Cell Complexes

JWR

Feb 6, 2005

1 Preliminaries

1.1. Unless otherwise specified, the word space means topological space andthe word map means continuous function between two topological spaces. Thenotation f : (X, A) → (Y, B) means f : X → Y , A ⊂ X, B ⊂ Y , and f(A) ⊂ B.It is often convenient to fix a base point x0 ∈ X. The pair (X,x0) is then calleda pointed space and the notation f : (X,x0) → (Y, y0) means f : X → Y ,x0 ∈ X, y0 ∈ Y , and f(x0) = y0.

1.2. The disjoint union of an indexed collection {Yα}α∈Λ is by definition theset ⊔

α∈Λ

Yα := {(y, α) : y ∈ Yα}.

There is a projection⊔

α∈Λ Yα →⋃

α∈Λ Yα : (y, α) 7→ y which is a bijection ifand only if the sets Yα are pairwise disjoint.

1.3. By a disk we understand a space homeomorphic to the standard closedunit disk

Dn := {x ∈ Rn : ‖x‖ ≤ 1}.We may on occasion use other models for the disk, e.g. the unit cube

In := {(x1, x2, . . . , xn) ∈ Rn : 0 ≤ xi ≤ 1}

or the standard simplex

∆n := {(x0, x1, . . . , xn) ∈ In+1 : x0 + x1 + · · ·+ xn = 1}.

The spaces Dn, In, and ∆n are homeomorphic; homeomorphisms can be con-structed using radial projection. We use the terms n-disk, n-cube, or n-simplexto indicate the dimension. When D is an n-disk we denote by ∂D the image of

Sn−1 := ∂Dn := {x ∈ Rn : ‖x‖ = 1}

under a homeomorphism Dn → D. By the Brouwer Invariance of DomainTheorem the subset ∂D ⊂ D is independent of the choice of the homeomorphismused to define it but for the definitions in this section the reader may take the

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term disk to signify one of Dn, In, or ∆n and define ∂D in each of these casesseparately. Thus

∂In :=n⋃

i=1

Ii−1 × {0, 1} × In−i

and

∂∆n :=n⋃

k=0

ιk(∆n−1)

where the inclusion ιk : ∆n−1 → ∆n is defined by

ιk(x1, . . . , xk) = (x1, . . . , xk, 0, xk+1, . . . , xn).

The set ∂D is called the boundary of the disk but the reader is cautionedthat in point set topology this term depends on the ambient space (in this caseRn). Thus (In point set topology the boundary of a set is the intersection ofits closure with the closure of its complement so the set theoretic boundary of∆n as a subset of Rn+1 is ∆n itself.) We may call a disk D a closed disk todistinguish it from the open disk D \ ∂D.

1.4. A path in X is a map γ : I→ X and a loop in X is a path with γ(0) = γ(1).The words are also used more generally, for example a map γ : [a, b] → X mightalso be called a path and a map γ : S1 → X might also be called a loop. Aspace X is called path connected iff for all x, y ∈ X there is a path γ : I→ Xwith γ(0) = x and γ(1) = y. A space is said to have a property locally iffevery point has arbitrarily small neighborhoods which have the property. Forexample, a space X is locally path connected iff for every x ∈ X and everyneighborhood V of x in X there is a path connected neighborhood U of x withU ⊂ V .

Exercise 1.5. For any space X define an equivalence relation by x ≡ y iff hereis a path γ : I → X with γ(0) = x and γ(1) = y. The equivalence classes arecalled the path components of X. Show that the following are equivalent.

(i) The space X is locally path connected, i.e. for every x0 ∈ X and everyneighborhood V of x0 there is an open set U with x0 ∈ U ⊂ V and suchthat any two points of U can be joined by a path in U .

(ii) For every x ∈ X and every neighborhood V of x in X there is a a neigh-borhood U of x with U ⊂ V and such that any two points of U can bejoined by a path in V .

(iii) The path components of every open set are open.

Solution. That (i) =⇒ (ii) is immediate: a path in U is a fortiori a path in V .Assume (ii). Choose an open set V and a path component C of V . (We will show C isopen.) Choose x0 ∈ C. By (ii) there is an open set U containing x0 such that any twopoints of U lie in the same path component of V . In particular any point of U lies inthe same path component of C as x0, i.e. U ⊂ C. Thus C is open. This proves (iii).Finally assume (iii). Choose x0 ∈ X and a neighborhood V of x0. Let U be the pathcomponent of V containing x0. By (iii), U is open. This proves (i).

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Remark 1.6. The whole space X is an open set so it follows from (iii) that ifX is locally path connected, then the path components are open.

Remark 1.7. It is a theorem of point set topology that a compact, connected,and locally connected metric space is path connected. (See [4] page 116.) How-ever, a compact metric space can be quite nasty. Consider for example theexercises in page 18 Chapter 0 of [3]. Pathological examples are important forunderstanding proofs, but for the most we will concentrate on nice spaces (finitecell complexes – See Definition 4.2 below).

1.8. A homotopy is a family of maps {ft : X → Y }t∈I such that the evaluationmap

X × I → Y : (x, t) 7→ ft(x)

is continuous. We do not distinguish between the homotopy and the corre-sponding evaluation map. We say two maps f, g : X → Y are homotopic andwrite f ' g iff there is a homotopy with f0 = f and f1 = g. When we say thatf, g : (X,A) → (Y,B) are homotopic it is understood that each stage ft of thehomotopy is also a map of pairs, i.e. ft(A) ⊂ B. If A ⊂ X we say f and g arehomotopic relative to A and write f ' g (rel A) iff there is a homotopy {ft}t

between f and g with ft(a) = f0(a) for all a ∈ A and t ∈ I.Remark 1.9. When X is compact and the space C(X, Y ) of all continuousmaps from X to Y is endowed with the compact open topology, two mapsf, g ∈ C(X,Y ) are homotopic if and only if they belong to the same pathcomponent of C(X,Y ). Thus (in this situation at least) a homotopy is truly apath of maps.

Definition 1.10. We say spaces X and Y are homotopy equivalent or havethe same homotopy type and write X ' Y iff there are maps f : X → Y andg : Y → X such that g ◦ f ' idX and f ◦ g ' idY .

Definition 1.11. A space X is called contractible iff it is homotopy equivalentto a point, i.e. iff there is a homotopy ft : X → X with f0 = idX and f1 aconstant.

Definition 1.12. Let X be a space. A subset A ⊂ X. is called a retract of Xiff there is a retraction of X to A, i.e. a left inverse to the inclusion A → X,i.e. a map r : X → A such that r(a) = a for a ∈ A. The subset A ⊂ X. is calleda deformation retract of X iff there is a deformation retraction of X toA, i.e. a homotopy {rt : X → X}t∈I such that r0(x) = x for x ∈ X, r1(X) = A,and rt(a) = a for a ∈ A and t ∈ I.Remark 1.13. If {x0} is a deformation retract of X for some x0 ∈ X, then cer-tainly X is contractible. A still stronger condition is that {x0} is a deformationretract of X for every x0 ∈ X. In general the reverse implications do not hold(see Exercises 6 and 7 on page 18 of [3]) but they do hold for cell complexes (seeCorollary 0.20 page 16 of [3]). See Section 4 for the definition of cell complex.

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Exercise 1.14. Construct explicit homotopy equivalences between the wedgeof circles

X =(

(−1, 0) + S1

)∪

((1, 0) + S1

),

the spectacles

Y =(

(−2, 0) + S1

)∪

((2, 0) + S1

)∪

([−1, 1]× 0

),

and the space

Z = S1 ∪(

0× [−1, 1])

obtained from a circle by adjoining a diameter.

Solution. Let p = (−1, 0), q = (1, 0), and C be the convex hull of X. It suffices toshow that each of the spaces X, Y , Z is homeomorphic to a deformation retract ofC \ {p, q}.

1. The space X is a deformation retract of C \ {p, q}. To see this radially projectthe insides of the two circles from their centers, radially project the portion ofE outside the circles and above the x-axis from (0, 2), and radially project theportion of E outside the circles and below the x-axis from (0,−2).

2. The space Y ′ := {(x, y) ∈ R2 : (x ± 1)2 + y2 = 1/4} ∪ �[−1/2, 1/2] × 0�

isa deformation retract of C \ {p, q}. To see this radially project the insides ofthe two circles from their centers, radially project the points of C outside thetwo circles and outside the square [−1, 1]2 from the corresponding center, andproject the remaining points vertically.

3. The space Z′ := ∂C ∪ �0× [−1, 1]�

is a deformation retract of C \ {p, q}. To seethis radially project the left half from p and the right half from q.

Each fiber of each of these retractions is a closed or half open interval so it is easy tosee that each of these retractions is a deformation retraction.

1.15. Suppose that E is a topological space, X is a set and g : E → X is asurjective map. The quotient topology on X is defined by the condition thatU ⊂ X is open iff g−1(U) ⊂ E is open. That this defines a topology is animmediate consequence of the identities

g−1

( ⋃

α∈Λ

Uα

)=

⋃

α∈Λ

g−1(Uα), g−1

( ⋂

α∈Λ

Uα

)=

⋂

α∈Λ

g−1(Uα)

for any indexed collection {Uα}α∈Λ of subsets of X. Since g is surjective, X canbe interpreted as the set of equivalence classes of an equivalence relation:

X = E/ ∼ where x ∼ y ⇐⇒ g(x) = g(y).

In most applications X is an identification space. Typically

E = Z t Y, X = Z tφ Y

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where Y t Z denotes the disjoint union and the notation means that there is acontinuous map φ : A → Z with A ⊂ Y and X = E/∼ where the equivalencerelation is the minimal equivalence relation satisfying y = φ(x) =⇒ y ∼ x forx ∈ A and y ∈ Z. Typically Y = Dn and A = ∂Dn := Sn−1. In this case we saythat X is obtained by attaching an n cell to Z along φ.

Remark 1.16. If A ⊂ X the notation X/A indicates the space X/∼ of equiv-alence classes where x ∼ y ⇐⇒ either x = y or else both x ∈ A and y ∈ A.If A 6= ∅, the space X/A has a distinguished point A/A. However, X/∅ = X ormore precisely, X/∅ =

{{x} : x ∈ X}.

2 Categories

2.1. A category C consists of

• a collection called the objects of the category;

• for each pair X, Y of objects a set C(X,Y ) called the morphisms from Xto Y ;

• for each object X a morphism idX ∈ C(X,X) called the identity mor-phism;

• for each triple X, Y , Z, of objects a map

C(X, Y )× C(Y, Z) → C(X, Z) : (f, g) 7→ g ◦ f

called composition;

satisfying

(associative law) (h ◦ g) ◦ f = h ◦ (g ◦ f), and

(identity laws) idY ◦ f = f ◦ idX = f for f ∈ C(X,Y ).

In most examples the objects are sets with some additional structure and themorphisms are maps which preserve that structure; we write f : X → Y insteadof f ∈ C(X, Y ).

2.2. A morphism g : Y → X is a left inverse (resp. right inverse) to themorphism f : X → Y iff g ◦ f = idX (resp. f ◦ g = idY ). An isomorphism is amorphism which has a left inverse and a right inverse. If g1 is a left inverse tof and g2 is a right inverse to f , then g1 = g2 and g1 is the only left inverse tof and the only right inverse. Hence an isomorphism f : X → Y has a uniqueinverse denoted f−1 : Y → X characterized by (either of) the equations

f−1 ◦ f = idX , f ◦ f−1 = idY .

2.3. Here are a few of the categories we will study.

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1. The category of groups and group homomorphisms.

2. The category of rings and ring homomorphisms.

3. The category of topological spaces and continuous maps. (The isomor-phisms are called homeomorphisms.)

4. The category of topological spaces and homotopy class of maps. (Theisomorphisms are called homotopy equivalences.)

5. The category of topological spaces with base point and continuous basepoint preserving maps f : (X,x0) → (Y, y0).

2.4. A covariant functor from a category C to a category D is an operationwhich assigns to each object X of C and object Φ(X) of D and to each morphismf ∈ C(X,Y ) of C a morphism Φ(f) ∈ D(Φ(X), Φ(Y )) such that

Φ(idX) = idΦ(X), Φ(g ◦ f) = Φ(g) ◦ Φ(f).

The notation f∗ = Φ(f) is often used. A contravariant functor from acategory C to a category D is an operation which assigns to each object X ofC and object Φ(X) of D and to each morphism f ∈ C(X,Y ) of C a morphismΦ(f) ∈ D(Φ(Y ), Φ(X)) such that

Φ(idX) = idΦ(X), Φ(g ◦ f) = Φ(f) ◦ Φ(g).

The notation f∗ = Φ(f) is often used.

Example 2.5. The operation which assigns to each pointed topological space(X, x0) the fundamental group π1(X,x0) is a covariant functor from the categoryof pointed topological spaces to the category of groups. For f : (X,x0) → (Y, y0)the induced map f∗ : π1(X,x0) → π1(Y, y0) sends the homotopy class of a loopγ to the homotopy class of f ◦ γ.

Example 2.6. The operation which assigns to each topological space X thering C(X,R) of continuous real valued functions on X is a contravariant functorfrom the category of topological spaces to the category of rings. For a continuousmap f : X → Y the induced morphism f∗ : C(Y,R) → C(X,R) is defined byf∗u = u ◦ f for u ∈ C(Y,R).

3 Surfaces

Exercise 3.1. Let P be a polygon with an even number of sides. Suppose thatthe sides are identified in pairs in any way whatsoever. Prove that the quotientspace is a compact surface. A surface is a space which is locally homeomorphicto R2. That the sides are identified in pairs means the following. There isan enumeration α1, β1, . . . , αn, βn of the edges of P (not necessarily in cyclicorder but without repetitions) and for each k = 1, 2 . . . , n a homeomorphismφk : αk → βk so that desired identification space S is obtained from P byidentifying x ∈ αk ⊂ ∂P with φk(x) ∈ βk ⊂ ∂P .

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Solution. Choose p ∈ S. We must construct a neighborhood U of p in S and ahomeomorphism h : U → R2. There are three cases.

Case 1. Assume p lies in the interior of P . Then there is an open disk in Pcentered at p. Any open disk is homeomorphic to R2.

Case 2. Assume that p = {a, b} where a ∈ α := αk, b ∈ β := βk, and φ(a) = bwhere φ := φk for some k = 1, 2 . . . , n. Abbreviate R := (−1, 1) × (−1, 1),R1 := (−1, 1) × [0, 1), R2 := (−1, 1) × (−1, 1], I := R1 ∩ R2. Let f and g bea homeomorphisms from R1 and R2 onto neighborhoods U1 and U2 of a andb in P respectively. Thus a ∈ f(I) ⊂ α and b ∈ g(I) ⊂ β. Shrinking andmodifying as necessary we may assume φ ◦ f(I) = g(I). Define ψ : I → Iby ψ(x, 0) = g−1(φ(f(x, 0)) and Ψ : R− → R− by Ψ(x, y) = ψ(x, 0) + (0, y).Replacing g by g ◦Ψ we may assume that φ ◦ f = g. Now define h : R → S byh(x, y) = f(x, y) if y ≥ 0 and h(x, y) = g(x, y) if y ≤ 0.

Case 3. Assume that p = {v1, v2, . . . , vk} is a set of vertices of P . Renamingand replacing some of the φj by φ−1

j we may assume that vj is the commonvertex of βj−1 and αj where β0 := βk. Let R be the open unit disk in C = R2

and write R = R1 ∪ · · · ∪Rk where Rj is the sector

Rj :={

r exp(iθ) : 0 ≤ r < 1,2(j − 1)π

k≤ θ ≤ 2jπ

k

}.

It is easy to construct homeomorphisms fj : Rj → Uj where Uj is a neighbor-hood of vj in P , for example we make take fj(r exp(iθ) = vj+cjr exp(i(ajθ+bj))where cj > 0 is small and aj and bj are judiciously chosen. Reflecting if neces-sary we can even achieve fj(Ij−1) ⊂ αj and fj(Ij) ⊂ βj where

Ij :={

r exp(iθ) : 0 ≤ r < 1, θ =2jπ

k

}.

To get a homeomorphism from R to a neighborhood of p in S we must find suchhomeomorphisms fj satisfying the additional condition that

φj ◦ fj+1|Ij = fj |Ij . (∗)For this we try

fj(r exp(iθ)) = vj + ρj(θ, r) exp(i(ajθ + bj)).

As in case 2 we modify fj+1|Ij to achieve (∗). This determines ρj(2(j−1)π/k, ·)and ρj(2π/k, ·). These last two maps are homeomorphisms (i.e. strictly increas-ing functions) from from the unit interval to two other intervals and we defineρj(θ, ·) for intermediate values of θ by linear interpolation.

Exercise 3.2. Let S′ be the surface obtained by modifying the surface S ofExercise 3.1 by replacing each homeomorphism φk : αk → βk by a differenthomeomorphism ψk : αk → βk. Assume each homeomorphism ψ−1

k ◦ φk : αk →αk fixes the endpoints of the side αk. Show that S and S′ are homeomorphic.

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Remark 3.3. The following representations are called standard:

(I) P is a square with boundary

∂P = α1 ∪ β1 ∪ α2 ∪ β2,

the sides are are enumerated and oriented in the clockwise direction, andthe identifications reverse orientation.

(II) P is a 4g-gon with boundary

∂P = α1 ∪ α2 ∪ β1 ∪ β2 ∪ · · · ∪ α2g−1 ∪ α2g ∪ β2g−1 ∪ β2g,

the sides are enumerated and oriented in the clockwise direction, and theidentifications reverse orientation.

(III) P is a 2k-gon with boundary

∂P = α1 ∪ β1 ∪ · · · ∪ αk ∪ βk.

the sides are enumerated and oriented in the clockwise direction, and theidentifications preserve orientation.

The Classification Theorem for Surfaces (See [2] Page 236, [6] page 9, or [5]page 204) implies that any compact surface is homeomorphic to one of these.One can transform any space S = P/∼ as in Exercise 3.1 to one of the standardrepresentations with a sequence of elementary moves. There are two kinds ofelementary moves: one which removes adjacent edges α and β that are identifiedreversing orientation and another which cuts a polygon in two along a diagonalproducing a new pair (α, β) and then reassembling along an old pair (α, β). Thisprocess produces a proof of the Classification Theorem from the assertion thatany compact surface is homeomorphic to some P/∼ (not necessarily a standardone). The process has nothing to do with proving that P/∼ is a compact surface.

Exercise 3.4. We say surface S is said to be the connected sum of surfacesS1 and S2 and write

S = S1 # S2

iff there are disks Di ⊂ Si (i = 1, 2) and a homeomorphism φ : ∂D2 → ∂D1

withS = (S1 \ E1) tφ (S2 \ E2)

where Ei := Di \∂Di is the interior of Di and ∂Di is the boundary of Di. Showthat

1. The connected sum of any surface S with the sphere S2 is homeomorphicto S.

2. A surface as in part (II) of Remark 3.3 is homeomorphic to a connectedsum

gT 2 := T 2 # T 2 # · · · # T 2

︸ ︷︷ ︸g

of g tori T 2 := S1 × S1.

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3. A surface as in part (III) of Remark 3.3 is homeomorphic to a connectedsum

kP 2 := P 2 # P 2 # · · · # P 2

︸ ︷︷ ︸k

of k projective planes.

Exercise 3.5. One says that the surface S # T 2 results from S by adding ahandle and that the surface S # P 2 results from S by adding a cross cap. Itis a consequence of the previous exercise that every compact surface is home-omorphic to the surface obtained from a sphere by adding handles and crosscaps. Given g, k > 0, find n so that (gT 2) # (kP 2) is homeomorphic to nP 2.

4 Cell Complexes

4.1. Throughout this section Φ = {Φα : Dα → X}α∈Λ denotes an indexedcollection of maps into a topological space X; the domain Dα of each map is a(space homeomorphic to the standard) closed disk of dimension nα := dim(Dα).Such a collection determines and is determined by a map

Φ :⊔

α∈Λ

Dα → X, Φ|Dα := Φα

from the disjoint union of the disks to X. For each integer n define

X(n) = X(k)(Φ) :=⋃

nα≤n

Φα(Dα).

Definition 4.2. Such a collection {Φα}α∈Λ as in 4.1 is called a cell structureon X iff it satisfies the following:

(1) The restriction of Φ to the disjoint union⊔

α∈Λ(Dα \ ∂Dα) of the interiorsof the disks Dα is a bijection onto X.

(2) Φ(∂Dα) ⊂ X(n−1) where n = nα.

(3) The space X has the quotient topology, i.e. a subset U ⊂ X is open if andonly if Φ−1

α (U) is an open subset of Dα for all α ∈ Λ.

A space X equipped with a cell structure is called a cell complex. The closedsubset X(n) ⊂ X is called the n-skeleton.

4.3. The following notations and terminology are used for cell complexes. Thesets X(n) give a filtration of X, i.e.

X =∞⋃

n=0

X(n), X(0) ⊂ X(1) ⊂ X(2) ⊂ · · · .

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The further notations

eα := Φα(Dα), eα := Φα(Dα \ ∂Dα), φα := Φα|∂Dα

are commonly used. Thus

X =⋃

α∈Λ

eα, eα ∩ eβ = ∅ for α 6= β,

and each eα is homeomorphic to an open disk. The sets eα are called cells, thesets eα are called closed cells, the maps Φα are called characteristic maps,and the maps φα are called attaching maps. Sometimes a cell is called anopen cell for emphasis, but a cell need not be an open subset of X. It iscustomary to write en

α to indicate that the dimension of the cell eα is n = nα so

X(n) =⋃

m≤n

emα =

⋃

m≤n

emα , and en

α \ enα ⊂ X(n−1).

A cell complex is called finite (resp. countable) iff there are only finitely many(resp. countably many) cells, i.e. iff the index set Λ is finite (resp. countable).A cell complex is called n-dimensional iff X(n) = X and X(n−1) 6= X and iscalled finite dimensional iff it is n-dimensional for some n. It is customary todenote a cell complex by the same letter X as its underlying topological spacerather than by the more correct notation {Φα : Dα → X}α∈Λ.

Example 4.4. Exercise 3.1 shows how to construct a finite cell complex whoseunderlying topological space X is a surface. This cell complex has only one2-cell. For the surface of type (II) (connected sum of g copies of the torus) inRemark 3.3 there is one 0-cell, one 2-cell, and 2g 1-cells, while for the surfaceof of type (III) (connected sum of k copies of the projective plane) there is one0-cell, one 2-cell, and k 1-cells.

Exercise 4.5. Construct an explicit homeomorphism between the sphere Sn

and the cell complex with one 0-cell, one n-cell, and no other cells.

Proposition 4.6. A map Φ as in 4.1 with Λ finite and which satisfies (1)and (2) also satisfies (3), i.e. it is a (finite) cell complex.

Proposition 4.7. For a cell complex the map Φ :⊔

α∈Λ Dα → X is proper,i.e. compact set in a cell complex intersects only finitely many cells of X. Inparticular, each closed cell en

α intersects only finitely many other cells emβ .

Proof. See [3] Proposition A.1 page 520.

Proposition 4.8. A subset of a cell complex X is closed if and only if it inter-sects each n-skeleton X(n) in a closed set.

Proof. See [3] Proposition A.2 page 521.

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Remark 4.9. Cell complexes were invented by G. H. C. Whitehead [11] whocalled them CW complexes. He defined a CW complex as a space X equippedwith a decomposition X =

⋃α eα which arises from some cell structure {Φα}α,

whereas for us a cell complex is a space X equipped with a cell structure {Φα}α.(Different cell structures can give rise to the same decomposition.) The C standsfor closure finite which signifies the conclusion of Proposition 4.7, i.e. thateach closed cell en

α intersects only finitely many other cells emβ with m ≤ n. The

W stands for weak topology, the property in the conclusion of Proposition 4.8,that a subset of X is closed if and only if it intersects each n-skeleton X(n) in aclosed set. Whitehead viewed a CW complex as being constructed inductively:the n-skeleton X(n) is constructed from the (n−1)-skeleton X(n−1) by attachingthe n-cells en

α along their boundary. This is why the maps φα : ∂Dnα → X(n−1)

are called attaching maps. The equivalence of Whitehead’s definition with theinductive definition is Proposition A.2 page 521 of [3]. Definition 4.2 is easilyseen to be equivalent to the inductive definition. Following [3] we use the termscell complex and CW complex synonymously, but other authors use the formerterm to signify a structure satisfying parts (1) and (2) of Definition 4.2 but notnecessarily (3). In these notes we are mostly concerned with finite cell complexeswhere part (3) is superfluous by Proposition 4.6.

Example 4.10. The Hawaiian earring

X =∞⋃

n=1

{(x, y) ∈ R2 : x2 − 2n−1x + y2 = 0}.

inherits a topology as a subset of R2 which is weaker (fewer open sets) thanits topology as a countable wedge sum of circles. The former topology satisfiesparts (1) and (2) of Definition 4.2 but not (3). The latter topology is its topologyas a cell complex. Calling the quotient topology the weak topology is confusingbecause of this example.

Definition 4.11. A map f : X → Y between cell complexes is called cellulariff it preserves skeletons, i.e. iff f(X(n)) ⊂ Y (n) for all n. The cell complexesand the cellular maps between them form a category: The identity map of acell complex is cellular, and the composition of cellular maps is cellular. Acellular isomorphism is an isomorphism in this category, i.e. a cellular mapf : X → Y such that there is a cellular map g : Y → X satisfying g ◦ f = idX

and f ◦ g = idY .

Proposition 4.12. A map f : X → Y between cell complexes is cellular if andonly if the image of each cell of X is a subset of a finite union of cells of Y ofthe same or lower dimension.

Proof. ‘If’ is immediate and ‘only if’ follows from the image of a closed set iscompact and therefore a subset of a finite subcomplex of Y .

Remark 4.13. A cellular map f need not satisfy the stronger condition that theimage of each cell of X is a subset of a cell of Y of the same or lower dimension.

11

If X = Y = I, X(0) = {0, 1}, Y 0 = {0, 1/2, 1}, and f is the identity map, thenf(X(0)) ⊂ Y (0) but f(X) is not contained in a cell of Y . The cell complexesand the cellular maps between them form a category somewhat analogous to thecategory simplicial complexes and the simplicial maps between them. However,for a simplicial map the image of a simplex is a subset of a simplex of the sameor lower dimension.

Proposition 4.14. A map f : X → Y between cell complexes is a cellularisomorphism if and only if f is a homeomorphism and maps each cell of X ontoa cell of Y (necessarily of the same dimension).

Proof. A cellular map between cell complexes is continuous so a cellular iso-morphism f is (in particular) a homeomorphism. For each k map f induces ahomeomorphism X(k)/X(k−1) → Y (k)/Y (k−1) and the map g := f−1 inducesthe inverse homeomorphism. But X(k)/X(k−1) is a wedge of spheres and be-comes disconnected if the wedge point is removed. Since the wedge point ofX(k)/X(k−1) is mapped to the wedge point of Y (k)/Y (k−1) the components ofthe complement are preserved, i.e. f maps each open k-cell of X homeomor-phicly onto a k-cell of Y and thus determines a bijection between the k-cellsof X and the k-cells of Y . (Homeomorphic cells must have the same dimen-sion by Brouwer’s Invariance of Domain Theorem but that theorem is not usedhere.)

Definition 4.15. A subcomplex of a cell complex X =⋃

α∈Λ enα is a closed

subset A of formA =

⋃

α∈Λ0

enα

where Λ0 ⊂ Λ. The condition that A is closed implies that A =⋃

α∈Λ0enα. Thus

A inherits a cell structure from X and the k-skeleton of A is A(k) = A ∩X(k).In particular, the inclusion map A → X is cellular. The k-skeleton X(k) of Xis a subcomplex. A cellular pair is a pair (X,A) consisting of a cell complexX and a subcomplex A ⊂ X.

Remark 4.16. The definitions do not require that a closed cell be a subcom-plex, but this is usually the case.

Proposition 4.17. A compact subset of a cell complex lies in a finite subcom-plex.

Proof. As in Proposition 4.7.

4.18. Certain operations on cell complexes produce new cell complexes as ex-plained on page 8 of chapter 0 of [3]. The product X×Y of two countable cellcomplexes is a cell complex. The quotient X/A obtained from a cellular pair(X, A) by collapsing A to a point is a cell complex. The cone CX on a cellcomplex X is a cell complex. The suspension SX of a cell complex is a cellcomplex. The join X ∗ Y of two cell complexes is a cell complex. The wedgesum X ∨ Y of two cell complexes is a cell complex (provided that the wedge

12

point is a zero cell of each). As noted in [3] the product topology is the quotienttopology (i.e. Item (3) of Definition 4.2 holds) for countable complexes but thiscan fail if one of the factors is uncountable.

Remark 4.19. There is an example of a product X × Y of two cell complexes(with X countable but Y not) which is not a cell complex in the product topol-ogy.

5 Some Homotopies

Lemma 5.1 (Mailed Fist Homotopy). Let (X, A) be a cullular pair. Then(X × 0) ∪ (A× I) is a deformation retract of X × I.Proof. We must show that there is a homotopy {Rt = (rt, τt) : X×I→ X×I}t∈Isatisfying r0(x) = x, rt(a) = a, r1(A) = A, and τ0(x) = 0, τt(a, s) = s, forx ∈ X, a ∈ A, t, s ∈ I. See Proposition 0.16 on page 26 of [3].

Corollary 5.2. Assume that (X,A) is a cellular pair and that A is contractible.Then the projection X → X/A is a homotopy equivalence.

Proof. Define K : (X × 0) ∪ (A × I) → (X × 0) ∪ (A × I) by K(x, 0) = x andK(a, t) = kt(a) for x ∈ X, a ∈ A, t ∈ I. Let {Rt : X× I → (X×0)∪ (A× I)}t∈Ibe as in the proof of Lemma 5.1. Then R1 : X × I →: (X × 0) ∪ (A × I) is aretraction, i.e. R1(x, 0) = (x, 0) and R1(a, t) = (a, t). Define ft : X → X byft(x) = K(R1(x, t)) so f0(x) = K(x, 0) = x and f1(x) = K(a, 1) = k1(a) = a0.Let π : X → X/A be the projection and define σ : X/A → X by σ(x) = f1(x)for x ∈ X \ A and σ({A}) = a0. Then σ ◦ π = f1 ' idX and π ◦ σ ' idX/A bythe homotopy ht(x) = π(ft(x)) for x ∈ X \A and ht({A}) = {A}.Corollary 5.3. Assume that (X, A) is a cellular pair and that f, g : A → Y arehomotopic. Then Y ∪f X ' Y ∪g X.

Proof. By hypothesis there is a map H : A × I → Y with H(·, 0) = f andH(·, 1) = g. There are inclusions

ι0 : Y tf X → Y tH (X × I), ι1 : Y tg X → Y tH (X × I)

induced from the inclusions X × 0 ⊂ X × I and X × 1 ⊂ X × I; it is enough toshow that ι0 and ι1 are homotopy equivalences. We prove the former; the sameargument (read 1− t for t) proves the latter.

Step 1. We define F : Y tH (X × I) → Y tf X. Let {Rt}t∈I be deformationretraction from X×I onto (X×0)∪(A×I) as in Lemma 5.1 and let Rt = (rt, τt).Define F : Y tH (X × I) → Y tf X by F (y) = y for y ∈ Y and

F (x, s) =

{H(R1(x, s)) ∈ Y if R1(x, s) ∈ A× I,r1(x) ∈ X if R1(x, s) ∈ X × 0.

13

The map F is well defined as H(R1(x, s)) = f(r1(x)) ∼ r1(x) for R1(x, s) ∈A× 0.

Step 2. We show F ◦ ι0 ' idY tf X . Define {Φt : Y tf X → Y tf X}t∈I byΦt(y) = y, Φt(x) = rt(x). The map Φt is well defined because rt(x) = x ∼ f(x)for x ∈ A. One easily checks that Φ0 = idY tf X and Φ1 = F ◦ ι0.

Step 3. We show ι0 ◦ F ' idY tH(X×I). Define {Ψt : Y tH X → Y tH X}t∈Iby Ψt(y) = y for y ∈ Y and Ψt(x, s) = Rt(x, s) ∈ X × I for (x, s) ∈ X × I. Themap Ψt is well defined because Rt(x, s) = (x, s) ∼ H(x, s) for (x, s) ∈ A × I.One easily checks that Ψ0 = idY tHX and Ψ1 = ι0 ◦ F .

Remark 5.4. Let Y = A, f = idA : A → Y and g : A → Y be a constant map:g(a) = a0. Clearly the space Y tf X is homeomorphic to the space X and thespace Y tg X is homeomorphic to the wedge sum A ∨ (X/A) where the pointa0 ∈ A is identified. Hence if A is contractible, then f and g are homotopicso X and A ∨ (X/A) are homotopy equivalent. By Corollary 5.6 below, {a0}is a deformation retract of A. Hence the wedge sum A ∨ (X/A) is homotopyequivalent to X/A. Thus Corollary 5.2 is a consequence of Corollary 5.3.

Theorem 5.5. Let (X, A) be a cellular pair. Then the inclusion map ι : A → Xis a homotopy equivalence if and only if A is a deformation retract of X.

Proof. (This is Corollary 0.20 page 16 of [3]. The following proof is from [12]page 25.) ’If’ is trivial. For ’only if’ let f : X → A be a homotopy inverse to ι.Let {φt : A → A}t∈I be a homotopy with φ1 = idA and φ0 = f ◦ ι. Extend toa homotopy {Φt : X → A}t∈I with Φ0 = f . Then Φ1 : X → A is a retraction.Hence, replacing f by Φ1, we may assume w.l.o.g. that the homotopy inverse fto ι is a retraction, i.e. that f |A = idA. Let {ψt : X → X}t∈I be a homotopyfrom ψ0 = idX to ψ1 = ι ◦ f . Let (Y, B) be the cellular pair defined by

Y = I×X, B = ({0, 1} ×X) ∪ (I×A).

Define h : ({0} × Y ) ∪ (I×B) → X by

h(0, t, x) = ψt(x)h(s, 0, x) = ψ0(x) = x

h(s, t, a) = ψ(1−s)t(a)h(s, 1, x) = ψ1−s(f(x))

for s, t ∈ I, x ∈ X, and a ∈ A. By Lemma 5.1, extend h to H : I× Y → X anddefine {gt : X → X}t∈I by gt(x) = H(1, t, x). Then

gt|A = idA, g0 = idX , g1 = f,

as gt(a) = H(1, t, a) = h(1, t, a) = ψ0(a) = a, g0(x) = H(1, 0, x) = h(1, 0, x) =ψ0(x) = x, and g1(x) = H(1, 1, x) = ψ0(f(x)) = f(x), for a ∈ A and x ∈ X.

Corollary 5.6. For a cell complex X the following are equivalent:

14

(i) X is contractible.

(ii) For some a0 ∈ X, {a0} is a deformation of X.

(iii) For every a0 ∈ X, {a0} is a deformation of X.

Remark 5.7. Of course, the implications (iii) =⇒ (ii) =⇒ (i) in Corollary 5.6hold for any space. Exercises 6 and 7 on page 18 of [3] show that the implications(ii) =⇒ (iii) and (i) =⇒ (ii) do not hold for general spaces.

Proposition 5.8. Let (X,A) be a cellular pair. Then there is a neighborhoodN of A in X such that A is a deformation retract of N .

Proof. This is Proposition A.5 page 523 of [3].

Corollary 5.9. A cell complex is locally contractible, i.e. for every x0 ∈ Xand every neighborhood V of x0 there is a neighborhood U ⊂ V of x0 and adeformation retraction of U onto {x0}.Problem 5.10. It is plausible that a retraction whose fibers are (homeomorphicto) intervals is a deformation retraction, but I doubt that this is true in com-plete generality. However, a theorem of Smale [9] combined with Whitehead’sTheorem (Theorem 4.5 page 346 of [3]) implies that map between finite cellcomplexes with contractible fibers is a homotopy equivalence. It then followsfrom Theorem 5.5 that if (X,A) is a cellular pair and r : X → A is a retrac-tion with contractible fibers, then A is a deformation retract of X. Prove (ordisprove) the following slightly stronger statement: If (X,A) is a cellular pairthen a retraction r : X → A with contractible fibers is a deformation retraction.

Exercise 5.11. Let X = D2× I be a solid cylinder at α : I→ X be a polygonalarc, possibly knotted, with α(0) = (0, 0, 1) ∈ D2×1 and α(1) = (0, 0, 0) ∈ D2×0.Show that the image A := α(I) is a deformation retract of X. (This examplecomes from Conley [1] page 26.)

Exercise 5.12. Given positive integers v, e, and f satisfying v − e + f = 2construct a cell complex homeomorphic to S2 having v 0-cells, e 1-cells, and f2-cells.

Solution. If v − e + f = 2 then

(v, e, f) = (1, 0, 1) + m(1,−1, 0) + n(0,−1, 1)

where m = v − 1 and n = f − 1. The sphere is a cell complex with v =f = 1 and e = 0 in the only way possible: the attaching map is constant. Toincrease m by one without changing n add a new vertex and edge and modifythe corresponding characteristic map by composing with complex square root.To increase n without changing m insert a loop at a vertex. The result followsby induction.

15

6 ∆-Complexes and Simplicial Complexes

Definition 6.1. Resume the notation of 4.1 and Definition 4.2. A cell complex{Φα : Dα → X}α∈Λ is called a ∆-complex iff each component Dα is a standardn simplex

∆n := {(x0, x1, . . . , xn) ∈ In+1 : x0 + x1 + · · ·+ xn = 1}

(as before, n = nα depends on α) and for each face ι : ∆n−1 → ∆n and eachcharacteristic map Φα the composition Φα ◦ ι is also a characteristic map. Theterm face here means

ι(x1, . . . , xn) = (x1, . . . , xk, 0, xk+1, . . . , xn)

for some k = 0, . . . , n, i.e. ι = ιk as in 1.3. The points of the 0-skeleton ofa ∆-complex are called vertices and the closed cells en

α := Φα(∆n) are calledsimplices. A ∆-complex is called a simplicial complex iff the characteristicmaps are determined by the vertices, i.e. whenever α, β ∈ Λ satisfy (nα = nβ

and) Φα(v) = Φβ(v) for each of the n + 1 vertices v = (0, . . . , 0, 1, 0, . . . , 0) of∆n, we have α = β. A simplicial complex {Φα : Dα → X}α∈Λ is called atriangulation of X. As was the case with cell complexes, it is customary todenote a ∆-complex by the same letter X as its underlying space rather thanby the more correct notation {Φα : Dα → X}α∈Λ.

Example 6.2. The simplex ∆n is a simplicial complex with a characteristicmap Φα : ∆m → ∆n for every subset α ⊂ {0, 1, . . . , n}. The map Φα is the map

Φα(x) =m∑

j=0

xjvij

for x = (x0, x1, . . . , xm) ∈ ∆m, α = {i0, i1, . . . , im} with i0 < i1 < · · · < im andwhere vi is the vertex of ∆n with 1 in the ith place and 0 elsewhere.

Example 6.3. The torus T 2 is the surface obtained from the unit square I2with the identifications (x, 0) ∼ (x, 1) and (0, y) ∼ (1, y). The Klein bottleK2 is the surface obtained from the unit square I2 with the identifications(x, 0) ∼ (x, 1) and (0, y) ∼ (1, 1− y). The projective plane P 2 is the surfaceobtained from the unit square I2 with the identifications (x, 0) ∼ (1− x, 1) and(0, y) ∼ (1, 1− y). For X = T 2,K2, P 2 there is a ∆-complex Φ as indicated inFigure 1. For X = T 2 and X = K2 there are one 0-simplex v, three 1-simplicesa, b, c, and two 2-simplices U and L. For X = P 2 we must adjoin another0-simplex w. In each case the bijections Φα is determined by the diagram inthe only way possible so that the corresponding maps to I2 are affine and thearrows on the edges match up. Note that had we reversed the arrows marked ain P 2 the vertices of triangle U would be cyclically (not linearly) ordered. Thefigure would still represent the projective plane but not a ∆-complex.

16

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U

La a

b

b

c

T 2

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-

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v

v

v

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b

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K2

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w

w

v

U

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b

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P 2

Figure 1: Three ∆-complexes

Example 6.4. Let T 2 = S1×S1 = R2/Z2 be the torus. There is a cell complexΦ with one 0-cell (the origin (0, 0) in R2), two 1-cells (the intervals I × {0}and {0} × I), and one 2-cell (the unit square I × I); the characteristic mapis Φ(x, y) = (x, y) mod Z2. If we add the diagonal we get the ∆-complex ofExample 6.3. There is an easy triangulation with vertices (p/3, q/3) mod Z2 forp, q = 0, 1, 2 and edges from (p/3, q/3) to

((p + 1)/3, q/3

),(p/3, (q + 1)/3

), and(

(p + 1)/3, (q + 1)/3). This triangulation has 9 vertices, 27 edges, and 18 faces.

By judiciously discarding a few vertices and edges we can achieve a triangulationwith 7 vertices. (See Exercise 6.6.)

6.5. It is easy to construct simplicial complexes whose geometric realizationsare homeomorphic to the sphere S2, the cylinder S1 × I, the Mobius band, thetorus, the Klein bottle, and the projective plane. (See [7] pages 16-19.)

Exercise 6.6. Show that for any triangulation of a compact surface, we have3f = 2e, e = 3(v − χ), and v ≥ (7 +

√49− 24χ )/2. In the case of the sphere,

projective plane and torus, what are the minimum values of the numbers v,e and t? (Here v,e and f denote the number of vertices, edges and trianglesrespectively; χ := v − e + f denotes the Euler characteristic.)

Solution. Let P be the set of pairs (E,F ) such that F is a face and E is an edgein the boundary of F , PE = {F : (E,F ) ∈ P}, and PF = {E : (E, F ) ∈ P}.Then #(PE) = 2 for each E so #(P ) = 2e. Also #(PF ) = 3 for each F so#(P ) = 3f . Hence 2e = 3f . From v − e + f = χ we get e = 3v − 3χ. Butclearly e ≤ (

v2

)= v(v−1)

2 so 0 ≤ v2 − 7v + 6χ so v ≥ 7+√

49−24χ2 . For the sphere

S2 we have χ = 2 and hence v ≥ 4 with equality for the tetrahedron. Forthe projective plane we have χ = 1 so v ≥ 6 with equality for the triangulationshown on page 15 of [6]. (This example is obtained by identifying opposite edgesof a hexagon and adding a triangle in the interior.) For the torus T 2 we haveχ = 0 so v ≥ 7 with equality in the following triangulation:

17

v1

v1 v1

v1

v2 v2

v3 v3

v4

v4

v5

v5

v6

v7

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@

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QQ

QQ

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Note that in all cases the lower bound is achieved when the 1-skeleton is thecomplete graph on v vertices. It is a consequence of the proof that this completegraph is the 1-skeleton of a surface of Euler characteristic χ only when the lowerbound is an integer.

6.7. Let {Φα : Dα → X}α∈Λ and {Ψβ : Dβ → Y }β∈Λ′ be ∆-complexes. A mapf : X → Y is called ∆-map iff for every α ∈ Λ there is a (necessarily unique)β ∈ Λ′ and a simplicial map fα : Dα → Dβ such that

Ψβ ◦ fα = f ◦ Φα.

That a map g : ∆n → ∆m is simplicial means that it sends each vertex vi of∆n to a vertex g(vi) of ∆m and that it is affine, i.e.

g

(n∑

i=0

xivi

)=

n∑

i=0

xig(vi).

With respect to the simplicial structure of Example 6.2 a map g : ∆n → ∆m issimplicial if and only if it is a ∆-map. A ∆-map between simplicial complexesis also called a simplicial map.

6.8. The ∆-complexes and ∆-maps between them form the objects and mor-phisms of a category, i,e, the identity map of of a ∆-complex is a ∆-map andthe composition of two ∆-maps is a ∆-map. The simplicial complexes form a(full) subcategory.

6.9. If (X, A) is a cellular pair and X is a ∆-complex, then the subcomplex Ais also a ∆-complex; in this case we call (X, A) a ∆-pair. The cell structureon X/A of 4.18 is then a ∆-complex and the projection X → X/A is a ∆-map. (This construction does not give a simplicial complex even when X is asimplicial complex.) As in 4.18 the operations of suspension, cone, join, andproduct also yield ∆-complexes (resp. simplicial complexes) when applied to∆-complexes (resp. simplicial complexes). The characteristic maps of the joinare determined by composition with the inverse of the canonical homomorphism

∆m ∗∆n → ∆m+n+1 : (x, y, t) 7→ tx + (1− t)y.

The characteristic maps of the product ∆-complex are given by compositionwith the maps

`σ : ∆m+n → ∆m ×∆n

18

described on page 277 of [3]. In the special case n = 1 this reduces to the prismstructure on ∆m × I described in the proof of Theorem 2.10 on page 111 of [3].As is the case for cell complexes, the product topology is the quotient topologyfor countable complexes but this can fail if one of the factors is uncountable.

7 Abstract Simplicial Complexes*

7.1. An abstract simplicial complex is a collection K of nonempty finite setscalled simplices such that every nonempty subset of a simplex is a simplex, i.e.

∅ 6= τ ⊂ σ ∈ K =⇒ τ ∈ K.

A subset of a simplex is called a face of the simplex. The elements of the set

V :=⋃

σ∈K

σ

are called vertices. Since v ∈ V ⇐⇒ {v} ∈ K It is customary not todistinguish between v ∈ V and {v} ∈ K. A subset of K that is itself a simplicialcomplex is called a subcomplex. If K1 and K2 are simplicial complexes withvertex sets V1 and V2, a map f : K1 → K2 is called a simplicial map if thereis a (necessarily unique) map f : V1 → V2 denoted by the same letter such that

f({v0, v1, . . . , vk}) = {f(v0), f(v1), . . . , f(vk)}for {v0, v1, . . . , vk} ∈ K1. Abstract simplicial complexes and simplicial mapsform the objects and morphisms of a category.

7.2. The dimension dim(σ) of a simplex σ is one less than its cardinality. Asimplex of dimension k is also called a k-simplex. The subcomplex

K(k) := {σ ∈ K : dim(σ) ≤ k}is called the k-skeleton of the simplicial complex K. As noted above we maydenote the set of vertices by

V = K(0).

A simplicial complex is called n-dimensional iff K(n) = K and K(n−1) 6= K; itis called finite dimensional iff it is n-dimensional for some n ∈ N. A simplicialcomplex is called finite if the set K is itself finite.

7.3. The geometric realization of an abstract simplicial complex K is theset

|K| :={

x ∈ [0, 1]V∣∣ ∑

v∈V

x(v) = 1 and supp(x) ∈ K

}.

Here V is the vertex set of K and supp(x) := {v ∈ V : x(v) 6= 0} is thesupport of the element x ∈ |K|. A simplicial map f : K → L induces a mapf : |K| → |L| called the geometric realization of f via the formula

f(x)(w) =∑

f(v)=w

x(v)

19

for x ∈ |K|. Geometric realization is functorial.

Theorem 7.4. Assume that K is a finite abstract simplicial complex of dimen-sion n. Let E = R2n+1 denote the Euclidean space of dimension 2n+1 and EV

denote the space of all maps from f : V → E. (The space EV has dimension#(V )·(2n+1).) Call an element f ∈ EV generic iff Whenever v0, v1, . . . , vk aredistinct points of V and k ≤ 2n + 1 the vectors f(v1)− f(v0), . . . , f(vk)− f(v0)are linearly independent. Then

(i) The set of generic elements is open and dense in EV .

(ii) If f is generic, the induced map f : |K| → E defined by

f(x) :=∑

v∈V

x(v)f(v)

is injective.

(iii) The topology on |K| induced by this embedding is independent of the choiceof the generic element f used to define it.

(iv) In this topology the geometric realization of a simplicial map between finitecomplexes is continuous.

Proof. For a sequence W = (v0, v1, . . . , vk) of distinct elements of V of form thematrix AW (f) ∈ R(2n+1)×k whose columns are the k vectors f(vi) − f(v0) fori = 1, . . . , k. For each k element subset R ⊂ {1, . . . , 2n + 1} let aR

W (f) be thedeterminant of the square matrix that results from AW (f) by deleting the rowswhose index is not in R. Then f is generic if and only if for every W there isan R such that aR

W (f) 6= 0. Since aRW (f) is a polynomial in the entries of f it

cannot vanish identically on an open subset of f ’s. This proves (i). Since |K|is compact, the map f : |K| → E is and embedding if and only if it is injective.But a generic map must be injective: if f(x) = f(y) then (as the supports of xand y are of cardinality at most n + 1) the union W of the supports of x and yhas cardinality at most 2n+2. But the equation f(x) = f(y) is a linear relationbetween the vectors {f(v) : v ∈ W}. As these vectors are independent we musthave x = y as required.

Remark 7.5. The induced map f : |K| → E can be injective even when f isnot generic. For countable K of dimension n, item (i) of Theorem 7.4 continuesto hold provided open and dense is weakened to residual (countable intersectionof open dense sets) so generic maps f exist by the Baire Category Theorem.Item (ii) will hold but item (iv) is false. As a subset of [0, 1]V the geometricrealization inherits a topology from the product topology of [0, 1]V . In thistopology a set A ⊂ |K| is closed if and only if A∩ |σ| is closed for every σ ∈ K,but (in the infinite case) this topology can be different from the one induced byan embedding in E.

20

7.6. An ordered simplicial complex is a simplicial complex equipped witha linear ordering for each simplex in such a way that if τ is a face of σ thenthe ordering assigned to τ is the restriction of the ordering assigned to σ. Asimplicial complex may be ordered in many ways. Generally we will choosean ordering for book keeping purposes and then prove that the choice doesn’tmatter.

Remark 7.7. The geometric realization |K| of an ordered simplicial complexis a ∆-complex

Φ := {Φσ : ∆dim(σ) → |σ|}σK

viaΦσ(x) = x0δv0 + x1δv1 + · · ·xnδvn

for x = (x0, x1, . . . , xn) ∈ ∆n (n = dim(σ)) and σ = {v0, v1, . . . , vn} wherev0 < v1 < · · · vn.

Problem 7.8. An abstract unordered ∆-complex is a disjoint union

W =⊔

α∈Λ

σα,

of nonempty finite subsets together with a function which assigns to each pair(w, α) with #(σα) > 1 w ∈ σα a bijection ι : σβ → σ \ {w}. (The index β ∈ Λis also a function of (α, w).) The elements of W are called vertices, the sets σα

are called simplices, and the maps ι are called attaching maps. An abstractordered ∆-complex is an abstract unordered ∆-complex together with a linearordering for each simplex σα such that each attaching map ι is the unique orderisomorphism from its domain to its target. Does every abstract unordered ∆-complex arise from an abstract ordered ∆-complex? (In other words given anabstract unordered ∆-complex can we order its simplices so that for each (w,α)the bijection ι is the unique order isomorphism?)

Remark 7.9. An abstract ∆-complex determines a cell complex Φ : D → Xwhere

D :=⊔

α∈Λ

|σα|

is the disjoint union of the geometric realizations of the simplices, X = D/∼,and Φ : D → X is the projection into the identification space. The equivalencerelation is generated by

x =∑

v∈β

xvδv ∈ |σβ | y =∑

v∈β

xvδι(v) ∈ |σα| =⇒ x ∼ y.

This cell complex is called the geometric realization of the abstract ∆-complex.

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References

[1] Charles Conley: Isolated Inveariant Sets and the Morse Index, CBMS Con-ferences in Math 38, AMS 1976.

[2] William Fulton: Algebraic Topology, A First Course Springer GTM 153,1995.

[3] Allen Hatcher: Algebraic Topology, Cambridge U. Press, 2002.

[4] John G. Hocking & Gail S. Young: Topology, Addisow Wesley, 1961.

[5] Morris W. Hirsch: Differential Topology, Springer GTM 33, 1976.

[6] William S. Massey: A Basic Course in Algebraic Topology, Springer GTM127, 1991.

[7] James R. Munkres: Elements of Algebraic Topology, Addison Wesley, 1984.

[8] S. Shelah: Can the fundamental group of a space be the rationals? Proc.Amer. Math. Soc. 103 (1988) 627–632.

[9] Stephen Smale: A Vietoris mapping theorem for homotopy. Proc. Amer.Math. Soc. 8 (1957), 604–610

[10] J. W. Vick: Homology Theory (2nd edition), Graduate Tests in Math 145,Springer 1994.

[11] J.H.C. Whitehead: Combinatorial homotopy II, Bull. A.M.S 55 (1949)453-496.

[12] George W. Whitehead: Elements of Homotopy Theory Graduate Tests inMath 61, Springer 1978.

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