Rational Computations of the Topological K-Theory of Classifying Spaces of Discrete Groups

Rational Computations of the Topological K -Theory of Classifying Spaces of Discrete

Groups

Wolfgang L¨ uck

Fachbereich Mathematik

Universit¨ at M¨ unster Einsteinstr. 62 48149 M¨ unster

Germany August 29, 2006

Abstract

We compute rationally the topological (complex) K-theory of the clas- sifying space BG of a discrete group provided that Ghas a cocompact G-CW-model for its classifying space for properG-actions. For instance word-hyperbolic groups and cocompact discrete subgroups of connected Lie groups satisfy this assumption. The answer is given in terms of the group cohomology ofGand of the centralizers of finite cyclic subgroups of prime power order. We also analyze the multiplicative structure.

Key words: topologicalK-theory, classifying spaces of groups.

Mathematics Subject Classification 2000: 55N15.

0. Introduction and Statements of Results

The main result of this paper is:

Theorem 0.1(Main result). LetGbe a discrete group. Denote byK^∗(BG)the topological (complex) K-theory of its classifying spaceBG. Suppose that there is a cocompactG-CW-model for the classifying spaceEGfor properG-actions.

∗email: lueck@math.uni-muenster.de

www: http://www.math.uni-muenster.de/u/lueck/

FAX: 49 251 8338370

Then there is aQ-isomorphism chⁿ_G:Kⁿ(BG)⊗_ZQ

∼=

−→

Y

i∈Z

H²ⁱ⁺ⁿ(BG;Q)

!

× Y

pprime

Y

(g)∈conp(G)

Y

i∈Z

H²ⁱ⁺ⁿ(BCGhgi;Qb^p)

! ,

wherecon_p(G)is the set of conjugacy classes (g) of elementsg∈Gof order p^d for some integerd≥1 andC_Ghgi is the centralizer of the cyclic subgroup hgi generated byg.

Theclassifying space EG for properG-actions is a proper G-CW-complex such that theH-fixed point set is contractible for every finite subgroupH ⊆G.

It has the universal property that for every properG-CW-complexX there is up to G-homotopy precisely one G-map f:X → EG. Recall that a G-CW- complex is proper if and only if all its isotropy groups are finite, and is finite if and only if it is cocompact. The assumption in Theorem 0.1 that there is a cocompact G-CW-model for the classifying space EGfor proper G-actions is satisfied for instance if G is word-hyperbolic in the sense of Gromov, if G is a cocompact subgroup of a Lie group with finitely many path components, if G is a finitely generated one-relator group, if G is an arithmetic group, a mapping class group of a compact surface or the group of outer automorphisms of a finitely generated free group. For more information aboutEGwe refer for instance to [8] and [23]. We will prove Theorem 0.1 in Section 4.

We will also investigate the multiplicative structure on Kⁿ(BG)⊗_ZQ in Section 5. If one is willing to complexify, one can show:

Theorem 0.2 (Multiplicative structure). Let Gbe a discrete group. Suppose that there is a cocompactG-CW-model for the classifying spaceEGfor proper G-actions.

Then there is aC-isomorphism chⁿ_G,C:Kⁿ(BG)⊗_ZC

∼=

−→

Y

i∈Z

H²ⁱ⁺ⁿ(BG;C)

!

× Y

pprime

Y

(g)∈conp(G)

Y

i∈Z

H²ⁱ⁺ⁿ(BCGhgi;Qb^p⊗_QC)

! ,

which is compatible with the standard multiplicative structure on K^∗(BG) and the one on the target given by

a, u_p,(g)

· b, v_p,(g)

= a·b,(a·v_p,(g)+b·u_p,(g)+u_p,(g)·v_p,(g)) for

(g) ∈ con_p(G);

a, b ∈ Y

i∈Z

H^2i+∗(BG;C);

u_p,(g), v_p,(g) ∈ Y

i∈Z

H^2i+∗(BCGhgi;Qb^p⊗_QC),

and the structures of a graded commutative ring on Q

i∈ZH^2i+∗(BG;C) and Q

i∈ZH^2i+∗(BCGhgi;Qbp ⊗_Q C) coming from the cup-product and the obvious Q

i∈ZH^2i+∗(BG;C)-module structure onQ

i∈ZH^2i+∗(BCGhgi;Qbp⊗_QC)coming from the canonical mapsBCGhgi →BGandC→Qbp⊗_QC.

We discuss in Remark 5.4 why it is necessary for the multiplicative structure to consider the complexified version and the isomorphism in Theorem 0.2 isnot compatible with the analogous multiplicative structure.

In Section 6 we will prove Theorem 0.1 and Theorem 0.2 under weaker finiteness assumptions than stated above.

IfGis finite, we get the following integral computation ofK^∗(BG). Through- out the paperR(G) will be the complex representation ring andIG be its aug- mentation ideal, i.e. the kernel of the ring homomorphism R(G) → Z send- ing [V] to dim_C(V). If Gp ⊆ G is a p-Sylow subgroup, restriction defines a map I(G) → I(Gp). Let Ip(G) be the quotient of I(G) by the kernel of this map. This is independent of the choice of thep-Sylow subgroup since two p- Sylow subgroups of G are conjugate. There is an obvious isomorphism from Ip(G)−^∼=→im(I(G)→I(G_p)). We will prove in Section 3

Theorem 0.3. (K-theory of BG for finite groups G). Let G be a finite group. For a prime p denote by r(p) = |con_p(G)| the number of conjugacy classes (g) of elements g ∈ G whose order |g| is p^d for some integer d ≥ 1.

Then there are isomorphisms of abelian groups K⁰(BG) ∼= Z× Y

pprime

Ip(G)⊗_ZZb^p

∼= Z× Y

pprime

(Zb^p)^r(p); K¹(BG) ∼= 0.

The isomorphismK⁰(BG)−^∼=→Z×Q

pprimeIp(G)⊗_ZZbp is compatible with the standard ring structure on the source and the ring structure on the target given by

(m, up⊗ap)·(n, vp⊗bp) = (mn,(mvp⊗bp+nup⊗ap+ (upvp)⊗(apbp)) form, n∈Z,up, vp ∈Ip(G) andap, bp ∈Zbp and the obvious multiplication in Z,Ip(G)andZbp.

The additive version of Theorem 0.3 has already been explained in [16, page 125]. Inspecting [15, Theorem 2.2] one can also derive the ring struc- ture. In [18] theK-theory ofBGwith coefficients in the field F_p ofpelements has been determined including the multiplicative structure. The proof of Theo- rem 0.3 we will present here is based on the ideas of this paper. We will and need to show a stronger statement about the pro-group{IG/(IG)ⁿ⁺¹}in Theorem 3.5 (b).

A version of Theorem 0.1 for topological K-theory with coefficients in the p-adic integers has been proved by Adem [1], [2] using the Atiyah-Segal comple- tion theorem for the finite groupG/G⁰ provided that Gcontains a torsionfree

subgroupG⁰ of finite index. Our methods allow to drop this condition, to deal withK^∗(BG)⊗_ZQdirectly and study systematically the multiplicative struc- ture forK^∗(BG)⊗_ZC. They are based on the equivariant cohomological Chern character of [22].

For integral computations of the K-theory and K-homology of classifying spaces of groups we refer to [17].

The paper is organized as follows:

1. Borel Cohomology and Rationalization 2. Some Preliminaries about Pro-Modules

3. The K-Theory of the Classifying Space of a Finite Group 4. Proof of the Main Result

5. Multiplicative Structures

6. Weakening the Finiteness Conditions 7. Examples and Further Remarks

References

The author wants to the thank the Max Planck Institute for Mathematics in Bonn for its hospitality during his stay from April 2005 until July 2005 when this paper was written.

1. Borel Cohomology and Rationalization

Denote byGROUPOIDS the category of small groupoids. Let Ω-SPECTRA be the category of Ω-spectra, where a morphismf:E→Fis a sequence of maps fn:En→Fn compatible with the structure maps and we work in the category of compactly generated spaces (see for instance [11, Section 1]). A contravari- ant GROUPOIDS-Ω-spectrum is a contravariant functor E: GROUPOIDS → Ω-SPECTRA.

Let E be a (non-equivariant) Ω-spectrum. We can associate to it a con- travariantGROUPOIDS-Ω-spectrum

EBor:GROUPOIDS→Ω-SPECTRA; G 7→ map(BG;E), (1.1) where BG is the classifying space associated to G and map(BG;E) is the ob- vious mapping space spectrum (see for instance [11, page 208 and Definition 3.10 on page 224]). In the sequel we use the notion of an equivariant cohomol- ogy theoryH^∗_? with values in R-modules of [22, Section 1]. It assigns to each (discrete) groupG a G-cohomology theoryH^∗_G with values in the category of R-modules on the category of pairs ofG-CW-complexes, where∗ runs through Z. LetH_?^∗(−,EBor) be the to EBor associated equivariant cohomology theory with values inZ-modules satisfying the disjoint union axiom, which has been constructed in [22, Example 1.8]. For a given discrete groupGand aG-CW-pair (X, A) andn∈Zwe get a natural identification

H_Gⁿ(X, A;EBor) = Hⁿ(EG×G(X, A);E), (1.2)

whereH^∗(−;E) is the (non-equivariant) cohomology theory associated toE. It is induced by the following composite of equivalences of Ω-spectra

map_Or(G) map_G(G/?, X)^G,map BG^G(G/H),E

→ map_Or(G) map_G(G/?, X)^G,map (EG×GG/?,E)

→ map map_G(G/?, X)⊗Or(G)EG×GG/?,E

→ map (EG×GX,E) using the notation of [22]. In the literatureHⁿ(EG×G(X, A);E) is calledthe equivariant Borel cohomology of (X, A) with respect to the (non-equivariant) cohomology theoryH^∗(−;E).

Our main example for E will be the topological K-theory spectrum K, whose associated (non-equivariant) cohomology theory H^∗(−;K) is topologi- calK-theoryK^∗.

There is a functor

Rat: Ω-SPECTRA→Ω-SPECTRA, E7→Rat(E) =E₍₀₎,

which assigns to an Ω-spectrumEits rationalizationE₍₀₎. The homotopy groups πk(E₍₀₎) come with a canonical structure of a Q-module. There is a natural transformation

i(E) :E → E(0) (1.3)

which induces isomorphisms

π_k(i(E)) :π_k(E)⊗_ZQ

∼=

−→ π_k(E₍₀₎). (1.4) Composing E_Bor with Rat yields a contravariant Or(G)-Ω-spectrum denoted by (EBor)₍₀₎. We obtain an equivariant cohomology theory with values in Q- modules byH_?^∗

−; (EBor)₍₀₎

. The map iinduces a natural transformation of equivariant cohomology theories

i^∗_?(−;E) :H_?^∗(−;EBor)⊗_ZQ → H_?^∗

−; (EBor)₍₀₎

. (1.5)

Lemma 1.6. IfGis a groupGand(X, A)is a relative finiteG-CW-pair, then iⁿ_G(X, A;E) :H_Gⁿ(X, A;EBor)⊗_ZQ→H_Gⁿ

X, A; (EBor)₍₀₎ is aQ-isomorphism for all n∈Z.

Proof. The transformationi^∗_G(−;E) is a natural transformation ofG-cohomology theories sinceQis flat over Z. One easily checks that it induces a bijection in the caseX =G/H, since then there is a commutative square with obvious iso- morphisms as vertical maps and the isomorphism of (1.4) as lower horizontal arrow

H_G^k (G/H;E_Bor)⊗_ZQ

i^∗_G(G/H;E)

−−−−−−−→ H_G^k

G/H; (E_Bor)₍₀₎

∼=



 y



 y

∼=

π_−k(map(BH,E))⊗_ZQ −−−−−−−−−−−−−→

π−k(i(map(BH,E)))

π_−k

(map(BH,E))₍₀₎ By induction over the number of G-cells using Mayer-Vietoris sequences one shows thati^∗_G(X, A) is an isomorphism for all relative finiteG-CW-pairs (X, A).

Remark 1.7. (Comparison of the various rationalizations). Notice that i^∗_G(X, A,E) of (1.5) is not an isomorphism for all G-CW-pairs (X, A) because the source does not satisfy the disjoint union axiom for arbitrary index sets in contrast to the target. The point is that− ⊗_ZQis compatible with direct sums but not with direct products.

Since H_?^∗

−; (K_Bor)₍₀₎

is an equivariant cohomology theory with values in Q-modules satisfying the disjoint union axiom, we can use the equivariant cohomological Chern character of [22] to computeH_G^∗

EG; (K_Bor)₍₀₎ for all groupsG.

This is also true for the equivariant cohomology theory with values in Q- modules satisfying the disjoint union axiomH_?^∗ −; K₍₀₎

Bor

. (Here we have changed the order of Bor and (0).) But this a much worse approximation of K^k(BG)⊗_ZQthanH_G^∗

BG; (EBor)₍₀₎

. Namely,iinduces using the universal property ofRata natural map of contravariantGROUPOIDS-Ω-spectra

(K_Bor)₍₀₎→ K₍₀₎

Bor

and thus a natural map H_G^k

X; (KBor)₍₀₎

→H_G^k X; K₍₀₎

Bor

but this map is in general not an isomorphism. Namely, it is not bijective for X =G/H for finite non-trivialH andk= 0. In this case the source turns out to be

π₀

(map(BH;K))₍₀₎

∼= K⁰(BH)⊗_ZQ ∼= Q× Y

p| |H|

(Qbp)^r(p) forr(p) the number of conjugacy classes (h) of non-trivial elementsh∈H ofp- power order, and the target isK⁰(BH;Q) which turns out to be isomorphic toQ since the rational cohomology ofBHagrees with the one of the one-point-space.

As mentioned before we want to use the equivariant cohomological Chern character of [22] to computeH_G^∗

X; (K_Bor)₍₀₎

. This requires a careful analysis of the contravariant functor

FGINJ→Q- MOD, H 7→H_G^k

G/H; (KBor)₍₀₎

=K^k(BH)⊗_ZQ,

from the categoryFGINJof finite groups with injective group homomorphisms as morphisms to the categoryQ- MOD of Q-modules. It will be carried out in Section 3 after some preliminaries in Section 2.

2. Some Preliminaries about Pro-Modules

It will be crucial to handle pro-systems and pro-isomorphisms and not to pass directly to inverse limits. In this section we fix our notation for handling pro-R-modules for a commutative ringR, where ring always means associative ring with unit. For the definitions in full generality see for instance [3, Appendix]

or [6,§2].

For simplicity, all pro-R-modules dealt with here will be indexed by the positive integers. We write{Mn, α_n} or briefly{Mn} for the inverse system

M₀←−^α1 M₁←−^α2 M₂←−^α3 M₃←−^α4 . . . .

and also writeα^m_n :=α_m+1◦ · · · ◦α_n:G_n→G_mforn > mand putαⁿ_n= id_Gn. For the purposes here, it will suffice (and greatly simplify the notation) to work with “strict” pro-homomorphisms{fn}:{Mn, αn} → {Nn, βn}, i.e. a collection of homomorphismsfn:Mn→Nn forn≥1 such thatβn◦fn=f_n−1◦αn holds for eachn≥2. Kernels and cokernels of strict homomorphisms are defined in the obvious way.

A pro-R-module{Mn, αn} will be calledpro-trivial if for eachm≥1, there is some n ≥ m such that α^m_n = 0. A strict homomorphism f: {Mn, αn} → {Nn, βn} is apro-isomorphism if and only if ker(f) and coker(f) are both pro- trivial, or, equivalently, for eachm≥1 there is somen≥msuch that im(β_n^m)⊆ im(f_m) and ker(f_n)⊆ker(α^m_n). A sequence of strict homomorphisms

{Mn, αn}−−−→ {M^{fn} _n⁰, α⁰_n}−→ {M^gn _n⁰⁰, α⁰⁰_n} will be calledexact if the sequences ofR-modulesMn

fn

−→Nn gn

−→M_n⁰⁰ is exact for eachn≥1, and it is calledpro-exact ifgn◦fn= 0 holds forn≥1 and the pro-R-module{ker(gn)/im(fn) is pro-trivial.

The following results will be needed later.

Lemma 2.1. Let 0 → {M_n⁰, α⁰_n} −−−→ {M^{fn} _n, α_n} −−−→ {M^{gn} _n⁰⁰, α⁰⁰_n} → 0 be a pro-exact sequence of pro-R-modules. Then there is a natural exact sequence

0→lim←−^n≥1M_n⁰

lim←−^n≥1fn

−−−−−−→lim←−^n≥1Mn

lim←−^n≥1gn

−−−−−−→lim←−^n≥1M_n⁰⁰−→^δ lim←−

1 n≥1M_n⁰

lim←−

1 n≥1f_n

−−−−−−→lim

←−

1 n≥1Mn

lim←−

1 n≥1g_n

−−−−−−→lim

←−

1

n≥1M_n⁰⁰→0.

In particular a pro-isomorphism {fn}:{Mn, αn} → {Nn, βn} induces isomor- phisms

lim←−^n≥1fn: lim←−^n≥1Mn

∼=

−→ lim←−^n≥1Nn; lim←−^1n≥1fn: lim←−^1n≥1Mn

∼=

−→ lim←−^1n≥1Nn.

Proof. If 0 → {M_n⁰, α⁰_n} −−−→ {M^{fn} n, αn} −→ {M^gn _n⁰⁰, α⁰⁰_n} → 0 is exact, the con- struction of the six-term sequence is standard (see for instance [31, Proposi- tion 7.63 on page 127]). Hence it remains to show for a pro-trivial pro-R-module {M_n, α_n}that lim

←−^n≥1M_n and lim

←−^1n≥1M_nvanish. This follows directly from the standard construction for these limits as the kernel and cokernel of the homo- morphism

Y

n≥1

M_n→ Y

n≥1

M_n, (x_n)_n≥1 7→(x_n−α_n+1(x_n+1))_n≥1.

Lemma 2.2. Fix any commutative Noetherian ring R, and any ideal I ⊆R.

Then for any exact sequenceM⁰ →M →M⁰⁰ of finitely generatedR-modules, the sequence

{M⁰/IⁿM⁰} → {M/IⁿM} → {M⁰⁰/IⁿM⁰⁰} of pro-R-modules is pro-exact.

Proof. It suffices to prove this for a short exact sequence 0 → M⁰ → M → M⁰⁰→0. RegardM⁰ as a submodule ofM, and consider the exact sequence

0→n_(InM)∩M⁰ IⁿM⁰

o→ {M⁰/IⁿM⁰} → {M/IⁿM} → {M⁰⁰/IⁿM⁰⁰} →0.

By [5, Theorem 10.11 on page 107], the filtrations{(IⁿM)∩M⁰} and {IⁿM⁰} ofM⁰ have “bounded difference”, i.e. there existsk >0 with the property that (I^n+kM)∩M⁰ ⊆IⁿM⁰ holds for all n≥ 1. The first term in the above exact sequence is thus pro-trivial, and so the remaining terms define a short sequence of pro-R-modules which is pro-exact.

3. The K-Theory of the Classifying Space of a Finite Group

Next we investigate the contravariant functor from the category FGINJ of finite groups with injective group homomorphisms as morphisms to the category Z- MOD ofZ-modules

FGINJ→Z- MOD, H 7→K^k(BH).

We need some input from representation theory. Recall thatR(G) denotes the complex representation ring. LetIGbe the kernel of the ring homomorphism res^{1}_G :R(G) → R({1} which is the same as the kernel of augmentation ring homomorphismR(G)→Zsending [V] to dim_C(V). We will frequently use the so calleddouble coset formula(see [28, Proposition 22 in Chapter 7 on page 58]).

It says for two subgroupsH, K ⊆G res^K_G◦ind^G_H = X

KgH∈K\G/H

ind_{c(g) :H∩g}−1Kg→K◦res^H∩g_H ^−1Kg, (3.1) wherec(g) is conjugation with g, i.e. c(g)(h) =ghg⁻¹, and ind and res denote induction and restriction. One consequence of it is that ind^G_H:R(H)→R(G) sendsIH toIG. Obviously res^H_G:R(G)→R(H) mapsIG toIH.

For an abelian group M letM_(p) be the localization of M at p. IfZ(p) is the subring ofQobtained fromZby inverting all prime numbers exceptp, then M_(p)=M ⊗_ZZ(p). Recall that the functor ?⊗_ZZ(p) is exact.

Lemma 3.2. Let Gbe a finite group. Let pbe a prime number and denote by Gp ap-Sylow subgroup of G. Then the composite

R(Gp)(p) ind^G_Gp

−−−−→R(G)(p) res^Gp_G

−−−→R(Gp)(p)

has the same image as

res^G_G^p:R(G)(p)→R(Gp)(p).

Proof. A subgroupH⊆Gis calledp-elementaryif it is isomorphic toC×Pfor a cyclic groupCof order prime topand ap-groupP. Let{C_i×P_i|i= 1,2, . . . , r}

be a complete system of representatives of conjugacy classes of p-elementary subgroups ofG. We can assume without loss of generalityPi⊆Gp. Define for i= 1,2, . . . , r a homomorphism of abelian groups

φi := X

G_p·g·(Ci×Pi)∈

Gp\G/(Ci×Pi)

ind_{c(g) :Pi∩g}⁻¹_Gpg→Gp◦res^Pi∩g

−1Gpg

P_i :R(Pi) → R(Gp).

Since the order ofC_i is prime top, we have (C_i×P_i)∩g⁻¹G_pg=P_i∩g⁻¹G_pg forg∈G. Hence the following diagram commutes (actually before localization) by the double coset formula

Lr

i=1R(Pi)(p)

Lr i=1ind^Gp_Pi

−−−−−−−−→ R(Gp)(p) Lr

i=1ind^Ci_Pi^×Pi



 y



 y^ind

G Gp

Lr

i=1R(Ci×Pi)_(p)

Lr i=1ind^Gp

Ci×Pi

−−−−−−−−−−→ R(G)_(p)

Lr

i=1res^Pi_Ci×Pi



 y



 y^res

Gp G

Lr

i=1R(Pi)_(p)

Lr i=1φi

−−−−−→ R(Gp)_(p)

The middle horizontal arrow Lr

i=1ind^G_C^p

i×Pi is surjective by Brauer’s Theo- rem [28, Theorem 18 in Chapter 10 on page 75]. The composite of the left lower vertical arrow and the left upper vertical arrowLr

i=1res^P_Cⁱ

i×Pi◦ind^C_P^i×Pi

i

isLr

i=1|Ci| ·id and hence an isomorphism. Now the claim follows from an easy diagram chase.

Lemma 3.3. Let pandq be different primes. Then the composition R(G_p)

ind^G_Gp

−−−−→R(G) ^res

Gq

−−−→G R(G_q) agrees with|Gq\G/Gp| ·ind^G_{1}^q ◦res^{1}_G

p .

Proof. This follows from the double coset formula (3.1) sinceGp∩g⁻¹Gqg={1}

for eachg∈G.

Lemma 3.4. Let G be a finite group and let IG ⊆R(G) be the augmentation ideal. Then the following sequence ofR(G)-modules is exact

0→ \

m≥1

(IG)^{m i}−→IG Q

pres^Gp_G

−−−−−−→ Y

p∈P(G)

im

res^G_G^p:IG→IG_p

→0,

whereiis the inclusion and P(G)is the set of primes dividing |G|.

Proof. The kernel ofQ

pres^G_G^p: R(G)→Q

pR(Gp) isT

m≥1(IG)^mby [4, Propo- sition 6.12 on page 269]. Hence it remains to show that

Y

p

res^G_G^p:IG→Y

p

im

res^G_G^p:IG→IG_p

is surjective. It suffices to show for a each prime numberq that its localization Y

p

res^G_G^p: (IG)_(q)→Y

p

im

res^G_G^p: (IG)_(q)→(IG_p)_(q)

is surjective. Next we construct the following commutative diagram L

p6=q(I^Gp)_(q)

Q

p6=qres^Gp_G ◦ind^G_Gp

−−−−−−−−−−−−→ Q

p6=qim

res^G_G^p: (I^G)_(q)→(I^Gp)_(q)

L

p6=qind^G_Gp





y ⁱ



 y (IG)_(q)

Q

pres^Gp_G

−−−−−−→ Q

pim

res^G_G^p: (IG)_(q)→(IG_p)_(q)

p1



 y

p2



 y coker

L

p6=qind^G_G

p

_f

−−−−→ im

res^G_G^q: (IG)_(q)→(IGq)_(q)

Here i is the inclusion and p1 and p2 are the obvious projections. Since the composition

M

p6=q

(IG_p)(q) L

p6=qind^G_Gp

−−−−−−−−→(IG)(q) Q

pres^Gp_G

−−−−−−→Y

p

im

res^G_G^p: (IG)(q)→(IG_p)(q)

p₂

−→im

res^G_G^q: (IG)_(q)→(IGq)_(q) agrees with

M

p6=q

res^G_G^q◦ind^G_G

p: M

p6=q

(IGp)_(q)→im

res^G_G^q: (IG)_(q)→(IGq)_(q) and hence is trivial by Lemma 3.3, there exists a map

f: coker



 M

p6=q

ind^G_Gp



→im

res^G_G^q: (IG)(q)→(IG_q)(q)

such that the diagram above commutes. Since p₂◦Y

p

res^G_G^p= res^G_G^q: (IG)_(q)→im

res^G_G^q: (IG)_(q)→(IGq)_(q) is by definition surjective, f is surjective. The upper horizontal arrow in the commutative diagram above is surjective by Lemma 3.2. Now the claim follows by an easy diagram chase.

Theorem 3.5(Structure of{IG/(IG)ⁿ⁺¹}). LetGbe a finite group. LetP(G) be the set of primes dividing|G|.

(a) There are positive integersa,b andc such that for each primepdividing the order of |G|

p^a·I^Gp ⊆ I²Gp; I^bG_p ⊆ p·I^Gp; IG·IG_p ⊆ I²G_p;

(I^Gp)^c ⊆ I^G·I^Gp;

(b) For a prime pdividing|G|letim(res^G_G^p)be the image ofres^G_G^p:IG→IG_p.

We obtain a sequence of pro-isomorphisms of pro-Z-modules {IG/(IG)ⁿ⁺¹}−^∼=→ Y

p∈P(G)

{im(res^G_G^p)/(IG)ⁿ·im(res^G_G^p)}

∼=

−→ Y

p∈P(G)

{im(res^G_G^p)/(IG_p)ⁿ·im(res^G_G^p)}

∼=

←− Y

p∈P(G)

{im(res^G_G^p)/(IG_p)^bn·im(res^G_G^p)}

∼=

−→ Y

p∈P(G)

{im(res^G_G^p)/pⁿ·im(res^G_G^p)}.

(c) There is an isomorphism of pro-Z-modules

{Z} ⊕ {IG/(IG)ⁿ}−^∼=→ {R(G)/(IG)ⁿ},

where{Z}denotes the constant inverse system Z←−^id Z←^id−. . . .

Proof. (a) The existence of the integers a, b and c for which the inclusions appearing in the statement of Theorem 3.5 hold follows from results of [4, The- orem 6.1 on page 265] and [7, Proposition 1.1 in Part III on page 277].

(b) These inequalities of assertion (a) imply that the second, third and fourth map of pro-Z-isomorphism appearing in the statement of Theorem 3.5 are in- deed well-defined pro-isomorphisms. The first map

{IG/(IG)ⁿ⁺¹}−^∼=→ Y

p∈P(G)

{im(res^G_G^p)/(IG)ⁿ·im(res^G_G^p)}

is a well-defined pro-isomorphism of pro-Z-modules by Lemma 2.2 and Lemma 3.4 providedn

T

m≥1(I^G)^m /IⁿG·

T

m≥1(I^G)^mo

is pro-trivial. The latter state- ment follows from Lemma 2.2 applied to the exact sequence

0 → \

m≥1

(IG)^m → IG → IG/ \

m≥1

(IG)^m → 0.

(c) Consider the isomorphism of finitely generated free abelian groups Z⊕IG

∼=

−→R(G), (m, x)7→x+m·[C].

It becomes an isomorphism of rings if we equip the source with the multiplication (m, x)·(n, y) = (mn, my+nx+xy). In particularIⁿG·(I^G⊕Z)⊆IⁿG⊕0 for n≥1. This finishes the proof of Theorem 3.5.

Now we can give the proof of Theorem 0.3.

Proof. In the sequel we abbreviate im(res^G_G^p) = im

res^G_G^p:IG→IG_p

. Notice that im(res^G_G^p)⊆R(G_p) is a finitely generated freeZ-module. We obtain from Lemma 2.1 and Theorem 3.5 an isomorphism

lim←−^n≥1R(G)/(IG)ⁿ ∼= Z× Y

p∈P(G)

lim←−^n≥1im(res^G_G^p)/pⁿ·im(res^G_G^p).

Now the Atiyah-Segal Completion Theorem [6] yields an isomorphisms lim←−^n≥1R(G)/(IG)ⁿ −^∼=→ lim←−^n≥1K⁰((BG)n) ←^∼=− K⁰(BG) (3.6) andK¹(BG) = 0. This implies

K⁰(BG) ∼= Z⊕ M

p∈P(G)

im

res^G_G^p: IG →IGp

⊗_ZZbp; K¹(BG) ∼= 0.

Next we show that the rank of the finitely generated free abelian group im

res^G_G^p:IG→IGp

⊆R(G_p) is the number r(p) of conjugacy classes (g) of elementsg∈Gwhose order |g|isp^d for some integerd≥1. This follows from the commutative diagram

C⊗_ZR(G) ^res

Gp

−−−−→G C⊗_ZR(G_p)

∼=



 y



 y

∼=

class_C(G) −−−−→

res^Gp_G

class_C(Gp)

where class_C(G) denotes the complex vector space of class functions on G, i.e.

functions G → C which are constant on conjugacy classes of elements, (and analogous forGp), the vertical isomorphisms are given by taking the character of a complex representation, and the lower horizontal arrow is given by restricting a functionG→Cto Gp.

Recall thatIp(G) is canonically isomorphic to im

res^G_G^p:IG →IG_p

. One easily checks that the isomorphisms obtained from the one appearing in Theorem 3.5 (b) and (c) by applying the inverse limit and the isomorphism (3.6) are compatible with the obvious multiplicative structures.

This finishes the proof of Theorem 0.3.

4. Proof of the Main Result

In this section we want to prove our main Theorem 0.1. We want to apply the cohomological equivariant Chern character of [22] to the equivariant cohomology

theoryH_?^∗

−; (KBor)₍₀₎

. This requires to analyze the contravariant functor FGINJ → Q- MOD, H 7→H_G^l

G/H; (K_Bor)₍₀₎

. (4.1)

From (1.2) and Lemma 1.6 we conclude that the contravariant functor (4.1) is naturally equivalent to the contravariant functor

FGINJ → Q- MOD, H 7→K^l(BH)⊗_ZQ. (4.2) Theorem 0.3 yields the contravariant functor (4.2) is trivial for odd l and is naturally equivalent to the contravariant functor

FGINJ → Q- MOD H 7→Q×Y

p

Ip(H)⊗_ZQbp (4.3) for even l, where the factor Qis constant inH and functoriality for the other factors is given by restriction.

Given a contravariant functorF:FGINJ→Q- MOD, define theQ[aut(H)]- module

THF(H) := ker



 Y

K(H

F(K ,→H) :F(H) → Y

K(H

F(K)



. (4.4) Next we compute TH K⁰(BH)⊗_ZQ

. Since TH is compatible with direct products, we obtain from (4.3) a canonicalQ[aut(H)]-isomorphism

TH K⁰(BH)⊗_ZQ

= TH(Q)×Y

p

TH(Ip(H)⊗_ZQb^p). (4.5)

SinceQis the constant functor, we get T_H(Q) :=

0 ifH 6={1};

Q ifH ={1}. (4.6)

Fix a prime numberp. Since for any finite groupH the map given by restriction to finite cyclic subgroups

R(H)→ Y

C⊆H Ccyclic

R(C)

is injective, we conclude

Lemma 4.7. For a finite groupH

T_H(Ip(H)) = 0, unlessH is a non-trivial cyclic p-group.

LetC be a non-trivial finite cyclicp-group. Then we get T_C(Ip(C)) = ker

res^C_C⁰:R(C)→R(C⁰)

, (4.8)

whereC⁰⊆Cis the unique cyclic subgroup of indexpinC.

Recall that taking the character of a rational representation of a finite group H yields an isomorphism

χ:R_Q(H)⊗_ZQ

∼=

−→class_Q(H),

where R_Q(H) is the rational representation ring of H and class_Q(H) is the rational vector space of functionsf:H→Qfor whichf(g₁) =f(g₂) holds if the cyclic subgroups generated byg₁andg₂are conjugate inH (see [28, page 68 and Theorem 29 on page 102]). Hence there is an idempotentθC∈R_Q(C)⊗_ZQwhich is uniquely determined by the property that its character sends a generator ofC to 1 and all other elements to 0. Denote its image under the change of coefficients mapR_Q(C)⊗_ZQ→R(C)⊗_ZQalso byθC. LetθC·R(C)⊗_ZQb^p⊆R(C)⊗_ZQb^p be the image of the idempotent endomorphismR(C)⊗_ZQb^p→R(C)⊗_ZQb^pgiven by multiplication withθC.

Lemma 4.9. For every non-trivial cyclic p-group C the inclusion induces a Q[aut(C)]-isomorphism

θ_C·R(C)⊗_ZQ

∼=

−→ T_C(Ip(C)⊗_ZQ).

Proof. Since the map res^C_C⁰: R(C)⊗_Z Q → R(C⁰)⊗_Z Q sends θC to zero, θC ·R(C)⊗_ZQ is contained in ker

res^C_C⁰:R(C)→R(C⁰)

⊗_ZQ. For x ∈ ker

res^C_C⁰:R(C)→R(C⁰)

⊗_ZQ one gets θ_C ·x−x = 0 by the calculation appearing in the proof of [21, Lemma 3.4 (b)].

Lemma 4.10. For every proper G-CW-complex X and n ∈ Z there is an isomorphism, natural inX,

chⁿ_G:H_Gⁿ

X; (K_Bor)₍₀₎ ∼=

−→ Y

i∈Z

H²ⁱ⁺ⁿ(G\X;Q) × Y

p

Y

(C)∈Cp(G)

Y

i∈Z

H_W²ⁱ⁺ⁿ

GC(CGC\X^C;θC·R(C)⊗_ZQb^p), whereCp(C)is the set of conjugacy classes of non-trivial cyclic p-subgroups of GandW_GC=N_GC/C_GC is considered as a subgroup ofaut(C)and thus acts onθ_C·R(C)⊗_ZQbp.

Proof. This follows from [22, Theorem 5.5 (c) and Example 5.6] using (4.6), Lemma 4.7 and Lemma 4.9.

For a generatort ∈C letCtbe theC-representation with Cas underlying complex vector space such thattoperates onCby multiplication with exp

2πi

|C|

. Let Gen(C) be the set of generators. Notice that aut(C) acts in an obvious way on Gen(C) such that the aut(C)-action is transitive and free, and acts onR(C) by restriction. In the sequel χV denotes for a complex representation V its character.

Lemma 4.11. Let C be a finite cyclic group. Then (a) The map

v(C) :θC·R(C)⊗_ZC

∼=

−→ Y

Gen(C)

C, [V]7→(χV(t))_t∈Gen(C) is aC[aut(C)]-isomorphism ifaut(C)acts on the target by permuting the factors. The mapv(C)is compatible with the ring structure on the source induced by the tensor product of representations and the product ring struc- ture on the target;

(b) There is an isomorphism of Q[aut(C)]-modules

u(C) :Q[Gen(C)] −^∼=→ θC·R(C)⊗_ZQ. Proof. (a) The map

R(C)⊗_ZC

∼=

−→ Y

g∈C

C, [V]7→(χ_V(g))_g∈C

is an isomorphism of rings. One easily checks that it is compatible with the aut(C) actions. Now the assertion follows from the fact that the character of θ_C sends a generator ofCto 1 and any other element of Cto 0.

(b) Obviously Q[Gen(C)] is Q[aut(C)]-isomorphic to the regular representa- tion Q[aut(C)] since Gen(C) is a transitive free aut(C)-set. It remains to show that θ_C·R(C)⊗_ZQ is Q[aut(C)]-isomorphic to the regular representa- tion Q[aut(C)]. By character theory it suffices to show that θ_C·R(C)⊗_ZC is C[aut(C)]-isomorphic to the regular representation C[aut(C)]. This follows from assertion (a).

Lemma 4.12. For every proper G-CW-complex X and n ∈ Z there is an isomorphism, natural inX,

ch

n G:H_Gⁿ

X; (KBor)_{(0) ∼=}

−→ Y

i∈Z

H²ⁱ⁺ⁿ(G\X;Q)×Y

p

Y

(g)∈con_p(G)

H²ⁱ⁺ⁿ(C_Ghgi\X^hgi;Qbp).

Proof. Fix a prime p. Let C be a cyclic subgroup ofG of order p^d for some integerd≥1. The obviousNGC-action onC given by conjugation induces an embedding of groupsWGC→aut(C). The obvious action of aut(C) on Gen(C) is free and transitive. Thus we obtain an isomorphism ofQbp[WGC]-modules

Qbp[Gen(C)]∼= Y

W_GC\Gen(C)

Qbp[W_GC].

This induces a natural isomorphism

H_W^k_GC(CGC\X^C;Qbp[Gen(C)]) −^∼=→ Y

W_GC\Gen(C)

H^k(CGC\X^C;Qbp), which comes from the adjunction (i^∗, i_!) of the functor restrictioni^∗ and coin- ductioni_! for the ring homomorphismi:Qbp→Qbp[W_GC] and the obvious iden- tificationi_!(Qbp) =Qbp[W_GC]. There is an obvious bijection between the sets

a

(C)∈Cp

WGC\Gen(C)∼= conp(G).

Now the claim follows from Lemma 4.10 and Lemma 4.11 (b).

Theorem 4.13(Computation ofKⁿ(EG×GX)⊗_ZQ). For every finite proper G-CW-complexX andn∈Zthere is a natural isomorphism

chⁿ_G:Kⁿ(EG×GX)⊗_ZQ

∼=

−→ Y

i∈Z

H²ⁱ⁺ⁿ(G\X;Q)×Y

p

Y

(g)∈con_p(G)

H²ⁱ⁺ⁿ(C_Ghgi\X^hgi;Qbp).

Proof. This follows from Lemma 1.6 and Lemma 4.12.

Lemma 4.14. Let Y 6= ∅ be a proper G-CW-complex such that Hep(Y;Q) vanishes for all p. Letf:Y → EG be a G-map. ThenG\f: G\Y →G\EG induces for allk isomorphisms

H_k(G\f;Q) :H_k(G\Y;Q) −^∼=→ H_k(G\EG;Q);

H^k(G\f;Q) :H^k(G\EG;Q) −^∼=→ H^k(G\Y;Q);

H^k(G\f;C) :H^k(G\EG;C) −^∼=→ H^k(G\Y,C);

H^k(G\f;Qb^p) :H^k(G\EG;Qb^p) −^∼=→ H^k(G\Y,Qb^p);

H^k(G\f;Qbp⊗_QC) :H^k(G\EG;Qbp⊗_QC) −^∼=→ H^k(G\Y,Qbp⊗_QC).

Proof. The mapC_∗(f)⊗_Zid_Q:C_∗(G\Y)⊗_ZQ→C_∗(EG)⊗_ZQisQ-chain map of projective QG-chain complexes and induces an isomorphism on homology.

Hence it is aQG-chain homotopy equivalence. This implies thatC_∗(f)⊗_QGM and hom_QG(C_∗(f), M) are chain homotopy equivalences and induce isomor- phisms on homology and cohomology respectively for everyQ-moduleM.

Now we can give the proof of Theorem 0.1.

Proof. We conclude from Lemma 4.14 that for any g ∈ conp(G) the up to CGhgi-homotopy uniqueCGhgi-map fg:ECGhgi →ECGhgiand the up toG- homotopy uniqueG-mapf: EG→EGinduce isomorphisms

H^k(G\f;Q) :H^k(G\EG;Q) −^∼=→ H^k(BG;Q); (4.15) H^k(CGhgi\fg;Qbp) :H^k(CGhgi\ECGhgi;Qbp) −^∼=→ H^k(BCGhgi,Qbp).(4.16) Now apply Theorem 4.13 toX =EGand use (4.15) and (4.16) together with the fact thatEG^hgiis a model forECGhgi.

5. Multiplicative Structures

In this section we want to deal with multiplicative structures and prove Theorem 0.2.

Remark 5.1. (Ring structures and multiplicative structures). Suppose that the Ω-spectrumEcomes with the structure of a ring spectrumµ:E∧E→ E. It induces a multiplicative structure on the (non-equivariant) cohomology theoryH^∗(−;E) associated toE. Thus the equivariant cohomology theory given by the equivariant Borel cohomologyH_?^∗(E?×?−;E) associated toE inherits a multiplicative structure the sense of [22, Section 6].

If the contravariantGROUPOIDS-Ω-spectrumFcomes with a ring structure of contravariant GROUPOIDS-Ω-spectra µ: F∧F → F, then the associated equivariant cohomology theory H_?^∗(−;F) inherits a multiplicative structure.

A ring structure on the Ω-spectrum E induces a ring structure of contravari- antGROUPOIDS-Ω-spectra onE_Bor. The induced multiplicative structure on H_?^∗(−;EBor) and the one on H_?^∗(E?×?−;E) are compatible with the natural identification (1.2).

A ring structure on the Ω-spectrumEinduces in a natural way a ring struc- ture on its rationalizationRat(E). Thus a ring structure on the contravariant GROUPOIDS-Ω-spectra onEBor induces a ring structure on the contravariant GROUPOIDS-Ω-spectra on (EBor)₍₀₎. The natural transformation of equiv- ariant cohomology theories appearing in (1.5) is compatible with the induced multiplicative structures.

In this discussion we are rather sloppy concerning the notion of a smash product. Since we are not dealing with higher structures and just want to take homotopy groups in the end, one can either use the classical approach in the sense of Adams or the more advanced new constructions such as symmetric spectra.

Lemma 5.2. The isomorphism appearing in Lemma 4.10 is compatible with the multiplicative structure on the source and the one on the target given by

(a, u_p,(C))·(b, v_p,(C)) = (a·b, a·v_p,(C)+b·u_p,(C)+u_p,(C)·v_p,(C)),

for

(C) ∈ C_p(G);

a, b ∈ H^∗(BG;Q);

u_p,(C), v_p,(C) ∈ H_W^∗_GC(CGC\X^C;θC·R(C)⊗_ZQb^p), and the structures of a graded commutative ring on Q

i∈ZH^2i+∗(BG;Q) and Q

i∈ZH_W^2i+∗

GC(CGC\X^C;θC·R(C)⊗_ZQbp) coming from the cup-product and the multiplicative structure onθ_C·R(C)⊗_ZQbpand the obviousQ

i∈ZH^2i+∗(BG;Q)- module structure onQ

i∈ZH_W^2i+∗

GC(C_GC\X^C;θ_C·R(C)⊗_ZQbp)coming from the canonical mapsC_GC\X →G\X andQ→Qbp.

Proof. The proof consists of a straightforward calculation which is essentially based on the following ingredients. In the sequel we use the notation of [22].

The equivariant Chern character of [22, Theorem 6.4] is compatible with the multiplicative structures.

In Theorem 0.3 we have analyzed for every finite groupH the multiplicative structure on

K⁰(BH) ∼= Z×Y

p

Ip(H)⊗_ZZbp.

Thus the Bredon cohomology group appearing in the target of the Chern char- acter whose source isH_G^∗

X; (K_Bor)₍₀₎

can be identified with Y

i∈Z

H^∗+2i(G\X;Q)

!

×Y

p

Y

i∈Z

H^2i+∗

QbpSub(G;F)(X;Ip(?)⊗_ZQbp)

with respect to the multiplicative structure analogously defined to the one ap- pearing in Theorem 0.3 taking the obvious multiplicative structures on the fac- tors and the module structures of the factor for p over Q

i∈ZH^∗+2i(G\X;Q) into account.

Fix a primep. TheQb^p[aut(C)]-map R(C)⊗_ZQb^p →θC·R(C)⊗_ZQb^p given by multiplication with the idempotentθC is compatible with the multiplicative structures. Using the identification of Lemma 4.9 we obtain for each cyclic p-group Ca retraction compatible with the multiplicative structures.

ρ_C:Ip(C)⊗_ZQbp → T_C(Ip(C)⊗_ZQbp)

Recall thatTK(Ip(K)⊗_ZQb^p) is trivial unlessK is a non-trivial cyclicp-group.

Use these retractions as the mapsρK in the definition of the isomorphismν of QbpSub(G;F)-modules forM =Ip(?)⊗_ZQbpin [22, (5.1)]. Then we obtain using the identification of Lemma 4.9 an isomorphism ofQbpSub(G;F)-modules

Ip(?)⊗_ZQbp

∼=

−→ Y

(C)∈C_p

i(C)_!(θ_C·R(C)⊗_ZQb^p),

which is compatible with the obvious multiplicative structure on the source and the one on the target given by the product of the multiplicative structures