PARTITION COMPLEXES, TITS BUILDINGS ANDwgd/Dvi/Partitions.And.Buildings.pdfPARTITION COMPLEXES, TITS...

PARTITION COMPLEXES, TITS BUILDINGS AND

SYMMETRIC PRODUCTS

G. Z. ARONE AND W. G. DWYER

Abstract. We construct a homological approximation to the par-tition complex, and identify it as the Tits building. This givesa homological relationship between the symmetric group and theaffine group, leads to a geometric tie between symmetric powers ofspheres and the Steinberg idempotent, and allows us to use the self-duality of the Steinberg module to study layers in the Goodwillietower of the identity functor.

1. Introduction

A partition complex is a geometric object associated to the posetof equivalence relations on a finite set; a Tits building is a geometricobject associated to the poset of subspaces of a vector space. The twoobjects are formally alike, but aside from that they don’t seem to havemuch in common. For instance, they have very different symmetries;permutation groups act on partition complexes, while affine groups orgeneral linear groups act on Tits buildings. Nevertheless, in this pa-per we describe a close relationship between the two, based upon theobservation that when we feed the partition complex to a particularhomological approximation machine, it is essentially a Tits buildingthat comes back out (§4-5) . This gives some surprising connectionsbetween the homology of symmetric groups and the homology of affinegroups or general linear groups (1.3, 1.6). In view of a tie we establishbetween the partition complex and symmetric powers of spheres (1.12),it also explains, in a geometric way, why the homology of these sym-metric powers is related to the Steinberg idempotent. Finally, althoughthe partition complex is not stably self-dual, the Tits building is. Therelationship between the two leads to a kind of shifted self-duality forthe partition complex, and this allows us to prove some conjectures ofArone and Mahowald about the layers in the Goodwillie tower of theidentity functor (1.17).

Date: January 10, 2000.Both authors were partially supported by the NSF.

1

2 G. Z. ARONE AND W. G. DWYER

In the course of the paper, we clarify some aspects of [1] and weave to-gether many threads from earlier work of Kuhn [11], Kuhn and Priddy[12], Mitchell [16], Mitchell and Priddy [17], and others. Part of theappeal of our approach is that it amounts to setting up a general ap-proximation technique and then just working out its consequences inone particular case.

We will now describe the results of the paper in more detail.

Partitions and subspaces. Let n denote the finite set {1, . . . , n}.An equivalence relation λ on n is said to be nontrivial if there is morethan one λ-equivalence class and at least two distinct elements of n areλ-equivalent to one another. The collection of nontrivial equivalencerelations on n is a poset (with the convention that λ1 ≤ λ2 if λ1 is arefinement of λ2) and the associated space Pn is called the partition

complex of n (see 1.19). The group Σn of permutations of n acts onPn in a natural way.

Suppose that n = pk (p a prime) and identify n in some way withthe vector space ∆ = (Fp)

k over the field Fp. Say that an equiva-lence relation λ on ∆ is linear if it is stable under translations (i.e.,v ∼λ w ⇒ v + x ∼λ w + x). Giving a linear equivalence relationamounts to specifying the subspace of all elements equivalent to 0, andso the poset of nontrivial linear equivalence relations is isomorphic tothe poset (under inclusion) of nontrivial proper subspaces of ∆. Theassociated space Tk is called the Tits building of ∆. The affine groupAffk,p = GLk(Fp) ⋉ ∆ acts on ∆ and on Tk. The identification of∆ with n gives a subgroup inclusion Affk,p → Σn, and treating linearequivalence relations as equivalence relations gives a map Tk → Pn

which is equivariant with respect to this inclusion.

An approximation theorem. Our first result states that the mapTk → Pn is in a certain sense an equivariant homology approximation.We need some notation to describe this. If G is a subgroup of a sym-metric group and M is a G-module, let M± denote the twist of M bythe sign representation of G; in particular, F

±p denotes the sign repre-

sentation of G on Fp. For a pair (X,Y ) of G-spaces, let HG∗ (X,Y ; M)

denote the relative Borel construction homology

H∗( (EG×X)/G, (EG× Y )/G; M ) .

This is relative homology with local coefficients. If X has a G-basepointx0, the reduced homology HG

∗ (X; M) is defined to be HG∗ (X, x0; M).

For any G-space X, the unreduced suspension X♦ of X is the pointedG-space obtained by collapsing the base of the cone on X to a pointand using this as the basepoint.

PARTITION COMPLEXES AND TITS BUILDINGS 3

1.1. Theorem. Suppose that n > 1 is an integer. If n is not a power

of p, then HΣn∗ (P♦

n ; F±p ) vanishes. If n = pk, then the map Tk → Pn

induces an isomorphism

HAffk,p

∗ (T♦k ; F

±p ) ∼= HΣn

∗ (P♦n ; F

±p ) .

1.2. Remark. The reader may ask why the local coefficients F±p are

necessary in 1.1. The explanation is that we find a Σn-model X for P♦n

and an Affk,p-model Y for T♦k such that Σn ∧Affk,p

Y is a type of Σn-approximation to X. (The space X is the cofibre of the map |P| → ∗from 5.1, and Y the cofibre of |E ′| → |E|.) In order to show that themap Σn ∧Affk,p

Y → X induces an isomorphism on reduced homologywith coefficients in some Σn-module M , we have to check that for eachisotropy subgroup G of the action of Σn on X, either

• p divides the order of ker(G→ Aut(M)), or• H∗(G; M) = 0 .

See 4.6 and 4.7. The isotropy subgroups of the action of Σn on X arethe nontrivial products Σn1

× · · · × Σnj⊂ Σn (4.1). If p is odd it is

possible for all of the nj’s to be less than p. In this case neither of theabove two conditions is satisfied if M is the trivial Σn-module Fp, butthe second condition is satisfied if M = F

±p .

Group homology interpretations. Theorem 1.1 leads to a surpris-ing group homology calculation. The spaces Tk and Pn above arespherical, in the sense that each is homotopy equivalent to a wedgeof spheres which all have the same dimension. For Tk this dimensionis k − 2, while for Pn it is n − 3. (See [20] [2] or [19, Th. 2] for theTits building, and [18, 4.109] for the partition complex.) The sus-pended spaces T♦

k and P♦n are also spherical, with the dimensions of

the spheres involved increased by one. The top group Hk−1(T♦k ; Z) is

a module over Affk,p which is called the Steinberg module Stk. The top

group L = Hn−2(P♦n ; Z) is a module over Σn without a standard name,

but since the Z-dual of the twisted module L± is sometimes denotedLien, we will denote L± itself by Lie∗n.

1.3. Theorem. Suppose that n > 1 is an integer. If n is not a power

of p, then H∗(Σn; Fp ⊗ Lie∗n) vanishes. If n = pk, then for i ≥ 0 there

are isomorphisms

Hi(Affk,p; Fp ⊗ St±k ) ∼= Hi−n+k+1(Σn; Fp ⊗ Lie∗n) .

1.4. Remark. In the above statement and in similar situations involv-ing group homology later on in the paper, Hi = 0 for i < 0. Thereis also a cohomological version of this theorem, which appears in 6.3.


The calculations of 6.3 show that Lien and Lie∗n are not in generalisomorphic as Σn-modules.

Since the subgroup (Fp)k of translations in Affk,p = GLk(Fp) ⋉ (Fp)

k

acts trivially on the complex Tk of linear equivalence relations, Stk isactually a module over GLk(Fp) = GLk,p. It turns out that Fp ⊗ Stk isprojective as a module over Fp[GLk,p], and that there is an idempotentǫStk ∈ Fp[GLk,p], called the Steinberg idempotent, such that for any

Fp[GLk,p]-module M there is an isomorphism

ǫStk ·M

∼= H0(GLk,p; M ⊗ Stk) .(1.5)

See [21], [17], or [10]. A little bit of calculation with a collapsingspectral sequence leads to the following consequence of 1.3.

1.6. Theorem. Suppose that n = pk > 1. Let ∆ = (Fp)k and let

GLk,p = Aut(∆) ⊂ Σn act on the group homology H∗(∆; F±p ) in the

natural way. Then for i ≥ 0 there are isomorphisms

ǫStk · Hi(∆; F±

p ) ∼= Hi−n+k+1(Σn; Fp ⊗ Lie∗n) .

1.7. Remark. In the above statement, the twist in F±p affects the action

of GLk,p on H∗(∆; F±p ) but does not affect the group homology itself.

A geometric interpretation. Theorem 1.1 also has a more geomet-ric interpretation. If X is a G-space, let XhG denote the Borel con-struction (EG×X)/G; if X has a G-basepoint x0, let XhG denote thereduced Borel construction XhG/(x0)hG = XhG/ BG.

1.8. Theorem. Suppose that n > 1, that ℓ is odd, and that Σn acts

on the sphere Sℓn by permuting the factors of Sℓn = (Sℓ)∧n. If n is

not a power of p, then the reduced mod p homology of (Sℓn ∧ P♦n )hΣn

vanishes. If n = pk, then the inclusion Tk → Pn induces a mod phomology isomorphism

(Sℓn ∧T♦k )h Affk,p

→ (Sℓn ∧P♦n )hΣn

.

In the course of proving 1.8 we also prove a dual result, which itturns out is needed in studying the Goodwillie tower. If X is a spaceor spectrum, let X# denote the Spanier-Whitehead dual of X (1.19).

1.9. Theorem. Suppose that n > 1, that ℓ is odd, and that Σn acts

on the sphere Sℓn by permuting the factors of Sℓn = (Sℓ)∧n. If n is not

a power of p, then the reduced mod p homology of (Sℓn ∧ (P♦n )#)hΣn

vanishes. If n = pk, then the inclusion Tk → Pn gives rise to a mod phomology isomorphism

(Sℓn ∧ (P♦n )#)hΣn

→ (Sℓn ∧ (T♦k )#)h Affk,p

.


1.10. Remark. Say that an equivalence relation on n is completely reg-

ular if all of the equivalence classes have the same size, and let Rn

denote the space associated to the poset of nontrivial completely reg-ular equivalence relations on n. If n = pk, the inclusion Tk → Pn

factors as Tk → Rn → Pn, and for odd ℓ Theorem 1.8 guarantees thatthe induced composite

(Sℓn ∧T♦k )h Affk,p

→ (Sℓn ∧R♦n )hΣn

→ (Sℓn ∧P♦n )hΣn

gives an isomorphism on mod p homology. It follows that the p-completionof (Sℓn∧P♦

n )hΣnis a retract of the p-completion of (Sℓn∧R♦

n )hΣn. This

answers a question of Kuhn [9]; the space (Sn ∧ R♦n )hΣn

is related tothe space Yk of [8, §5].

1.11. Remark. Theorems 1.8 and 1.9 also hold (in the appropriate sta-ble sense) for negative odd values of ℓ. If p = 2, the assumption that ℓis odd can be removed from both theorems.

Relationship to symmetric powers. If X is a space, let SPn(X)denote the n’th symmetric power of X, i.e., the quotient space Xn/Σn.Choice of a basepoint gives an inclusion SPn−1(X)→ SPn(X).

1.12. Theorem. Suppose that n ≥ 1 and that Sℓ is the ℓ-sphere. Then

in the stable range with respect to ℓ there is an equivalence

SPn(Sℓ)/ SPn−1(Sℓ) ≃ Sℓ ∧ (Sn ∧P♦n )hΣn

.

1.13. Remark. A map is an equivalence “in the stable range with re-spect to ℓ” if it is an m-equivalence for m = 2ℓ − ǫ, where ǫ is asmall constant. We leave it to the reader to determine whether in anyparticular case that comes up in this paper ǫ should be 1, 2, or 3.

Passing to the limit and making a homology calculation leads to thefollowing result.

1.14. Theorem. Suppose that n > 1 and that S0 is the stable zero

sphere. Then there is an equivalence

SPn(S0)/ SPn−1(S0) ≃ Σ∞(Sn ∧P♦n )hΣn

.

The i’th mod p homology group of this spectrum is isomorphic to

Hi−2n+2(Σn; Fp ⊗ Lie∗n) .

1.15. Remark. The statement that SPn(S0)/ SPn−1(S0) is the suspen-sion spectrum of a space appears in a paper of Lesh [13]; in the courseof our arguments we recover her identification of the space involved asthe classifying space of a particular family of subgroups of Σn (7.4).


Combining the above result with Theorem 1.6 gives a connection be-tween the homology of symmetric power spectra and certain images ofthe Steinberg idempotent. Relationships like this first appeared in apaper of Mitchell and Priddy [17]. Theorems 1.8 and 1.14 give a directgeometric explanation for such a relationship.

1.16. Goodwillie layers. As described in [1], evaluating the n’thlayer in the Goodwillie tower for the identity functor at a pointed spaceX yields the spectrum

Dn(X) = Map∗(S1 ∧P♦

n , Σ∞X∧n)hΣn.

The following theorem was conjectured by Arone and Mahowald; itis a type of duality statement. The proof depends upon combiningthe relationship described above between the Tits building and thepartition complex with the fact that the Tits building is stably self-dual.

1.17. Theorem. Suppose that X is an odd sphere and that n = pk.

Then after p-completion there is an equivalence

S2(k−1)+1 ∧Dn(X) ≃p

(Σ∞P♦n ∧X∧n)hΣn

.

Comparing this with 1.14 gives the following result, which was firstproved by N. Kuhn with another technique.

1.18. Theorem. Suppose that S0 is the stable zero sphere and that

n = pk. Then after p-completion there is an equivalence

S2(k−1)+1 ∧Dn(S1) ≃p

SPn(S0)/ SPn−1(S0) .

We also obtain a retraction (9.6) involving the p-completion of Dn(X),where X is an odd sphere and n = pk. The existence of a retractionlike this was conjectured by Mahowald.

Organization of the paper. In §2 we set up a general scheme forapproximating G-spaces (G a finite group) and then in §3 study itshomological properties. Section 4 describes a particular approximationto the partition complex, and §5 identifies it more or less as a Titsbuilding. (To get the Tits building, it is actually necessary to work ina relative sense with respect to the approximation of a one-point space.)Section 6 has proofs of the main homological results and §7 deals withsymmetric powers. The next two sections establish the stable self-duality of the Tits building and then combine this with earlier resultsto study layers in the Goodwillie tower of the identity. Finally, §10has some algebraic material which is needed to deal with the Spanier-Whitehead duality which appears in 1.9.


1.19. Notation and terminology. As a technical convenience, wetake the term space on its own to mean simplicial set [15] [6]. A mapof spaces is an equivalence or weak equivalence if it becomes an ordinaryweak equivalence of topological spaces after geometric realization. Theunion of a space X with a disjoint basepoint is X+; the suspensionspectrum associated to a pointed space X is Σ∞X.

If G is a finite group, a map f : X → Y of G-spaces is said tobe a G-equivalence or weak G-equivalence if f induces a weak equiv-alence XK → Y K for each subgroup K of G. If M is a G-module,f is said to be an HG

∗ (– ; M)-equivalence if it induces an isomorphismH∗(XhG; M) → H∗(YhG; M). Note that the coefficients here are lo-cal coefficients. We will sometimes encounter G-spectra, by which wemean ordinary spectra with an action of G (these are called naive G-spectra in [14]). For instance, if X is a pointed space then Σ∞X∧n isa Σn-spectrum. If X is a G-spectrum, the spectrum XhG is defined bystabilizing the usual Borel construction [14, I, 3.7]. A map of G-spectrais said to be an HG

∗ (– ; M)-equivalence if the induced map of Borel con-structions gives an isomorphism on homology with local coefficientsin M . (The local coefficient homology of a stable Borel construction isdefined by stabilizing the reduced local coefficient homology of unstableBorel constructions.)

The equivariant cohomology H∗G(X; M) of a G-space is the cohomol-

ogy of XhG with local coefficients in M . Equivariant cohomology ofa G spectrum, reduced equivariant cohomology of a pointed G-space,etc., are defined in obvious ways. Note that we use the canonical an-tiautomorphism g 7→ g−1 of G to switch back and forth if necessarybetween right G-modules and left G-modules, so that the same modulemight be used as coefficients both for equivariant homology and forequivariant cohomology.

If X is a spectrum or a pointed space, then X# = Map∗(X,S0) de-notes the Spanier-Whitehead dual of X; here S0 is the sphere spectrum.If X is a G-spectrum, so is X#.

If P is a poset, then |P| denotes the space associated to P ; this is thesimplicial set whose nondegenerate m-simplices correspond to (m+1)-tuples of elements of P which are totally ordered by the partial orderrelation. More precisely, this is the nerve [3, XI §2] of the categorywhose objects are the elements of P , and in which there is exactlyone morphism x → y if y ≤ x. (Note the reversal, which is just forconvenience; it does not affect the homotopy type of the nerve [4, 2.10]but allows the functor O∗(C)

op → C of 2.10 to be covariant.) If P hasa maximal element or a minimal element then |P| is contractible [4,2.6].


We are grateful to N. Kuhn for reading through an initial version ofthis manuscript and making very useful suggestions. In particular, hehelped us to formulate some of our results in a much better way. Wealso thank J. Grodal for many helpful comments.

2. Approximations

Suppose that G is a finite group and that X is a G-space (in otherwords, a simplicial set with an action of G). Recall from [5, 1.3] that acollection C of subgroups of G is a set of subgroups closed under conju-gation. In this section we construct for each such C an approximationXC of X and a G-map aC : XC → X. The space XC is in a sense thebest approximation to X which can be built up from the orbits G/K,K ∈ C. We give examples of the approximations, and describe someproperties of the construction.

2.1. Remark. A family F of subgroups of G is a collection which isclosed under inheritance, in the sense that if K ∈ F and H ⊂ K thenH ∈ F . Some of the collections we look at are families, and some ofthem aren’t.

For the rest of this section, C denotes a fixed collection of subgroupsof G.

Constructing the approximation. For a G-space X, let Iso(X) de-note the collection of all subgroups of G which appear as isotropy sub-groups of simplices of X. A G-space X is said to have C-isotropy ifIso(X) ⊂ C. A G-map X → Y is said to be a C-equivalence if theinduced map XK → Y K is an equivalence of spaces for each K ∈ C.

2.2. Definition. Suppose that X is a G-space. A G-map f : Y → Xis said to be a C-approximation to X if Y has C-isotropy and f is aC-equivalence.

2.3. Proposition. Let X be a G-space. Then there exists a func-

torial C-approximation to X, denoted aC : XC → X. Any two C-approximations to X are canonically G-equivalent.

The uniqueness part of the proposition depends on the followingwell-known result (see for instance [4, 4.1]).

2.4. Lemma. Suppose that the G-map f : X → Y is a C-equivalence

for C = Iso(X) ∪ Iso(Y ). Then f is a G-equivalence.

Proof of 2.3. We sketch one construction, which is related to ideas from[7] in the same way as in [5] the subgroup decomposition is relatedto the centralizer decomposition. Let O∗(C) be the category whose


objects consists of pairs (G/K, c), where K ∈ C and c ∈ G/K; amap (G/K, c) → (G/K ′, c′) is a G-map f : G/K → G/K ′ such thatf(c) = c′. For any G-space X there is a functor from O∗(C)

op to thecategory of spaces given by

ΦCX(G/K, c) = MapG(G/K,X) ∼= XK .

We define XC to be the homotopy colimit [3] of ΦCX . Evaluating ele-

ments of ΦCX(G/K, c) at c leads to a map aC : XC → X. This map is

G-equivariant, where the action of G on XC is induced by the action ofG on O∗(C) defined by letting g ∈ G send (G/K, c) to (G/K, gc). It isclear by inspection that Iso(XC) ⊂ C. It is also easy to see that for anysubgroup K ⊂ G, (XC)

K is the homotopy colimit of ΦCX over the full

subcategory of O∗(C)op containing the objects (G/H, c) with the prop-erty that K fixes c. If K ∈ C, then this subcategory has (G/K, eK) asa terminal object, and so the homotopy colimit of ΦC

X over this subcat-egory is equivalent to ΦC

X(G/K, eK) = XK . The construction of XC

is clearly functorial. If Y → X is another C-approximation of X thenthere is a commutative diagram

YC −−−→ XC

y

y

Y −−−→ X

in which by 2.4 the upper arrow and the left vertical arrow are G-equivalences.

2.5. Remark. In later sections we will mostly be interested in the col-lection E of all nontrivial elementary abelian p-subgroups of G. TheE-approximation to a G-space X is closely related to the Henn’s workin [7]. Henn builds an approximation to the Borel construction onX; we essentially work with the fibres over BG and build XE as thecorresponding approximation to X itself.

We will now give some examples and properties of the C-approximationconstruction.

2.6. Elementary examples. If C is the collection of all subgroupsof G, then XC is G-equivalent to X. If C = {G}, then XC is XG.If C = {{e}}, then XC is EG × X. If C is the collection of all non-trivial subgroups of G, then XC is G-equivalent to the G-singular set

SingG(X), i.e., to the subspace of X consisting of all simplices whichhave a nontrivial isotropy subgroup (to see this note that SingG(X) hasC-isotropy and that the inclusion SingG(X)→ X is a C-equivalence).

2.7. The universal space E C. Let ∗ be the trivial one-point G-space.


2.8. Definition. The universal space for C, denoted E C, is the C-approximation (∗)C to ∗. The classifying space for C, denoted BC, isthe quotient (E C)/G.

2.9. Remark. If C = {{e}}, then E C is EG and BC = BG. Moregenerally, if C is a family of subgroups of G, then E C is the ordinaryuniversal space [14] of C and BC is the associated classifying space.

2.10. Homotopy type of E C. The collection C can be treated as a posetunder subgroup inclusion, and the associated space (1.19) is denoted|C|. (This is not the same as BC !) The action of G on C by conjugationinduces an action of G on |C|. The space E C is isomorphic to the nerveof the category O∗(C)

op, and so there is a map E C → |C| induced by thefunctor O∗(C)

op → C which sends (G/K, c) to the isotropy subgroupGc. This map is G-equivariant and is a weak equivalence of spaces [5,3.8] [4, 4.12]. In particular E C is weakly equivalent to a finite complex.

2.11. Fixed point sets of E C. If K is a subgroup of G we let C ↓ Kdenote the set {H ∈ C | H ⊂ K} and K ↓ C the set {H ∈ C | K ⊂ H}.Both of these sets are sub-posets of C; the first one is also a collectionof subgroups of K. For any subgroup K of G the fixed point set (E C)K

is equivalent to |K ↓ C| [4, 2.14]. Note that if K ∈ C then K ↓ C hasK as a minimal element and so (E C)K is contractible, as it must be.

By construction Iso(E C) = C. If C′ is a collection of subgroups of Gwith C′ ⊂ C, then there is a natural G-inclusion E C′ → E C.

2.12. Building XC for general X. Suppose that K is a subgroupof G, and that C′ = C ↓ K. If X is a K-space with C ′-isotropy, thenG ×K X is a G-space with C-isotropy. If the K-map Y → X is a C ′-equivalence, then G×K Y → G×K X is a C-equivalence. Both of thesestatements are easy to prove by inspection. If follows that if Y → Xis a C′-approximation of X as K-space, then G×K Y → G×K X is aC-approximation of G×K X as a G-space.

In particular (G/K)C can be identified up to G-equivalence withG ×K (∗)C′ = G ×K E C′, and so (G/K)C is weakly equivalent as aspace to the finite complex G ×K |C

′|. It is clear that (G/K ×∆[n])Cis isomorphic to (G/K)C ×∆[n]. Since the functor X 7→ XC commuteswith homotopy pushouts and directed colimits, this calculation for theG-cells G/K ×∆[n] gives an inductive approach to understanding theG-homotopy type of XC for any G-space X.

2.13. Preservation of equivalences. Say that a G-map f : X → Yis a (G, n)-equivalence if it induces an n-equivalence XK → Y K foreach subgroup K of G; more generally, f is a (C, n)-equivalence if it


induces an n-equivalence XK → Y K for each K ∈ C. The followingstatement is easy to prove by the inductive method used in the proofof [4, 4.1].

2.14. Proposition. If X → Y is a (C, n)-equivalence, then the in-

duced map XC → YC is a (G, n)-equivalence.

2.15. Preservation of dimension. It follows from 2.7 and 2.12 thatif X is a finite G-space, that is, one with a finite number of nondegen-erate simplices, then XC is equivalent as a space to a finite complex.Moreover, there is a integer dC such that the homotopy dimension ofXC is bounded above by dim(X) + dC. The integer dC can be taken tobe the dimension of |C|.

Note however that XC is not necessarily G-equivalent to a finitecomplex, even if X = ∗ (see 2.6).

3. Homological properties of approximations

Let G be a finite group, C a collection of subgroups of G, and M afixed Fp[G]-module. In this section we study the faithfulness of the C-approximation of a G-space (§2) from the point of view of equivariantM -homology. In particular, we look for ways to identify G-spaces Xsuch that the map aC : XC → X or its Spanier-Whitehead dual a#

C :X# → XC

# are HG∗ (– ; M)-equivalences (1.19).

Testing with a forward arrow. Before looking at whether the mapXC → X is an HG

∗ (– ; M)-equivalence, we will establish some terminol-ogy.

3.1. Definition. If X is a G-space, the collection C is M-ample for

X if the map XC → X is an HG∗ (– ; M)-equivalence.

If there is more than one group around, we will sometime emphasizethe role of G by saying that C is (G,M)-ample for X.

3.2. Proposition. Suppose that X is a G-space and that, for each

K ∈ Iso(X), C is (G,M)-ample for G/K. Then C is (G,M)-ample for

X.

This is proved by a Mayer-Vietoris argument (2.12).

3.3. Proposition. Let K be a subgroup of G. If C ↓ K is (K,M)-ample for ∗, then C is (G,M)-ample for G/K.

Proof. See [7, 2.8]. Let C′ = C ↓ K. Since (G/K)C ∼= G ×K (∗)C′ (cf.2.12) the result follows from the form of Shapiro’s lemma which says ifY is a K-space, then (G×K Y )hG is equivalent to YhK .


The following proposition is the starting point we will use in applyingthe previous two results.

3.4. Proposition. Let E be the collection of nontrivial elementary

abelian p-subgroups of G. If p divides the order of the kernel of the

action map G→ Aut(M), then E is M-ample for ∗.

Proof. By the remarks in 2.10, E is M -ample for ∗ if and only if themap |E| → ∗ is an HG

∗ (– ; M)-equivalence. The proposition then followsfrom [5, §8].

3.5. Remark. See [5, 1.4] for examples involving other collections.

Testing with a reversed arrow. Our second measure of the effec-tiveness of C-approximation uses Spanier-Whitehead duality.

3.6. Definition. Let X be a G-space and M a G-module. The collec-tion C is reverse M-ample for X (or reverse (G,M)-ample for X) if the

map (XC)#+ ← X#

+ induced by XC → X is an HG∗ (– ; M)-equivalence.

3.7. Proposition. Suppose that X is a G-space with finite skeleta

such that, for each K ∈ Iso(X), C is reverse (G,M)-ample for G/K.

Assume in addition that the spaces X and XC have the homotopy type

of finite complexes. Then C is reverse (G,M)-ample for X.

Proof. Let Z be the cofibre of the map XC → X and Z(n) the cofibreof the map (skn X)C → skn X. (Here skn X is the n-skeleton of X.) LetdZ be the homotopy dimension of Z (as a space, not as a G-space); byassumption, dZ is finite. Let D = Z#, D(n) = Z(n)#. By an inductiveMayer-Vietoris argument (cf. 2.12) it is easy to see that C is reverse(G,M)-ample for skn X, or equivalently that HG

∗ (D(n); M) vanishes;we want to prove that HG

∗ (D; M) also vanishes. The homology groupsof the spectra D and D(n) are concentrated in dimensions ≤ 0. Thegroup Hi D vanishes unless 0 ≥ i ≥ −dZ , and Hi D(n) vanishes unless0 ≥ i ≥ −(n + dC) (see 2.15). In fact, if n >> dZ , then by 2.14 thereare isomorphisms

Hi D(n) =

Hi D i ≥ −dZ

0 −dZ > i > −n

0 i < −(n + dC + 1)

Now there is a spectral sequence

E2i,j(D) = Hi(G; Hj(D; M))⇒ HG

i+j(D; M)

which maps to parallel spectral sequences Eri,j(D(n)). The E2-page

E2i,j(D) is concentrated in the horizontal band 0 ≥ j ≥ −dZ , while


E2i,j(D(n)) is concentrated in an upper band 0 ≥ j ≥ −dZ and a

lower band (−n) ≥ j ≥ −(n + dC + 1); the map D → D(n) induces anisomorphism on E2-pages in the upper band. Since Er(D(n)) convergesto zero, we can conclude that Er(D) converges to zero if we can pickn in such a way that no nontrivial differential in Er(D(n)) jumps fromthe lower band to the upper band (remember that these are fourthquadrant homology spectral sequences, so differentials move up and tothe left). Choose n1 and n2 so that n1 > n2 + dC + 1 >> dZ . Thenthe natural map E2(D(n1))→ E2(D(n2)) is an isomorphism on upperbands but zero on lower bands, since the lower bands in these two E2-pages lie in different strips in the plane. By naturality all differentialsin Er(D(n1)) from the lower band to the upper band are zero, whichas above implies that Er(D) converges to zero.

3.8. Proposition. Let K be a subgroup of G. If C ↓ K is reverse

(K,M)-ample for ∗, then C is reverse (G,M)-ample for G/K.

Proof. Given 10.2, this follows from the proof of 3.3 .

3.9. Proposition. Let E be the collection of nontrivial elementary

abelian p-subgroups of G. Suppose that p divides the order of G, that

M is a finite Fp[G]-module, and that every element of order p in G acts

trivially on M . Then E is reverse M-ample for ∗.

This will be proved below in §10.

4. An approximation to the partition complex

In this section we apply the approximation machinery of §2 in aspecial case. The finite group in question is the symmetric group Σn,with its collection E of nontrivial elementary abelian subgroups. TheΣn-space X to be approximated is a model for the partition complexPn, where we use the term model to mean that there is a Σn-mapX → Pn which is a weak equivalence of spaces. Let M denote thesign representation F

±p of Σn on Fp. We find a model X such that E is

both M -ample and reverse M -ample for X. It will later turn out thatif n = pk, the space XE can be described in terms of the Tits buildingTk.

For the rest of this section we let Σ denote Σn and M the Σ-moduleF±p .

4.1. A model for Pn. For each equivalence relation λ on n, let Kλ

denote the subgroup of Σ consisting of all σ ∈ Σ which preserve λ in thestrong sense that for each j ∈ n, σ(j) ∼λ j. Let P denote the collectionconsisting of all the subgroups Kλ for nontrivial equivalence relations λ.


Up to conjugacy, the elements of P are exactly the subgroups of Σ ofthe form Σn1

× · · · ×Σnjwith j > 1,

∑

ni = n, and ni > 1 for at leastone i. The association λ 7→ Kλ is bijective and order preserving, andso |P| is isomorphic as a Σ-space to the partition complex Pn.

By 2.10, there is a Σ-map EP → |P| = Pn which is an equivalenceof spaces. The space EP is the model for Pn that we will work with.

4.2. Another description. There is another way to construct EPwhich gives the same result up to Σ-equivalence. Let F denote the fam-ily of all non-transitive subgroups of Σ, i.e. the family of all subgroupsof Σ which do not act transitively on the set n, and let F◦ ⊂ F bethe subcollection obtained by deleting the trivial subgroup {e}. SinceP ⊂ F◦, there is a Σ-inclusion EP → EF◦.

4.3. Lemma. The inclusion EP → EF◦ is a Σ-equivalence.

Proof. By 2.4, it is enough to show that for each H ∈ F◦ the map(EP)H → (EF◦)H is an equivalence. By 2.11 (see [4, 2.12]), thisamounts to showing that the inclusion |H ↓ P| → |H ↓ F◦| is an equiv-alence, or, what is the same thing (2.11), that |H ↓ P| is contractible.Let λ(H) be the equivalence relation on n in which the equivalenceclasses are the H-orbits, and let K = Kλ(H). Then K ∈ P and K is aminimal element of H ↓ P , so the result follows.

4.4. Remark. By the remarks in 2.7 there is a commutative diagramof Σ-spaces

EP −−−→ EF◦

y

y

|P| −−−→ |F◦|

in which the vertical arrows are weak equivalences. By 4.3 the upperarrow is a Σ-equivalence, and so the lower arrow is a weak equivalence.The construction in the proof of 4.3 gives for each H ∈ F◦ an elementKλ(H) ∈ P. The assignment H 7→ Kλ(H) is order-preserving and Σ-equivariant; it induces a Σ-map |F◦| → |P| which is a left inverse tothe lower arrow in the above square, and so is a weak equivalence.

The approximation. Let E ′ denote the collection of subgroups of Σgiven by E ∩ F◦. This is the collection obtained from E by deletingall elementary abelian p-subgroups of Σ which act transitively on n.In terms of E ′, we can now give a preliminary calculation of the E-approximation (EP)E ≃ (EF◦)E .

4.5. Lemma. The natural map E(E ′)→ EF◦ is an E-approximation.


Proof. By construction the Σ-space E(E ′) has E ′-isotropy, and so afortiori it has E-isotropy. It remains to show that the indicated map isan E-equivalence. However, the V -fixed points of both spaces involvedare contractible if V ∈ E ′ and empty if V ∈ E \ E ′.

Ampleness and reverse ampleness. Recall that M is the sign rep-resentation of Σ on Fp.

4.6. Lemma. If K is an element of P, then the collection E ↓ K of

subgroups of K is (K,M)-ample for the trivial K-space ∗.

Proof. The collection E ↓ K is the collection of nontrivial elementaryabelian p-subgroups of K. If p divides the order of K it is clear thatp also divides the order of the kernel of the action map K → Aut(M),and so by 3.4 the collection E ↓ K is (K,M)-ample for ∗. If p doesnot divide the order of K then p must be odd and by calculation thegroup Hi(K; M) vanishes for i ≥ 0. By inspection the poset E ↓ K isempty. For trivial reasons, then, the map |E ↓ K| → ∗ is an HK

∗ (– ; M)-equivalence, and so (by the proof of 3.4) E ↓ K is (K,M)-ample forthe space ∗.

4.7. Proposition. The collection E is M-ample for EP.

Proof. This follows from 4.6, 3.3, 3.2 and the fact that Iso(EP) =P .

4.8. Lemma. If K is an element of P, then E ↓ K is reverse (K,M)-ample for the trivial K-space ∗.

Proof. This is very similar to the proof of 4.6; it depends on 3.9.

4.9. Proposition. The collection E is reverse M-ample for EP.

Proof. This follows from 3.7, 3.8, and 4.8. It is necessary to checkthat the Σ-spaces E(E) and ∗ have finite skeleta, but this is clear byinspection. It is also necessary to check that EP , (EP)E and (∗)E allhave the homotopy types of finite complexes. By 2.10 and 4.5, thesespaces are equivalent, respectively, to the finite simplicial complexes|P|, |E ′|, and |E|.

4.10. Remark. Similar arguments show that the collection E is bothM -ample and reverse M -ample for the trivial Σ-space ∗.

5. Computing with the approximation

In this section we draw some specific conclusions from the approx-imation construction in §4. We continue to use the notation of thatsection.


We are going to concentrate on the commutative square

|E ′| −−−→ |E|

y

y

|P| −−−→ ∗

(5.1)

of Σ-spaces. The left vertical arrow here requires a little comment:it is constructed by composing the inclusion E ′ → F◦ with the posetretraction F◦ → P from 4.4. Recall that M denotes the Σ-module F

±p .

We are interested in proving the following three statements.

5.2. Proposition. Both of the vertical arrows in 5.1 are HΣ∗ (– ; M)-

equivalences.

5.3. Proposition. If n is not a power of p, then the homotopy cofibre

of the upper horizontal arrow in 5.1 is contractible.

Note that the homotopy cofibre of the lower horizontal arrow in 5.1is P♦

n (see 4.1).

5.4. Proposition. Suppose that n = pk. Let C be the homotopy cofi-

bre of the upper horizontal arrow in 5.1. Then there is a Σ-map

C → Σ+ ∧Affk,pT♦

k

which is a weak equivalence of spaces. Under this equivalence the map

C → P♦n induced by the square 5.1 corresponds to the map

Σ+ ∧Affk,pT♦

k → P♦n

induced by the Affk,p-equivariant inclusion Tk → Pn.

Proof of 5.2. It is enough show that in the square

|E ′| −−−→ |E|

y

y

|F◦| −−−→ ∗

both of the vertical arrows are HΣ∗ (– ; M)-equivalences. Consider the

square

E(E ′) −−−→ E(E)

y

y

EF◦ −−−→ ∗

Both vertical arrow are E-approximations, the right by constructionand the left by 4.5. By 4.7 and 4.10, both vertical arrows are HΣ

∗ (– ; M)-equivalences. The proposition follows from the construction of 2.10,


which produces a Σ-map from the second square to the first giving aweak equivalence at each of the four corners.

Proof of 5.3. This is obvious; E ′ = E because for cardinality reasonsno elementary abelian subgroup of Σ can act transitively on n.

Suppose that n = pk. It is not hard to see that up to conjugacy thereis a unique elementary abelian subgroup of Σ which acts transitivelyon n. Such a subgroup is obtained using an identification of (Fp)

k withn and then letting this additive group act on itself by translation. Let∆ denote a chosen one of these subgroups. The normalizer N(∆) of ∆in Σ is isomorphic to the semidirect product Aut(∆)⋉∆ ∼= Affk,p, andthe quotient N(∆)/∆ is isomorphic to Aut(∆) ∼= GLk,p.

5.5. Lemma. Suppose that n = pk. Let ∆ and N be as above. Then

there is a homotopy pushout square of Σ-spaces

Σ×N |E′ ↓ ∆| −−−→ Σ×N |E ↓ ∆|

y

y

|E ′| −−−→ |E|

.

Proof. By inspection this is a pushout square, and the upper horizontalarrow is an inclusion.

Proof of 5.4. Consider the square of 5.5. The poset E ′ ↓ ∆ is the posetof all subgroups of ∆ other than the trivial subgroup and ∆ itself, andso |E ′ ↓ ∆| is the Tits building Tk. The poset E ↓ ∆ is the poset ofall subgroups of ∆ other than the trivial subgroup. This poset has ∆as a maximal element, and so |E ↓ ∆| is contractible; in fact, |E ↓ ∆|is combinatorially the cone on |E ′ ↓ ∆|. Both horizontal maps in thissquare thus have homotopy cofibre Σ+ ∧N T♦

k . It is easy to check thatthe map from this space to P♦

n given by 5.1 is the obvious one.

6. The homological results

In this section we give proofs of the main homological results fromthe introduction, namely, 1.1, 1.3, 1.6, 1.8, and 1.9. All of these resultsfollow directly from the calculations in §5. We continue to denote thesymmetric group Σn by Σ.

Proof of 1.1. Let M be the Σ-module F±p . By 5.2, both of the vertical

arrows in the square 5.1 are HΣ∗ (– ; M)-equivalences. By a long exact

sequence argument, the induced map of horizontal homotopy cofibresis also an HΣ

∗ (– ; M)-equivalence. The proof is finished by identifyingthe map between these cofibres (5.3, 5.4), and, if n = pk, by applyingShapiro’s lemma.


Proof of 1.3. If G is a finite group, X is a G-space, and M is a G-module, then there is a Serre spectral sequence

E2i,j = Hi(G; Hj(X

♦; M))⇒ HGi+j(X

♦; M) .

If X is spherical and has homotopy dimension d, the spectral sequencecollapses at E2 into isomorphisms

HGi (X♦; M) ∼= Hi−d−1(G; Hd(X; M)) .

The theorem is proved by starting with 1.1 and applying this collapseobservation to the spherical space Pn and, if n = pk, to the sphericalspace Tk.

Proof of 1.6. If M is a module over Affk,p, then associated to the groupextension ∆→ Affk,p → GLk,p is a Serre spectral sequence

E2i,j = Hi(GLk,p; Hj(∆; M))⇒ Hi+j(Affk,p; M) .

In the special case M = Fp⊗ St±k , the action of ∆ on M is trivial, andso the spectral sequence takes the form

E2i,j = Hi(GLk,p; Hj(∆; Fp)⊗ St±k )⇒ Hi+j(Affk,p; M) ,

where the action of GLk,p on Hj(∆; Fp)⊗St±k is a diagonal one. This ac-tion can be identified with the diagonal action of GLk,p on Hj(∆; F±

p )⊗Stk. Since Fp ⊗ Stk is projective as a module over GLk,p, the groupHi(GLk,p; N ⊗ Stk) vanishes for any i > 0 and any Fp[GLk,p]-moduleN . By formula 1.5, then, the above spectral sequence collapses intoisomorphisms

Hi(Affk,p; Fp ⊗ St±k ) ∼= ǫStk · Hi(∆; F±

p )

and the proof is finished by combining these isomorphisms with 1.3.

Proof of 1.8. Since ℓ is odd, the action of Σ on Hℓn(Sℓn; Fp) gives theΣ-module F

±p . Essentially by the Thom isomorphism theorem (or by a

Serre spectral sequence argument) there are natural isomorphisms

Hi((Sℓn ∧P♦

n )hΣ; Fp) ∼= HΣi−ℓn(P♦

n ; F±p )

as well as similar natural isomorphisms for H∗((Sℓn ∧ T♦

k )h Affk,p; Fp).

The theorem is proved by combining these isomorphisms with 1.1.

Proof of 1.9. Let M denote the Σ-module F±p . Consider the square

|E ′|#+ ←−−− |E|#+

x

x

|P|#+ ←−−− (∗)#+

(6.1)


obtained by applying Spanier-Whitehead duality to 5.1. The line ofargument in the proof of 5.2 shows that the vertical arrows in thissquare are HΣ

∗ (– ; M)-equivalences; it is only necessary to replace theappeal to 4.7 by an appeal to 4.9. By a long exact sequence argument,it follows that the induced map of horizontal homotopy fibres is anHΣ

∗ (– ; M)-equivalence. The homotopy fibre of the lower horizontal mapis (P♦

n )#. Let C be the homotopy fibre of the upper map. The kindof Thom isomorphism that figures in the proof of 1.8 shows that theinduced map

(Sℓn ∧ (P♦n )#)hΣ → (Sℓn ∧ C)hΣ(6.2)

gives an isomorphism on mod p homology. If n is not a power of pthen C is contractible (5.3). If n = pk then, by 5.4, C is equivalent to(Σ+∧Affk,p

T♦k )#, which by 10.2 is in turn equivalent to Σ+∧Affk,p

(T♦k )#.

Thus if n is not a power of p the target of the map 6.2 is contractible,while if n = pk it follows from the stable form of Shapiro’s lemma thatthe target of 6.2 is equivalent to (Sℓn ∧ (T♦

k )#)h Affk,p.

6.3. Remark. Theorem 1.9 leads to a cohomological version of 1.3.Suppose that n = pk. The spectrum (P♦

n )# is spherical; its only non-trivial mod p homology group is in dimension (2−n) and is isomorphicto Fp ⊗ Lie±n as a module over Σn. Similarly, (T♦

k )# is spherical; itsonly nontrivial mod p homology group is in dimension 1 − k and isisomorphic to Hom(Stk, Fp) as a module over Affk,p. By the collaps-ing spectral sequence argument in the above proof of 1.3, the mod phomology equivalence of 1.9 gives isomorphisms

Hi(Affk,p; Hom(St±k , Fp)) ∼= Hi+n−k−1(Σn; Fp ⊗ Lien) .(6.4)

If G is a finite group and M is a module over Fp[G], there is a naturalisomorphism between H∗(G; Hom(M, Fp)) and the Fp-dual of H∗(G; M)(see the proof of 10.1). By this duality, 6.4 gives isomorphisms

Hi(Affk,p; Fp ⊗ St±k ) ∼= Hi+n−k−1(Σn; Fp ⊗ Lie∗n) .

There is another consequence of 6.4. By the self-duality of Fp ⊗ Stk

described in §8, Hom(St±k ; Fp) is isomorphic as an Affk,p-module toFp ⊗ St±k . In combination with 1.3, then, 6.4 gives isomorphisms

Hi−n+k+1(Σn; Fp ⊗ Lie∗n) ∼= Hi+n−k−1(Σn; Fp ⊗ Lien) .

It is not hard to see that the groups involved are not all zero (1.14),and it follows that if n is a power of p then Fp⊗Lie∗n and Fp⊗Lien arenot isomorphic as Σn-modules.

The above arguments also show that if n is not a power of p thenthe groups H∗(Σn; Fp ⊗ Lie∗n) and H∗(Σn; Fp ⊗ Lien) vanish.


7. Symmetric powers

The goal of this section is to prove Theorems 1.12 and 1.14. Thereare three auxiliary propositions. We continue to use the notation ofsection 4, so that in particular Σ = Σn, F◦ is the collection of allnontrivial subgroups of Σ which do not act transitively on n, and Pis the collection of subgroups of Σ described in 4.1. In the first twostatements below, Σ acts on the sphere Sℓn by permuting the factorsof Sℓn = (Sℓ)∧n.

7.1. Proposition. The space SPn(Sℓ)/ SPn−1(Sℓ) is homeomorphic to

the quotient space Sℓn/Σ

Let P be the collection of subgroups of Σ given by adding to P thetrivial subgroup {e}. Recall from 2.7 that BP is defined to be thequotient space (E P)/Σ.

7.2. Proposition. In the stable range with respect to ℓ there is an

equivalence Sℓn/Σ ≃ Sℓ ∧ BP♦

The following proposition is interesting because it relates an orbitspace (BP) on the left to a homotopy orbit space on the right. In thestatement, Σ acts on Sn = (S1)∧n by permuting smash factors.

7.3. Proposition. There is an equivalence of spaces

S1 ∧ BP♦ ≃ S1 ∧ (Sn ∧ EP♦)hΣ

Theorem 1.12 is now proved by combining 7.1, 7.2, and 7.3. Theorem1.14 is proved by first checking that the maps of 1.12 are compatiblewith suspension and then copying the spectral sequence calculationfrom the proof of 1.3.

7.4. Remark. Recall from 4.1 that F is the family of nontransitivesubgroups of Σ. It is clear that P ⊂ F , and the argument of 4.3 showsthat the induced inclusion E P → EF is a Σ-equivalence. It followsthat the quotient map BP → BF is an equivalence. Combining thiswith 7.1 and 7.2, and passing to the limit in ℓ gives the followingcalculation of Lesh.

7.5. Corollary. [13] Let S0 be the stable zero sphere, and let F be the

family of nontransitive subgroups of Σ. Then there is an equivalence

SPn(S0)/ SPn−1(S0) ≃ Σ∞ BF♦ .

Proposition 7.1 is clear by inspection. We now go on to the proofsof 7.2 and 7.3. The first proof depends on a few lemmas.


7.6. Lemma. Suppose that G is a finite group, that X is a G-space,

and that C′ is a collection of subgroups of G. Let C = C′ ∪ {G}. Then

the natural G-map XG → X gives rise to a homotopy pushout diagram

of G-spaces

(XG)C′ −−−→ (XG)C

y

y

XC′ −−−→ XC

Proof. Assume G /∈ C′, since otherwise the statement is trivial. By2.4, it is only necessary to check that for each K ∈ C the square be-comes a homotopy pushout square when the fixed point functor (–)K

is applied. This is obvious if K ∈ C′, since the horizontal arrows areC′-equivalences. The remaining case is K = G. The G-fixed point setsof the spaces on the left are empty, since these spaces have C ′-isotropy.Since G ∈ C, the G-fixed points of the spaces on the right are bothequivalent to XG.

7.7. Remark. In the situation of 7.6, it is easy to see that (XG)C′ isisomorphic to XG × (∗)C′ , and similarly that (XG)C is isomorphic toXG × (∗)C. This last product is G-equivalent to XG, because ∗ hasC-isotropy.

The following lemma can be proved by the inductive technique usedto prove [4, 4.1]. The notion of (G, n)-equivalence is from 2.13.

7.8. Lemma. Suppose that G is a finite group. If X → Y is a (G, n)-equivalence of G-spaces, then the quotient map X/G → Y/G is an

n-equivalence of spaces.

Proof of 7.2. Let X = Sℓn and let C = P ∪ {Σ}. Consider the map ofsquares

(XΣ)P −−−→ (XΣ)C

y

y

XP −−−→ XC

−→

(XΣ)P −−−→ (XΣ)C

y

y

(∗)P −−−→ Y

.

Here the map XP → (∗)P is induced by X → ∗ and Y is definedso that the right-hand square is a homotopy pushout square. Themap XC → Y is well-defined because (7.6) the left hand square is ahomotopy pushout square. If K is a subgroup of Σ then XK is asphere So(K)ℓ, where o(K) is the number of orbits of the action of Kon n. If K ∈ P , then o(K) ≥ 2, so that XK → (∗)K = ∗ is a (2ℓ− 1)-equivalence. By 2.14, the map XP → (∗)P is a (Σ, 2ℓ− 1)-equivalence.


It follows that XC → Y is also a (Σ, 2ℓ − 1)-equivalence. Since Xhas C-isotropy, the map XC → X is a Σ-equivalence. The fixed pointset XΣ is Sℓ, so by 7.7 the right-hand square can be identified up toΣ-equivalence as the square

Sℓ × E P −−−→ Sℓ

y

y

E P −−−→ Y

.

It is easy to argue by naturality that the upper arrow and the left-handvertical arrow are the obvious projections. This implies that Y is Σ-equivalent to the join Sℓ# E P , which is itself Σ-equivalent to Sℓ∧E P♦.Since X is Σ-equivalent to Y in the stable range with respect to ℓ, itfollows from 7.8 that X/Σ is equivalent to Y/Σ ≃ Sℓ ∧ BP♦ in thesame stable range.

The proof of 7.3 also depends on a few lemmas. Recall (2.6) that if Gis a finite group and X is a G-space, SingG(X) is the G-subspace of Xconsisting of all simplicies which have a nontrivial isotropy subgroup.

7.9. Lemma. Suppose that G is a finite group, X is a pointed G-

space, and Y ⊂ X is any G-subspace of X containing SingG(X). Then

the diagram

YhG −−−→ XhG

y

y

Y/G −−−→ X/G

is a homotopy pushout diagram.

Proof. The statement is clear if X = Y . The fact that SingG(X) ⊂ Yimplies that X is obtained from Y by adding free G-cells of the formCk = (G × ∆[k], G × ∂∆[k]). For any one of these cells Ck the map(Ck)hG → Ck/G is clearly an equivalence of pairs. The lemma is provedby induction on the number of added cells, if this number is finite, andthen in general by passage to a sequential colimit.

7.10. Lemma. If k ≥ 2 the quotient Sk/Σk is contractible.

Proof. We will give a topological argument. Take S1 to be the unitcircle in the complex plane, and use the basepoint 1 ∈ S1 to obtaininclusions SPi(S1) → SPj(S1), i < j. Then Sk/Σk is homeomorphicto SPk(S1)/ SPk−1(S1) (by 7.1), so to obtain the desired contractibilityit is enough to show that the map S1 = SP1(S1) → SPk(S1) is anequivalence. Now SPk(S1) is equivalent to SPk(C \ {0}), which, by


the fundamental theorem of algebra, is homeomorphic to the space ofmonic polynomials of degree k with complex coefficients and nonzeroconstant term. The result follows easily from the fact that using thecoefficients of the polynomials as coordinates gives a homeomorphismSPk(C \ {0}) ∼= C

k−1 × (C \ {0}).

7.11. Lemma. If X is a pointed Σ-space with Iso(X) ⊂ P∪{Σ}, then

the quotient (X ∧ Sn)/Σ is contractible.

Proof. The statement is clear if X = ∗. By induction on the numberof Σ-cells of X (and eventual passage to a sequential colimit) it isenough to show that if (Y ∧ Sn)/Σ is contractible and X is obtainedfrom Y by adding a single Σ-cell of the form Σ/K × (∆[m], ∂∆[m]),K ∈ P ∪ {Σ}, then (X ∧Sn)/Σ is contractible. There is a (homotopy)pushout diagram

∂∆[m]+ ∧ (Sn/K) −−−→ (Y ∧ Sn)/Σ

y

y

∆[m]+ ∧ (Sn/K) −−−→ (X ∧ Sn)/Σ

where K is of the form Σn1×· · ·×Σnj

, with∑

ni = n and at least oneinteger ni greater than one. By 7.10 the quotient Sn/K ∼= Sn1/Σn1

∧· · · ∧ Snj/Σnj

is contractible, and the lemma follows.

7.12. Lemma. The fixed point inclusion S1 = (Sn)Σ → Sn induces a

Σ-equivalence

EP♦ ∧ S1 → EP♦ ∧ Sn .

Proof. As in 7.4, it is enough to prove the result with E P replaced byEF . We check that for each subgroup K of Σ the fixed point map(EF♦ ∧ S1)K → (EF♦ ∧ Sn)K is an equivalence. If K acts transitivelyon n, this is the map (∅♦) ∧ S1 → (∅♦) ∧ (Sn)K = (∅♦) ∧ S1 and soit is a homeomorphism. If K does not act transitively on n, the space(EF)K is contractible, hence (EF♦)K is contractible, and hence bothdomain and range of the above fixed point map are contractible.

Proof of 7.3. Let X denote the pointed Σ-space E P♦. Then SingΣ(X)is EP♦. It is clear that SingΣ(X ∧ Sn) is contained in SingΣ(X) ∧ Sn,and so by 7.9 there is a homotopy pushout diagram

(EP♦ ∧ Sn)hΣ −−−→ (X ∧ Sn)hΣ

y

y

(EP♦ ∧ Sn)/Σ −−−→ (X ∧ Sn)/Σ

.


Since Iso(EP♦) = P∪{Σ}, the lower left-hand space is contractible by7.11. The space X is contractible because E P is contractible, wherethis last follows (cf. 2.10) from the fact that the poset P containsthe minimal element {e}; it follows that the upper right-hand spaceis also contractible. The homotopy pushout diagram then shows that(X ∧ Sn)/Σ is equivalent to S1 ∧ (EP♦ ∧ Sn)hΣ. By 7.12, (X ∧ Sn)/Σ

is equivalent to BP ∧ S1.

8. duality

It is known that mod p reduction of the Steinberg module is self-dual.In this section we describe an explicit duality isomorphism and use itto construct a geometric duality map for the suspension spectrum ofthe Tits building.

Fix n = pk, and assume k > 2; the cases with k ≤ 2 are simplerbut require some adjustments in the notation. Let ∆ be the elemen-tary abelian p-group (Fp)

k, Tk the associated Tits building, and G thegroup GLk,p = Aut(∆). Let C∗ denote the reduced normalized sim-

plicial chain complex of T♦k with coefficients in Fp. We will use the

simplicial model for T♦k from [15, 27.6] (adjusted in an evident way to

omit basepoint identifications), so that the nondegenerate m-simplicesof T♦

k , m > 0, correspond bijectively to the nondegenerate (m − 1)-simplices of Tk; there are two zero-simplices. Thus

Cm∼=

{

⊕

I Fp[G/PI ] 0 < m ≤ k − 1

0 m > k − 1

where I ranges through the set of ordered partitions of k with (m + 1)constituents and PI is the standardparabolic subgroup of G associatedwith I. Recall that if I = 〈i1, i2, . . . , im+1〉 with

∑

ij = k and ij ≥ 1,then PI is the subgroup of G which preserves the flag

(Fp)i1 ⊂ (Fp)

i1+i2 ⊂ · · · ⊂ (Fp)P

ij = ∆ .

In particular, Ck−1∼= Fp[G/B], where B is the Borel subgroup of upper

triangular matrices, and Ck−2∼=

⊕k−1i=1 Fp[G/Pi], where Pi is the stan-

dard parabolic subgroup associated with the partition 〈1, . . . , 2, . . . , 1〉with (k − 1) constituents and 2 in the i-th place. The boundary map∂k−1 : Ck−1 → Ck−2 is the sum of components (−1)idi, where the mapdi : Fp[G/B] → Fp[G/Pi] is induced by the inclusion B → Pi. Notethat there are only (k − 1) terms in the sum for ∂k−1 because oneof the simplicial face operators induces the zero map on normalizedchains. The homology of C∗ is concentrated in degree k − 1 and


Hk−1(C∗) ∼=⋂k−1

i=1 ker(di) ∼= Hk−2(Tk; Fp) is the mod p Steinberg repre-sentation of G.

Let C∗ be the cochain complex dual to C∗. Each group Cj is a directsum of permutation modules and so has a basis preserved by G. Itfollows that Cj = Hom(Cj, Fp) is isomorphic to Cj as a module over G.The cohomology of C∗ is concentrated in degree k− 1 and Hk−1(C∗) isabstractly isomorphic to Hk−1(C∗), at least as a vector space over Fp.The group Hk−1(C∗) is the dual of the mod p Steinberg module. Thecoboundary map ∂k−1 : Ck−2 → Ck−1 is given by the alternating sumof maps di, where di is dual to di. The map di can be interpreted as atransfer associated with the inclusion B → Pi.

We use permutation bases to identify Cj with Cj. To emphasize this,we will use the notation C(j) for both of these groups. Thus we viewthe boundary and cobundary maps ∂j and ∂j as maps C(j)→ C(j−1)and C(j − 1)→ C(j) respectively.

Consider the composite homomorphism

S : Hk−1(C∗)→ C(k − 1)→ Hk−1(C∗)

where the first map is the inclusion of the kernel of ∂k−1 and the secondmap is projection to the cokernel of ∂k−1. Clearly, S is a G-equivariantmap.

8.1. Lemma. The map S is an isomorphism between the mod p Stein-

berg module and its dual.

Proof. Since the source and the target of S are finite dimensional vectorspaces over Fp of the same dimension, it is enough to show that S issurjective. To do this, we consider the maps ei : C(k − 1)→ C(k − 1)defined by ei = didi for i = 1, . . . , k − 1. Let wi be the transposition(i, i + 1) in the symmetric group Σk, which is the Weyl group of thestandard split (B,N)-pair structure on G. It is not hard to see thatei can be identified with 1 + fwi

, where fwiis the G-endomorphism of

Fp[G/B] referred to in [10, 2.7]. Since (fwi)2 = −fwi

, it follows that(ei)

2 = ei; this also follows directly from the fact that the index of Bin Pi is congruent to 1 mod p. The endomorphisms fwi

satisfy braidrelations due to Iwahori [10, 2.4], and (as a consequence of the factthat (fwi

)2 = −fwi) the maps ei satisfy the same relations:

eiej = ejei if |i− j| ≥ 2eiei+1ei = ei+1eiei+1

.

To prove that S is surjective it is enough to prove that for eachu ∈ C(k − 1) there exists an element v in the image of ∂k−1 suchthat u + v ∈ ker(∂k−1). Let u ∈ C(k − 1). Let ei = 1 − ei and let


Ei = e1e2 · · · ei. Define

w = Ek−1Ek−2 · · · E1u

and let v = w − u.Clearly, v is in the subspace of C(k − 1) generated by the images of

the maps ei, and thus v is in the image of ∂k−1. (To derive this lastconclusion, note that djdi = 0 if i 6= j, so that ei = ±∂k−1diei.) Theidempotents ei satisfy the same braid relations that the idempotents ei

do, and from this it is easy to see that eiw = w and thus that eiw = 0for i = 1, . . . , k−1. By inspection, eix = 0 only if dix = 0, so it followsthat w ∈ ker(∂k−1).

Our next step will be to construct a map of spectra that realizes themap S on homology. Consider the map

α : Σ∞T♦k −→ Σ∞(Sk−1 ∧ (G/B)+)

given by collapsing the (k − 2)-skeleton of T♦k to a point. The mod p

homology of both spectra is concentrated in dimension (k − 1) andit is clear that the induced homology map in this dimension is theinclusion of the Steinberg module in C(k − 2). In the case of theSpanier-Whitehead dual map

α# :(

Sk−1 ∧ (G/B)+

)#−→

(

T♦k

)#

the mod p homology of both spectra is concentrated in degree (−k+1)and the induced homology map in this dimension is the projection ofC(k − 1) on the dual of the Steinberg module. By 10.2 there is aG-equivariant map

β : Σ∞(Sk−1 ∧ (G/B)+) −→ S2(k−1) ∧(

Sk−1 ∧ (G/B)+

)#

inducing on mod p homology the isomorphism Ck−1∼= Ck−1.

Now let Stop be the composed map

(S2(k−1) ∧ α#) · β · α : Σ∞T♦k −→ S2(k−1) ∧

(

T♦k

)#.

8.2. Theorem. The map Stop is a GLk,p-equivariant map that induces

an isomorphism in mod p homology.

Proof. It is clear that Stop is equivariant and that it induces the al-gebraic map S on the only non-trivial homology group. The theoremfollows from lemma 8.1.


9. Layers in the Goodwillie tower of the identity

In this section we prove Theorems 1.17 and 1.18. We begin with animmediate consequence of Theorem 8.2. Let ∆ = (Fp)

k, and observethat the map Stop from 8.2 is equivariant with respect to Affk,p =GLk,p ⋉∆, where ∆ as usual acts trivially on the Tits building and itsdual.

9.1. Proposition. For any Affk,p-spectrum W, the map(

Σ∞T♦k ∧W

)

h Affk,p→

(

S2(k−1) ∧ (T♦k )# ∧W

)

h Affk,p

induced by Stop is a mod p homology isomorphism.

Now suppose that n = pk, and embed Affk,p in Σn in the usual way(§1, §5). Let X be a pointed space, so that X∧n has a natural actionof Σn, and let W be the spectrum Σ∞X∧n with the induced action ofAffk,p. Since (T♦

k )# ∧ W ≃ Map∗(T♦k ,W), Proposition 9.1 becomes

the following statement.

9.2. Corollary. For any pointed space X, the map(

Σ∞T♦k ∧X∧n

)

h Affk,p→ S2(k−1) ∧Map∗

(

T♦k , Σ∞X∧n

)

h Affk,p

induced by Stop is a mod p homology isomorphism.

9.3. Remark. If W is a spectrum with an action of Affk,p, there is anequivalence Wh Affk,p

≃ (Wh∆)h GLk,p(cf. [4, 8.5]). Let X be as in 9.2,

and let U denote Σ∞(X∧n)h∆. It follows that the map of 9.2 can beinterpreted as the mod p homology isomorphism

(Σ∞T♦k ∧ U)h GLk,p

→ S2(k−1) ∧Map∗(T♦k ,U)h GLk,p

(9.4)

induced by Stop. Let B ⊂ GLk,p be the group of upper-triangularmatrices. By way in which Stop is constructed in §8, the map in 9.4factors through the spectrum

(Sk−1 ∧ (GLk,p /B)+ ∧ U)h GLk,p≃ Sk−1 ∧ UhB .(9.5)

This implies that after p-completion the spectra in 9.2 are retracts ofthe p-completion of the spectrum in 9.5.

Proof of 1.17. Let X be an odd-dimensional sphere. In this case, by1.8 and 1.9 there are mod p equivalences

(

Σ∞T♦k ∧X∧n

)

h Affk,p→

(

Σ∞P♦n ∧X∧n

)

hΣn

and

Map∗(P♦n , Σ∞X∧n)hΣn

→ Map∗

(

T♦k , Σ∞X∧n

)

h Affk,p.

Combining these equivalences with 9.2 we obtain 1.17.


Let G = GLk,p. The Steinberg idempotent ǫStk ∈ Fp[G] (1.5) lifts

to an element ǫStk ∈ Z[G] = π0Σ

∞(G)+ and in this way can be madeto act on any G-spectrum. A straightforward telescope constructionshows that after p-completion a G-spectrumW can be split (nonequiv-ariantly) as a wedge W1 ∨ W2, where H∗(W1; Fp) ∼= ǫSt

k · H∗(W ; Fp).The spectrum W1 is well-defined up to homotopy and we will denoteit ǫSt

k ·W. A homology calculation using the fact that the Tits buildingis spherical shows that up to p-completion there is an equivalence

ǫStk · W ≃

pS1−k ∧ (T♦

k ∧W)hG .

In view of 9.2 and the remarks above in 9.3, the proof of 1.17 gives thefollowing statement.

9.6. Corollary. Let X be an odd-dimensional sphere and let n = pk.

Then up to p-completion there is an equivalence

Sk ∧Dn(X) ≃p

ǫStk · Σ

∞(X∧n)h∆ .

Note that since X is a sphere, (X∧n)h∆ is the Thom space of a vectorbundle over B∆.

Proof of 1.18. By Theorem 1.14 there is an equivalence of spectra(

Σ∞P♦n ∧ Sn

)

hΣ≃ SPn(S0)/ SPn−1(S0)

On the other hand, by 1.17 the left-hand side is equivalent after p-completion to S2(k−1)+1 ∧Dpk(S1). This proves 1.18.

9.7. Remark. That Corollary 9.6 and Theorem 1.18 are true had beensuggested by M. Mahowald to the first-named author a few years priorto the writing of this paper. Theorem 1.18 and Corollary 9.6 (in thecase X = S1) have been demonstrated by N. Kuhn, using differentmethods. He shows that the mod p cohomologies of the spectra in-volved are isomorphic as modules over the Steenrod algebra. In thisparticular case it turns out that the homotopy types of the spectra aredetermined up to p-completion by their mod p cohomology modules.

10. Reverse ampleness

The purpose of this section is to prove 3.9. We will follow a line ofargument similar to the one used in [5] to prove the results in section 8of that paper. We first describe a kind of acyclicity which impliesreverse ampleness, prove a general acyclicity theorem, and then applyit to the collection of nontrivial elementary abelian subgroups. For therest of the section G denotes a particular finite group.


We first need some constructions from [5]. Recall that a coefficient

system H for G [5, 4.1] is a functor from the category of Fp[G]-modulesto the category of vector spaces over Fp which preserves arbitrary di-rect sums. If X is a G-space, the Bredon homology of X with coeffi-cients inH, denoted HBrG

∗ (X;H), is the homology of the chain complexCBrG

∗ (X;H) obtained by the following three-step process:

1. Apply the free Fp-module functor dimensionwise to X, to obtaina simplicial Fp[G]-module Fp[X].

2. Apply H dimensionwise to Fp[X] to obtain a simplicial vectorspace H(Fp[X]).

3. Normalize H(Fp[X]) to obtain CBrG∗ (X;H).

A G-space X is said to be acyclic for H if the map X → ∗ induces anisomorphism HBrG

∗ (X;H)→ HBrG∗ (∗;H).

If M is a module over Fp[G], let HiM denote the coefficient system

for G which assigns to an Fp[G]-module A the Fp-module Hi(G; A ⊗Hom(M, Fp)). The following proposition is an analog of [5, 6.2].

10.1. Proposition. Suppose that M is a finite Fp[G]-module. If the

G-space |C| is acyclic for the coefficient systems HiM , i ≥ 0, then C is

reverse M-ample for ∗.

10.2. Lemma. Suppose that K is a subgroup of G and that X is a

pointed K-space. Then there is a G-map

G+ ∧K (X#)→ (G+ ∧K X)#

which is an equivalence of spectra.

Proof. There is a G-module isomorphism Z[G]→ Hom(Z[G], Z) whichtakes the basis element e of Z[G] to the map Z[G] → Z obtained by

sending∑

x∈G cxx to ce. This extends easily to a G-map Σ∞G+ → G#+

which is a weak equivalence of spectra; note that both of these spectraare wedges of S0; their respective zero-dimensional homology groupsare isomorphic respectively to Z[G] and Hom(Z[G], Z). The proof isfinished by observing that because Σ∞G+ is a finite K-spectrum thereis for any K-spectrum Y an equivalence

Σ∞G+ ∧K Y # ∼= G#+ ∧K Y # → (Σ∞G+ ∧K Y )# .

10.3. Remark. It is an algebraic reflection of the above lemma that ifK is a subgroup of G, then Hj(G; M⊗Z[G/K]) is naturally isomorphicto Hj(K; M). We note for future reference that if H ⊂ K, then underthis isomorphism the projection G/H → G/K induces the cohomo-logical transfer map Hj(H; M) → Hj(K; M). For example, the mapH0(G; M ⊗Z[G/H])→ H0(G; M ⊗Z[G/K]) induced by G/H → G/K


is the map MH →MK given by averaging an element of MH over cosetrepresentatives of H in K.

Proof of 10.1. By 2.10, it is enough to show that under the given as-sumptions the map S0 = (∗)#

+ → |C|#+ induced by |C| → ∗ is an

HG∗ (– ; M)-equivalence. For any Fp[G]-module A there is a natural iso-

morphism HomG(A, Hom(M ; Fp)) ∼= Hom(A⊗Fp[G] M, Fp); this impliesthat for any G-space or G-spectrum Y , there are isomorphisms

H∗G(Y ; Hom(M, Fp)) ∼= Hom(HG

∗ (Y ; M), Fp) .

Note that left and right module structures are being implicitly switchedhere (1.19). Given this duality formula, it is enough to show that the

map S0 → |C|#+ induces an isomorphism on H∗G(– ; Hom(M, Fp)).

Let K be a subgroup of G. By inspection (see 10.3 and 10.2) thereare natural isomorphisms

HBrGi (G/K;Hj

M) ∼=

{

HjG((G/K)#

+; Hom(M, Fp)) i = 0

0 i > 0.

Dualizing the skeletal filtration of |C| thus gives a spectral sequence

Ei,j2 = HBrG

i (|C| ;HjM)⇒ Hj−i

G (|C|#+ ; Hom(M, Fp)) .

This spectral sequence converges because the skeletal filtration of |C| isfinite. Under the assumed acyclicity condition, the spectral sequencecollapses onto the j-axis and gives the desired isomorphism.

The next proposition is like [5, 6.8], but a little more awkward toformulate. In the statement, H is a coefficient system for G. If K isa subgroup of G, H|K is the coefficient system for K which assigns tothe K-module N the abelian group H(Z[G]⊗Z[K] N).

10.4. Proposition. Let X be a G-space, K a subgroup of G of index

prime to p, and Y a subspace of X which is closed under the action of

K. Assume that Y is acyclic for H|K, and that for any x ∈ X \ Y the

following three conditions hold:

1. the map H(Fp[G/Kx])→ H(Fp[G/Gx]) is zero,

2. the map H(Fp[G/Kx])→ H(Fp[G/G]) = H(Fp) is zero, and

3. for any y ∈ Y with Kx ⊂ Ky, the mapH(Fp[G/Kx])→ H(Fp[G/Ky])is zero.

Then X is acyclic for H.

Proof. Compare this with [5, proof of 6.8], but note that we use slightlydifferent notation. For instance, what we denote CBrG

∗ (X;H) is de-noted CG

∗ (X;H) in [5].


The transfers associated to the maps q : G ×K Xn → Xn provide amap t : CBrG

∗ (X;H) → CBrK(X;H|K) [5, 4.1]. By [5, 5.10] this mapcommutes with differentials, and there is a commutative diagram

CBrG∗ (X;H)

t−−−→ CBrK

∗ (X;H|K)q

−−−→ CBrG∗ (X;H)

u

y

v

y

u

y

CBrG∗ (∗;H) −−−→ CBrK

∗ (∗;H|K) −−−→ CBrG∗ (∗;H)

in which the lower arrows are induced by similar transfers and projec-tions. The index assumption assures that the horizontal composites areisomorphisms. Let D be the graded subspace of CBrK

∗ (X;H|K) whichin dimension m is given by H|K(Fp[Xm \ Ym]). By assumption (3), Dis actually a subcomplex; the quotient complex Q = CBrK

∗ (X;H|K)/Dis clearly isomorphic to CBrK

∗ (Y ;H|K). By assumption (1) the map qin the above diagram factors through a map Q→ CBrG

∗ (X;H), and byassumption (2) the map v factors through a map Q→ CBrK

∗ (∗;H|K).This implies that the homology map HBrG

∗ (X;H) → HBrG∗ (∗;H) is a

retract of the isomorphism HBrK∗ (Y ;H|K) → HBrK

∗ (∗;H|K), and thetheorem follows from the fact that a retract of an isomorphism is anisomorphism.

This is a variation on Webb’s theorem (see [22] or [5, 6.0]), but inthis cohomological situation the hypotheses are stronger.

10.5. Proposition. Let X be a G-space, P a Sylow p-subgroup of G,

and M a module over Fp[G]. Suppose that for any nonidentity subgroup

Q of P the fixed point set XQ is contractible. Assume in addition that

for each simplex x ∈ X the order of Gx is divisible by p, and that every

element of order p in G acts trivially on M . Then X is acyclic for the

functors Hj(G; M ⊗ –), j ≥ 0.

Proof. We will use the remarks in 10.3. Let Y be the P -subspace of Xconsisting of simplices which are fixed by a nonidentity element of P .By [5, 4.7] the map Y → ∗ is a P -equivalence, and so by [5, 4.8] thespace Y is acyclic for the functors Hj(P ; M ⊗ –). Let x be a simplexof X which is not in Y , so that Px = {e}. We now check the threeconditions of 10.4. For (1), the map

Hj({e}; M) = Hj(Px; M)→ Hj(Gx; M)

is trivial for j > 0 because the domain group vanishes, and for j = 0because it can be identified with the norm or transfer map

∑

g∈Gx

g : M →MGx .


This norm map vanishes because M is a vector space over Fp and thereis an element of order p in Gx which acts trivially on M . For similarreasons the map Hj({e}; M) → Hj(G; M) vanishes (condition (2)), aswell as the maps Hj({e}; M) → Hj(Py; M) for each y ∈ Y (condition(3)).

Proof of 3.9. By 10.1, it is enough to prove that the G-space |E| isacyclic for the coefficient systems Hi

M . This will be a consequenceof 10.5 if we can check that for every simplex x ∈ |E|, Gx has orderdivisible by p, and that for every nonidentity p-subgroup Q of G, |E|Q

is contractible. Both of these conditions are verified in [5, §8], wherethe space |E| is denoted Xδ

E .

References

1. G. Arone and M. Mahowald, The Goodwillie tower of the identity functor and

the unstable periodic homotopy of spheres, Invent. Math. 135 (1999), no. 3,743–788.

2. A. Bjorner and M. Wachs, Bruhat order of Coxeter groups and shellability, Adv.in Math. 43 (1982), no. 1, 87–100.

3. A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations,Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 304.

4. W. G. Dwyer, Homology decompositions for classifying spaces of finite groups,Topology 36 (1997), no. 4, 783–804.

5. , Sharp homology decompositions for classifying spaces of finite groups,Group representations: cohomology, group actions and topology (Seattle, WA,1996), Amer. Math. Soc., Providence, RI, 1998, pp. 197–220.

6. P. G. Goerss and J. F. Jardine, Simplicial homotopy theory,http://www.math.uwo.ca/∼jardine/papers/simp-sets/.

7. H.-W. Henn, Centralizers of elementary abelian p-subgroups, the Borel con-

struction of the singular locus and applications to the cohomology of discrete

groups, Topology 36 (1997), no. 1, 271–286.8. N. J. Kuhn, A Kahn–Priddy sequence and a conjecture of G. W. Whitehead,

Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 3, 467–483.9. , Corrigenda: “A Kahn–Priddy sequence and a conjecture of G. W.

Whitehead”, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 189–190.10. , The modular Hecke algebra and Steinberg representation of finite

Chevalley groups, Jour. of Algebra 91 (1984), 125–141.11. , Chevalley group theory and the transfer in the homology of symmetric

groups, Topology 24 (1985), no. 3, 247–264.12. N. J. Kuhn and S. B. Priddy, The transfer and Whitehead’s conjecture, Math.

Proc. Cambridge Philos. Soc. 98 (1985), no. 3, 459–480.13. K. Lesh, Identification of infinite loop spaces arising from group theory,

preprint, 1997.14. L. G. Lewis, Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant

stable homotopy theory, Springer-Verlag, Berlin, 1986.15. J. P. May, Simplicial objects in algebraic topology, University of Chicago Press,

Chicago, IL, 1992, Reprint of the 1967 original.


16. S. A. Mitchell, Finite complexes with A(n)-free cohomology, Topology 24

(1985), no. 2, 227–246.17. S. A. Mitchell and S. B. Priddy, Stable splittings derived from the Steinberg

module, Topology 22 (1983), no. 3, 285–298.18. P. Orlik and H. Terao, Arrangements of hyperplanes, Springer-Verlag, Berlin,

1992.19. D. Quillen, Finite generation of the groups Ki of rings of algebraic integers,

(1973), 179–198. Lecture Notes in Math., Vol. 341.20. L. Solomon, The Steinberg character of a finite group with BN -pair, Theory

of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Ben-jamin, New York, 1969, pp. 213–221.

21. R. Steinberg, Prime power representations of finite linear groups, Canad. J.Math. 8 (1956), 580–591.

22. P. Webb, A split exact sequence of Mackey functors, Comment. Math. Helvetici66 (1991), 34–69.

Department of Mathematics, University of Chicago, Chicago, IL

60637 USA

Department of Mathematics, University of Notre Dame, Notre Dame,

IN 46556 USA

E-mail address: [email protected] address: [email protected]

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