2. Using Theorem 62.21, the

correspondence Ai 7→ Ai and v 7→ u thus provides a graded ring isomorphism

H2∗(Nn3 (ℓ))

≈−→ H2∗(BCn2 (ℓ)) ≈ H2∗(Nn2 (ℓ))

which divides the degrees by half. This suggests that Nn3 (ℓ) is a conjugation space,

which we shall prove below. The involution τ on R3 given by the reflection throughthe horizontal plane induces an involution, still called τ , on Nn

3 (ℓ), with fixed pointset equal to Nn

2 (ℓ).

Proposition 62.67. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. Then, thespace Nn

3 (ℓ) endowed with the involution τ is a conjugation manifold.

The proofs of this proposition use some Hamiltonian geometry, directly or indi-rectly, so their understanding requires some knowledge in the subject, as presentedin e.g. [11, Chapters II and III]. Recall that, if ℓ is generic, Nn

d (ℓ) is a symplecticmanifold [116, 109, 86, 87]

Lemma 62.68. Let ℓ = (ℓ1, . . . , ℓn) be a generic length vector. Then,

(a) Nnd (ℓ) is simply connected.

(b) τ induces the multiplication by (−1) on H2(Nn3 (ℓ); Z).

(c) τ is anti-symplectic.

Proof. The proof of (a) and (b) proceeds by induction on n. When n = 3,there are two chambers up to permutation, represented by ℓ0 = (1, 1, 2) andℓ1 = (1, 1, 1). As N 3

3 (ℓ0) = ∅ and N 33 (ℓ1) = N 3

2 (ℓ1) = pt, (a) and (b) are true. Forthe induction step, we consider the diagonal-length function δ : Nn

3 (ℓ) → R givenby δ(z) = |ℓnzn − ℓn−1zn−1|. By Lemma 62.6, we can slightly change ℓn−1 with-out modifying the O(2)-diffeomorphism type of Cn3 (ℓ) and thus the τ -equivariantdiffeomorphism type of Nn

3 (ℓ). Therefore, we can assume that ℓn−1 6= ℓn, in whichcase δ is a smooth map. Using [81, Theorem 3.2], we deduce that δ is Morse-Bott function. The critical points are of even index and are isolated, except pos-sibly for the two extrema. The preimage of the maximum is either a point or

63. EQUIVARIANT CHARACTERISTIC CLASSES 387

Nn−13 (ℓ1, . . . , ℓn−2, ℓn−1 + ℓn) and the preimage of the minimum is either a point

or Nn−13 (ℓ1, . . . , ℓn−2, |ℓn−1 − ℓn|). This proves (a) by induction.The restriction δ′ of δ to Nn

2 (ℓ) ⊂ Nn−13 (ℓ) is also Morse-Bott with Crit δ′ =

Crit δ but the index of each critical point is divided in half. Thus, passing a criticalpoint of δ index 2 corresponds to add a conjugation 2-cell. This proves (b) byinduction on n.

To prove (c), we use that Nn−13 (ℓ) is the SO(3)-symplectic reduction at 0 of∏n

i=0 S2ℓi

where S2ℓi

is the standard 2 sphere equipped with the SO(3)-homogeneoussymplectic form with symplectic volume equal to 2ℓi (see [87]). The involutionτ is clearly anti-symplectic on S2

ℓiand this property descends to the symplectic

reduction.

Proof of Proposition 62.67. We give below three proofs. The first one isthat indicated in [85, Example 8.7].

1st proof. By induction on n, using the function δ : Nn3 (ℓ) → R of the proof

of Lemma 62.68. For n = 3, N 33 (ℓ) is either empty or a point, which starts the

induction. The induction step uses that δ is the moment map of an S1-Hamiltonianaction on Nn

3 (ℓ) [116] and called the bending flow [109]. It may be visualizedas a rotation of zn−1 and zn at constant speed around of axis ℓnzn + ℓn−1zn−1,leaving the other other zi’s fixed. As a moment map for a circle action, it satisfies

Critδ = (Nn3 (ℓ))S

1

. We have seen that the critical points of δ are isolated or

polygon space with fewer edges. Hence, by induction hypothesis, (Nn3 (ℓ))S

1

is aconjugation space. The involution τ is anti-symplectic by Lemma 62.68 and satisfiesτ(γz) = γ−1τ (z) for all z ∈ N 3

3 (ℓ) and γ ∈ S1. That Nn3 (ℓ) is a conjugation

manifold thus follows from [85, Theorem 8.3].

2nd proof. by a symplectic reduction for the Hamiltonian action of a maximal torus.The complex conjugation on Gr(2; Cn) is anti-symplectic, with fixed point set equalto Gr(2; Rn), descends to the involution τ on Nn

3 (ℓ). The manifold Gr(2; Cn) withthe complex conjugation is a conjugation space (see p. 361 or Remark 57.14). ThatNn

3 (ℓ) is a conjugation space thus follows from [85, Theorem 8.12].

3rd proof. By Theorem 62.66, Hodd(Nn3 (ℓ)) = 0 and H∗(Nn

3 (ℓ)) is generated byH2(Nn

3 (ℓ)). By Lemma 62.68, Nn3 (ℓ) is simply connected and H2τ is multiplication

by (−1). Therefore, Nn3 (ℓ) is as conjugation manifold by results of V. Puppe (see

[163, Theorem 5 and Remark 2]).

Remark 62.69. The quotient BNn3 (ℓ) = BCn3 (ℓ)/SO(2) is called in [87] the

abelian polygon space (being an S1-symplectic reduction while Nn3 (ℓ) is an SO(3)-

symplectic reduction). In [87, Theorem 6.4], the integral cohomology ring ofBNn

3 (ℓ) is computed: the statement is as that of Theorem 62.66 but with therelators of Corollary 62.15. The involution τ is defined on BNn

3 (ℓ) with fixed pointset BCn2 (ℓ). The 4th proof of Proposition 62.67 may be easily adapted to show thatτ is a conjugation on BNn

3 (ℓ).

63. Equivariant characteristic classes

Let Γ be a topological group and let X be a Γ-space. Let ξ = (p : E → X) be aΓ-equivariant vector bundle overX . Recall from Lemma 43.9 that ξ is then inducedby i : X → XΓ from the vector bundle ξΓ over XΓ. The Stiefel-Whitney classes of


ξΓ are called the equivariant Stiefel-Whitney classes of ξ. Hence, w(ξ) = ρ(w(ξΓ))where ρ : H∗Γ(X)→ H∗(X) is the forgetful homomorphism.

An important example is given by the the tautological bundle ξj (j = 1, . . . , r)over the flag manifold Fl(n1, . . . , nr) (see § 55.2), which is an O(n)-equivariantvector bundle of rank nj .

Proposition 63.1. Let R be a closed subgroup of O(n). Then, as an H∗(BR)-algebra, H∗R(Fl(n1, . . . , nr)) is generated by the equivariant Stiefel-Whitney classeswi((ξj)R) (j = 1 . . . , r, i = 1, . . . , nj) of the tautological bundles. In particular,Fl(n1, . . . , nr) is R-equivariantly formal.

Proof. We have noticed above that wi(ξj) = ρ(wi((ξj)R)). By Theorem 55.24,H∗(Fl(n1, . . . , nr)) is additively generated by the monomials in the wi(ξj)’s. Sucha monomial is the image by ρ of the corresponding monomial in the wi((ξj)R)’s(and Fl(n1, . . . , nr) is thus R-equivariantly formal). By the Leray-Hirsch theo-rem 28.26, the monomials in the wi((ξj)R)’s generate H∗R(Fl(n1, . . . , nr)) as anH∗(BR)-module, whence the proposition.

We specialize to the Grassmannian Gr(k; Rn) = Fl(k, n− k) with R = T2, themaximal 2-torus of diagonal matrices inO(n). By (43.21), H∗(BT2) ≈ Z2[u1, . . . , un],with deg(ui) = 1. The tautological bundles are ζ = ξ1 and ζ⊥ = ξ2 (see Exam-ple 55.30). As Gr(k; Rn) is T2-equivariantly formal, the restriction to the fixedpoints r : H∗T2

(Gr(k; Rn))→ H∗T2(Gr(k; Rn)T2 ) is injective by Theorem 44.17. The

fixed point set Gr(k; Rn)T2 is discrete and in bijection with [nk ], the set of binarywords λ = λ1 · · ·λn such that

∑λi = k. We identify Gr(k; Rn)T2 with [nk ] via this

bijection, which associates to λ the coordinate k-plane

Πλ = (t1, . . . , tn) ∈ Rn | ti = 0 if λi = 0 .Note that Πλ is the “center” of the Schubert cell CFλ for the standard complete flagF in Rn (see § 55.3).

Proposition 63.2. For λ = λ1 · · ·λn ∈ [nk ], the restriction homomorphism

rλ : H∗T2(Gr(k; Rn))→ H∗T2

(λ) ≈ Z2[u1, . . . , un]

satisfies

rλ(w(ζT2)) =

∏

λi=1

(1 + ui) and rλ(w((ζ⊥)T2) =

∏

λi=0

(1 + ui) .

We may note that the classes rλ(w(ζT2)) and rλ(w(ζ⊥

T2)) satisfy the GKM-

conditions (see p. 397).

Proof. Let E(ζ)λ be the fiber of ζ over πλ, seen as a T2-equivariant vectorbundle over λ. One has a T2-equivariant isomorphism

E(ζ)λ ≈⊕

λi=1

L(ui)

where L(ui) is the the T2-equivariant line bundle over λ, on which T2 acts viathe homomorphism dia(δ1, . . . , δn) 7→ δi. This homomorphism is associated to uiunder the bijection of (43.18). By Lemma 43.19, w(L(ui)T2) = 1 + ui and, usingLemma 43.12 and (54.4), we get

rλ(w(ζT2)) = w((E(ζ)λ)T2) =

∏

λi=1

(1 + ui) .


The proof of the assertion for ζ⊥ is similar, since

E(ζ⊥)λ = (t1, . . . , tn) ∈ Rn | ti = 0 if λi = 1 .

Remark 63.3. Proposition 63.2 implies that the relation

w(ζT2)w((ζ⊥)T2

) =

n∏

i=1

(1 + ui)

holds true in H∗T2(Gr(k; Rn). In fact, this provides a presentation of H∗T2

(Gr(k; Rn)(see Corollary 64.22 and Remark 64.24).

Example 63.4. Consider Gr(1; R3) ≈ RP 2. The T 2-fixed points are in bijec-tion with 100, 010, 001. As a Z2[u1, u2, u3]-algebra, H∗T2

(RP 2) is generated by

w1 = w1(ζT2) and wi = wi(ζ

⊥T2

) (i = 1, 2). The table of rλ(−) for these classes is

w1 w1 w2

100 u1 u2 + u3 u2u3

010 u2 u1 + u3 u1u3

001 u3 u1 + u2 u1u2

One checks that the relations

(63.5) w1 + w1 = σ1 , w2 + w1w1 = σ2 , w1w2 = σ3 .

are satisfied. For a generalization to Gr(1; Rn), see Example 64.25.

Example 63.6. The Z2[u1, . . . , u4]-algebraH∗T2(Gr(2; R4)) is generated by wi =

wi(ζT2) and wi = wi(ζ

⊥T2

) (i = 1, 2) (i = 1, 2). The table of rλ(−) for these classesis

w1 w1 w2 w2

1100 u1 + u2 u3 + u4 u1u2 u3u4

1010 u1 + u3 u2 + u4 u1u3 u2u4

0110 u2 + u3 u1 + u4 u2u3 u1u4

1001 u1 + u4 u2 + u3 u1u4 u2u3

0011 u3 + u4 u1 + u2 u3u4 u1u2

The following relations are thus satisfied (compare Example 64.27).

(63.7)

w1 + w1 = σ1

w2 + w1w1 + w2 = σ2

w2w1 + w1w2 = σ3

w2w2 = σ4 .

The above results have their analogues in the complex case. The same proofas for Proposition 63.1 gives the following proposition.

Proposition 63.8. Let R be a closed subgroup of U(n). Then, as an H∗(BR)-algebra, H∗R(FlC(n1, . . . , nr)) is generated by the equivariant Chern classes ci((ξj)R)(j = 1 . . . , r, i = 1, . . . , nj) of the tautological bundles. In particular, FlC(n1, . . . , nr)is R-equivariantly formal.


As in the real case, we specialize to the Grassmannian Gr(k; Cn) = Fl(k, n−k)and their tautological bundles ζ = ξ1 and ζ⊥ = ξ2, with R = T , the maximal2-torus of diagonal matrices in U(n). As seen in (43.29), H∗(BT ) ≈ Z2[v1, . . . , vn],with deg(vi) = 2. The associated 2-torus is the maximal 2-torus T2 of diagonalmatrices in O(n), and the action of T2 on Gr(k; Rn) shows that Gr(k; Rn)T =Gr(k; Rn)T2 . As Gr(k; Cn) is T -equivariantly formal, the restriction to the fixedpoints r : H∗T2

(Gr(k; Rn))→ H∗T2(Gr(k; Rn)T2 ) is injective by Theorem 44.29. The

same proof of Proposition 63.2 gives the analogous formulae

rλ(w(ζT )) =∏

λi=1

(1 + vi) and rλ(w(ζ⊥T

)) =∏

λi=0

(1 + vi) .

We now study the equivariant characteristic classes of a rigid Γ-bundle (con-versations with T. Holm were useful for this part). Recall that a Γ-equivariant

vector bundle ξ = (Ep−→ X) is called rigid if the Γ-action on X is trivial (see

p. 254). Since then XΓ ≈ BΓ × X , the equivariant Stiefel-Whitney class w(ξΓ)belongs to H∗(XΓ) ≈ H∗(BΓ) ⊗ H∗(X). If η is a Γ-equivariant vector bundleover a space Y which is Γ-equivariantly formal, and if Γ is a 2-torus for instance,then r(w(ηΓ)) = w(η|Y Γ), r : H∗Γ(Y ) → H∗(Y Γ) is injective (see Theorem 44.17)and η|Y Γ is rigid. Hence, in such cases, rigid equivariant vector bundles play animportant role.

Let ξ = (Ep−→ X) be a rigid Γ-equivariant vector bundle, where Γ is a 2-torus.

Let χ : Γ → O(1) ≈ ±1 be a homomorphism. We call ξ a weight Γ-bundle withrespect to χ (or just a χ-weight Γ-bundle) if γ · v = χ(γ)v for all γ ∈ Γ and v ∈ E.

Lemma 63.9. If Γ as 2-torus, then any rigid Γ-equivariant vector bundle overa locally contractible space decomposes into a Whitney sum ξ =

⊕χ∈hom(Γ,O(1)) ξ

χ

of weight subbundles.

Proof. Let ξ = (Ep−→ X) and let x ∈ X . Let ϕ : p−1(U)

≈−→ U × Rr bea trivialization of ξ over an open set U ⊂ X . Such trivialization is of the formϕ(v) = (p(v), ϕ2(v)), where ϕ2 : p−1(U)→ Rr is a continuous map which is a linearisomorphism on each fiber, and there is a bijection from the set of trivializationsof ξ over U and such maps. A continuous map y 7→ Aϕy from U to O(r) is thus

defined by the equation ϕ2(γv) = Aϕp(v)(γ)ϕ2(v), required to be valid for all γ ∈ Γ

and v ∈ p−1(U). If U retracts by deformation onto b ∈ U , a classical folklore factabout representation theory says that there is a continuous map y 7→ gy from U toO(r), with gb = id, such that Aϕy (γ) = gyA

ϕb (γ)g−1

y (see e.g. [77, Lemma 1.2]). Wecan thus construct a new trivialization ϕ(v) = (p(v), ϕ2(v) of ξ over U by settingϕ2(v) = g−1

p(v)ϕ2(v). One checks that

(63.10) ϕ2(γv) = Aϕb (γ)v ,

in other words, the map y 7→ Aϕy is constant over U . As X is locally contractible,there are such trivializations (Ux, ϕx) as above around each x ∈ X . Define

Eχ = v ∈ E | γ · v = χ(γ)v, ∀ γ ∈ Γ .As Γ as 2-torus, the vector space Ex decomposes into a direct sum of weight sub-spaces

(63.11) Ex =⊕

χ∈hom(Γ,O(1))

Eχx


(see the proof of Lemma 43.23). Clearly, Eχ ∩ p−1(x) = Eχx and, using (63.10),Eχ is the total space of a χ-weight subbundle of ξ. By (63.11), this proves theproposition.

By Lemma 43.12, ξΓ is the Whitney sum of the bundles ξχΓ . Hence, to computethe Stiefel-Whitney classes of rigid Γ-equivariant vector bundles, it suffices to knowthose of χ-weight bundles. The following proposition provides the answer. Weidentify hom(Γ, O(1)) with H1(BΓ) via the bijection of (43.18).

Proposition 63.12. Let Γ be a 2-torus and let χ ∈ H1(BΓ). Let ξ be a χ-weight Γ-bundle of rank r over X. Then, in H∗Γ(X) ≈ H∗(BΓ) ⊗ H∗(X), onehas

(63.13) w(ξΓ) =r∑

k=0

[(1 + χ)k × wr−k(ξ)

].

Proof. Let χ = (R → pt) be the χ-weight line bundle over a point and letχX = q∗χ, where q : X → pt is the constant map. Hence, χX is a χ-weight bundleoverX whose underlying bundle is a trivial line bundle. Note that any bundle η overX may be endowed with a structure of a χ-weight bundle using the isomorphismη ≈ η ⊗ χX. In fact, using the definition of the tensor product bundle of (43.11),

we see that any χ-weight bundle ξ is obtained this way: ξ ≈Γ ξ ⊗ χX , where ξ isthe bundle ξ endowed with the trivial Γ-action. By Lemma 43.12, one has

(63.14) ξΓ ≈ ξΓ ⊗ (χX)Γ .

Consider the two projections πBΓ : XΓ → BΓ and πX : XΓ → X (using that XΓ ≈BΓ × X). As Γ-acts trivially on E(ξ), one checks easily that ξΓ = π∗

Xξ and thus

w(ξΓ) = 1×w(ξ). By (43.10), (χX)Γ = π∗BΓχΓ and, by Lemma 43.19, w(χΓ) = 1+χ.

Hence, w((χX)Γ) = (1 + χ)× 1. As (χX)Γ is a line bundle, the proposition followsfrom (63.14) together with Lemma 56.9.

Example 63.15. Let ξ be a vector bundle of rank r over X . We let G = 1, τact on ξ by τ(v) = −v. Hence, ξ is an u-weight for u the generator of H1(BG).Using the identification H∗G(X) ≈ H∗(X)[u], Formula (63.13) becomes

w(ξG) =

r∑

k=0

[wr−k(ξ)(1 + u)k

].

Thus,wj(ξG) = wj(ξ) + wj−1(ξ)u + · · ·+ w1(ξ)u

j−1 + uj .

In particular, the evaluation ev1 at u = 1 of the equivariant Euler class eG(ξ) =e(ξG) satisfies

(63.16) ev1(eG(ξ)) = ev1(e(ξG)) = w(ξ) .

Proposition 63.12 has its analogue for T -equivariant complex vector bundles,where T is a torus. Let χ ∈ hom(T, U(1)), giving rise to κ(χ) ∈ H2(BT ) (see(43.25)). Then, if ξ be a χ-weight complex vector T -bundle of rank r over X , itsequivariant Chern class c(ξT ) satisfies

(63.17) c(ξT ) =

r∑

k=0

[(1 + κ(χ))k × cr−k(ξ)

].


in H∗T (X) ≈ H∗(BT ) ⊗ H∗(X). The proof is the same as for Proposition 63.12,using at the end Lemma 57.24 instead of Lemma 56.9.

64. The equivariant cohomology of certain homogeneous spaces

The title of the section paraphrases that of A. Borel’s famous paper [14]. Themain goal is to prove and exemplify Theorem 64.1 below, due to A. Knutson (un-published). In addition, at the end of the section, we study the so-called GKM-conditions for the flag manifolds.

Theorem 64.1 (A. Knutson). (1) Let Γ1,Γ2 be two closed subgroups ofa compact Lie group Γ. Suppose that Γ/Γ1 or Γ/Γ2 is Γ-equivariantlyformal. Then, there is an GrA-isomorphism

(64.2) ΞΓ1,Γ,Γ2 : H∗Γ1(Γ/Γ2)

≈−→ H∗(BΓ1)⊗H∗(BΓ) H∗(BΓ2) .

(2) Let (Γ,Γ1,Γ2) and (Γ′,Γ′1,Γ′2) be two data as in (1). Let Φ: Γ→ Γ′ be a

continuous homomorphism such that Φ(Γi) ⊂ Γ′i. Denote by Φi : Γi → Γ′ithe restriction of Φ. Set Ξ = ΞΓ1,Γ,Γ2 and Ξ′ = ΞΓ′1,Γ

′,Γ′2. Then the

following diagram

(64.3)

H∗Γ′1(Γ′/Γ′2)

Ξ′

≈//

Φ∗

H∗(BΓ′1)⊗H∗(BΓ′) H∗(BΓ′2)

Φ∗1⊗Φ∗2

H∗Γ1(Γ/Γ2)

Ξ

≈// H∗(BΓ1)⊗H∗(BΓ) H

∗(BΓ2)

is commutative (the vertical arrows are induced by Φ and Φi, using thefunctorialities of the Borel construction).

Remark 64.4. Part (3) says that ΞΓ1,Γ,Γ2 is an isomorphism of H∗(BΓ1)-modules with respect to the isomorphism ΞΓ1,Γ,Γ. With the obvious vertical iden-tification in the following commutative diagram

H∗Γ1(Γ/Γ)

ΞΓ1,Γ,Γ

≈// H∗(BΓ1)⊗H∗(BΓ) H

∗(BΓ)

≈

H∗(BΓ1)

≈

OO

ΞΓ1

≈// H∗(BΓ1)

,

the isomorphism ΞΓ1,Γ,Γ is identified with a GrA-automorphism ΞΓ1 of H∗(BΓ1).We do not know in general whether ΞΓ1 coincides with the identity. This is howeverthe case in the following cases:

(1) Γ1 = ±1, since H∗(B±1) ≈ Z2[u].(2) Γ1 a 2-torus. One uses (1), the naturality of ΞΓ1 and that H1(BΓ1) ≈

hom(Γ1, ±1 (see (43.18)).(3) Γ1 = O(n). One uses (2), the naturality of ΞΓ1 and that H∗(BO(n)) →

H∗(BT2) is injective, where T2 is a maximal 2-torus of O(n) (see Theo-rem 56.1).

(4) Γ1 a torus or U(n). The argument is analogous to (1)–(3) above.

64. THE EQUIVARIANT COHOMOLOGY OF CERTAIN HOMOGENEOUS SPACES 393

Proof of Theorem 64.1. Let Γ1 × Γ2 acts on Γ by (γ1, γ2) · γ = γ1γγ−12 .

The kernel of the projection Γ1 × Γ2 → Γ1 acts freely on Γ. By Lemma 40.10, onehas a GrA-isomorphism

(64.5) H∗Γ1(Γ/Γ2)

≈−→ H∗Γ1×Γ2(Γ) .

Let Γ1 × Γ × Γ2 acts on Γ × Γ by (γ1, β, γ2) · (γ′, γ′′) = (γ1γ′β−1, βγ′′γ−1

2 ). ThenΓ = 1×Γ×1 acts freely on Γ×Γ and the multiplication µ : Γ×Γ→ Γ coincides withthe quotient map Γ×Γ→ Γ×Γ)/Γ. Hence, as above, one has a GrA-isomorphism

(64.6) µ∗ : H∗Γ1×Γ2(Γ)

≈−→ H∗Γ1×Γ×Γ2(Γ× Γ) .

The kernel of the projection Γ1×Γ×Γ2 → Γ acts freely on Γ×Γ. By Lemma 40.10again, one has a GrA-isomorphism

(64.7) H∗Γ(Γ1\Γ× Γ/Γ2)≈−→ H∗Γ1×Γ×Γ2

(Γ× Γ) .

Now, Γ1\Γ and Γ/Γ2 being closed smooth manifolds, they are equivalent to finiteΓ-CW-complexes (see [105]). If Γ1\Γ or Γ/Γ2 is Γ-equivariantly formal, then theequivariant Kunneth theorem 42.13 holds true, telling us that the strong equivariantcross product gives a GrA-isomorphism

(64.8) H∗Γ(Γ1\Γ)⊗H∗Γ(pt) H∗Γ(Γ/Γ2)

≈−→ H∗Γ(Γ1\Γ× Γ/Γ2) .

Using Example 40.4, we get a final GrA-isomorphism

(64.9) H∗Γ(Γ1\Γ)⊗H∗Γ(pt) H∗Γ(Γ/Γ2) ≈ H∗(BΓ1)⊗H∗(BΓ) H

∗(BΓ2) .

Combining the isomorphisms (64.5)–(64.9) provides the GrA-isomorphism ΞΓ1,Γ,Γ2 .Point (2) comes from the functoriality of the above constructions and Point (3) isjust an observation about Diagram (64.3).

Remark 64.10. As the right member of (64.2) is symmetric in Γ1 and Γ2, onehas a GrA-isomorphism

H∗Γ1(Γ/Γ2) ≈ H∗Γ2

(Γ/Γ1) .

This can be more easily deduced from (64.5) and thus, does not require Γ/Γ1 orΓ/Γ2 being Γ-equivariantly formal.

The first two applications of Theorem 64.1 concern the extreme cases Γ1 = Γand Γ1 = 1. The space Γ/Γ = pt is Γ-equivariantly formal. Therefore, for anyclosed subgroup Γ2 of Γ, one has the GrA-isomorphism

(64.11) H∗Γ(Γ/Γ2) ≈ H∗(BΓ)⊗H∗(BΓ) H∗(BΓ2) ≈ H∗(BΓ2) .

Note that one already has an identification H∗Γ(Γ/Γ2) ≈ H∗(BΓ2) established inExample 40.4. We do not know whether these two identifications are the same (itcan be proved for special cases, as in Remark 64.4). In the case Γ1 = 1, we mustassume that Γ/Γ2 is Γ-equivariantly formal. We then get GrA-isomorphism(64.12)H∗(Γ/Γ2) ≈ Z2 ⊗H∗(BΓ) H

∗(BΓ2) ≈ H∗(BΓ2)/image

(H∗(BΓ)→ H∗(BΓ2)

).

We now concentrate on flag manifolds. The real flag manifold is O(n)-equivar-iantly formal by Proposition 63.1. The complex flag manifold is U(n)-equivariantlyformal by Proposition 63.8. Here below a choice of examples.


Example 64.13. Let β1 : Γ1 → O(n) denote the inclusion of the closed sub-group Γ1 in O(n). Theorem 64.1 together with Proposition 63.1 implies that

H∗Γ1(Fl(n1, . . . , nr)) ≈ H∗(BΓ1)⊗H∗(BO(n)) H

∗(BO(n1)× · · · ×BO(nr)) .

Recall from Theorem 55.17 that H∗(BO(n)) ≈ Z2[w1, . . . , wn], where wi = wi(ζn),the Stiefel-Whitney classes of the tautological bundle ζn. Also, H∗(BO(nj)) ≈Z2[w1(ζnj

), . . . , wnj(ζnj

)]. Using the Kunneth formula, one has

H∗(BO(n1)× · · · ×BO(nr)) ≈ Z2[wi(ζnj)] (j = 1, . . . , r , i = 1, . . . , nj) .

By Theorem 56.3, the homomorphism H∗(BO(n))→ H∗(BO(n1)× · · · ×BO(nr))sends wi to wi(ζn1 × · · · × ζnr

). Hence

(64.14) H∗Γ1(Fl(n1, . . . , nr)) ≈ H∗(BΓ1)[wi(ζnj

) | j = 1, . . . , r , i = 1, . . . , nj ]/I ,


w∗(ζn1 × · · · × ζnr) +H∗β1(w∗) (w∗ = 1 + w1 + w2 + · · · ) .

In the particular case of the complete flag manifold Fl(1, . . . , 1), Isomorphism (64.14)takes the form

(64.15) H∗Γ1(Fl(1, . . . , 1)) ≈ H∗(BΓ1)[x1, . . . , xn]

/(σi(x1, . . . , xn) = H∗β1(wi)) ,

where σi is the i-th elementary symmetric polynomial in the variables xj (seeExample 55.29).

Example 64.16. Let Γ1 = T2 ≈ O(1) × · · · × O(1) be the maximal 2-torusof the diagonal matrices in O(n). By Theorem 56.1 and its proof, H∗(BT2) ≈Z2[u1, . . . , un], with deg(ui) = 1, and H∗β1(wi) = σi(u1, . . . , un), where σi denotesthe i-th elementary symmetric polynomial. Also, O(n)/T2 ≈ Fl(1, . . . , 1) is O(n)-equivariantly formal by Proposition 63.1. Therefore, for any closed subgroup Γ2 inO(n), Theorem 64.1 provides the isomorphism

(64.17) Ξ = ΞT2,O(n),Γ2: H∗T2

(O(n)/Γ2)≈−→ Z2[u1, . . . , un]⊗H∗(BO(n)) H

∗(BΓ2) .

For instance, (64.14) implies that(64.18)

H∗T2(Fl(n1, . . . , nr)) ≈ Z2[u1, . . . , un][wi(ζnj

) | j = 1, . . . , r , i = 1, . . . , nj]/I ,

where I is the ideal generated by wi(ζn1 ⊕· · ·⊕ ζnr)−σi(u1, . . . , un) (i = 1, . . . , n).

In the particular case of the full flag manifold Fl(1, . . . , 1), we thus get that

(64.19) H∗T2(Fl(1, . . . , 1)) ≈ Z2[u1, . . . , un, x1, . . . , xn]

/I ,


σi(u1, . . . , un, ) = σi(x1, . . . , xn, ) (i = 1, . . . , n) .

Another example is given by the Stiefel manifold Stief(k,Rn) = O(n)/O(n−k)of orthonormal k-frames in Rn. Here, H∗(BO(n)) → H∗(BO(n − k)) is just theobvious epimorphism Z2[w1 . . . , wn] → Z2[w1 . . . , wn−k]. Hence, (64.17) impliesthat

(64.20) H∗T2(Stief(k,Rn)) ≈ Z2[u1, . . . , un]

/(σn−k+1, . . . , σn) .

Note that Stief(k,Rn)) is not O(n)-equivariantly formal. At the contrary,

ρ : H∗T2(Stief(k,Rn))→ H∗(Stief(k,Rn))


is the zero homomorphism in positive degrees.

It is reasonable to conjecture that the isomorphism ΞΓ1,O(n),O(n1)×···×O(nr) ofTheorem 64.1 identifies the equivariant Stiefel-Whitney class wi((ξj)Γ1) (see § 63)with 1⊗ wi(ζj). The following proposition proves it for the maximal 2-torus T2:

Proposition 64.21. Under the isomorphism Ξ = ΞT2,O(n),O(n1)×···×O(nr), onehas

Ξ(wi((ξj)T2)) = 1⊗ wi(ζj) .Proof. We start with the Grassmannian Gr(k; Rn) = Fl(k, n − k). For the

inclusion Φ: (T2, O(n), 1×O(n−k))→ (T2, O(n), O(k)×O(n−k)), Diagram (64.3)has the form

H∗T2(Gr(k; Rn)) Ξ

≈//

Φ∗

H∗(BT2)⊗H∗(BO(n)) H∗(BO(k) ×BO(n− k))

id⊗Φ∗2

H∗T2(Stief(k,Rn)) Ξ

≈// H∗(BT2)⊗H∗(BO(n)) H

∗(BO(n − k))

(we do not write the indices to the isomorphisms Ξ). The O(n − k)-principalbundle Stief(k,Rn) → Gr(k; Rn) is ξ1, so Stief(k,Rn)T2 → Gr(k; Rn)T2 is (ξ1)T2 .Hence, w1((ξ1)T2) ∈ kerΦ∗. Using (64.20), one sees that ker(id ⊗ Φ∗2) is the Z2-vector space generated by 1 ⊗ w1(ζ1). Therefore, Ξ(w1((ξ1)T2)) = 1 ⊗ w1(ζ1). Asymmetric argument shows that Ξ(w1((ξ2)T2)) = 1⊗ w1(ζ2).

This starts an induction argument on k to prove the proposition for Gr(k; Rk+1) =Fl(k, 1). The induction step uses Diagram (64.3) for the inclusion

Φ: (T2, O(k + 1), O(k)×O(1))→ (T2, O(k + 2), O(k + 1)×O(1))

which looks like

H∗T2(Gr(k + 1; Rk+2))

Ξ

≈//

Φ∗

H∗(BT2)⊗H∗(BO(k+2)) H∗(BO(k + 1)×BO(1))

id⊗Φ∗2

H∗T2(Gr(k; Rk+1))

Ξ

≈// H∗(BT2)⊗H∗(BO(k+1)) H

∗(BO(k) ×BO(1))

.

By induction hypothesis, Ξ(wi((ξ1)T2)) = 1⊗wi(ζ1) for i ≤ k and Ξ(w1((ξ2)T2)) =1 ⊗ w1(ζ2). By Proposition 63.1, wk+1((ξ1)T2) is in kerΦ∗. On the other hand,ker(id ⊗ Φ∗2) is, in degree k + 1, the Z2-vector space generated by 1 ⊗ wk+1(ζ1).This proves that Ξ(wk+1((ξ1)T2)) = 1⊗ wk+1(ζ1).

We now prove the the proposition for Gr(k; Rn) = Fl(k, n − k), by inductionon n. The previous argument starts the induction for n = k + 1. The inductionstep proceeds as above, using Diagram (64.3) for the inclusion

Φ: (T2, O(n), O(k) ×O(n− k))→ (T2, O(n+ 1), O(k)×O(n+ 1− k))and checking kerΦ∗ and ker(id ⊗ Φ∗2) in degree n− k + 1.

Finally, consider the map πj : Fl(n1, . . . , nr)→ Gr(nj ; Rn) defined, in the mu-tually orthogonal subspaces presentation of Fl(n1, . . . , nr) given in (2) p. 310, byπj(W1, . . . ,Wr) = Wj . Take the simplest permutation matrix σ ∈ O(n) so thatσ(0× Rnj × 0) = Rnj × 0. The conjugation with σ gives an homomorphism

Φ: (T2, O(n), O(n1)× · · · ×O(nr))→ (T2, O(n), O(n1), O(nj)×O(n− nj))


such that Φ2(1 × O(nj) × 1) = O(nj) × 1 and, for k 6= j, Φ2(1 × O(nk) × 1) ⊂1×O(n− nj). Diagram (64.3) for Φ has the form

H∗T2(Gr(nj ; Rn))

Ξ

≈//

π∗j

H∗(BT2)⊗H∗(BO(nj)) H∗(BO(nj)×BO(n − nj))

id⊗Φ∗2

H∗T2(Fl(n1, . . . , nr))

Ξ

≈// H∗(BT2)⊗H∗(BO(n)) H

∗(BO(n1)× · · · ×BO(nr))

Therefore,

Ξ(wi((ξj)T2)) = Ξπ∗j (wi((ξ1)T2)) since ξj = π∗j ξ1

= (id ⊗ Φ∗2)Ξ(wi((ξ1)T2))

= (id ⊗ Φ∗2)(1⊗ wi(ζ1)) (case Gr(nj ; Rn) done above)

= 1⊗ wi(ζj) since ξj = π∗j ξ1

Using (64.18), Proposition 64.21 has the following corollary.

Corollary 64.22. One has an isomorphism of Z2[u1, . . . , un]-algebra

H∗T2(Fl(n1, . . . , nr)) ≈ Z2[u1, . . . , un][wi((ξj)T2) | j = 1, . . . , r , i = 1, . . . , nj ]

/I ,


(64.23) wi((ξ1)T2 ⊕ · · · ⊕ ξr)T2 )− σi(u1, . . . , un) (i = 1, . . . , n) .

Remark 64.24. The vanishing of the generators of I is equivalent to the rela-tion

w((ξ1)T2) · · ·w((ξr)T2) =n∏

i=1

(1 + ui)

holding in H∗T2(Fl(n1, . . . , nr)). This relation may be obtained in the following way.

The trivializing map of (55.15) provides a morphism of T2-equivariant bundles

E(ξ1 ⊕ · · · ⊕ ξr) //

Rn

Fl(n1, . . . , nr)

f // pt

where T2 acts on Rn via the standard action of O(n). Using Example 43.22, onehas

w((ξ1)T2 ⊕ · · · ⊕ (ξr)T2) = f∗(w((Rn)T2) =

n∏

i=1

(1 + ui) .

Example 64.25. Let Γ = O(n) and Γ2 = O(1)×O(n− 1), so Γ/Γ2 ≈ RPn−1.Then H∗(BO(1)) ≈ Z2[w1] and H∗(BO(n − 1)) ≈ Z2[w1, . . . , wn−1], where wiand wi are the Stiefel-Whitney classes of the tautological bundles ζ1 and ζn−1. IfΓ1 = T2, we get, as in (64.19) that H∗T2

(RPn−1) is the quotient of

Z2[u1, . . . , un][w1, w1, . . . , wn−1]


by the relations

(64.26) w1 + w1 = σ1 , wk + w1wk−1 = σk (k = 2, . . . , n− 1) , w1wn−1 = σn .

These relations are the same as those of (63.5), no wonder given Corollary 64.22.As RPn−1 is T2-equivariantly formal, H∗(RPn−1) is the quotient of H∗T2

(RPn−1)

by the relations ui = 0. Relations (64.26) becomes wk = wk1 and wn1 = 0. Thus,H∗(RPn−1) ≈ Z2[w1]/(w

n1 ) as expected.

Example 64.27. Let Γ = O(4) and Γ2 = O(2) × O(2), so Γ/Γ2 ≈ Gr(2; R4).As in Example 64.25, we see that H∗T2

(Gr(2; R4)) is isomorphic to the quotient ofZ2[u1, . . . , u4][w1, w2, w1, w2] by the relations

(64.28)

w1 + w1 = σ1

w2 + w1w1 + w2 = σ2

w2w1 + w1w2 = σ3

w2w2 = σ4 .

These relations are the same as those of (63.7) which is coherent with Corol-lary 64.22. As in Example 64.25, we get a presentation of H∗(Gr(2; R4)) by settingui = 0. This presentation is equivalent to that of Example 55.35.

Example 64.29. In the case Γ1 = 1 (the trivial subgroup), one hasH∗1(Fl(n1, . . . , nr)) ≈ H∗(Fl(n1, . . . , nr)) and the above coincides with some the-

orems of § 55.2 (e.g. Theorem 55.24 and Example 55.29).

Example 64.30. The analogues of the above examples works for the complexflag manifolds FlC(n1, . . . , nr), where O(n) is replaced by U(n), T2 is replaced by themaximal torus T ≈ (S1)n of the diagonal matrices in U(n) and the Stiefel-Whitneyclasses are replaced by the Chern classes. The variables xi and yi have degree 2instead of 1. The analogue of Proposition 64.21 says that Ξ(ci((ξj)T2)) = 1⊗ci(ζj).One can show that these results are valid for the cohomology with coefficients inany field.

Example 64.31. Consider the case where Γ = S3 ⊂ C2 and Γ1 = Γ2 = S1 ⊂R2. Thus, Γ/Γ2 ≈ S2. By (7) p. 230, the inclusion i : S2 → (S2)Γ is homotopyequivalent to a principal bundle with structure group S3. By Proposition 28.36,H2iis an isomorphism and therefore S2 is S3-equivariantly formal. Now, BS1 ≈ CP∞

and BS3 ≈ HP∞. Hence, H∗(BΓ1) ≈ Z2[x], H∗(BΓ2) ≈ Z2[y] and H∗(BΓ) ≈

Z2[p], where x and y are of degree 2 and p of degree 4. The inclusion αi : Γi → Γsatisfies H∗Bα1(p) = x2 and H∗Bα2(p) = y2 (see Proposition 35.14). Therefore,using Theorem 64.1, one gets

H∗S1(S2) ≈ Z2[x, y]/(x2 = y2) .

We finish this section by studying the GKM-conditions for the flag manifolds.Viewing Fl(n1, . . . , nr) ⊂ SM(n) using (55.3), the fixed point set Fl(n1, . . . , nr)

T2

is clearly formed by the diagonal matrices. If ∆ = dia(x1 . . . , xn) is a diagonalmatrix and σ ∈ Symn, we set ∆σ = dia(xσ(1) . . . , xσ(n)). We say that a class

a ∈ H∗T2(Fl(n1, . . . , nr)

T2) satisfies the GKM-conditions if, for all transpositionτ = (i, j), the class a∆ − a∆τ is a multiple ui − uj. This is an ad hoc formulationfor Fl(n1, . . . , nr) (in the spirit of [117]) of the conditions introduced in [69] byM. Goresky, R. Kottwitz and R. MacPherson (whence the initials GKM). Theimportance of the GKM-conditions is illustrated in the following result.


Proposition 64.32. The image of

r : H∗T2(Fl(n1, . . . , nr))→ H∗T2

(Fl(n1, . . . , nr)T2)

is the set of classes satisfying the GKM-conditions.

Proof. We first prove that the classes in the image of r satisfy the GKM-conditions. Let τ = (i, j) with i < j. The GKM-condition for τ is trivial if∆τ = ∆ (i.e, when xi = xj). We may thus assume that xi 6= xj . In the proofof Proposition 57.4, we have introduced an embedding rij : SO(2) → SO(n). Theorbit of ∆ under the action of rij(SO(2)) on Fl(n1, . . . , nr) by conjugation is a circleCij joining ∆ to ∆τ . This circle is T2-invariant and is actually an (ui − uj)-circlein the sense of Example 44.28, with fixed points ∆,∆τ. One has a commutativediagram

H∗T2

(Fl(n1, . . . , nr))r //

H∗T2

(Fl(n1, . . . , nr)T2)

≈ //M

Fl(n1,...,nr)T2

Z2[u1, . . . , un]

H∗T2

(Cij)r // H∗

T2(CT2

ij )≈ //

M

∆,∆τ

Z2[u1, . . . , un]

where all the vertical arrows are induced by inclusions. The right vertical arrow isjust the projection. Therefore, the GKM-condition for τ comes from Example 44.28.

For the converse, we use the weighted trace f(M) =∑n

j=1 j Mjj which is,

by Proposition 55.5, a Morse function with Critf = Fl(n1, . . . , nr)T2 . Let Wx =

f−1(−∞, x] and let Tx be the set of those transpositions τ such that ∆ and ∆τ

are in Wx. We claim that a class a ∈ H∗T2(WT2

x ) is in the image of r : H∗T2(Wx)→

H∗T2(WT2

x ) if and only if it satisfies the GKM-conditions for Tx. The “only if” partis proven as above since, if (i, j) ∈ Tx, then Cij ⊂ Wx. The proof of the “if” partproceeds by induction on the number nx of critical values of f in Wx, startingtrivially if nx = 0 or 1. For the induction step, choose z < y such that nz = ny− 1.Let Mz,y = Wy −Wz . As in (44.22), one has the commutative diagram

(64.33)

0 // H∗T2(Wy,Wz)

α //

rz,y

H∗T2(Wy)

β //

ry

H∗T2(Wz) //

rz

0

0 // H∗T2(Mz,y)

α // H∗T2(WT2

y )β // H∗T2

(WT2z ) // 0

where all arrows are induced by the inclusions and where the horizontal lines areexact.

If a ∈ H∗T2(WT2

y ) satisfies the GKM-conditions for Ty, so does a = β(a) for

Tz. By induction hypothesis, a = rz(b) for some b ∈ H∗T2(Wz). As β is surjective,

there exists b ∈ H∗T2(Wy) such that a − ry(b) = α(c) for some c ∈ H∗T2

(WT2z,y). By

the “only if” part the class ry(b) satisfies the GKM-conditions for Ty, and then sodoes a − ry(b). let D ∈ MT2

z,y. Let TD be the set of transpositions in Ty such thatDτ 6= D. For each (i, j) ∈ TD, the class (a− ry(b))D is a multiple of ui − uj (since(a − ry(b))∆ = 0 when ∆ 6= D). Since α is injective, the class cD is a multiple ofui − uj for each (i, j) ∈ TD. By the proof of Proposition 55.5, the negative normal

65. THE KERVAIRE INVARIANT 399

bundle ν−(D) for f at D is the Whitney sum

ν−(D) =⊕

(i,j)∈TD

TDCij .

As Z2[u1, . . . , un] is a unique factorization domain, the class cD is a multiple of∏

(i,j)∈TD

(ui − uj) =∏

(i,j)∈TD

e(TDCij) = e(ν−(D)) .

This can be done for any D ∈MT2z,y. Using Diagram (44.21), we deduced that there

exists c ∈ H∗T2(Wy ,Wz) such that rz,y(c) = c. Hence, a = ry(α(c) + b).

Analogous GKM-relations hold true for the T -equivariant cohomology of thecomplex flag manifold FlC(n1, . . . , nr). Here, T is the maximal torus of diagonalmatrix in U(n). It is naturally isomorphic to U(1)n and thus H∗(BT ) is isomorphicto Z2[v1, . . . , vn] where vi ∈ H2(BT ) is the class associated, under the map κof (43.25), to the projection of U(1)n onto its i-th factor. As in the real case, wesee FlC(n1, . . . , nr) ⊂ HM(n) (using (57.2)), then FlC(n1, . . . , nr)

T is formed bythe diagonal matrices. We say that a class a ∈ H∗T (FlC(n1, . . . , nr)

T ) satisfies theGKM-conditions if, for all transposition τ = (i, j), the class a∆ − a∆τ is a multiplevi − vj .

Proposition 64.34. The image of injective homomorphism of Z2[v1, . . . , vn]-algebras

r : H∗T (FlC(n1, . . . , nr))→ H∗T (FlC(n1, . . . , nr)T )

is the set of classes satisfying the GKM-conditions.

Proof. We use the injective homomorphism rij : SU(2) → U(n), introducedin the proof of Proposition 57.4. The orbit of ∆ under the action of rij(SU(2)) onFlC(n1, . . . , nr) by conjugation is a 2-sphere Sij (diffeomorphic to SU(2)/U(1) ≈S2), whose intersection with FlC(n1, . . . , nr)

T is ∆,∆τ. This sphere is T -invariantand is actually a χ-sphere in the sense of Example 44.30, with κ(χ) = vi − vj .Note that FlC(n1, . . . , nr)

T = FlC(n1, . . . , nr)T2 . The proof of Proposition 64.34

then goes as that of Proposition 64.32, replacing Cij by Sij and the material ofProposition 55.5 and Example 44.28 by that of Proposition 57.4 and Example 44.30.

65. The Kervaire invariant

In 1960, Michel Kervaire introduced an invariant for framed manifolds whichenabled him to construct the first topological manifold admitting no smooth struc-ture (see Theorem 65.19 below). The computation of the Kervaire invariant thenled to one of the most important problems in homotopy theory (see Theorem 65.18).In this section, we give a survey of the geometric side of the Kervaire invariant, us-ing the surgery point of view of Wall (we belive that such a presentation is new inthe literature). The stable homotopy aspect of the invariant only briefly mentioned,being beyond the scope of this book. I was helped by the notes of Cl. Weber [201].We start with the invariant introduced by C. Arf [9] in order to classify quadraticforms in characteristic 2.


Let V be a finite dimensional Z2-vector space. A quadratic form on V is a mapq : V → Z2 such that the expression

(65.1) B(x, y) = q(x) + q(y) + q(x+ y)

defines a bilinear form B on V , the bilinear form associated to q. It is obviouslysymmetric and alternate, i.e. B(x, x) = 0.

For i ∈ Z2, let αi(q) = ♯ q−1(i). The majority (or democratic) invariant of q isthe element maj(q) ∈ Z2 defined by

maj(q) =

1 if α1(q) > α0(q)

0 otherwise.

In fact, α0(q) 6= α1(q) when B is non-degenerate (see Proposition 65.2 below).Suppose that the associated bilinear form B is non-degenerate. Since it is

alternate, there exists a symplectic basis a1, . . . , ak, b1 . . . , bk of V for B (see theproof of Lemma 58.2). The Arf invariant Arf(q) ∈ Z2 of q is defined by

Arf(q) =

k∑

i=1

q(ai)q(bi) .

That Arf(q) is independent of the choice of the symplectic basis follows fromPoint (1) of the following proposition. Two quadratic forms q and q′ on V areequivalent if there is an automorphism h of V such that q′ = qh.

Proposition 65.2. Let V be a finite dimensional Z2-vector space. Let q and q′

be two quadratic forms on V with non-degenerate associated bilinear forms. Then

(1) α0(q) 6= α1(q).(2) Arf(q) = maj(q).(3) q and q′ are equivalent if and only if Arf(q) = Arf(q′).

Proof. (Compare [23, § III.1].) Suppose first that dimV = 2. Let A = a, bbe a symplectic basis forB on V with which we compute Arf(q) = q(a)q(b). Supposethat Arf(q) = 0. There are then two cases:

(i) q(a) + q(b) = 0, hence maj(q) = 0.(ii) q(a) + q(b) = 1. By (65.1), one has q(a + b) = 0 and maj(q) = 0. By

symmetry, one may assume that q(a) = 0 and q(b) = 1. Then, thechanging of basis a′ = a+ b and b′ = b makes (b) equivalent to (a).

If Arf(q) = 1, then q(a) = q(b) = 1, thus q(a+ b) = 1 and maj(q) = 1. Points (1),(2) and (3) are thus proven when dimV = 2. We denote by qi the quadratic formfor which Arf(qi) = i.

We now prove (1) and (2) by induction on dimV . Let B = a1, . . . , ak, b1 . . . , bkbe a basis of V which is symplectic for B. One has V = V ⊕ V where V is generatedby aj, bj | j ≤ k − 1 and V is generated by ak, bk. The restriction of q to V

(respectively: V ) is denoted by q (respectively: q). We have

Arf(q) = Arf(q) + Arf(q) using the basis B= maj(q) + maj(q) by induction hypothesis.


It thus suffices to prove that maj(q) + maj(q) = maj(q). Suppose that q = q1. Onehas α1(q1) = 3 and α0(q1) = 1. Therefore,

α1(q) = 3α0(q) + α1(q)α0(q) = 3α1(q) + α0(q) .

By induction hypothesis, α0(q) 6= α1(q). Therefore, maj(q) 6= maj(q), which impliesthat

maj(q) = maj(q) + 1 = maj(q) + maj(q1) .

If q = q0, a similar argument shows that maj(q) = maj(q) = maj(q) + maj(q0).Point (1) is thus established.

It remains to prove Point (3). If q and q′ are equivalent, it is obvious thatmaj(q) = maj(q′). Conversely, we easily deduce from above that q is an orthogonalsum of q0’s and q1’s and that Arf(q) is the numbers of q1’s mod 2. That q and q′

are equivalent when Arf(q) = Arf(q′) then comes from the equivalence

q1 ⊞ q1 ≃ q0 ⊞ q0

which is achieved by the automorphism

(65.3) h(a1) = a1 + a2 , h(a2) = a2 , h(b1) = b1 , h(b2) = b1 + b2 .

We now make some preparation for the Kervaire invariant. Let ξ and ξ′ be twovector bundles over the same space X and let ηr = (prX : X×Rr → X) denote thetrivial product vector bundle of rank r over X . A stable isomorphism from ξ to ξ′

is a family of isomorphisms hr,r′ : ξ ⊕ ηr → ξ′ ⊕ ηr′ for each r, r′ sufficiently large,such that if s ≥ r and s′ ≥ r′, the diagram

ξ ⊕ ηrhr,r′

≈//

ξ′ ⊕ ηr′

ξ ⊕ ηs

hs,s′

≈// ξ′ ⊕ ηs′

is commutative, where the vertical arrows are the inclusion morphisms. A stabletrivialization of a vector bundle ξ overX is a stable isomorphism of ξ with a productbundle. A vector bundle admitting a stable trivialization is called stably trivial.

A framed manifold is a smooth manifold M together with a smooth stabletrivialization of its tangent bundle TM , called a stable framing of M . Two framedmanifolds M1 and M2 of the same dimension m and with BdM1 = BdM2 areframed cobordant if there is exists a framed manifold Wm+1 such that

BdW = M1 ∪M2 and M1 ∩M2 = BdM1 = BdM2

and whose stable framing extends those ofM1 andM2. The set of framed cobordismclasses of framed closed manifolds of dimensionm is denoted by Ωfrm . It is an abeliangroup for the disjoint union.

Let Mm be a closed manifold. Let νkM be the normal bundle of an embeddingof M into Rn+k. If k > m, such an embedding is unique up to isotopy, so the stableisomorphism class of νkM is well defined. There is a canonical stable trivialization hM


of TM⊕νkM since the latter is the restriction of TRn to M . If h : TM⊕ηr ≈−→ ηm+r

is represents a stable isomorphism, the stable isomorphism represented by

ηm+k+r oo hM

≈TM ⊕ νkM ⊕ ηr ≈

// TM ⊕ ηr ⊕ νkMh

≈// ηm+r ⊕ νkM

represents a stable trivialization of νkM . For k large enough, such a stable trivial-ization gives a vector bundle morphism

E(νkM ) //

Rk

M // pt

.

Applying the Pontryagin-Thom construction (see p. 354) to this morphism gives anelement of πm+k(S

k). This produces an isomorphism Ωfrm ≈ πSm from the framedcobordism onto the m-stem

πSm = lim−→

k

πm+k(Sk)

(see [149, § 7]).

Example 65.4. Let (Sn, F ) be the standard sphere equipped with a stableframing F . The comparison between F and the standard stable framing F0 (ex-tending to Dn+1) takes the form F = λF · F0, where λF : Sn → SO is a smoothmap, whose class [λF ] ∈ πn(SO) is unique (compare Lemma 51.3). The correspon-dence λF 7→ [Sn, F ] ∈ Ωnfr ≈ πSn gives a map Jn : πn(SO) → πSn which coincides

with the J-homomorphism of Whitehead [203].

We now describe framed surgery, following the point of view of Wall [200] (foranother approach, see p. 408). Let β : Sj ×Dm−j →Mm (0 ≤ j ≤ m) be a smoothembedding. We consider the (m+ 1)-dimensional manifold

Wβ = M × [0, 1] ∪β Dj+1 ×Dm−j

where β is seen having image in M × 1. The corners of Wβ may be smoothedin a canonical way (see [16, Appendix, Theorem 6.2]) and thus W is a smoothcobordism between M and Mβ where

Mβ = M − int(imβ) ∪β|Sj×Sm−j−1 Dj+1 × Sm−j−1 .

The manifold Mβ is said being obtained from M by a surgery using β. If M isendowed with a stable framing F which extends to a stable framing of Wβ , we saythat the surgery on β is a stably framed surgery (for the framing F ).

LetMm be a manifold and let α : Sj →M be a continuous map, with j < m−1.We wish to perform a surgery on M using a smooth embedding β : Sj×Dm−j →Mso that the restriction of β to Sj×0 is homotopic to α. Let Imm(Sj×Dm−j ,M)be the set of regular homotopy classes of smooth immersions from Sj ×Dm−j intoM . The restriction to Sj × 0 provides a map

ρ : Imm(Sj ×Dm−j ,M)→ [Sj ,M ] .

Proposition 65.5. Let Mm be a manifold and let j < m− 1. Then


(a) a stable framing F of M provides a map

φF : [Sj ,M ]→ Imm(Sj ×Dm−j ,M)

which is a section of ρ, i.e. ρφF (a) = a for all a ∈ [Sj ,M ].(b) Suppose that φF (a) contains an embedding β. Then, the surgery on β is

a stably framed surgery for the framing F .

Proof. Let α : Sj → M be a continuous map, which we extend to α1 : Sj ×Dm−j → M by α1(x, z) = α(x). The framing F of M gives rise to a stabletrivialization of α∗1TM . On the other hand, T (Sj ×Dm−j) has a canonical stabletrivialization. Comparing these two trivializations gives a stable isomorphism αS1from T (Sj×Dm−j) to α∗1TM . Since j < m−1, πj(GL(m; R))→ πj(GL(m+N ; R))is an isomorphism and thus αS1 is induced by a unique isomorphism from T (Sj ×Dm−j) to α∗1TM , giving rise to an injective bundle map α′1 : T (Sj×Dm−j)→ TM .Assertion (a) then follows from the classification of immersions [91, § 5], saying thatImm(Sj ×Dm−j ,M) is (by the tangent map) in bijection with the set of injectivebundle maps from T (Sj ×Dm−j) into TM (this also uses that j < m− 1).

If β is as in (b), the equation Tβ = α′1 is satisfied. This is exactly whatis needed to extend the stable framing F over Wβ . For more details, see [200,Theorem 1.1].

Proposition 65.6 (Surgery below the middle dimension). A stably framedcompact m-dimensional manifold Mm is stably framed cobordant to a manifold M ′

which is ([m/2]− 1)-connected (i.e. πi(M′) = 0 for i ≤ m/2− 1).

Proof. Let F be the stable framing of M and let a ∈ [Sk,M ]. If k < m/2,general position implies that φF (a) contains an embedding α : Sk × Dm−k → Musing which, by Proposition 65.5, a stably framed surgery may be performed on M .Up to homotopy equivalence, Wα is obtained from M by attaching a (k + 1)-celland from Mα by by attaching a (m − k)-cell. Hence, the inclusions i : M → Wα

and j : Mα →Wα satisfy

• π∗i : πp(M) → πp(Wα) is an isomorphism for p ≤ k − 1. For p = k, it issurjective and kills [α].• π∗j : πp(Mα)→ πp(Wα) is an isomorphism for p ≤ m− k − 1.

Therefore, if k ≤ m/2 − 1, then πp(Mα) is isomorphic to πp(M) for p ≤ k − 1and, if [α] 6= 0, πk(Mα) is isomorphic to a strict quotient of πk(M). In particu-lar, if M is (k − 1)-connected, so is Mα and, in addition, the class [α] has been“killed”. Therefore, by a finite sequence of framed surgeries, one may obtain astably framed manifold M ′ which is [m/2]− 1-connected. For more details, see e.g.[200, Theorem 1.2].

We now treat the middle dimensional surgery for an (k − 1)-connected stablyframed closed manifold Mm with m = 2k (k odd). Consider the Hurewicz homo-morphisms h : πk(M) → Hk(M ; Z) and h2 : πk(M) → Hk(M) (sending [γ : Sk →M ] to H∗γ([S

k])); since k ≥ 3, we do not worry about base points). By theHurewicz theorem [80, Theorem 4.32], h is an isomorphism and, the universal coef-ficient theorem [80, Theorem 3B.5], Hk(M) ≈ Hk(M ; Z)⊗Z2. Hence, h2 descendsto an isomorphism

(65.7) πk(M)/2 πk(M)

≈−→ Hk(M) .


Let β : V v → Nn be an immersion of smooth manifolds. The self-intersectionSI(β) of β is defined by

SI(β) = x ∈ β(V ) | ♯β−1(x) > 1 .When β is in general position and 3v < 2n, then ♯β−1(x) ≤ 2 and SI(β) is a(2v − n)-dimensional submanifold of N (see [75, Theorem 2.5])

Let a ∈ πk(M). Chose an immersion α : Sk × Dk → M in general positionrepresenting φF (a) (where F is the stable framing of M). Let α0 be the restrictionsof α to Sk × 0. The self-intersection SI(α0) is thus a finite number of points.Define

q(α) = ♯SI(α0) mod 2 .

Proposition 65.8. Let m = 2k ≥ 6 with k odd. Let Mm be (k − 1)-connectedmanifold endowed with a stable framing F . Then

(a) the above correspondence a 7→ q(α) induces, via (65.7), a well defined map

q = qM : Hk(M)→ Z2 .

(b) for all a, b ∈ Hk(M) one has

q(a+ b) = q(a) + q(b) + a · bwhere a · b denotes the (absolute) intersection form (see § 32.3).

(c) q(a) = 0 if and only if φF (a) contains an embedding.

Proof. Let α and α be two immersions representing φF (a) which are in generalposition. There are thus joined by an immersion A : Sk×Dk×I →M×I which wealso assumed to be in general position. Then, SI(A0) is a compact 1-manifold withboundary SI(α0) ∪SI(α0). As an arc has two ends, one has q(α) = q(α). Thus,q(α) depends only on a ∈ πk(M), so we can write q(a). In order to prove Point (a),it remains to establish that q(2a) = 0, which will be done together with the proofof (b).

Let a, b ∈ πk(M). Let α ∈ φF (a) and β ∈ φF (b) be two immersions in generalposition. Then, φf (a + b) may be represented by an immersion γ obtained byconnected sum of α and β along a tube Sk−1×Dk × I, disjoint from the images ofα and β except at its ends. Let B0(α, β) ∈ Z2 defined by

B0(α, β) = ♯ [(α0(Sk) ∩ β0(S

k)] mod 2 .

Obviously, one has

(65.9) q(a+ b) = q(a) + q(b) +B0(α, β) .

We claim that

(65.10) B0(α, β) = a · b .Indeed, if α0 and β0 are embeddings, this is just Corollary 33.17. We claim that aand b may be represented by embeddings α0 and β0 such that α0(S

k) ∩ β0(Sk) =

α0(Sk)∩ β0(S

k). Suppose first that ♯SI(α0) is even. The points of SI(α0) may bebe pairwise eliminated by Whitney procedure (since M is simply connected: see[206, Theorem 4 and its proof]). This produces a regular homotopy αt0 : Sk → Mwith α0

0 = α0, so that α0 = α10 is an embedding. The control on the Whitney

process guarantees that the intersection of αt0(Sk) with β(Sk) is constant in t. If

♯SI(α0) is odd, we use that there exists an immersion µ : Sk → Rm with ♯SI(µ) = 1(see [206, § I.2]). We can compose µ with a chart Rm → M whose range is away


from α0(Sk) ∪ β0(S

k), obtaining an immersion µ′ : Sk → M which, as a map, isnull-homotopic. Let α : Sk → M be the immersion obtained by connected sum ofα0 and µ′. Then, α0 represents a and ♯SI(α0) is even, so we can proceed as in theprevious case. The whole process may be independently applied to β0. This provesthe claim and then Equation (65.10).

Equations (65.9) and (65.10) imply that q(2a) = 0, because

a · a = 〈PD(a) PD(a), [M ]〉 = 0 .

Indeed, each x ∈ Hk(M) satisfies x x = Sqk(x) = x vk(M), where vk(M)is the Wu class. But, as TM is stably trivial, its Stiefel-Whitney class satisfiesw(TM) = 1, which implies that v(M) = 1 by the Wu formula. This proves (a)and, thus, Equations (65.9) and (65.10) imply (b).

For Point (c), suppose that q(a) = 0 and let α ∈ φF (a) be an immersion ingeneral position. The self-intersection SI(α0) then consists of an even numberof double points. These points can then be pairwise eliminated by the Whitneyprocedure as explained above.

By Point (b) of Proposition 65.8, qM : Hk(M) → Z2 is a quadratic form asso-ciated to the absolute intersection form of M . If BdM is either empty or a Z2-homology sphere, the intersection form is non-degenerate (see Proposition 32.17).Its Arf invariant Arf(qM ) is thus defined and is called the Kervaire invariant c(M)of M . The case where BdM is a Z2-homology sphere is too general for our purposeso we introduce the following definition: a compact manifold M is almost closed ifBdM is either empty or a homotopy sphere.

Proposition 65.11. Let m = 2k ≥ 6 (k odd). Let M0 and M1 be two (k− 1)-connected stably framed almost closed manifolds of dimension m. If M0 and M1

are stably framed cobordant, then c(M0) = c(M1).

Proof. Let Wm+10 be a stably framed cobordism between M0 and M1. If

BdM0 (and thus BdM1) is empty, we remove a tube Dm × I out of W0, getting astably framed manifold Wm+1 whose boundary is the connected sum M = M0♯M1.If BdM0 (and thus BdM1) is a homotopy sphere, we set W = W0 and M = BdW .The manifold M is a (k− 1)-connected stably framed closed manifold and, clearly,c(M) = c(M0) + c(M1). It thus suffices to prove that c(M) = 0. By surgerybelow the middle dimension (see Proposition 65.6), we may assume that W is(k − 1)-connected. To prove that c(M) = 0, it is enough to show that q(B) = 0where B = ker(Hk(M) → Hk(W )). Indeed, Proposition 32.13 and Kroneckerduality imply that 2 dimB = dimHk(M). The vanishing of q(B) thus implies,using Proposition 65.2, that c(M) = Arf(q) = maj(q) = 0.

Let h∗ and h∗ denote the integral (co)homology. As M is (k − 1)-connectedand almost closed, the Hurewicz theorem, the integral Poincare duality and theuniversal coefficient theorem imply that

(65.12) πk(M) ≈ hk(M) ≈ hk(M,BdM) ≈ hk(M) ≈ hom(hk(M); Z) .

Therefore, all the groups in (65.12) are free abelian and the isomorphism hk(M) ≈hk(M) sends BZ = ker(hk(M)→ hk(W )) onto BZ = Image

(hk(W )→ hk(M)

). As

hk(M) is free abelian, the same proof as for Proposition 32.13 shows that BZ is adirect summand of hk(M) and rankhk(M) = 2 rankBZ (all groups in the analogueof Diagram (32.14) are free abelian). Hence, BZ is a direct summand of hk(M).


The homomorphism BZ/2BZ → B is then injective, where B = Image(Hk(W ) →

Hk(M)). By Proposition 32.13, dimHk(M) = 2 dimB, so B and BZ/2BZ have

the same dimension. This shows that the homomorphism BZ → B is surjective.Consider the commutative diagram

πk+1(W ) //

πk+1(W,M) //

πk(M) //

≈

πk(W )

≈

hk+1(W ) // hk+1(W,M) // hk(M) // hk(W )

whose rows are exact. The bijectivities and surjectivity seen on vertical arrowscome from the Hurewicz-Whitehead theorem [175, Ch. 7, Section 5, Theorem 9],since M and W are (k− 1)-connected. By five-lemma’s arguments, we deduce thatπk+1(W,M)→ BZ → B is surjective.

Let b ∈ B. By the above, there is a map β : Sk → M , representing b, whichextends to γ : Dk+1 → W . Using the stable framing of W , the pair if map (γ, β)

determines, in its homotopy class, a pair of immersion (γ, β) : (Dk+1, Sk)→ (W,M)which we may assume to be in general position. The self-intersection SI(γ0) is a

compact 1-dimensional manifold whose boundary is SI(β0). As an arc has two

ends, one has q(b) = q(β0) = 0.

Example 65.13. The sphere Sk has a standard stable framing. Let M = Sk×Sk (k odd), with the product stable framing. The manifold M is (k− 1)-connectedand, by the Kunneth formula, Hk(M) ≈ Hk(Sk) ⊗ H0(Sk) ⊕ H0(Sk) ⊗ Hk(Sk),with generators a = [Sk]⊗1 and b = 1⊗ [Sk]. As M is the boundary of Sk×Dk+1

and of Dk+1×Sk, one has c(M) = 0 and q(a) = q(b) = 0 by Proposition 65.11 andits proof. As a · b = 1, one has q(a+ b) = 1. Note that a+ b is represented by thediagonal manifold of Sk × Sk.

Proposition 65.8 permits us to define the Kervaire invariant for any stablyframed almost closed manifold M2k (k odd), as c(M) = c(M ′) where M ′ is a(k − 1)-connected manifold stably framed cobordant to M . For closed manifolds,this gives a map

c : Ωfr2k → Z2 (k odd) .

which is a homomorphism. Indeed, the sum in Ωfr2k may be represented by theconnected sum. If M1 and M2 are (k−1) connected stably framed closed manifolds,then M = M1♯M2 is (k − 1)-connected and c(M) = c(M1) + c(M2).

Proposition 65.14. Let m = 2k ≥ 6 (k odd). Let Mm be a stably framedalmost closed manifold. Then, c(M) = 0 if and only if M is stably framed cobordantto a contractible manifold (if BdM is not empty) or to a homotopy sphere (if BdMis empty).

Proof. The “if” part is obvious since a contractible manifold of a homotopysphere is (k − 1)-connected and its middle dimensional homology vanish.

The proof of the converse uses the integral (co)homology, denoted by h∗ and h∗.By surgery below the middle dimension, we may suppose that M is (k − 1)-connected. As seen in (65.12), hk(M) is free abelian. Consider the integral in-tersection form on hk(M) given by a · b = 〈PD(a) PD(b), [M ]Z〉 (for some choiceof a generator [M ]Z ∈ Hm(M,BdM)). This form is unimodular since M is almost


closed (same proof as for Proposition 32.17, or see [95, p. 58]). As k is odd, theintegral intersection form is alternate. Hence, hk(M) admits a skew-symplectic

basis, i.e. a basis a1, b1, . . . , ap, bb such that ai · aj = bi · bj = 0 and ai · bj = ±δij(same proof as that of Lemma 58.2, or see [153, IV.1]). Under the isomorphism

hk(M)/2hk(M) ≈ Hk(M) the basis ai, bi gives a symplectic basis ai, bi for theZ2-intersection form.

By changing the basis ai, bi, we may assume that q(a1) = 0. Indeed, this can

be achieved if q(a1)q(b1) = 0 (by exchanging a1 with b1 if necessary). Otherwise,as Arf(q) = c(M) = 0, there exists j 6= 1 such that q(aj)q(bj) = 1 (say j = 2). Thebasis change of (65.3) then does the job.

We are thus in position to perform a stably framed surgery on an embeddingα : Sk ×Dk →M representing a1, giving a stably framed manifold M ′. Let M0 =M − int(α(Sk ×Dk)), contained in both M and M ′.

By excision, hj(M,M0) ≈ hj(Sk × Dk, Sk × Sk−1) vanishes except for j = k

where it is infinite cyclic. As M is (k − 1)-connected, the integral homology exactsequence of the pair (M,M0) yields to the exact sequence

(65.15) 0→ hk(M0)→ hk(M)h∗j−−→ hk(M,M0)→ hk−1(M0)→ 0

where j : (M, ∅) → (M.M0) denotes the pair inclusion. Since M is almost closed,Poincare and Kronecker dualities provide the commutative diagram

hk(M)h∗j //

OO[M ]≈

hk(M,M0) oo ≈hk(S

k ×Dk, Sk × Sk−1)OO[Sk×Dk]≈

hk(M,BdM)≈ // hk(M)

h∗α //

k≈

hk(Sk ×Dk)

k≈

hom(hk(M); Z)(h∗α)♯

// hom(hk(Sk ×Dk); Z)

As hk(M) is free abelian, h∗α : hk(Sk × Dk) → hk(M) is injective, so h∗α is

surjective. Hence, h∗j is surjective and, by (65.15), hk−1(M0) = 0. Note thathi(M0) ≈ hi(M) = 0 for i < k and, since k ≥ 2, van Kampen’s theorem applied to

M ≈M0 ∪ (Sk ×Dk) and M0 ∩ (Sk ×Dk) ≈ Sk × Sk−1

implies that M0 is simply connected. Therefore, M0 is (k − 1)-connected. Also,from (65.15), we deduce that hk(M0) is free with rankhk(M0) = rankhk(M)− 1.

From similar considerations with the pair (M ′,M0), we deduce that M ′ is(k−1)-connected and that hk(M

′) is free with rankhk(M′) = rankhk(M)−2. The

above process may be repeated with M ′ showing that, thanks to a finite number ofstably framed surgeries, M is stably framed cobordant to M which is k-connected.By Poincare duality, h∗(M) = 0 if ∗ ≤ m− 1. If BdM is not empty, then h∗(M) ≈h∗(pt) and, as M is simply connected, M is contractible by the Hurewicz-Whiteheadtheorem [80, Proposition 4.74]. If M is closed, then h∗(M) ≈ h∗(S

m) and by theHurewicz theorem, πm(M) ≈ hm(M) ≈ Z. If γ : Sm → M represents a generatorof πm(M), then γ is a homotopy equivalence by the Hurewicz-Whitehead theorem.Hence M is a homotopy sphere.


Corollary 65.16. Let m = 2k ≥ 6 with k odd. Let Mm be a compact stablyframed manifold whose boundary is a homotopy sphere. Suppose that c(M) = 0.Then Σ = BdM is diffeomorphic to the standard sphere Sm−1.

Proof. By Proposition 65.14, the homotopy sphere Σ is also equal to Bd Mwhere M is contractible. Since Σ is simply connected, it is a consequence of theh-cobordism that M is diffeomorphic to Dm (see [148, Proposition A, p. 108]).Hence, Σ is diffeomorphic to Sm−1.

We have so far presented the stably framed surgery under the point of view ofWall [200], using the theory of immersions. An earlier approach was introducedby Milnor in [145] and developed in [111, 23] (for a presentation of the Kervaireinvariant in this framework, see [130, 118]). With this method, in order to performa stably framed surgery on a class a ∈ πj(Mm), we first represent a by an embeddingα : Sj → M (this is possible when m > 2j and when m = 2j if M is simplyconnected [75]). The stable framing of M gives a trivialization of ηN ⊕ TM whereηN is the trivial bundle of rank N (N large). We thus get a trivialization F ofηN ⊕ α∗TM ≈ ηN ⊕ TSj ⊕ να. The vector bundle η1 ⊕ TSj is the restriction ofthe tangent bundle to Rj+1 and therefore has a canonical field of of orthonormalframes. Thus, ηN ⊕ TSj = η1 ⊕ ηN−1⊕ TSj has a canonical field of of orthonormalframes. Together with the above trivialization F , this gives an element [α] of thehomotopy group πj(Stief(j + N,Rm+N)) of the Stiefel manifold Stief(j+N,Rm+N)which is shown to be the obstruction to perform a stably framed surgery on theclass a. Homotopy groups of Stiefel manifold are known (see [177, Theorem 25.6]).For 2j < m, πj(Stief(j + N,Rm+N)) = 0, whence the surgery below the middledimension. For m = 2k with k odd, one has πk(Stief(k + N,R2k+N)) = Z2. Thisgives the quadratic form q : Hk(M) → Z2, by q(a) = [α]. Consider the principalbundle

P =(SO(k)→ SO(2k +N)→ Stief(k +N,R2k+N )

)

and its homotopy exact sequence

πk(SO(2k + N))j∗−→ πk(Stief(k + N,R2k+N))

∂−→ πk−1(SO(k))

i∗−→ πk−1(SO(2k + N)) .

Changing the stable framing of M adds to q(a) an element in the image of j∗. TheSO(k)-principal bundle α∗P is the bundle of orthonormal frames in νk. Hence,∂(q(a)) ∈ πk−1(SO(k)) ≈ πk(BSO(k)) classifies να. If k is odd and k 6= 1, 3, 7,then ker i∗ ≈ Z2 (see [23, Corollary IV.1.11]) and thus ∂ is injective. This provesthe following result.

Lemma 65.17. Let M be a (k − 1)-connected 2k-dimensional stably framedmanifold, with k odd and k 6= 1, 3, 7. Then, a class a ∈ Hk(M) satisfies q(a) = 0 ifand only if it is representable by a k-sphere in M with trivial normal bundle.

The Kervaire invariant has several other definitions (see, e.g. [22, 121] andalso the proof of Theorem 65.19 below). These more homotopic descriptions weremuch used for computing the image of the Kervaire invariant c : Ωmfr ≈ πSm → Z2,an outstanding problems in stable homotopy theory. One of the main advancewas due to Browder [22]. An almost complete solution (except for m = 126) wasprovided by Hill, Hopkins and Ravenel in 2009, who proved the following theorem(see [90]).


Theorem 65.18. Let m = 2k with k odd. Then, the Kervaire invariantc : Ωmfr → Z2 is surjective if m = 2, 6, 14, 30, 62 and possibly 126. In all otherdimensions, the Kervaire invariant vanishes.

The vanishing of the Kervaire invariant in dimension m implies the existenceof a closed PL-manifold Km which has not the homotopy type of a closed smoothmanifolds. Here below a description of the construction of Km, originally due toM. Kervaire [112].

Let pi : Di → Sk (i = 1, 2) be two copies of the unit disk bundle associated tothe tangent disk bundle to Sk (k ≥ 3). Let A be a closed k-disk in Sk. Choose

trivializations µi : p−1i (A)

≈−→ A × Dk ≈ Dk × Dk over A (they are unique up toisotopy: see Lemma 51.3). Let K0 be the quotient space of D1 ∪D2 under theidentification µ1(x, y) = µ2(y, x). After rounding the corners (see [16, Appendix,Theorem 6.2]), K0 is a smooth compact manifold of dimension m = 2k, withboundary Σ0 = Σm−1

0 . This is an example of the so called plumbing technique (see[95, § 8], [118]).

The boundary manifold Σ0 is a homotopy sphere. Indeed, K0 is (k − 1)-connected and hk(K0) is free abelian. The integral intersection form clearly induces

an isomorphism hk(K0)≈−→ hom(hk(K0),Z). By the analogue for the integral

homology of Proposition 32.17, we deduce that h∗(Σ0) ≈ h∗(Sm−1) (compare [95,p. 58]). Moreover, K0 is a thickening of Sk ∨Sk. By general position (since k ≥ 3),one has π1(Σ0) ≈ π1(K0) = 1. Hence, Σ0 is a homotopy sphere (see the proof ofProposition 65.14).

We claim that c(K0) = 1. Indeed, Hk(K0) has a symplectic basis a, b repre-sented by the two copies of Sk. Recall that TSk is isomorphic to the normal bundleof the diagonal sphere in Sk × Sk (see Lemma 33.21). It then follows from Exam-ple 65.13 (or from Lemma 65.17 if k 6= 1, 3, 7) that qK0(a) = qK0(b) = 1. Thereforec(K0) = 1.

The smooth structure on K0 determines a unique PL-structure (see (3) onp. 170). The homotopy sphere Σ0 is PL-isomorphic to the standard sphere bySmale’s theorem [173]. Let K be the PL-manifold obtained by gluing to K0 thecone over Σ0. The homotopy sphere Σ0 = Σm−1

0 is called the Kervaire sphere whilethe PL-manifold K = Km is called the Kervaire manifold.

Theorem 65.19. Let m = 2k ≥ 10 with k odd. Then the following assertionsare equivalent.

(a) The Kervaire invariant c : Ωmfr → Z2 vanishes.

(b) The Kervaire sphere Σm−10 is not diffeomorphic to the standard sphere.

(c) The Kervaire manifold Km has not the homotopy type of a smooth closedmanifold.

Thus, according to Theorem 65.18, (b) and (c) are true for 10 ≤ m 6= 14, 30, 62and possibly 126. Theorem 65.19 goes back to Kervaire [112] who proved it in 1960form = 10, constructing the first example in history of a topological closed manifoldnot admitting any smooth structure (even up to homotopy type). Together withthe discovery by J. Milnor in 1956 of several smooth structures on the 7-sphere,Kervaire’s result, was quite influential in the history of differential topology (see[51]).


Proof. (c) ⇒ (b). Suppose that (b) is not true, so there is a diffeomorphismϕ : Sm−1 → Σ0. Then K0 ∪ϕ Dm is a smooth closed manifold which is homeomor-phic to K. This contradicts (c).

(b) ⇒ (a). If (a) is not true, there is a closed smooth stably framed manifoldNm with c(N) = 1. Let N0 be N with an open m-disk removed. Thus, N0

is a smooth almost closed stably framed manifold with c(N0) = 1. Let P bethe boundary connected sum of N0 and K0. The boundary of P is Σ0 ♯ S

m−1,diffeomorphic to Σ0. But P is a smooth almost closed stably framed manifold withc(P ) = c(N0) + c(K0) = 0. By Corollary 65.16, Σ0 is diffeomorphic to Sm−1.

(a) ⇒ (c). This is more complicated and we just sketch the idea of the proof.Suppose that (c) is not true, so there is a smooth closed manifoldM and a homotopyequivalence f : Km → M . Hence, M is of dimension m and is (k − 1)-connected.Let x be the cone point of K and y = f(x). Let D be an open m-disk in M aroundy and let M0 = M −D. Using boundary collars in K0 and M0, one can constructcontinuous maps of pairs fK : (K0,Σ0) → (K,x) and fM : (M0, S

m−1) → (M, y).These maps induce isomorphism

(65.20)

Hk(K)H∗f

≈//

H∗fK≈

Hk(M)

H∗fM≈

Hk(K0) oo Ψ

≈Hk(M0)

where Ψ is defined to make the diagram commutative. For the sake of this proof, anm-dimensional relative smooth manifold (X,Y ) is called acceptable if X is (k− 1)-connected and Hi(X,Y ;G) = 0 for k < i < n for all coefficient groups G. Theabove pairs (K,x), (M, y), (K0,Σ0) and (M0, S

m−1) are acceptable. If (X,Y )is an acceptable pair, Kervaire and Milnor in [111, pp. 531–534] defined a mapψX,Y : Hk(X) → Z2 (this is the map ψ0 : Hk(X ; Z) → Z2 of [111, p. 534] whichdescents to Hk(X)). The following two properties hold true.

(i) Let g : (X,Y )→ (X ′, Y ′) be a continuous map between acceptable pairs.If g induces an isomorphism H∗(X ′, Y ′; Z)→ H∗(X,Y ; Z), then

ψX,Y H∗g = ψX′,Y ′ .

(ii) Let (X,Y ) be an acceptable pair with X stably parallelizable. ThenψX,Y (a) = 0 if and only if a is representable by an embedded k-spherewith trivial normal bundle [111, Lemma 8.3 and p. 534].

Another fact is that, as M is homotopy equivalent to K, it is stably paral-lelizable (see [24, Theorem A2.1]). Therefore, the quadratic form qM and qM0 aredefined. Since (a) is true and m ≥ 10, one has m 6= 14 by Theorem 65.18 and thusLemma 65.17 applies. Therefore, for a ∈ Hk(M0) ≈ Hk(M), one has

qM (a) = qM0(a)

= ψM,M0(a) by (ii) and Lemma 65.17

= ψK,K0 Ψ(a) by (65.20) and (i)

= qK0 Ψ(a) by (ii) and Lemma 65.17.

Hence

c(M) = maj(qM ) = maj(qK0) = 1 ,


which contradicts Assertion (a).

Besides giving the Kervaire invariant for a stably framed manifold, the Arfinvariant of a quadratic form determines the surgery obstruction group L2(π) ≈Z2 for π of order ≤ 2 [23, 199]. It has also applications in classical and highdimensional knot theory (see [201] for a survey on these works).

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Index

actionwith scattered fixed point set, 357

code associated to an, 358semi-free, 359

acyclic carrier, 44

Adem relations, 285admissible sequence, 287

affine order, 12Alexander duality, 185

almost closed manifold, 405

annihilator, 188, 222Arf invariant, 400

axial map, 207axioms of a cohomology theory, 105

barycenter, 14

barycentric subdivision, 14Betti number, 23

total, 238

binomial coefficients mod 2, 208Bockstein homomorphism, 289

Borel construction, 218, 230Borsuk-Ulam theorem, 206

boundary of a simplex, 10

boundary operatorcellular, 87

in a chain complex, 16ordered, 46

simplicial, 15

singular, 54bouquet, 73

cohomology ring of, 114bundle, 144

induced, 153, 293of finite type, 147

pair, 148

principal, 229, 249, 295trivialization, 295

universal, 229, 250vector, 157–159, 293–295

bundle characteristic map, 159

bundle gluing map, 159

C1-triangulation, 170

cap productfunctoriality, 128, 129in relative simplicial cohomology, 129simplicial, 127singular, 131

categoryCW, 96GrA, 111GrA[u], 220GrV, 29RCW, 98Simp, 11Top, 11Top2, 61TopG, 218TopΓ, 230Lusternik-Schnirelmann, 124

cellattaching a, 83characteristic map of a, 84convex, 12homology, 92open, 83

cellular(co)chain, 87chain, 88cochain, 88cohomology, 88homology, 88map, 91

chaincellular, 87, 88complex, 16

morphism of, 18homotopy, 32, 64map, 18ordered, 45simplicial, 14singular, 53

chain space, 366

big, 367chamber of a length vector, 368characteristic class

of a 2-fold covering, 120

421

422 INDEX

characteristic map

global, 85of a 2-fold covering, 119

characteristic map of a cell, 84

Chern classesmod2, 336

uniqueness, 338action of the Steenrod algebra on, 338

integral, 336

circuit, 22cobordant manifolds, 353

cobordism, 355coboundary operator

cellular, 88

in a chain complex, 17ordered, 46

simplicial, 15cochain

cellular, 87, 88

complex, 16morphism of, 18

map, 18ordered, 45

simplicial, 14

singular, 53unit, 21, 55

cocyclessimplicial, 15

singular, 54

code, 358cofibrant pair, 69

cohomologycellular, 88

connecting homomorphism, 33

simplicial, 38singular, 59

extension of the fiber, 145of a cochain complex, 18

operation, 269

ordered, 46positive parts, 114

reducedsingular, 56

relative


simplicial, 16

singular, 54theory, 105

natural transformation, 106colouring definitions

of cellular (co)chains, 88

of ordered (co)chains, 45of simplicial (co)chains, 14

of singular (co)chains, 53combinatorial manifold, 170

complex

chain, 16

cochain, 16

convex-cell, 12full, 24

simplicial, 9

componentof a simplicial complex, 11

coneon a simplicial complex, 23

on a topological space, 65

conjugationcell, 361

complex, 361equation, 360

universal, 363

manifold, 364space, 360

conjugation space, 336connected sum, 115

cohomology ring of, 115

connecting1-cochain, 55, 120

homomorphismcohomology, 33

homology, 36

connecting homomorphismsimplicial, 40

contiguous simplicial maps, 32contractible space, 65

convex cell, 12

convex-cell complex, 122-fold covering

characteristic class of, 120deck involution, 120

transfer exact sequence, 123

transfer homomorphism, 122cross product

associativity, 140equivariant, 244

strong, 244

functoriality, 133in cohomology, 132

in homology, 134in relative cohomology, 133

of maps, 131

reduced, 142equivariant, 248

cross-square map, 275

equivariant, 280cup product

commutativity, 110commutativity of, 112

functoriality, 111, 113

in relative simplicial cohomology, 111in relative singular cohomology, 113, 131


cup-square map, 117, 209, 292

CW

INDEX 423

complex, 83cartesian product, 85equivariant, 217, 240finite, 91homology-cell complex, 93regular, 89weak topology, 83

pair, 83space, 83structure, 83

on Sn, 85on projective spaces, 85perfect, 91

cyclessimplicial, 15singular, 54

deck involutionof a 2-fold covering, 120

degreelocal, 78

degree of a mapbetween manifolds, 172between pseudomanifold, 30between spheres, 78

democratic invariant, 400dimension

of a simplex, 9of a simplicial complex, 9topological invariance, 81

disjoint union axiom, 106duality

Alexander, 185Kronecker, 16–21

ordered, 46singular, 54

Lefschetz, 178Poincare, 171

Eilenberg-McLane space, 101cohomology of, 289

equivariantcross product, 244CW-complex, 217, 240Stiefel-Whitney classes, 388vector bundle, 252

equivariant cohomologyfor a pair with involution, 220in general, 231

equivariantly formal, 222, 233Euclidean bundle, 157Euler

characteristicof a finite CW-complex, 91of a finite simplicial complex, 25

of a manifold, 171, 183, 193class, 154, 157

equivariant, 253exact sequence

cohomology, 34homology, 36of cochain complexes, 33simplicial (co)homology, 39, 41

of a triple, 42singular (co)homology, 60

of a triple, 62exactness axiom, 105excision

axiom, 105property, 68simplicial, 50

excisive couple, 113extension of a cochain


faceexterior algebra, 163front, back, 112in a regular CW-complex, 89inclusion, 54of a simplex, 9space, 163

fiber inclusion, 144finite (co)-homology type, 23, 63, 76finite type (graded vector space), 23flag complex, 163flag manfold

tautological bundle over..., 314flag manifold, 310

complete, 311, 334complex, 333

tautological bundle over..., 335forgetful homomorphism (in equiv.

cohomology), 221, 233framed

cobordism, 401manifold, 401

framed bundle (of a K-vector bundle), 252framed bundle (of a vector bundle), 296framing (stable), 401full complex, 24fundamental class

of a polyhedral homology manifold, 171of a pseudomanifold, 22of a relative homology manifold, 179

G-contractible, 217generic length vector, 368genus

of a nonorientable surface, 28of an orientable surface, 28

geometricopen simplex, 10

simplex, 10geometric realization

of a simplicial complex, 10metric topology, 10

424 INDEX

weak topology, 10

G-homotopy, 217

Giambelli formula, 329GKM-conditions, 397

complex case, 399

global characteristic map, 85

G-map, 217good pair, 68

GrA, 111

graded algebras

category of, 111GrA[u], 220

graph

simplicial, 9

Grassmannian, 310complex, 333

infinite, 314, 335

tautological bundle over, 314, 335

GrV, 29Gubeladze’s theorem, 164

Gysin

exact sequence, 155, 156

homomorphism, 189

ham sandwich theorem, 206Hawaiian earring, 107

homogeneous space, 310

homology

cell, 92complex, 93

cellular, 88

connecting homomorphism, 36

of a chain complex, 17ordered, 46

reduced

singular, 56

relativesimplicial, 39

singular, 59

simplicial, 16

singular, 54sphere, 185

homology manifold, 170

relative, 178

homotopic maps, 64

homotopy, 64axiom, 105

chain, 32, 64

equivalence, 65

equivalent pairs or spaces, 65property (in singular (co)homology), 64

quotient, 218, 230

relative, 293

type, 65Hopf

bundle, 202–205

vector, 205

invariant, 209, 292

map, 203

intersection form, 183invariance of dimension, 81

inverse problem, 383

joinsimplicial, 10

topological, 144Jordan Theorem, 82

Kervaireinvariant, 405, 406

manifold, 409sphere, 409

Kirwan surjectivity theorems, 265–268Klein bottle

cohomology algebra of, 117triangulation of, 27

Kroneckerduality, 16–21

ordered, 46singular, 54

pair, 17morphism of, 19

pairing

extended, 55on (co)chains, 15

on (co)homology, 18on cellular (co)chains, 88

on ordered (co)chains, 46on singular (co)chains, 54

Kunneth theorem, 135equivariant, 244

reduced, 142relative, 138, 150

Lefschetz duality, 178length vector, 366

chamber of a, 368dominated, 380

generic, 368lopsidedness of a, 381

normal, 384Leray-Hirsch Theorem, 145, 149

2nd version, 147link, 11

linking number, 194localization

principle, 187

theorem, 237, 240long subset, 368

lopsidedness, 381Lusternik-Schnirelmann category, 124

m-involution, 357

majority invariant, 400manifold

combinatorial, 170

INDEX 425

PL, 170

polyhedral homology, 170

relative polyhedral homology, 178

topolgical, 81, 115map

cellular, 91

of simplicial triple, 41

of topological pairs, 61

piecewise affine, 10

mapping cylinderbundle pair, 153

neighbourhood, 69

mapping torus, 160

exact sequence, 160

maximal simplex, 9

Mayer-Vietoris

connecting homomorphismssimplicial, 44

singular, 75

data, 75

sequence

simplicial, 44

singular, 75meridian sphere, 195

Morse function, 259

Morse-Bott

function, 259

critical manifold, 259

perfect, 260self-indexed, 311

inequalities, 259

lacunary principle, 260

polynomial, 259

natural transformation, 96

NDR-pair, 70

negative normal bundle, 259nilpotency class, 125

non-singular map, 206, 292

null-homotopic, 104

octonionic projective plane

cohomology algebra of OP n, 204

odd map, 205

open simplex, 10

orderaffine, 12

simplicial, 12

ordered

chain, 45

cochain, 45

cohomology, 46homology, 46

simplex, 45

orientation

of a vector bundle, 297

of a vector space, 297

transport along a path, 302

pairCW, 83good, 68Kronecker, 17simplicial, 38topological, 59

map of, 61perfect

CW-structure, 91Morse-Bott function, 260

piecewise affine map, 10Pieri formula, 328PL-manifold, 170Poincare dual

functoriality, 188of a homology class, 183of a submanifold, 186

Poincare duality, 171Poincare series/polynomial, 23, 63, 76

pointed space, 73well, 73, 141

polygon space, 367abelian, 387free, 367spatial, 386

polyhedral homologymanifold, 170

relative, 178Pontryagin-Thom construction, 354product

cap, see also cap productin relative simplicial cohomology, 129

cup, see also cup productin relative singular cohomology, 131

of CW-complexes, 85slant, 339tensor, 132

projective spaceRP∞ as Eilenberg-space, 104cohomology algebra of CP n, 203cohomology algebra of HP n, 203cohomology algebra of RP n, 124, 201complex, 202octonionic (projective plane), 204quaternionic, 203real, 85simplicial cohomology algebra of RP 2,

116standard CW-structure on RP n, 85tautological line bundle, 302triangulation of RP 2, 26

proper map, 259pseudomanifold, 22

fundamental class of, 22fundamental cycle of, 22

quadratic form, 400

reduced

426 INDEX

equivariant cohomology, 223, 233

singular (co)homology, 56

suspension, 142reduced symmetric square, 345

regular

CW-complex, 89

value (topological), 78

relativecohomology

ordered, 49

simplicial, 38

singular, 59homology

simplicial, 39

singular, 59

singular cycle, 60representable functor, 102

retraction, 68, 81

by deformation, 68

Riemannian metric, see also Eclideanbundle, 157

robot arm map, 367

scattered fixed point set, 357

Schubert

cell, 321variety, 322

Schubert calculus, 324–329

complex, 338

Giambelli formula, 329Pieri formula, 328

Segre class, 338

self-intersection, 404

semi-free action, 359

short subset, 49, 368Simp, 11

simplex, 9

boundary of a, 10

geometric, 10geometric open, 10

maximal, 9

ordered, 45

singular, 53small, 65

standard, 53

simplicial

(co)homology

exact sequence, 39, 41, 42category, 11

chain, 14

cochain, 14

cocycles, 15cohomology, 16

cohomology connecting homomorphism,38

complex, 9

finite, 9

cycles, 15

graph, 9

homology, 16homology connecting homomorphism, 40

map, 11

of simplicial pairs, 41order, 12

for the barycentric subdivision, 14pair, 38

pairs

map of, 41relative

cohomology, 38homology, 39

suspension, 50

triad, 43triple, 41

map of, 41simplicial complex

component of, 11

connected, 11Euclidean realization, 10

geometric realization, 10locally finite, 10

simplicial excision, 50

singularchain, 53

cochain, 53cocycles, 54

cohomology, 54

cycles, 54homology, 54

reduced (co)homology, 56relative

cohomology, 59

homology, 59simplex, 53

small, 65skeleton

of a CW-complex, 83

of a simplicial complex, 10slant product, 339

slice inclusion, 139small

map, 65

singular simplex, 65smash product, 141

Smith inequality, 239

spin structure, 305splitting principle, 331, 337

for complex bundles, 337generalized, 330, 337

star, 11

open, 11Steenrod

algebra, 286squares, 276

characterisation, 290

Stiefel manifold, 314, 335, 394

INDEX 427

Stiefel-Whitney classes, 307

action of the Steenrod algebra on, 332

dual, 320equivariant, 388

first, 299

of projective spaces, 342–343

second, 303uniqueness, 331

Stiefel-Whitney numbers, 353

subcomplex

of a CW-complex, 83of a simplicial complex, 9

subdivision, 13

barycentric, 14

operator, 176subset definitions

of (co)chains, 14

of cellular (co)chains, 88

of ordered (co)chains, 45of singular (co)chains, 53

Sullivan conjecture, 202, 292

surface, 25

cohomology algebra of, 116, 117genus

of a nonorientable surface, 28

of an orientable surface, 28

nonorientable, 28orientable, 28

surgery, 402

suspension, 72

isomorphism, 73reduced, 142

simplicial, 50

symplectic basis, 339

tautological vector bundle, 205, 302, 314

complex, 335tensor product, 132

Thom

class, 152

isomorphism theorem, 153, 157space, 350

Thom-Pontryagin construction

see Pontryagin-Thom construction, 354

Top, 11

Top2, 61TopG, 218

TopΓ, 230

topological

complexity, 126pair, 59

regular value, 78

2-torus (2-elementary abelian group), 254

torus (Lie group), 257associated 2-torus, 257

torus (manifold)

cohomology algebra of T 2, 117

triangulation of T 2, 26

transfer, 122exact sequence, 123, 156

transport of an orientation, 302triangulable

pair, 178space, 12

triangulation, 12, 178triple

topological, 62map of, 62

umkehr homomorphism, 189unit cochain, 21, 47, 55

vector bundle, 157–159, 293–295associated framed bundle, 252, 296complex, 298equivariant, 252

rigid, 254, 390Euclidean, 157, 297induced, 158, 293isomorphism of, 293orientable, 297product, 293stable isomorphism of, 401structures on, 296tensor product, 253

for line bundles, 300, 337Thom space, 350trivial, 293trivialization, 293

stable, 401Whitney sum, 158

equivariant, 252vertex,vertices, 9

Wang exact sequence, 160weak homotopy equivalence, 105weak topology (for CW-complexes), 83wedge, see also buquet, 73weight

bundle, 390weighted trace, 311, 334well cofibrant pair, 70

equivariant, 234presentation of, 70

well pointed space, 73, 141equivariant, 247

Wuclass, 338formula, 339

Mod Two Homology and Cohomology (Jean Claude)

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