Chern characters for proper equivariant homology theories and applications to

angewandte Mathematik

(Walter de Gruyter BerlinNew York 2002

Chern characters for proper equivariant homology theories and applications to

K- and L-theory

ByWolfgang LuÈck* at MuÈnster

Abstract. We construct for an equivariant homology theory for proper equiv- ariant CW-complexes an equivariant Chern character, provided that certain conditions are satis®ed. This applies for instance to the sources of the assembly maps in the Farrell- Jones Conjecture with respect to the family F of ®nite subgroups and in the Baum- Connes Conjecture. Thus we get an explicit calculation in terms of group homology of Qn_ZKn…RG† and Qn_ZLn…RG† for a commutative ring R with QHR, provided the Farrell-Jones Conjecture with respect to F is true, and of Qn_ZK_n^topÿ

C_r…G;F† F ˆR;C, provided the Baum-Connes Conjecture is true. for

0. Introduction and statements of results

In this paper we want to achieve the following two goals. Firstly, we want to construct an equivariant Chern character for a proper equivariant homology theory H^? which takes values in R-modules for a commutative ring R with QHR. The Chern character identi®es H_n^G…X† with the associated Bredon homology, which is much easier to handle and can often be simpli®ed further. Secondly, we apply it to the sources of the assembly maps appearing in the Farrell-Jones Conjecture with respect to the family F of ®nite subgroups and in the Baum-Connes Conjecture. The target of these assembly maps are the groups we are interested in, namely, the rationalized algebraic K- and L- groups QnZKn…RG† and QnZLn…RG† of the group ring RG of a (discrete) group G with coe½cients in R and the rationalized topological K-groups Qn_ZK_n^topÿ

C_r…G;F† of the reduced group C-algebra of G over F ˆR;C. These conjectures say that these assembly maps are isomorphisms. Thus combining them with our equivariant Chern character yields explicit computations of these rationalized K- and L-groups in terms of group homology and theK-groups andL-groups of the coe½cient ringR orF (see Theo- rem 0.4 and Theorem 0.5).

Throughout this paper all groups are discrete andRwill denote a commutative asso- ciative ring with unit. A properG-homology theoryH^G assigns to anyG-CW-pair…X;A†

which is proper, i.e. all isotropy groups are ®nite, a Z-graded R-module H^G…X;A† such

thatG-homotopy invariance, excision and the disjoint union axiom hold and there is a long exact sequence of a properG-CW-pair. An equivariant proper homology theoryH^?assigns to any group G a proper G-homology theory H^G, and these are linked for the various groups G by an induction structure. An example is equivariant bordism for smooth ori- ented manifolds with cocompact proper orientation preserving group actions. The main examples for us will be given by the sources of the assembly maps appearing in the Farrell- Jones Conjecture with respect to F and in the Baum-Connes Conjecture. These notions will be explained in Section 1.

To any equivariant proper homology theoryH^?we will construct in Section 3 another equivariant proper homology theory, the associated Bredon homologyBH^?. The point is thatBH^? is much easier to handle thanH^?. Although we will not deal with equivariant spectra in this paper, we mention that the equivariant Bredon homologyBH^? is given by a product of equivariant Eilenberg-MacLane spaces, whose homotopy groups are given by the collection of theR-modulesH_q^G…G=H†, and that the equivariant Chern character can be interpreted as a splitting of certain equivariant spectra into products of equivariant Eilenberg-MacLane spectra. We will construct an isomorphism of equivariant homology theories ch^?:BH^? !^G H^? in Section 4, provided that a certain technical assumption is ful®lled, namely, that the covariant RSub…G;F†-module H_q^G…G=?†GH_q^?…† is ¯at for allqAZand all groupsG. The construction of ch^G for a given groupGrequires that H^? is de®ned for all groups, not only for G. There are some favourite situations, where the technical assumption above is automatically satis®ed, and the Bredon homologyBH^?can be computed further. Let FGINJ be the category of ®nite groups with injective group homomorphisms as morphisms. The equivariant homology theory de®nes a covariant functor H_q^?…†: FGINJ!RÿMOD which sendsH to H_q^H…†. Functoriality comes from the induction structure. Suppose that this functor can be extended to a Mackey functor.

This essentially means that one also gets a contravariant structure by restriction and the induction and restriction structures are related by a double coset formula (see Section 5). An important example of a Mackey functor is given by sending H to the rational, real or complex representation ring.

Theorem 0.1. Let R be a commutative ring with QHR. Let H^? be a proper equi- variant homology theory with values in R-modules. Suppose that the covariant functor H_q^?…†: FGINJ!RÿMOD extends to a Mackey functor for all qAZ. Then there is an isomorphism of proper homology theories

ch^?:BH^? !^G H^?:

Theorem 0.1 is the equivariant version of the well-known result (explained in Exam- ple 4.1) that for a (non-equivariant) homology theoryH with values inR-modules and a CW-pair…X;A†there are natural isomorphisms

L

p‡qˆnHpÿ

X;A;H^q…†

GHn…X;A†:

The associated Bredon homology can be decomposed further. De®ne for a ®nite groupH

SHÿ

H_q^H…†

:ˆcoker L

KHHK3H

ind_K^H: L

KHHK3H

H_q^K…† !H_q^H…†

! :

For a subgroupHHG we denote by NGH the normalizer and byCGH the centralizer of H in G. Let HCGH be the subgroup of NGH consisting of elements of the formhc for hAH and cACGH. Denote by WGH the quotient NGH=HCGH. Notice that WGH is

®nite ifHis ®nite.

Theorem 0.2. Consider the situation and assumptions of Theorem 0.1. Let I be the set of conjugacy classes…H† of ®nite subgroups H of G.Then there is for any group G and any proper G-CW-pair…X;A†a natural isomorphism

BH_n^G…X;A†G L

p‡qˆn

L

…H†AIHpÿ

CGHn…X^H;A^H†;R

n_R‰WGHŠSHÿ

H_q^H…† :

Theorem 0.1 and Theorem 0.2 reduce the computation of H_n^G…X;A†to the compu- tation of the singular or cellular homology R-modules Hpÿ

CGHn…X^H;A^H†;R

of the CW-pairs CGHn…X^H;A^H† including the obvious right WGH-operation and of the left R‰WGHŠ-modulesSHÿ

H_q^H…†

which only involve the valuesH_q^G…G=H† ˆH_q^H…†.

Suppose that H^? comes with a restriction structure as explained in Section 6. Then it induces a Mackey structure on H_q^?…† for all qAZand a preferred restriction structure on BH^? so that Theorem 0.1 applies and the equivariant Chern character is compatible with these restriction structures. IfH^? comes with a multiplicative structure as explained in Section 6, thenBH^?inherits a multiplicative structure and the equivariant Chern character is compatible with these multiplicative structures (see Theorem 6.3).

If we have the following additional structure, which will be available in the examples we are interested in, then we can simplify the Bredon homology further. Namely, we assume that the Mackey functor H_q^H…† is a module over the Green functor Qn_ZRQ…?† which assigns to a ®nite groupHthe rationalized ring of rationalH-representations. This notion is explained in Section 7. In particular it yields for any ®nite group H the structure of a Qn_ZRQ…H†-module onH_q^H…†. Let classQ…H†be the ring of functions f:H!Qwhich satisfy f…h1† ˆf…h2†if the cyclic subgroupshh1iandhh2igenerated byh1andh2are con- jugate inH. Taking characters yields an isomorphism of rings

w:Qn_ZRQ…H† !^G classQ…H†:

Given a ®nite cyclic group C, there is the idempotent y_C^C AclassQ…C† which assigns 1 to a generator of C and 0 to the other elements. This element acts on H_q^C…†. The image imÿ

y_C^C:H_q^C…† !H_q^C…†

of the map given by multiplication with the idempotent y_C^C is a direct summand inH_q^C…†and will be denoted byy_C^CH_q^C…†.

Theorem 0.3. Let R be a commutative ring with QHR. Let H^? be a proper equi- variant homology theory with values in R-modules. Suppose that the covariant functor FGINJ!RÿMODsending H toH_q^H…†extends to a Mackey functor for all qAZ,which is a module over the Green functor Qn_ZRQ…?† with respect to the inclusion Q!R. Let J be the set of conjugacy classes…C† of ®nite cyclic subgroups C of G. Then there is an iso- morphism of proper homology theories

ch^?:BH^? !^G H^?:

Moreover,for any group G and any proper G-CW-pair…X;A†there is a natural isomorphism BH_n^G…X;A† ˆ L

p‡qˆn

L

…C†AJHpÿ

CGCn…X^C;A^C†;R

n_R‰WGCŠÿ

y_C^CH_q^C…† : Since Qn_ZKq…R?†, Qn_ZLq…R?† and Qn_ZK_q^topÿ

C_r…?;F†

are Mackey functors and come with module structures over the Green functor Qn_ZRQ…?† as explained in Section 8, Theorem 0.3 implies

Theorem 0.4. Let R be a commutative ring withQHR.Denote by F the ®eldRorC.

Let G be a(discrete)group.Let J be the set of conjugacy classes…C†of ®nite cyclic subgroups C of G. Then the rationalized assembly map in the Farrell-Jones Conjecture with respect to the family F of ®nite subgroups for the algebraic K-groups Kn…RG† and the algebraic L-groups Ln…RG† and in the Baum-Connes Conjecture for the topological K-groups K_n^topÿ

C_r…G;F†

can be identi®ed with the homomorphisms L

p‡qˆn

L

…C†AJHp…CGC;Q†n_Q‰WGCŠy_C^Cÿ

Qn_Z Kq…RC†

!Qn_ZKn…RG†;

L

p‡qˆn

L

…C†AJHp…CGC;Q†n_Q‰WGCŠy_C^Cÿ

Qn_ZLq…RC†

!Qn_ZLn…RG†;

L

p‡qˆn

L

…C†AJHp…CGC;Q†n_Q‰WGCŠy_C^Cÿ

Qn_ZK_q^topÿ

C_r…C;F†

!Qn_ZK_n^topÿ

C_r…G;F† : In the L-theory case we assume that R comes with an involution R!R, r7!r and that we use on RG the involution which sends P

gAGrgg to P

gAGrgg^ÿ1.

If the Farrell-Jones Conjecture with respect toFand the Baum-Connes Conjecture are true,then these maps are isomorphisms.

Notice that in Theorem 0.3 and hence in Theorem 0.4 only cyclic groups occur. The basic input in the proof is essentially the same as in the proof of Artin's theorem that any character in the complex representation ring of a ®nite groupHis rationally a linear com- bination of characters induced from cyclic subgroups. Moreover, we emphasize that all the splitting results are obtained after tensoring with Q, no roots of unity are needed in our construction. In the special situation that the coe½cient ringRis a ®eldFof characteristic zero and we tensor withFn_Z?for an algebraic closureF ofF, one can simplify the expres- sions further as carried out in Section 8. As an illustration we record the following partic- ular nice case.

Theorem 0.5. Let G be a(discrete)group. Let T be the set of conjugacy classes…g†of elements gAG of ®nite order.There is a commutative diagram

L

p‡qˆn

L

…g†ATHp…CGhgi;C†n_ZKq…C† ƒƒƒ! Cn_ZKn…CG†

??

?y

??

?y L

p‡qˆn

L

…g†ATHp…CGhgi;C†n_ZK_q^top…C† ƒƒƒ! Cn_ZK_n^topÿ

C_r…G†

where CGhgi is the centralizer of the cyclic group generated by g in G and the vertical arrows come from the obvious change of ring and of K-theory maps Kq…C† !K_q^top…C† and Kn…CG† !K_n^topÿ

C_r…G†

. The horizontal arrows can be identi®ed with the assembly maps occuring in the Farrell-Jones Conjecture with respect to F for Kn…CG† and in the Baum- Connes Conjecture for K_n^topÿ

C_r…G†

after applyingCn_Zÿ.If these conjectures are true for G, then the horizontal arrows are isomorphisms.

Notice that Theorem 0.5 and the results of Section 8 show that the computation of theK- andL-theory ofRGseems to split into one part, which involves only the group and consists essentially of group homology, and another part, which involves only the coe½- cient ring and consists essentially of itsK-theory. Moreover, a change of rings or change of K-theory map involves only the coe½cient ring R and not the part involving the group.

This seems to suggest to look for a proof of the Farrell-Jones Conjecture which works for all coe½cients simultaneously. We refer to Example 1.5 and to [3], [9], [12], [13], [14] and [15] for more information about the Farrell-Jones and the Baum-Connes Conjectures and about the classes of groups, for which they have been proven.

We mention that a di¨erent construction of an equivariant Chern character has been given in [2] in the case, where H^G is equivariant K-homology after applying Cn_Zÿ.

Moreover, the lower horizontal arrow in Theorem 0.5 has already been discussed there.

The computations of K- and L-groups integrally and with RˆZ as coe½cients are much harder (see for instance [18]).

I would like to thank Tom Farrell for a lot of fruitful discussions of the Farrell-Jones Conjecture and related topics and the referee for his very detailed and very helpful report.

1. Equivariant homology theories

In this section we describe the axioms of a (proper) equivariant homology theory.

The main examples for us are the source of the assembly map appearing in the Farrell- Jones Conjecture with respect to the familyFof ®nite subgroups for algebraicK- andL- theory and the equivariantK-homology theory which appears as the source of the Baum- Connes assembly map and is de®ned in terms of Kasparov's equivariantKK-theory.

Fix a discrete group G and an associative commutative ring R with unit. A G-CW- pair …X;A† is a pair of G-CW-complexes. It is called proper if all isotropy groups of X are ®nite. Basic informations aboutG-CW-pairs can be found for instance in [16], Section 1 and 2. A G-homology theory H^G with values in R-modules is a collection of covariant functors H_n^G from the category ofG-CW-pairs to the category of R-modules indexed by nAZtogether with natural tranformationsq_n^G…X;A†:H_n^G…X;A† !H_nÿ1^G …A†:ˆH_nÿ1^G …A;j†

fornAZsuch that the following axioms are satis®ed:

(a) G-homotopy invariance.

If f0and f1areG-homotopic maps…X;A† ! …Y;B†ofG-CW-pairs, then H_n^G…f₀† ˆH_n^G…f₁† fornAZ.

(b) Long exact sequence of a pair.

Given a pair…X;A†ofG-CW-complexes, there is a long exact sequence

. . .^Hƒƒƒ!^n‡1G…j†H_n‡1^G …X;A†ƒƒƒ^qn‡1G!H_n^G…A† ƒƒƒ^HnG!^i† H_n^G…X†ƒƒƒ!^HnG…j† H_n^G…X;A†ƒƒƒ!^qnG . . .; wherei:A!X and j:X ! …X;A†are the inclusions.

(c) Excision.

Let …X;A† be a G-CW-pair and let f:A!B be a cellular G-map of G-CW- complexes. Equip …XWfB;B† with the induced structure of a G-CW-pair. Then the canonical map…F; f†:…X;A† ! …XWfB;B†induces an isomorphism

H_n^G…F;f†:H_n^G…X;A† !^G H_n^G…XWfB;B†:

(d) Disjoint union axiom.

Let fXijiAIg be a family of G-CW-complexes. Denote by ji:Xi! `

iAIXi the canonical inclusion. Then the map

L

iAIH_n^G…j_i†:L

iAIH_n^G…Xi† !^G H_n^G `

iAIXi

is bijective.

IfH^G is de®ned or considered only for properG-CW-pairs…X;A†, we call it aproper G-homology theoryH^G with values in R-modules.

Leta:H !G be a group homomorphism. Given anH-spaceX, de®ne theinduction of X withato be theG-space indaX which is the quotient of GX by the rightH-action …g;x† h:ˆÿ

ga…h†;h^ÿ1x

for hAH and …g;x†AGX. If a:H !G is an inclusion, we also write ind_H^G instead of inda.

A (proper) equivariant homology theory H^? with values in R-modules consists of a (proper) G-homology theory H^G with values inR-modules for each group G together with the following so called induction structure: given a group homomorphism a:H !G and anH-CW-pair…X;A†such that ker…a†acts freely onX, there are for allnAZnatural isomorphisms

inda:H_n^H…X;A† !^G H_n^Gÿ

inda…X;A†

…1:1†

satisfying:

(a) Compatibility with the boundary homomorphisms.

q_n^Gindaˆindaq_n^H.

(b) Functoriality.

Let b:G!K be another group homomorphism such that ker…ba† acts freely on X. Then we have fornAZ

indbaˆH_n^K…f₁† indbinda:H_n^H…X;A† !H_n^Kÿ

indba…X;A†

; where f1: indbinda…X;A† !^G indba…X;A†, …k;g;x† 7!ÿ

kb…g†;x

is the natural K- homeomorphism.

(c) Compatibility with conjugation.

FornAZ,gAG and a (proper)G-CW-pair…X;A†the homomorphism indc…g†:G!G:H_n^G…X;A† !H_n^Gÿ

indc…g†:G!G…X;A†

agrees with H_n^G…f₂† for the G-homeomorphism f2:…X;A† !ind_c…g†:G!G…X;A† which sendsxto…1;g^ÿ1x†inGc…g†…X;A†.

This induction structure links the various homology theories for di¨erent groups G.

It will play a key role in the construction of the equivariant Chern character even if we want to carry it out only for a ®xed groupG. We will later need

Lemma 1.2. Consider ®nite subgroups H;KHG and an element gAG with gHg^ÿ1HK.Let R_g^ÿ1:G=H !G=K be the G-map sending g⁰H to g⁰g^ÿ1K and c…g†:H !K be the homomorphism sending h to ghg^ÿ1.Letpr:…ind_c…g†:H!K† ! be the projection. Then the following diagram commutes:

H_n^H…† ƒƒƒƒƒƒƒƒ!^{HnK…pr†indc…g†} H_n^K…†

indH^G

??

??

y^{G indKG}

??

?? y^G H_n^G…G=H† ƒƒƒƒƒƒƒ!^H

nG…R_gÿ1†

H_n^G…G=K†:

Proof. De®ne a bijectiveG-map f1: ind_c…g†:G!Gind_H^G !ind_K^Gind_c…g†:H!Kby send- ing …g1;g2;† in G_c…g†GH to …g1gg2g^ÿ1;1;† in GKK_c…g†. The condition that induction is compatible with composition of group homomorphisms means precisely that the composite

H_n^H…†ƒ!^indHG H_n^G…ind_H^G† ƒƒƒƒƒ!^{indc…g†:G!G} H_n^G…ind_c…g†:G!Gind_H^G†ƒƒƒƒ!^HnG…f1† H_n^G…ind_K^Gind_c…g†:H!K† agrees with the composite

H_n^H…†ƒƒƒƒƒ!^{indc…g†:H!K} H_n^K…ind_c…g†:H!K†^indƒ!^KG H_n^G…ind_K^Gind_c…g†:H!K†:

Naturality of induction implies H_n^G…ind_K^Gpr† ind_K^G ˆind_K^GH_n^K…pr†. Hence the follow- ing diagram commutes:

H_n^H…† ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ^{HnK…pr†indc…g†:H!K} ! H_n^K…†

indH^G

??

?y

??

?y^indKG H_n^G…G=H† ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ!

Hn^G…indK^Gpr†Hn^G…f1†indc…g†:G!G

H_n^G…G=K†:

By the axioms the homomorphism indc…g†:G!G:H_n^G…G=H† !H_n^G…indc…g†:G!GG=H†agrees with H_n^G…f₂† for the map f2:G=H !indc…g†:G!GG=H which sends g⁰H to …g⁰g^ÿ1;1H†

inG_c…g†G=H. Since the composite…ind_K^Gpr† f1f2is justR_g^ÿ1, Lemma 1.2 follows. r Example 1.3. Let K be a homology theory for (non-equivariant) CW-pairs with values in R-modules. Examples are singular homology, oriented bordism theory or topologicalK-homology. Then we obtain two equivariant homology theories with values inR-modules by the following constructions:

H_n^G…X;A† ˆKn…GnX;GnA†;

H_n^G…X;A† ˆKnÿ

EGG…X;A†

:

The second one is called theequivariant Borel homology associated toK. In both casesH^G inherits the structure of a G-homology theory from the homology structure on K. Let a:HnX !^G Gn…GaX†be the homeomorphism sendingHxtoG…1;x†. De®ne

b:EHH X !EGGGaX by sending …e;x† to ÿ

Ea…e†;1;x

for eAEH, xAX and Ea:EH!EG the a-equivariant map induced by a. Induction for a group homomorphism a:H!G is induced by these mapsaand b. If the kernel ker…a†acts freely on X, the mapb is a homotopy equivalence and hence in both cases indais bijective.

Example 1.4. Given a properG-CW-pair…X;A†, one can de®ne theG-bordism group W_n^G…X;A†as the abelian group ofG-bordism classes of maps f:…M;qM† ! …X;A†whose sources are oriented smooth manifolds with orientation preserving proper smoothG-actions such thatGnM is compact. The de®nition is analogous to the one in the non-equivariant case. This is also true for the proof that this de®nes a proper G-homology theory. There is an obvious induction structure coming from induction of equivariant spaces. It is well- de®ned because of the following fact. Let a:H !G be a group homomorphism. Let M be an oriented smooth H-manifold with orientation preserving proper smooth H-action such thatHnM is compact and ker…a†acts freely. Then indaM is an oriented smoothG- manifold with orientation preserving proper smoothG-action such thatGnM is compact.

The boundary of indaM is indaqM.

Our main example will be

Example 1.5. Let R be a commutative ring. There are equivariant homology theories H^? such that H_n^G…† is the rationalized algebraic K-group Qn_ZKn…RG† or the rationalized algebraicL-groupQn_ZLn…RG†of the group ringRGor such that H_n^G…† is the rationalized topologicalK-theoryQn_ZK_n^topÿ

C_r…G;R†

orQn_ZK_n^topÿ

C_r…G;C†

of the

reduced real or complex C-algebra of G. Denote by E…G;F† the classifying space of G with respect to the familyF of ®nite subgroups ofG. This is a G-CW-complex whose H-®xed point set is contractible forH AFand is empty otherwise. It is unique up to G- homotopy because it is characterized by the property that for anyG-CW-complexXwhose isotropy groups belong to F there is up to G-homotopy precisely one G-map from X to E…G;F†. TheG-spaceE…G;F†agrees with the classifying spaceEGfor properG-actions.

De®ne E…G;VC† for the family VC of virtually cyclic subgroups analogously. The as- sembly map in the Farrell-Jones Conjecture with respect to F and in the Baum-Connes Conjecture are the maps induced by the projectionE…G;F† !

H_n^Gÿ

E…G;F†

!H_n^G…†;

…1:6†

where one has to choose the appropriate homology theory among the ones mentioned above. The Baum-Connes Conjecture says that this map is an isomorphism (even without rationalizing) for the topological K-theory of the reduced group C-algebra. The Farrell- Jones Conjecture with respect toFis the analogous statement.

It is important to notice that the situation in the Farrell-Jones Conjecture is more complicated. The Farrell-Jones Conjecture itself is formulated with respect to the family VC, i.e. it says that the projectionE…G;VC† ! induces an isomorphism (even without rationalizing)

H_n^Gÿ

E…G;VC†

!H_n^G…†:

…1:7†

For the version of the Farrell-Jones Conjecture with respect to VC no counterexamples are known, whereas the version forFis not true in general. In other words, the canonical mapE…G;F† !E…G;VC†does not necessarily induce an isomorphism

H_n^Gÿ

E…G;F†

!H_n^Gÿ

E…G;VC†

:

This is due to the existence of Nil-groups. However, if for instance R is a ®eld of charac- teristic zero, this map is bijective for algebraicK-theory. Hence the Farrell-Jones Conjec- ture for Qn_ZKn…FG† for a ®eld F of characteristic zero is true with respect to F if and only if it is true with respect toVC. At the time of writing not much is known about this conjecture forKn…FG† for a ®eld F of characteristic zero, since most of the known results are for the algebraicK-theory forZG. The situation inL-theory is better since the change of rings map Qn_ZLn…ZG† !Qn_ZLn…QG† is bijective for any group G. The Farrell- Jones Conjecture for both Qn_ZLn…ZG† and Qn_ZLn…QG† is true with respect to both Fand VC if Gis a cocompact discrete subgroup of a Lie group with ®nitely many path components [9], if G is a discrete subgroup of GLn…CG† [10], or if G is an elementary amenable group [11].

The target of the assembly map for F in (1.6) is Qn_ZKn…RG†, Qn_ZLn…RG† or QnZK_n^topÿ

C_r…G;F†

forF ˆR;C. These are the groups we would like to compute. The source of the assembly map forFin (1.6) is the part which is better accessible for compu- tations. We will apply the equivariant Chern character for proper equivariant homology theories to it which is possible since E…G;F† is proper (in contrast to E…G;VC† and ).

Thus we get computations of the rationalizedK- andL-groups, provided the Farrell-Jones Conjecture with respect toFand the Baum-Connes Conjecture are true.

For more informations about the relevant G-homology theories H^G mentioned above we refer to [3], [5], [9]. It is not hard to construct the relevant induction structures so that they yield equivariant homology theories H^?. We remark that one can construct for them also restriction structures and multiplicative structures in the sense of Section 6.

2. Modules over a category

In this section we give a brief summary about modules over a category as far as needed for this paper. They will appear in the de®nition of the source of the equivariant Chern character.

Let C be a small category and let R be a commutative associative ring with unit.

A covariant RC-module is a covariant functor from C to the category RÿMOD of R- modules. De®ne a contravariant RC-module analogously. Morphisms of RC-modules are natural transformations. Given a groupG, letG^be the category with one object whose set of morphisms is given byG. Then a covariantRG-module is the same as a left^ RG-module, whereas a contravariant RG-module is the same as a right^ RG-module. All the construc- tions, which we will introduce for RC-modules below, reduce in the special case CˆG^ under the identi®cation above to their classical versions forRG-modules. The reader should have this example in mind.

The category RCÿMOD of (covariant or contravariant) RC-modules inherits the structure of an abelian category from RÿMOD in the obvious way, namely objectwise.

For instance a sequence 0!M!N !P!0 ofRC-modules is called exact if its evalu- ation at each object in C is an exact sequence in RÿMOD. The notion of a projective RC-module is now clear. Given a familyBˆ …ci†iAI of objects of C, thefree RC-module with basis Bis

RC…B†:ˆL

iAIRmorC…ci;?†:

The name free with basis B refers to the following basic property. Given a covariant RC-moduleN, there is a natural bijection

homRCÿ

RC…B†;N

!^G Q

iAIN…ci†; f 7!ÿ

f…ci†…idci†

iAI: …2:1†

ObviouslyRC…B†is a projectiveRC-module. AnyRC-moduleM is a quotient of some free RC-module. For instance, there is an obvious epimorphism fromRC…B† toM if we take B to be the family of objects indexed by `

cAOb…C†M…c†, where we assign c to mAM…c†.

Therefore an RC-module M is projective if and only if it is a direct summand in a free RC-module. The analogous considerations apply to the contravariant case.

Given a contravariant RC-module M and a covariant RC-module N, one de®nes theirtensor product over RCto be the followingR-moduleMn_RCN. It is given by

Mn_RCN ˆ L

cAOb…C†M…c†n_RN…c†=@;

where@ is the typical tensor relationmf nnˆmnfn, i.e. for each morphism f:c!d inC,mAM…d†andnAN…c†we introduce the relationM…f†…m†nnÿmnN…f†…n† ˆ0.

The main property of this construction is that it is adjoint to the homR-functor in the sense that for anyR-moduleLthere are natural isomorphisms ofR-modules

homR…Mn_RCN;L† !^G homRCÿ

M;homR…N;L†

; …2:2†

homR…Mn_RCN;L† !^G homRCÿ

N;homR…M;L†

: …2:3†

Consider a functor F:C!D. Given a covariant or contravariant RD-module M, de®ne its restriction with F to be resFM :ˆMF. Given a covariant RC-module M, its induction with Fis the covariantRD-module indFMgiven by

…indFM†…??†:ˆRmorDÿ

F…?†;??

n_RCM…?†:

Given a contravariantRC-moduleM, itsinduction with Fis the contravariantRD-module indFM given by

…indFM†…??†:ˆM…?†n_RCRmorDÿ

??;F…?†

: Restriction withFcan be written in the covariant case as

resFN…?† ˆhomRDÿ

RmorDÿ

F…?†;??

;N…??†

and in the contravariant case as resFN…?† ˆhomRDÿ

RmorDÿ

??;F…?†

;N…??†

because of (2.1). We conclude from (2.3) that induction and restriction form an adjoint pair, i.e.

for two RC-modules M and N, which are both covariant or both contravariant, there is a natural isomorphism ofR-modules

homRD…indFM;N† !^G homRC…M;resFN†:

…2:4†

Given a contravariant RC-module M and a covariant RD-module N, there is a natural R-isomorphism

…indFM†n_RDN !^G Mn_RC…resFN†:

…2:5†

It is explicitly given by ÿ

f:??!F…?†

nmnn7!mnN…f†…n† or can be obtained for- mally from (2.2) and (2.4). One easily checks

indFRmorC…c;?† ˆRmorDÿ

F…c†;??

…2:6†

for cAOb…C†. This shows that indF respects direct sums and the properties free and projective.

Next we explain how one can reduce the study of projectiveRC-modules to the study of projective Raut…c†-modules, where aut…c† is the group of automorphisms of an object cinC. Given a covariantRC-module M, we obtain for each objectcinC a leftRaut…c†- moduleRcM :ˆM…c†. Given a leftRaut…c†-moduleN, we obtain a covariant RC-module EcN by

EcN…?†:ˆRmorC…c;?†n_Raut…c†N: …2:7†

Notice thatEcresp.Rcis induction resp. restriction with the obvious inclusion of categories aut…c† !d C. Hence Ec and Rc form an adjoint pair by (2.4). In particular we get for any covariantRC-moduleMan in Mnatural homomorphism

ic…M†:EcM…c† !M …2:8†

by the adjoint of id:RcM!RcM. Explicitly ic…M† maps …f:c!?†nm to M…f†…m†.

Given a covariant RC-module M, de®ne M…c†_s to be the R-submodule of M…c† which is spanned by the images of all R-maps M…f†:M…b† !M…c†, where f runs through all morphisms f:b!c with target c which are not isomorphisms in C. Obviously M…c†_s is anRaut…c†-submodule ofM…c†. De®ne a leftRaut…c†-moduleScM by

ScM :ˆM…c†=M…c†_s: …2:9†

We callC anEI-categoryif any endomorphism inC is an isomorphism. Notice that EcmapsRaut…c†toRmorC…c;?†. Provided thatCis an EI-category,

ScRmorC…d;?†G_Raut…c†Raut…c†; if cGd;

andScRmorC…d;?† ˆ0 otherwise. This implies for a freeRC-module M ˆL

iAIRmorC…ci;?†;

L

…c†AIs…C†EcScMGRCM;

where Is…C† is the set of isomorphism classes …c† of objects c in C. This splitting can be extended to projective modules as follows.

Let M be an RC-module. We want to check whether it is projective or not. Since Sc is compatible with direct sums and each projective module is a direct sum in a free RC-module, a necessary (but not su½cient) condition is thatScM is a projectiveRaut…c†- module. Assume thatScM is Raut…c†-projective for all objectsc inC. We can choose an Raut…c†-splittingsc:ScM !M…c†of the canonical projection

M…c† !ScM ˆM…c†=M…c†_s.

Then we obtain after a choice of representativescA…c† for any…c†AIs…C†a morphism of RC-modules

T: L

…c†AIs…C†EcScM ƒƒƒƒƒƒƒƒ!

…c†AIs…C†Ecsc

L

…c†AIs…C†EcM…c†ƒƒƒƒƒƒƒƒ!

…c†AIs…C† ic…M†

M;

…2:10†

whereic…M†has been introduced in (2.8).

The length l…c†ANWfygof an object cis the supremum over all natural numbers l for which there exists a sequence of morphismsc0!^f1 c1!^f2 c2!^f3 . . .!^fl cl such that no fi

is an isomorphism andcl ˆc. If each objectchas lengthl…c†<y, we say thatChas ®nite length.

Theorem 2.11. Let C be an EI-category of ®nite length. Let M be a covariant RC-module such that the Raut…c†-module ScM is projective for all objects c in C. Let sc:ScM!M…c†be an Raut…c†-section of the canonical projection M…c† !ScM.Then the map introduced in(2.10)

T: L

…c†AIs…C†EcScM!M

is surjective. It is bijective if and only if M is a projective RC-module.

Proof. We show by induction over the length l…d† that T…d† is surjective for any object d in C. For any object d and Raut…d†-module N there is an in N natural aut…d†- isomorphism N…d† !^G SdEdN which sends n to the class of …id:d!d†nn. If d1 and d2

are non-isomorphic objects in C, then Sd1Ed2N ˆ0. This implies that SdT is an iso- morphism for all objects dAC. Hence it su½ces for the proof of surjectivity of T…d† to show that each element ofM…d†_s is in the image ofT…d†. It is enough to verify this for an element of the form M…f†…x† for xAM…d⁰† and a morphism f:d⁰!d which is not an isomorphism inC. SinceC is an EI-category, l…d⁰†<l…d†. By induction hypothesis T…d⁰† is surjective and the claim follows.

Suppose that Tis injective. Then Tis an isomorphism of RC-modules. Its source is projective sinceEcsends projective Raut…c†-modules to projectiveRC-modules. Therefore M is projective. We will not need the other implication that for projectiveM the map T is bijective in this paper. Therefore we omit its proof but refer to [16], Theorem 3.39 and Corollary 9.40. r

Given a contravariantRC-moduleMand a leftRaut…c†-moduleN, there is a natural isomorphism

Mn_RCEcNGM…c†n_Raut…c†N: …2:12†

It is explicitly given by mn…f:c!?†nn7!M…f†…m†nn. It is due to the fact that tensor products are associative. For more details about modules over a category we refer to [16], Section 9A.

3. The associated Bredon homology theory

Given a (proper)G-homology theory resp. equivariant homology theory with values inR-modules, we can associate to it another (proper)G-homology theory resp. equivariant homology theory with values inR-modules called Bredon homology, which is much simpler.

The equivariant Chern character will identify this simpler proper homology theory with the given one.

Before we give the construction we have to organize the coe½cients of aG-homology theory H^G. The smallest building blocks of G-CW-complexes or G-spaces in general are the homogeneous spacesG=H. The book keeping of all the valuesH^G…G=H†is organized using the following two categories.

The orbit category Or…G† has as objects homogeneous spaces G=H and as mor-

phismsG-maps. Let Sub…G†be the category whose objects are subgroupsHofG. For two subgroups H and K of G denote by conhomG…H;K† the set of group homomorphisms f:H!K, for which there exists an element gAG with gHg^ÿ1HK such that f is given by conjugation with g, i.e. f ˆc…g†:H !K, h7!ghg^ÿ1. Notice that c…g† ˆc…g⁰† holds for two elementsg;g⁰ AG withgHg^ÿ1HK and g⁰H…g⁰†^ÿ1HK if and only if g^ÿ1g⁰ lies in the centralizerCGH ˆ fgAGjghˆhgfor allhAHgofHinG. The group of inner auto- morphisms of K acts on conhomG…H;K† from the left by composition. De®ne the set of morphisms

morSub…G†…H;K†:ˆInn…K†nconhomG…H;K†:

There is a natural projection pr: Or…G† !Sub…G† which sends a homogeneous spaceG=H toH. Given aG-map f:G=H !G=K, we can choose an element gAG with gHg^ÿ1HK and f…g⁰H† ˆg⁰g^ÿ1K. Then pr…f† is represented by c…g†:H !K. Notice that morSub…G†…H;K† can be identi®ed with the quotient morOr…G†…G=H;G=K†=CGH, where gACGH acts on morOr…G†…G=H;G=K† by composition with R_g^ÿ1:G=H !G=H, g⁰H 7!g⁰g^ÿ1H. We mention as illustration that for abelian G, morSub…G†…H;K† is empty if H is not a subgroup of K, and consists of precisely one element given by the inclusion H !K ifHis a subgroup inK.

Denote by Or…G;F†HOr…G† and Sub…G;F†HSub…G† the full subcategories, whose objects G=H and H are given by ®nite subgroups HHG. Both Or…G;F† and Sub…G;F†are EI-categories of ®nite length.

Given a proper G-homology theory H^G with values in R-modules we obtain for nAZa covariantROr…G;F†-module

H_n^G…G=?†: Or…G;F† !RÿMOD; G=H 7!H_n^G…G=H†:

…3:1†

Let …X;A† be a pair of proper G-CW-complexes. Then there is a canonical identi-

®cationX^H ˆmap…G=H;X†^G. Thus we obtain contravariant functors Or…G;F† !CW ÿPAIRS; G=H 7! …X^H;A^H†;

Sub…G;F† !CW ÿPAIRS; G=H 7!CGHn…X^H;A^H†;

where CW ÿPAIRS is the category of pairs of CW-complexes. Composing them with the covariant functor CW ÿPAIRS!RÿCHCOM sending …Z;B† to its cellular chain complex with coe½cients in R yields the contravariant ROr…G;F†-chain complex C^Or…G;F†…X;A† and the contravariant RSub…G;F†-chain complex C^Sub…G;F†…X;A†. Both chain complexes are free. Namely, if Xn is obtained from Xnÿ1WAn by attaching the equivariant cellsG=HiDⁿforiAIn, then

C_n^Or…G;F†…X;A†GL

iAIn

RmorOr…G;F†…G=?;G=Hi†;

…3:2†

C_n^Sub…G;F†…X;A†GL

iAIn

Rmor_Sub…G;F†…?;Hi†:

…3:3†

Given a covariantROr…G;F†-module M, the equivariant Bredon homology (see [4]) of a pair of properG-CW-complexes…X;A†with coe½cients inMis de®ned by

H_n^Or…G;F†…X;A;M†:ˆHnÿ

C^Or…G;F†…X;A†n_ROr…G;F†M : …3:4†

This is indeed a properG-homology theory. Hence we can assign to a properG-homology theory H^G another proper G-homology theory which we call the associated Bredon homology

BH_n^G…X;A†:ˆ L

p‡qˆnH_p^Or…G;F†ÿ

X;A;H_q^G…G=?†

: …3:5†

There is a canonical homomorphism ind_prC^Or…G;F† …X;A† !^G C^Sub…G;F†…X;A† which is bi- jective (see (2.6), (3.2), (3.3)). Given a covariantRSub…G;F†-moduleM, it induces using (2.5) a natural isomorphism

H_n^Or…G;F†…X;A;resprM† !^G Hnÿ

C^Sub…G;F†…X;A†n_RSub…G;F†M : …3:6†

This will allow to view modules over the category Sub…G;F† which is smaller than the orbit category and has nicer properties from the homological algebra point of view. In particular we will exploit the following elementary lemma.

Lemma 3.7. Suppose that the covariant RSub…G;F†-module M is ¯at, i.e. for any exact sequence0!N1!N2!N3!0of contravariant RSub…G;F†-modules the induced sequence of R-modules

0!N1n_RSub…G;F†M!N2n_RSub…G;F†M!N3n_RSub…G;F†M !0 is exact. Then the natural map

Hnÿ

C^Sub…G;F†…X;A†

n_RSub…G;F†M!^G Hnÿ

C^{Sub…G; F†}…X;A†n_RSub…G;F†M is bijective.

Suppose, we are given a proper equivariant homology theory H^? with values inR- modules. We get from (3.1) for each groupGandnAZa covariantRSub…G;F†-module

H_n^G…G=?†: Sub…G;F† !RÿMOD; H7!H_n^G…G=H†:

…3:8†

We have to show that forgACGHtheG-mapR_g^ÿ1:G=H !G=H,g⁰H !g⁰g^ÿ1H induces the identity on H_n^G…G=H†. This follows from Lemma 1.2. We will denote the covariant Or…G;F†-module obtained by restriction with pr: Or…G;F† !Sub…G;F† from the Sub…G;F†-moduleH_n^G…G=?†of (3.8) again byH_n^G…G=?†as introduced already in (3.1).

Next we show that the collection of the G-homology theories BH^G…X;A† de®ned in (3.5) inherits the structure of a proper equivariant homology theory. We have to specify the induction structure.

Let a:H !G be a group homomorphism and …X;A† be an H-CW-pair such that ker…a†acts freely on X. We only explain the case, where ais injective. In the general case one has to replace Fby the smaller familyF…X† of subgroups ofH which occur as sub- groups of isotropy groups ofX. Induction withayields a functor denoted in the same way

a: Or…H;F† !Or…G;F†; H=K 7!inda…H=K† ˆG=a…K†:

There is a natural isomorphism of Or…G;F†-chain complexes indaC^Or…H;F†…X;A† !^G C^Or…G;F†ÿ

inda…X;A†

and a natural isomorphism (see (2.5)) ÿindaC^Or…H;F†…X;A†

n_ROr…G;F†H_q^G…G=?† !^G C^Or…H;F†…X;A†n_ROr…H;F†ÿ

resaH_q^G…G=?†

:

The induction structure onH^?yields a natural equivalence of ROr…H;F†-modules H_q^H…H=?† !^G resaH_q^G…G=?†:

The last three maps can be composed to a chain isomorphism C^{Or…H; F†}…X;A†n_ROr…H;F†H_q^H…H=?† !^G Cÿ

inda…X;A†

n_ROr…G;F†H_q^G…G=?†;

which induces a natural isomorphism inda:H_p^Or…H;F†ÿ

X;A;H_q^H…H=?†

!^G H_p^Or…G;F†ÿ

inda…X;A†;H_q^G…G=?†

: Thus we obtain the required induction structure.

Remark 3.9. For anyG-homology theoryH^G with values inR-modules for a com- mutative ringRthere is an equivariant version of the Atiyah-Hirzebruch spectral sequence.

It converges toH_p‡q^G …X;A†and itsE²-term isE_p;q² ˆHp^Or…G†ÿ

X;A;H_q^G…G=?†

. If…X;A†is proper, theE²-term reduces toHp^Or…G;F†ÿ

X;A;H_q^G…G=?†

. Existence of a bijective equiv- ariant Chern character amounts to saying that this spectral sequence collapses completely for properG-CW-pairs…X;A†.

4. The construction of the equivariant Chern character

In this section we want to construct the equivariant Chern character. It is motivated by the following non-equivariant construction.

Example 4.1. Consider a (non-equivariant) homology theory H with values in R-modules for QHR. Then a (non-equivariant) Chern character for aCW-complex X is given by the following composite:

chn: L

p‡qˆnHpÿ

X;Hq…† _G

a

L

p‡qˆnHp…X;R†n_RHq…†

ƒƒƒƒƒƒƒƒƒ

p‡qˆn hurnid G

L

p‡qˆnp_p^s…X‡;†n_ZRn_RHq…†ƒƒƒƒƒƒ!

p‡qˆn Dp;q

Hn…X†:

Here the canonical map a is bijective, since any R-module is ¯at over Z because of the assumption QHR. The second bijective map comes from the Hurewicz homomorphism.

The map Dp;q is de®ned as follows. For an elementanbAp_p^s…X‡;†n_ZHq…† choose a representative f:S^p‡k!S^k5X‡ ofa. De®neDp;q…anb†to be the image ofbunder the composite

Hq…† !^s Hp‡q‡k…S^p‡k;† ƒƒƒƒƒ^Hp‡q‡k!^f† Hp‡q‡k…S^k5X‡;†ƒ!^sÿ1 Hp‡q…X†;

where sdenotes the suspension isomorphism. This map turns out to be a transformation of homology theories and induces an isomorphism for X ˆ . Hence it is a natural equi- valence of homology theories. This construction is due to Dold [7].

Let…X;A†be a properG-CW-pair. LetRbe a commutative ring withQHR. LetH^? be an equivariant homology theory with values inR-modules. LetG be a group. Consider a ®nite subgroupHHG. We want to construct anR-homomorphism

ch^G_p;q…X;A†…H†:Hpÿ

CGHn…X^H;A^H†;R

nRH_q^G…G=H† !H_p‡q^G …X;A†;

…4:2†

where Hpÿ

CGHn…X^H;A^H†;R

is the cellular homology of the CW-pair CGHn…X^H;A^H† withR-coe½cients. For (notational) simplicity we give the details only forAˆj. The map is de®ned by the following composite:

Hp…CGHnX^H;R†n_RH_q^G…G=H†

Hp…pr1;R†nRid

x?

??^G

Hp…EGCGHX^H;R†n_RH_q^G…G=H†

hur…EG_CGHX^H†nRindH^G

x?

??^G p_p^sÿ

…EGCGHX^H†_‡

n_ZRn_RH_q^H…†

Dp;q^H…EG_CGHX^H†

??

?y

H_p‡q^H …EGCGHX^H†

indpr:CGHH!H

x?

??^G

H_p‡q^CGHH…EGX^H†

ind_mH

??

?y^G

H_p‡q^G …indmHEGX^H†

Hp‡q^G…ind_mHpr2†

??

?y

H_p‡q^G …indmHX^H†

Hp‡q^G…vH†

??

?y H_p‡q^G …X†:

Some explanations are in order. We have a leftCGH-action onEGX^H by g…e;x† ˆ …eg^ÿ1;gx†

forgACGH, eAEG and xAX^H. The map pr₁:EGCGHX^H !CGHnX^H is the canon- ical projection. It induces an isomorphism

Hp…pr₁;R†:Hp…EGCGHX^H;R† !^G Hp…CGHnX^H;R†

by the following argument. Each isotropy group of theCGH-spaceX^H is ®nite. The pro- jection induces an isomorphism Hp…BL;R†GHp…;R† for pAZ and any ®nite group L because by assumption the order of L is invertible in R. Hence Hp…pr₁;R† is bijective if X^H ˆCGH=Lfor some ®niteLHCGH. Now apply the usual Mayer-Vietoris and colimit arguments.

For any space Y let hur…Y†:p_p^s…Y‡†n_ZR!Hp…Y;R† be the Hurewicz homo- morphism. It is bijective since QHR and therefore hur is a natural tranformation of (non-equivariant) homology theories which induces for the one-point space Y ˆ an isomorphismp_p^s…‡†n_ZRGHp…;R†for pAZ.

Given a spaceZ and a ®nite groupH, considerZas an H-space by the trivial action and de®ne a map

D_p;^H_q…Z†:p_p^s…Z‡†n_ZH_q^H…† ˆp_p^s…Z‡†n_ZRn_RH_q^H…† !H_p‡q^H …Z†

as follows. For an elementanbAp_p^s…Z‡†n_ZH_q^H…†choose a representative f:S^p‡k !S^k5Z‡

ofa. De®neD_p;q^H …Z†…anb†to be the image ofbunder the composite

H_q^H…† !^s H_p‡q‡k^H …S^p‡k;†ƒƒƒƒƒ!^H

p‡q‡kH …f†

H_p‡q‡k^H …S^k5Z‡;† ƒ^s!^ÿ1 H_p‡q^H …Z†;

where s denotes the suspension isomorphism. Notice thatH is ®nite so that anyH-CW- complex is proper.

The group homomorphism pr:CGHH!H is the obvious projection and the group homomorphism mH:CGHH!G sends …g;h† togh. Notice that theCGHH- action onEGX^H comes from the givenCGH-action and the trivialH-action and that the kernels of the two group homomorphisms above act freely onEGX^H. So the induction isomorphisms on homology for these group homomorphisms exists for theCGHH-space EGX^H.

We denote by pr₂:EGX^H !X^H the canonical projection. TheG-map