Chern characters for proper equivariant homology theories and applications to K- and L-theory

Wolfgang Lück^{∗} Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany

February 15, 2001

Abstract

We construct for an equivariant homology theory for proper equivariant CW -complexes an equivariant Chern character, provided that certain conditions are satisfied. This applies for instance to the sources of the assembly maps in the Farrell-Jones Conjecture with respect to the family F of finite subgroups and in the Baum-Connes Conjecture. Thus we get an explicit calculation in terms of group homology of ℚ ⊗_{ℤ} K_{n}(RG) and ℚ ⊗_{ℤ} L_{n}(RG) for a commutative ring R with ℚ ⊂ R, provided the Farrell-Jones Conjecture with respect to F is true, and of ℚ ⊗_{ℤ} K_{n}^{top}(C_{r}^{∗}(G, F)) for F = ℝ, ℂ, provided the Baum-Connes Conjecture is true. Key words: equivariant homology theory, Chern character, K- and L-groups. Mathematics Subject Classification 2000: 55N91, 19D50, 19G24, 46L80

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 ℚ ⊂ R. The Chern character identifies H_{n}^{G}(X) with the associated Bredon homology, which is much easier to handle and can often be simplified 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 finite 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 ℚ ⊗_{ℤ} K_{n}(RG) and ℚ ⊗_{ℤ} L_{n}(RG) of the group ring RG of a (discrete) group G with coefficients in R and the rationalized topological K-groups ℚ ⊗_{ℤ} K_{n}^{top}(C_{r}^{∗}(G, F)) of the reduced group C^{∗}-algebra of G over F = ℝ, 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 the K-groups and L-groups of the coefficient ring R or F (see Theorem 0.4 and Theorem 0.5).

∗email: wolfgang.lueck@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370

Throughout this paper all groups are discrete and R will denote a commutative associative ring with unit. A proper G-homology theory H_{∗}^{G} assigns to any G-CW -pair (X, A) which is proper, i.e. all isotropy groups are finite, a ℤ-graded R-module H_{∗}^{G}(X, A) such that G-homotopy invariance, excision and the disjoint union axiom hold and there is a long exact sequence of a proper G-CW -pair. An equivariant proper homology theory H_{∗}^{?} 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 oriented 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 theory H_{∗}^{?} we will construct in Section 3 another equivariant proper homology theory, the associated Bredon homology BH_{∗}^{?}. The point is that BH_{∗}^{?} is much easier to handle than H_{∗}^{?}. Although we will not deal with equivariant spectra in this paper, we mention that the equivariant Bredon homology BH_{∗}^{?} is given by a product of equivariant Eilenberg-MacLane spaces, whose homotopy groups are given by the collection of the R-modules H_{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_{∗}^{?} ⟶^{≅} H_{∗}^{?} in Section 4, provided that a certain technical assumption is fullfilled, namely, that the covariant RSub(G, F)-module H_{q}^{G}(G/?) ≅ H_{q}^{?}(∗) is flat for all q ∈ ℤ and all groups G. The construction of ch_{∗}^{G} for a given group G requires that H_{∗}^{?} is defined for all groups, not only for G. There are some favourite situations, where the technical assumption above is automatically satisfied, and the Bredon homology BH_{∗}^{?} can be computed further. Let FGINJ be the category of finite groups with injective group homomorphisms as morphisms. The equivariant homology theory defines a covariant functor H_{q}^{?}(∗) : FGINJ→ R − MOD which sends H 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 ℚ ⊂ R. Let H_{∗}^{?} be a proper equivariant homology theory with values in R-modules. Suppose that the covariant functor H_{q}^{?}(∗) : FGINJ → R − MOD extends to a Mackey functor for all q ∈ ℤ. Then there is an isomorphism of proper homology theories

ch_{∗}^{?} : BH_{∗}^{?} ⟶^{≅} H_{∗}^{?}.

Theorem 0.1 is the equivariant version of the well-known result (explained in Example 4.1) that for a (non-equivariant) homology theory H_{∗} with values in R-modules and a CW -pair (X, A) there are natural isomorphisms

⨁_{p+q=n} H_{p}(X, A; H^{q}(∗)) ≅ H_{n}(X, A).

The associated Bredon homology can be decomposed further. Define for a finite group H

S_{H} (H_{q}^{H}(∗)) := coker ( ⨁_{\begin{matrix}K⊂H \\ K≠H \end{matrix}} ind_{K}^{H} : ⨁_{\begin{matrix}K⊂H \\ K≠H \end{matrix}} H_{q}^{K}(∗) → H_{q}^{H}(∗)) .

For a subgroup H ⊂ G we denote by N_{G}H the normalizer and by C_{G}H the centralizer of H in G. Let H · C_{G}H be the subgroup of N_{G}H consisting of elements of the form hc for h ∈ H and c ∈ C_{G}H. Denote by W_{G}H the quotient N_{G}H/H · C_{G}H. Notice that W_{G}H is finite if H is finite.

Theorem 0.2 Consider the situation and assumptions of Theorem 0.1. Let I be the set of conjugacy classes (H) of finite 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) ≅ ⨁_{p+q=n} ⨁_{(H)∈I} H_{p}(C_{G}H\(X^{H}, A^{H}); R) ⊗_{R[W_{G}H]} S_{H} (H_{q}^{H}(∗)) .

Theorem 0.1 and Theorem 0.2 reduce the computation of H_{n}^{G}(X, A) to the computation of the singular or cellular homology R-modules H_{p}(C_{G}H\(X^{H}, A^{H}); R) of the CW -pairs C_{G}H\(X^{H}, A^{H}) including the obvious right W_{G}H-operation and of the left R[W_{G}H]-modules S_{H} (H_{q}^{H}(∗)) which only involve the values H_{q}^{G}(G/H) = H_{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 q ∈ ℤ and a preferred restriction structure on BH_{∗}^{?} so that Theorem 0.1 applies and the equivariant Chern character is compatible with these restriction structures. If H_{∗}^{?} comes with a multiplicative structure as explained in Section 6, then BH_{∗}^{?} 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 ℚ ⊗_{ℤ} R_{ℚ}(?) which assigns to a finite group H the rationalized ring of rational H-representations. This notion is explained in Section 7. In particular it yields for any finite group H the structure of a ℚ ⊗_{ℤ} R_{ℚ}(H)-module on H_{q}^{H}(∗). Let class_{ℚ}(H) be the ring of functions f : H → ℚ which satisfy f(h_{1}) = f(h_{2}) if the cyclic subgroups ⟨h_{1}⟩ and ⟨h_{2}⟩ generated by h_{1} and h_{2} are conjugate in H. Taking characters yields an isomorphism of rings

χ : ℚ ⊗_{ℤ} R_{ℚ}(H) ⟶^{≅} class_{ℚ}(H).

Given a finite cyclic group C, there is the idempotent θ_{C}^{C} ∈ class_{ℚ}(C) which assigns 1 to a generator of C and 0 to the other elements. This element acts on H_{q}^{C}(∗). The image im (θ_{C}^{C} : H_{q}^{C}(∗) → H_{q}^{C}(∗)) of the map given by multiplication with the idempotent θ_{C}^{C} is a direct summand in H_{q}^{C}(∗) and will be denoted by θ_{C}^{C} · H_{q}^{C}(∗).

Theorem 0.3 Let R be a commutative ring with ℚ ⊂ R. Let H_{∗}^{?} be a proper equivariant homology theory with values in R-modules. Suppose that the covariant functor FGINJ → R − MOD sending H to H_{q}^{H}(∗) extends to a Mackey functor for all q ∈ ℤ, which is a module over the Green functor ℚ⊗_{ℤ} R_{ℚ}(?) with respect to the inclusion ℚ → R. Let J be the set of conjugacy classes (C) of finite cyclic subgroups C of G. Then there is an isomorphism of proper homology theories

ch_{∗}^{?} : BH_{∗}^{?} ⟶^{≅} H_{∗}^{?}.

Moreover, for any group G and any proper G-CW -pair (X, A) there is a natural isomorphism

BH_{n}^{G}(X, A) = ⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C\(X^{C}, A^{C}); R) ⊗_{R[W_{G}C]} (θ_{C}^{C} · H_{q}^{C}(∗)) .

Since K_{q}(R?), L_{q}(R?) and K_{q}^{top}(C_{r}^{∗}(?, F)) are Mackey functors and come with module structures over the Green functor ℚ ⊗_{ℤ} R_{ℚ}(?) as explained in Section 8, Theorem 0.3 implies

Theorem 0.4 Let R be a commutative ring with ℚ ⊂ R. Denote by F the field ℝ or ℂ. Let G be a (discrete) group. Let J be the set of conjugacy classes (C) of finite cyclic subgroups C of G. Then the rationalized assembly map in the Farrell-Jones Conjecture with respect to F for the algebraic K-groups K_{n}(RG) and the algebraic L-groups L_{n}(RG) and in the Baum Connes Conjecture for the topological Kgroups K_{n}^{top}(C_{r}^{∗}(G, F)) can be identified with the homomorphisms

⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}(RC)) → ℚ ⊗_{ℤ} K_{n}(RG);
⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}^{C} · (ℚ ⊗_{ℤ} L_{q}(RC)) → ℚ ⊗_{ℤ} L_{n}(RG);
⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}^{top}(C_{r}^{∗}(C, F))) → ℚ ⊗_{ℤ} K_{q}^{top}(C_{r}^{∗}(G, F)).

In the L-theory case we assume that R comes with an involution R → R, r ↦ \overline{r} and that we use on RG the involution which sends ∑_{g∈G} r_{g} · g to ∑_{g∈G} \overline{r_{g}} · g^{−1}.

If the Farrell-Jones Conjecture with respect to F and 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 finite group H is rationally a linear combination 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 coefficient ring R is a field F of characteristic zero and we tensor with \overline{F} ⊗_{ℤ}? for an algebraic closure \overline{F} of F, one can simplify the expressions further as carried out in Section 8. As an illustration we record the following particular nice case.

Theorem 0.5 Let G be a (discrete) group. Let T be the set of conjugacy classes (g) of elements g ∈ G of finite order. There is a commutative diagram

⨁_{p+q=n} ⨁_{(g)∈T} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}(ℂ) ⟶ ℂ ⊗_{ℤ} K_{n}(ℂG)
↓  ↓
⨁_{p+q=n} ⨁_{(g)∈T} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}^{top}(ℂ) ⟶ ℂ ⊗_{ℤ} K_{n}^{top}(C_{r}^{∗}(G))

where C_{G}⟨g⟩ 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 K_{q}(ℂ) → K_{q}^{top}(ℂ) and K_{n}(ℂG) → K_{n}^{top}(C_{r}^{∗}(G)). The horizontal arrows can be identified with the assembly maps occuring in the Farrell-Jones Conjecture with respect to F for K_{n}(ℂG) and in the Baum-Connes Conjecture for K_{n}^{top}(C_{r}^{∗}(G)) after applying ℂ ⊗_{ℤ} −. 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 the K- and Ltheory of RG seems to split into one part, which involves only the group and consists essentially of group homology, and another part, which involves only the coefficient ring and consists essentially of its K-theory. Moreover, a change of rings or change of K-theory map involves only the coefficient 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 coefficients 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 different construction of an equivariant Chern character has been given in [2] in the case, where H_{∗}^{G} is equivariant K-homology after applying ℂ ⊗_{ℤ} −. Moreover, the lower horizontal arrow in Theorem 0.5 has already been discussed there.

The computations of K- and L-groups integrally and with R = ℤ as coefficients are much harder (see for instance [18]).

The paper is organized as follows

1. Equivariant homology theories
2. Modules over a category
3. The associated Bredon homology theory
4. The construction of the equivariant Chern character
5. Mackey functors
6. Restriction structures and multiplicative structures
7. Green functors
8. Applications to K- and L-theory

References

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 family F of finite subgroups for algebraic K- and L-theory and the equivariant K-homology theory which appears as the source of the Baum-Connes assembly map and is defined in terms of Kasparov’s equivariant KK-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 finite. Basic informations about G-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 of G-CW -pairs to the category of R-modules indexed by n ∈ ℤ together with natural tranformations ∂_{n}^{G}(X, A) : H_{n}^{G}(X, A) → H_{n−1}^{G}(A) := H_{n−1}^{G}(A, ∅) for n ∈ ℤ such that the following axioms are satisfied:

(a) G-homotopy invariance If f_{0} and f_{1} are G-homotopic maps (X, A) → (Y, B) of G-CW -pairs, then H_{n}^{G}(f_{0}) = H_{n}^{G}(f_{1}) for n ∈ ℤ;

(b) Long exact sequence of a pair Given a pair (X, A) of G-CW -complexes, there is a long exact sequence

… ⟶^{H_{n+1}^{G}(j)} H_{n+1}^{G}(X, A) ⟶^{∂_{n+1}^{G}} H_{n}^{G}(A) ⟶^{H_{n}^{G}(i)} H_{n}^{G}(X) ⟶^{H_{n}^{G}(j)} H_{n}^{G}(X, A) ⟶^{∂_{n}^{G}} …,

where i : 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 (X ∪_{f} B, B) with the induced structure of a G-CW -pair. Then the canonical map (F, f) : (X, A) → (X ∪_{f} B, B) induces an isomorphism

H_{n}^{G}(F, f) : H_{n}^{G}(X, A) ⟶^{≅} H_{n}^{G}(X ∪_{f} B, B);

(d) Disjoint union axiom Let {X_{i} | i ∈ I} be a family of G-CW -complexes. Denote by j_{i} : X_{i} → ∐_{i∈I} X_{i} the canonical inclusion. Then the map

⨁_{i∈I} H_{n}^{G}(j_{i}) : ⨁_{i∈I} H_{n}^{G}(X_{i}) ⟶^{≅} H_{n}^{G} (∐_{i∈I} X_{i})

is bijective.

If H_{∗}^{G} is defined or considered only for proper G-CW -pairs (X, A), we call it a proper G-homology theory H_{∗}^{G} with values in R-modules.

Let α : H → G be a group homomorphism. Given an H-space X, define the induction of X with α to be the G-space ind_{α} X which is the quotient of G × X by the right H-action (g, x) · h := (gα(h), h^{−1}x) for h ∈ H and (g, x) ∈ G × X. If α : H → G is an inclusion, we also write ind_{H}^{G} instead of ind_{α}.

A (proper) equivariant homology theory H_{∗}^{?} with values in R-modules consists of a (proper) G-homology theory H_{∗}^{G} with values in R-modules for each group G together with the following so called induction structure: given a group homomorphism α : H → G and a H-CW -pair (X, A) such that ker(α) acts freely on X, there are for each n ∈ ℤ natural isomorphisms

ind_{α} : H_{n}^{H}(X, A) ⟶^{≅} H_{n}^{G}(ind_{α}(X, A))  (1.1)

satisfying

(a) Compatibility with the boundary homomorphisms ∂_{n}^{G} ∘ ind_{α} = ind_{α} ∘∂_{n}^{H};

(b) Functoriality Let β : G → K be another group homomorphism such that ker(β ∘ α) acts freely on X. Then we have for n ∈ ℤ

ind_{β∘α} = H_{n}^{K}(f_{1}) ∘ ind_{β} ∘ ind_{α} : H_{n}^{H}(X, A) → H_{n}^{K}(ind_{β∘α}(X, A)),

where f_{1} : ind_{β} ind_{α}(X, A) ⟶^{≅} ind_{β∘α}(X, A), (k, g, x) ↦ (kβ(g), x) is the natural K-homeomor-phism;

(c) Compatibility with conjugation For n ∈ ℤ, g ∈ G and a (proper) G-CW -pair (X, A) the homomorphism ind_{c(g):G→G} : H_{n}^{G}(X, A) → H_{n}^{G}(ind_{c(g):G→G}(X, A)) agrees with H_{n}^{G}(f_{2}) for the G-homeomorphism f_{2} : (X, A) → ind_{c(g):G→G}(X, A) which sends x to (1, g^{−1}x) in G ×_{c(g)} (X, A).

This induction structure links the various homology theories for different 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 fixed group G. We will later need

Lemma 1.2 Consider finite subgroups H, K ⊂ G and an element g ∈ G with gHg^{−1} ⊂ K. Let R_{g^{−1}} : G/H → G/K be the G-map sending g′H to g′g^{−1}K and c(g) : H → K be the homomorphism sending h to ghg^{−1}. Let pr : (ind_{c(g):H→K} ∗) → ∗ be the projection. Then the following diagram commutes

H_{n}^{H}(∗) ⟶^{H_{n}^{K}(pr)∘ind_{c(g)}} H_{n}^{K}(∗)
ind_{H}^{G} ↓≅  ^{ind}_{K}^{G} ↓≅
H_{n}^{G}(G/H) ⟶^{H_{n}^{G}(R_{g^{−1}} )} H_{n}^{G}(G/K)

\underline{\underline{Proof}} : Define a bijective G-map f_{1} : ind_{c(g):G→G} ind_{H}^{G} ∗ → ind_{K}^{G} ind_{c(g):H→K} ∗ by sending (g_{1}, g_{2}, ∗) in G ×_{c(g)} G ×_{H} ∗ to (g_{1}gg_{2}g^{−1}, 1, ∗) in G ×_{K} K ×_{c(g)} ∗. The condition that induction is compatible with composition of group homomorphisms means precisely that the composite

H_{n}^{H}(∗) ⟶^{ind_{H}^{G}} H_{n}^{G}(ind_{H}^{G} ∗) ⟶^{ind_{c(g):G→G}} H_{n}^{G}(ind_{c(g):G→G} ind_{H}^{G} ∗) ⟶^{H_{n}^{G}(f_{1})} H_{n}^{G}(ind_{K}^{G} ind_{c(g):H→K} ∗)

agrees with the composite

H_{n}^{H}(∗) ⟶^{ind_{c(g):H→K}} H_{n}^{K}(ind_{c(g):H→K} ∗) ⟶^{ind_{K}^{G}} H_{n}^{G}(ind_{K}^{G} ind_{c(g):H→K} ∗).

Naturality of induction implies H_{n}^{G}(ind_{K}^{G} pr)∘ind_{K}^{G} = ind_{K}^{G} ∘H_{n}^{K}(pr). Hence the following diagram commutes

H_{n}^{H}(∗)  ⟶^{H_{n}^{K}(pr)∘ind_{c(g):H→K}}  H_{n}^{K}(∗)
ind_{H}^{G} ↓  ↓^{ind}_{K}^{G}

H_{n}^{G}(G/H)) ⟶ H_{n}^{G}(ind_{K}^{G} pr)∘H_{n}^{G}(f_{1})∘ind_{c(g):G→G} H_{n}^{G}(G/K)

By the axioms the homomorphism ind_{c(g):G→G} : H_{n}^{G}(G/H) → H_{n}^{G}(ind_{c(g):G→G} G/H) agrees with H_{n}^{G}(f_{2}) for the map f_{2} : G/H → ind_{c(g):G→G} G/H which sends g′H to (g′g^{−1}, 1H) in G ×_{c(g)} G/H. Since the composite (ind_{K}^{G} pr) ∘ f_{1} ∘ f_{2} is just R_{g^{−1}}, Lemma 1.2 follows.

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 topological K-homology. Then we obtain two equivariant homology theories with values in R-modules by the following constructions

H_{n}^{G}(X, A) = K_{n}(G\X, G\A);
H_{n}^{G}(X, A) = K_{n}(EG ×_{G} (X, A)).

The second one is called the equivariant Borel homology associated to K. In both cases H_{∗}^{G} inherits the structure of a G-homology theory from the homology structure on K_{∗}. Let a : H\X ⟶^{≅} G\(G ×_{α} X) be the homeomorphism sending Hx to G(1, x). Define b : EH ×_{H} X → EG ×_{G} G ×_{α} X by sending (e, x) to (Eα(e), 1, x) for e ∈ EH, x ∈ X and Eα : EH → EG the α-equivariant map induced by α. Induction for a group homomorphism α : H → G is induced by these maps a and b. If the kernel ker(α) acts freely on X, the map b is a homotopy equivalence and hence in both cases ind_{α} is bijective.

Example 1.4 Given a proper G-CW -pair (X, A), one can define the G-bordism group Ω_{n}^{G}(X, A) as the abelian group of G-bordism classes of maps f : (M, ∂M) → (X, A) whose sources are oriented smooth manifolds with orientation preserving proper smooth G-actions such that G\M is compact. The definition is analogous to the one in the non-equivariant case. This is also true for the proof that this defines a proper G-homology theory. There is an obvious induction structure coming from induction of equivariant spaces. It is well-defined because of the following fact. Let α : H → G be a group homomorphism. Let M be an oriented smooth H-manifold with orientation preserving proper smooth H-action such that H\M is compact and ker(α) acts freely. Then ind_{α} M is an oriented smooth G-manifold with orientation preserving proper smooth G-action such that G\M is compact. The boundary of ind_{α} M is ind_{α} ∂M.

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 ℚ ⊗_{ℤ} K_{n}(RG) or the rationalized algebraic L-group ℚ ⊗_{ℤ} L_{n}(RG) of the group ring RG or such that H_{n}^{G}(∗) is the rationalized topological K-theory ℚ ⊗_{ℤ} K_{n}^{top}(C_{r}^{∗}(G; ℝ)) or ℚ ⊗_{ℤ} K_{n}^{top}(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 family F of finite subgroups of G. This is a G-CW -complex whose H-fixed point set is contractible for H ∈ F and is empty otherwise. It is unique up to G-homotopy because it is characterized by the property that for any G-CW -complex X whose isotropy groups belong to F there is up to G-homotopy precisely one G-map from X to E(G, F). The G-space E(G, F) agrees with the classifying space \underline{E}G for proper G-actions. Define E(G; VC) for the family VC of virtually cyclic subgroups analogously. The assembly map in the Farrell-Jones Conjecture with respect to F and in the Baum-Connes Conjecture are the maps induced by the projection E(G, F) → ∗

H_{n}^{G}(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 to F is 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 projection E(G, VC) → ∗ induces an isomorphism (even without rationalizing)

H_{n}^{G}(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 for F is not true in general. In other words, the canonical map E(G, F) → E(G, VC) does not necessarily induce an isomorphism H_{n}^{G}(E(G, F)) → H_{n}^{G}(E(G, VC)). This is due to the existence of Nil-groups. However, if for instance R is a field of characteristic zero, this map is bijective for algebraic K-theory. Hence the Farrell-Jones Conjecture for ℚ ⊗_{ℤ} K_{n}(F G) for a field F of characteristic zero is true with respect to F if and only if it is true with respect to VC. At the time of writing not much is known about this conjecture for K_{n}(F G) for a field F of characteristic zero, since most of the known results are for the algebraic K-theory for ℤG. The situation in L-theory is better since the change of rings map ℚ ⊗_{ℤ} L_{n}(ℤG) → ℚ ⊗_{ℤ} L_{n}(ℚG) is bijective for any group G. The Farrell-Jones Conjecture for both ℚ⊗_{ℤ} L_{n}(ℤG) and ℚ⊗_{ℤ} L_{n}(ℚG) is true with respect to both F and VC if G is a cocompact discrete subgroup of a Lie group with finitely many path components [9], if G is a discrete subgroup of GL_{n}(ℂG) [10], or if G is an elementary amenable group [11].

The target of the assembly map for F in (1.6) is ℚ ⊗_{ℤ} K_{n}(RG), ℚ ⊗_{ℤ} L_{n}(RG) or ℚ ⊗_{ℤ} K_{n}^{top}(C_{r}^{∗}(G, F)) for F = ℝ, ℂ. These are the groups we would like to compute. The source of the assembly map for F in (1.6) is the part which is better accessible for computations. 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 rationalized K- and L-groups, provided the Farrell-Jones Conjecture with respect to F and 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 definition 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 R − MOD of R-modules. Define a contravariant RC-module analogously. Morphisms of RC-modules are natural transformations. Given a group G, let Ĝ be the category with one object whose set of morphisms is given by G. Then a covariant RĜ-module is the same as a left RG-module, whereas a contravariant RĜ-module is the same as a right RG-module. All the constructions, which we will introduce for RC-modules below, reduce in the special case C = Ĝ under the identification above to their classical versions for RG-modules. The reader should have this example in mind.

The category RC − MOD of (covariant or contravariant) RC-modules inherits the structure of an abelian category from R − MOD in the obvious way, namely objectwise. For instance a sequence 0 → M → N → P → 0 of RC-modules is called exact if its evaluation at each object in C is an exact sequence in R − MOD. The notion of a projective RC-module is now clear. Given a family B = (c_{i})_{i∈I} of objects of C, the free RC-module with basis B is

RC(B) := ⨁_{i∈I} R mor_{C}(c_{i},?).

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

hom_{RC}(RC(B), N) ⟶^{≅} ∏_{i∈I} N(c_{i}),  f ↦ (f(c_{i})(id_{c_{i}}))_{i∈I}.  (2.1)

Obviously RC(B) is a projective RC-module. Any RC-module is a quotient of some free RC-module. For instance, there is an obvious epimorphism from RC(B) to M if we take B to be the family of objects indexed by ∐_{c∈Ob(C)} M(c), where we assign c to m ∈ M(c). Therefore a 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 defines their tensor product over RC to be the following R-module M ⊗_{RC} N. It is given by

M ⊗_{RC} N = ⨁_{c∈Ob(C)} M(c) ⊗_{R} N(c)/ ∼,

where ∼ is the typical tensor relation mf ⊗ n = m ⊗ fn, i.e. for each morphism f : c → d in C, m ∈ M(d) and n ∈ N(c) we introduce the relation M(f)(m) ⊗ n − m ⊗ N(f)(n) = 0. The main property of this construction is that it is adjoint to the hom_{R}-functor in the sense that for any R-module L there are natural isomorphisms of R-modules

hom_{R}(M ⊗_{RC} N, L) ⟶^{≅} hom_{RC}(M, hom_{R}(N, L));  (2.2)
hom_{R}(M ⊗_{RC} N, L) ⟶^{≅} hom_{RC}(N, hom_{R}(M, L)).  (2.3)

Consider a functor F : C → D. Given a covariant or contravariant RD-module M, define its restriction with F to be res_{F} M := M ∘ F. Given a covariant RC-module M, its induction with F is the covariant RD-module ind_{F} M given by

(ind_{F} M)(??) := R mor_{D}(F (?),??) ⊗_{RC} M(?).

Given a contravariant RC-module M, its induction with F is the contravariant RD-module ind_{F} M given by

(ind_{F} M)(??) := M(?) ⊗_{RC} R mor_{D}(??, F (?)).

Restriction with F can be written in the covariant case as res_{F} N(?) = hom_{RD}(R mor_{D}(F (?),??), N(??)) and in the contravariant case as res_{F} N(?) = hom_{RD}(R mor_{D}(??, 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 of R-modules

hom_{RD}(ind_{F} M, N) ⟶^{≅} hom_{RC}(M, res_{F} N).  (2.4)

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

(ind_{F} M) ⊗_{RD} N ⟶^{≅} M ⊗_{RC} (res_{F} N).  (2.5)

It is explicitly given by (f :?? → F (?)) ⊗ m ⊗ n ↦ m ⊗ N(f)(n) or can be obtained formally from (2.2) and (2.4). One easily checks

ind_{F} R mor_{C}(c,?) = R mor_{D}(F (c),??)  (2.6)

for c ∈ Ob(C). This shows that ind_{F} respects direct sums and the properties free and projective.

Next we explain how one can reduce the study of projective RC-modules to the study of projective R aut(c)-modules, where aut(c) is the group of automorphism of an object c in C. Given a covariant RC-module M, we obtain for each object c in C a left R aut(c)-module R_{c}M := M(c). Given a left R aut(c)- module N, we obtain a covariant RC-module E_{c}N by

E_{c}N(?) := R mor_{C}(c,?) ⊗_{R aut(c)} N.  (2.7)

Notice that E_{c} resp. R_{c} is induction resp. restriction with the obvious inclusion of categories { \atop aut(c)} → C. Hence E_{c} and R_{c} form an adjoint pair by (2.4). In particular we get for any covariant RC-module M an in M natural homomorphism

i_{c}(M) : E_{c}M(c) → M  (2.8)

by the adjoint of id : R_{c}M → R_{c}M. Explicitly i_{c}(M) maps (f : c →?) ⊗ m to M(f)(m). Given a covariant RC-module M, define M(c)_{s} to be the R-submodule of M(c) which is spanned by the images of all Rmaps 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 an R aut(c)-submodule of M(c). Define a left R aut(c)-module S_{c}M by

S_{c}M := M(c)/M(c)_{s}.  (2.9)

We call C an EI-category if any endomorphism in C is an isomorphism. Notice that E_{c} maps R aut(c) to R mor_{C}(c,?). Provided that C is an EI-category, S_{c}R mor_{C}(d,?) ≅_{R aut(c)} R aut(c), if c ≅ d, and S_{c}R mor_{C}(d,?) = 0 otherwise. This implies for a free RC-module M = ⨁_{i∈I} R mor_{C}(c_{i},?)

⨁_{(c)∈Is(C)} E_{c}S_{c}M ≅_{RC} M,

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 S_{c} is compatible with direct sums and each projective module is a direct sum in a free RC-module, a necessary (but not sufficient) condition is that S_{c}M is a projective R aut(c)-module. Assume that S_{c}M is R aut(c)-projective for all objects c in C. We can choose a R aut(c)-splitting σ_{c} : S_{c}M → M(c) of the canonical projection M(c) → S_{c}M = M(c)/M(c)_{s}. Then we obtain after a choice of representatives c ∈ (c) for any (c) ∈ Is(C) a morphism of RC-modules

T : ⨁_{(c)∈Is(C)} E_{c}S_{c}M \underrightarrow{⨁_{(c)∈Is(C)} E_{c}σ_{c}} ⨁_{(c)∈Is(C)} E_{c}M(c) \underrightarrow{⨁_{(c)∈Is(C)} i_{c}(M)} M,  (2.10)

where i_{c}(M) has been introduced in (2.8).

The length l(c) ∈ ℕ ∪ {∞} of an object c is the supremum over all natural numbers l for which there exists a sequence of morphisms c_{0} ⟶^{f_{1}} c_{1} ⟶^{f_{2}} c_{2} ⟶^{f_{3}} … ⟶^{f_{l}} c_{l} such that no f_{i} is an isomorphism and c_{l} = c. If each object c has length l(c) < ∞, we say that C has finite length.

Theorem 2.11 Let C be an EI-category of finite length. Let M be a covariant RC-module such that the R aut(c)-module S_{c}M is projective for all objects c in C. Let σ_{c} : S_{c}M → M(c) be an R aut(c)-section of the canonical projection M(c) → S_{c}M. Then the map introduced in (2.10)

T : ⨁_{(c)∈Is(C)} E_{c}S_{c}M → M

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

\underline{\underline{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 R aut(d)-module N there is an in N natural aut(d)-isomorphism N(d) ⟶^{≅} S_{d}E_{d}N which sends n to the class of (id : d → d) ⊗ n. If d_{1} and d_{2} are non-isomorphic objects in C, then S_{d_{1}}E_{d_{2}}N = 0. This implies that S_{d}T is an isomorphism for all objects d ∈ C. Hence it suffices for the proof of surjectivity of T (d) to show that each element of M(d)_{s} is in the image of T (d). It is enough to verify this for an element of the form M(f)(x) for x ∈ M(d′) and a morphism f : d′ → d which is not an isomorphism in C. Since C is an EI-category, l(d′) < l(d). By induction hypothesis T (d′) is surjective and the claim follows.

Suppose that T is injective. Then T is an isomorphism of RC-modules. Its source is projective since E_{c} sends projective R aut(c)-modules to projective RC-modules. Therefore M is projective. We will not need the other implication that for projective M the map T is bijective in this paper. Therefore we omit its proof but refer to [16, Theorem 3.39 and Corollary 9.40].

Given a contravariant RC-module M and a left R aut(c)-module N, there is a natural isomorphism

M ⊗_{RC} E_{c}N ≅ M(c) ⊗_{R aut(c)} N.  (2.12)

It is explicitly given by m ⊗ (f : c →?) ⊗ n ↦ M(f)(m) ⊗ n. 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 in R-modules, we can associate to it another (proper) G-homology theory resp. equivariant homology theory with values in Rmodules 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 coefficients of a G-homology theory H_{∗}^{G}. The smallest building blocks of G-CW -complexes or G-spaces in general are the homogeneous spaces G/H. The book keeping of all the values H_{∗}^{G}(G/H) is organized using the following two categories.

The orbit category Or(G) has as objects homogeneous spaces G/H and as morphisms G-maps. Let Sub(G) be the category whose objects are subgroups H of G. For two subgroups H and K of G denote by conhom_{G}(H, K) the set of group homomorphisms f : H → K, for which there exists an element g ∈ G with gHg^{−1} ⊂ K such that f is given by conjugation with g, i.e. f = c(g) : H → K, h ↦ ghg^{−1}. Notice that c(g) = c(g′) holds for two elements g, g′ ∈ G with gHg^{−1} ⊂ K and g′H(g′)^{−1} ⊂ K if and only if g^{−1}g′ lies in the centralizer C_{G}H = {g ∈ G | gh = hg for all h ∈ H} of H in G. The group of inner automorphisms of K acts on conhom_{G}(H, K) from the left by composition. Define the set of morphisms

mor_{Sub (G)}(H, K) := Inn(K)\ conhom_{G}(H, K).

There is a natural projection pr : Or(G) → Sub(G) which sends a homogeneous space G/H to H. Given a G-map f : G/H → G/K, we can choose an element g ∈ G with gHg^{−1} ⊂ K and f(g′H) = g′g^{−1}K. Then pr(f) is represented by c(g) : H → K. Notice that mor_{Sub (G)}(H, K) can be identified with the quotient mor_{Or (G)}(G/H, G/K)/C_{G}H, where g ∈ C_{G}H acts on mor_{Or (G)}(G/H, G/K) by composition with R_{g^{−1}} : G/H → G/H, g′H ↦ g′g^{−1}H. We mention as illustration that for abelian G mor_{Sub (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 if H is a subgroup in K.

Denote by Or(G, F) ⊂ Or(G) and Sub(G, F) ⊂ Sub(G) the full subcategories, whose objects G/H and H are given by finite subgroups H ⊂ G. Both Or(G, F) and Sub(G, F) are EI-categories of finite length.

Given a proper G-homology theory H_{∗}^{G} with values in R-modules we obtain for n ∈ ℤ a covariant ROr(G, F)-module

H_{n}^{G}(G/?) : Or(G, F) → R − MOD, G/H ↦ H_{n}^{G}(G/H).  (3.1)

Let (X, A) be a pair of proper G-CW -complexes. Then there is a canonical identification X^{H} = map(G/H, X)^{G}. Thus we obtain contravariant functors

Or(G, F) → CW − PAIRS,  G/H ↦ (X^{H}, A^{H});
Sub(G, F) → CW − PAIRS,  G/H ↦ C_{G}H\(X^{H}, A^{H}),

where CW − PAIRSis the category of pairs of CW -complexes. Composing them with the covariant functor CW − PAIRS→ R − CHCOM sending (Z, B) to its cellular chain complex with coefficients 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 X_{n} is obtained from X_{n−1} ∪ A_{n} by attaching the equivariant cells G/H_{i} × D^{n} for i ∈ I_{n}, then

C_{n}^{Or (G,F)}(X, A) ≅ ⨁_{i∈I_{n}} R mor_{Or (G,F)}(G/?, G/H_{i});  (3.2)
C_{n}^{Sub (G,F)}(X, A) ≅ ⨁_{i∈I_{n}} R mor_{Sub (G,F)}(?, H_{i}).  (3.3)

Given a covariant ROr(G, F)-module M, the equivariant Bredon homology (see [4]) of a pair of proper G-CW -complexes (X, A) with coefficients in M is defined by

H_{n}^{Or (G,F)}(X, A; M) := H_{n}(C_{∗}^{Or (G,F)}(X, A) ⊗_{ROr (G,F)} M).  (3.4)

This is indeed a proper G-homology theory. Hence we can assign to a proper G-homology theory H_{∗}^{G} another proper G-homology theory which we call the associated Bredon homology

BH_{n}^{G}(X, A) := ⨁_{p+q=n} H_{p}^{Or (G,F)}(X, A; H_{q}^{G}(G/?)).  (3.5)

There is a canonical homomorphism ind_{pr} C_{∗}^{Or (G,F)}(X, A) ⟶^{≅} C_{∗}^{Sub (G,F)}(X, A) which is bijective (see (2.6), (3.2), (3.3)). Given a covariant RSub(G, F)-module M, it induces using (2.5) natural isomorphisms

H_{n}^{Or (G,F)}(X, A; res_{pr} M) ⟶^{≅} H_{n}(C_{∗}^{Sub (G,F)}(X, A) ⊗_{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 flat, i.e. for any exact sequence 0 → N_{1} → N_{2} → N_{3} → 0 of contravariant RSub(G, F)-modules the induced sequence of R-modules 0 → N_{1} ⊗_{RSub (G,F)} M → N_{2} ⊗_{RSub (G,F)} M → N_{3} ⊗_{RSub (G,F)} M → 0 is exact. Then the natural map

H_{n} (C_{∗}^{Sub (G,F)}(X, A)) ⊗_{RSub (G,F)} M ⟶^{≅} H_{n} (C_{∗}^{Sub (G,F)}(X, A) ⊗_{RSub (G,F)} M)

is bijective.

Suppose, we are given a proper equivariant homology theory H_{∗}^{?} with values in R-modules. We get from (3.1) for each group G and n ∈ ℤ a covariant RSub(G, F)-module

H_{n}^{G}(G/?) : Sub(G, F) → R − MOD, H ↦ H_{n}^{G}(G/H).  (3.8)

We have to show that for g ∈ C_{G}H the G-map R_{g^{−1}} : G/H → G/H, g′H → g′g^{−1}H 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)-module H_{n}^{G}(G/?) of (3.8) again by H_{n}^{G}(G/?) as introduced already in (3.1).

Next we show that the collection of the G-homology theories BH_{∗}^{G}(X, A) defined in (3.5) inherits the structure of a proper equivariant homology theory. We have to specify the induction structure.

Let α : H → G be a group homomorphism and (X, A) be a H-CW -pair such that ker(α) acts freely on X. We only explain the case, where α is injective. In the general case one has to replace F by the smaller family F(X) of subgroups of H which occur as subgroups of isotropy groups of X. Induction with α yields a functor denoted in the same way

α : Or(H, F) → Or(G, F), H/K ↦ ind_{α}(H/K) = G/α(K).

There is a natural isomorphism of Or(G, F)-chain complexes

ind_{α} C_{∗}^{Or (H,F)}(X, A) ⟶^{≅} C_{∗}^{Or (G,F)}(ind_{α}(X, A))

and a natural isomorphism (see (2.5))

(ind_{α} C_{∗}^{Or (H,F)}(X, A)) ⊗_{ROr (G,F)} H_{q}^{G}(G/?) ⟶^{≅} C_{∗}^{Or (H,F)}(X, A) ⊗_{ROr (H,F)} (res_{α} H_{q}^{G}(G/?)) .

The induction structure on H_{∗}^{?} yields a natural equivalence of ROr(H, F)-modules

H_{q}^{H}(H/?) ⟶^{≅} res_{α} H_{q}^{G}(G/?).

The last three maps can be composed to a chain isomorphism

C_{∗}^{Or (H,F)}(X, A) ⊗_{ROr (H,F)} H_{q}^{H}(H/?) ⟶^{≅} C_{∗}(ind_{α}(X, A)) ⊗_{ROr (G,F)} H_{q}^{G}(G/?),

which induces a natural isomorphism

ind_{α} : H_{p}^{Or (H,F)}(X, A; H_{q}^{H}(H/?)) ⟶^{≅} H_{p}^{Or (G,F)}(ind_{α}(X, A); H_{q}^{G}(G/?)).

Thus we obtain the required induction structure.

Remark 3.9 For any G-homology theory H_{∗}^{G} with values in R-modules for a commutative ring R there is an equivariant version of the Atiyah-Hirzebruch spectral sequence. It converges to H_{p+q}^{G}(X, A) and its E^{2}-term is E_{p,q}^{2} = H_{p}^{Or (G)}(X, A; H_{q}^{G}(G/?)). If (X, A) is proper, the E^{2}-term reduces to H_{p}^{Or (G,F)}(X, A; H_{q}^{G}(G/?)). Existence of a bijective equivariant Chern character amounts to saying that this spectral sequence collapses completely for proper G-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 ℚ ⊂ R. Then a (non-equivariant) Chern character for a CW -complex X is given by the following composite

\begin{aligned}ch_{n} : ⨁_{p+q=n} H_{p}(X; H_{q}(∗)) & ⟵_{α}^{≅} & ⨁_{p+q=n} H_{p}(X; R) ⊗_{R} H_{q}(∗) \\ & ⨁_{p+q=n} hur ⊗ id ; ⟵ ; ≅ & ⨁_{p+q=n} π_{p}^{s}(X_{+}, ∗) ⊗_{ℤ} R ⊗_{R} H_{q}(∗) & \underrightarrow{⨁_{p+q=n} D_{p,q}} & H_{n}(X). \end{aligned}

Here the canonical map α is bijective, since any R-module is flat over ℤ because of the assumption ℚ ⊂ R. The second bijective map comes from the Hurewicz homomorphism. The map D_{p,q} is defined as follows. For an element a ⊗ b ∈ π_{p}^{s}(X_{+}, ∗) ⊗_{ℤ} H_{q}(∗) choose a representative f : S^{p+k} → S^{k} ∧ X_{+} of a. Define D_{p,q}(a ⊗ b) to be the image of b under the composite

H_{q}(∗) ⟶^{σ} H_{p+q+k}(S^{p+k}, ∗) ⟶^{H_{p+q+k}(f)} H_{p+q+k}(S^{k} ∧ X_{+}, ∗) ⟶^{σ^{−1}} H_{p+q}(X),

where σ denotes the suspension isomorphism. This map turns out to be a transformation of homology theories and induces an isomorphism for X = ∗. Hence it is a natural equivalence of homology theories. This construction is due to Dold [7].

Let (X, A) be a proper G-CW -pair. Let R be a commutative ring with ℚ ⊂ R. Let H_{∗}^{?} be an equivariant homology theory with values in R-modules. Let G be a group. Consider a finite subgroup H ⊂ G. We want to construct an R-homomorphism

\underline{ch}_{p,q}^{G}(X, A)(H) : H_{p}(C_{G}H\(X^{H}, A^{H}); R) ⊗_{R} H_{q}^{G}(G/H) → H_{p+q}^{G}(X, A),  (4.2)

where H_{p}(C_{G}H\(X^{H}, A^{H}); R) is the cellular homology of the CW -pair C_{G}H\(X^{H}, A^{H}) with R-coefficients. For (notational) simplicity we give the details only for A = ∅. The map is defined by the following composite

H_{p}(C_{G}H\X^{H}; R) ⊗_{R} H_{q}^{G}(G/H)
H_{p}(pr_{1};R)⊗_{R}id↑≅
H_{p}(EG ×_{C_{G}H} X^{H}; R) ⊗_{R} H_{q}^{G}(G/H)
hur(EG×_{C_{G}H}X^{H})⊗_{R}ind_{H}^{G} ↑≅
π_{p}^{s}((EG ×_{C_{G}H} X^{H})_{+}) ⊗_{ℤ} R ⊗_{R} H_{q}^{H}(∗)
D_{p,q}^{H}(EG×_{C_{G}H}X^{H})↓
H_{p+q}^{H}(EG ×_{C_{G}H} X^{H})
ind^{pr:C_{G}H×H→H} ↑≅
H_{p+q}^{C_{G}H×H}(EG × X^{H})
ind^{m_{H}} ↓≅
H_{p+q}^{G}(ind_{m_{H}} EG × X^{H})
H_{p+q}^{G}(ind_{m_{H}} pr_{2})↓
H_{p+q}^{G}(ind_{m_{H}} X^{H})
H_{p+q}^{G}(v_{H})↓
H_{p+q}^{G}(X)

Some explanations are in order. We have a left C_{G}H-action on EG × X^{H} by g(e, x) = (eg^{−1}, gx) for g ∈ C_{G}H, e ∈ EG and x ∈ X^{H}. The map pr_{1} : EG ×_{C_{G}H} X^{H} → C_{G}H\X^{H} is the canonical projection. It induces an isomorphism

H_{p}(pr_{1}; R) : H_{p}(EG ×_{C_{G}H} X^{H}; R) ⟶^{≅} H_{p}(C_{G}H\X^{H}; R)

by the following argument. Each isotropy group of the C_{G}H-space X^{H} is finite. The projection induces an isomorphism H_{p}(BL; R) ≅ H_{p}(∗; R) for p ∈ ℤ and any finite group L because by assumption the order of L is invertible in R. Hence H_{p}(pr_{1}; R) is bijective if X^{H} = C_{G}H/L for some finite L ⊂ C_{G}H. Now apply the usual Mayer-Vietoris and colimit arguments.

For any space Y let hur(Y) : π_{p}^{s}(Y_{+}) ⊗_{ℤ} R → H_{p}(Y; R) be the Hurewicz homomorphism. It is bijective since ℚ ⊂ R and therefore hur is a natural tranformation of (non-equivariant) homology theories which induces for the one-point space Y = ∗ an isomorphism π_{p}^{s}(∗_{+}) ⊗_{ℤ} R ≅ H_{p}(∗; R) for p ∈ ℤ.

Given a space Z and a finite group H, consider Z as an H-space by the trivial action and define a map

D_{p,q}^{H}(Z) : π_{p}^{s}(Z_{+}) ⊗_{ℤ} H_{q}^{H}(∗) = π_{p}^{s}(Z_{+}) ⊗_{ℤ} R ⊗_{R} H_{q}^{H}(∗) → H_{p+q}^{H}(Z)

as follows. For an element a ⊗ b ∈ π_{p}^{s}(Z_{+}) ⊗_{ℤ} H_{q}^{H}(∗) choose a representative f : S^{p+k} → S^{k} ∧ Z_{+} of a. Define D_{p,q}^{H}(Z)(a ⊗ b) to be the image of b under the composite

H_{q}^{H}(∗) ⟶^{σ} H_{p+q+k}^{H}(S^{p+k}, ∗) ⟶^{H_{p+q+k}^{H}(f)} H_{p+q+k}^{H}(S^{k} ∧ Z_{+}, ∗) ⟶^{σ^{−1}} H_{p+q}^{H}(Z),

where σ denotes the suspension isomorphism. Notice that H is finite so that any H-CW -complex is proper.

The group homomorphism pr : C_{G}H × H → H is the obvious projection and the group homomorphism m_{H} : C_{G}H × H → G sends (g, h) to gh. Notice that the C_{G}H × H-action on EG × X^{H} comes from the given C_{G}H-action and the trivial H-action and that the kernels of the two group homomorphisms above act freely on EG×X^{H}. So the induction isomorphisms on homology for these group homomorphisms exists for the C_{G}H × H-space EG × X^{H}.

We denote by pr_{2} : EG × X^{H} → X^{H} the canonical projection. The G-map v_{H} : ind_{m_{H}} X^{H} = G ×_{m_{H}} X^{H} → X sends (g, x) to gx.

Lemma 4.3 Let G be a group and let X be a proper G-CW -complex. Then

(a) The map \underline{ch}_{p,q}^{G}(X)(H) is natural in X;

(b) Consider H, K ⊂ G and g ∈ G with gHg^{−1} ⊂ K. Let L_{g^{−1}} : X^{K} → X^{H} and \overline{L_{g^{−1}}} : C_{G}K\X^{K} → C_{G}H\X^{H} be the map induced by left multiplication with g^{−1}. Let R_{g^{−1}} : G/H → G/K be given by right multiplication with g^{−1}. Then the following square commutes

H_{p}(C_{G}K\X^{K}; R) ⊗_{R} H_{q}^{G}(G/H) ⟶^{H_{p}(\overline{L_{g^{−1}}} ;R)⊗_{R}id} H_{p}(C_{G}H\X^{H}; R) ⊗_{R} H_{q}^{G}(G/H)
id ⊗_{R}H_{q}^{G}(R_{g^{−1}} )↓  ↓^{\underline{ch}_{p,q}^{G}(X)(H)}
H_{p}(C_{G}K\X^{K}; R) ⊗_{R} H_{q}^{G}(G/K) ⟶_{\underline{ch}_{p,q}^{G}(X)(K)}  H_{p+q}^{G}(X)

(c) Consider a G-map f : G/H → X. Let u ∈ π_{0}(C_{G}H\X^{H}) ⊂ H_{0}(C_{G}H\X^{H}; R) be the element represented by f(eH). Then the map

H_{q}^{G}(G/H) → H_{q}^{G}(X), v ↦ \underline{ch}_{0,q}^{G}(X)(H)(u ⊗_{R} v)

agrees with the map H_{q}^{G}(f).

\underline{\underline{Proof}} : (a) is obvious.

(b) Since gHg^{−1} ⊂ K we can define a group homomorphism c(g^{−1}) : C_{G}K → C_{G}H by mapping g′ to g^{−1}g′g. The map

R_{g} × L_{g^{−1}} : EG × X^{K} → EG × X^{H}, (e, x) ↦ (eg, g^{−1}x)

is (c(g^{−1}) : C_{G}K → C_{G}H) equivariant with respect to the C_{G}K-action on EG × X^{K} given by g′ · (e, x) = (eg′^{−1}, g′x) and the analogous C_{G}H-action on EG × X^{H}. It induces a map

\overline{R_{g} × L_{g^{−1}}} : EG ×_{C_{G}K} X^{K} → EG ×_{C_{G}H} X^{H}.

If we extend the C_{G}H-action on EG × X^{H} and the C_{G}K-action on EG × X^{K} to a C_{G}H × H-action and a C_{G}K × H-action in the trivial way, we also get C_{G}H × H-maps

\begin{matrix}{ \atop R_{g} × L_{g^{−1}}} : ind_{c(g^{−1}×id):C_{G}K×H→C_{G}H×H} EG × X^{K} = (C_{G}H × H) ×_{c(g^{−1})×id} EG × X^{K} & → & EG × X^{H} \\ (c, h, e, x) & ↦ & (egc^{−1}, cg^{−1}x) \end{matrix}

and

{ \atop L_{g^{−1}}} : ind_{c(g^{−1})×id:C_{G}H×H→C_{G}H×H} X^{K} = (C_{G}H × H) ×_{c(g^{−1})×id} X^{K} → X^{H}, (c, h, x) ↦ (cg^{−1}x).

In the sequel the maps p_{i} denote the canonical projections. They are of the form Y × K/gHg^{−1} → Y. The maps f_{i} denote canonical equivariant homeomorphisms which describe the natural identifications of ind_{β∘α} Z with ind_{β} ind_{α} Z. One easily checks using the axioms of an induction structure that the following three diagrams commute:

H_{p}(C_{G}K\X^{K}; R)  ⟶^{H_{p}(\overline{L_{g^{−1}}} ;R)}  H_{p}(C_{G}H\X^{H}; R)
≅↑^{H_{p}(pr_{1};R)}  ^{H_{p}(pr_{1};R)}↑≅
H_{p}(EG ×_{C_{G}K} X^{K}; R) ⟶^{H_{p}(\overline{R_{g}×L_{g^{−1}}} ;R)} H_{p}(EG ×_{C_{G}H} X^{H}; R)
≅↑^{hur(EG×_{C_{G}K}X^{K})}  ^{hur(EG×_{C_{G}H}X^{H})}↑≅
π_{p}^{s}((EG ×_{C_{G}K} X^{K})_{+}) ⊗_{ℤ} R ⟶^{π_{p}^{s}(\overline{R_{g}×L_{g^{−1}}} )} π_{p}^{s}((EG ×_{C_{G}H} X^{H})_{+}) ⊗_{ℤ} R

and

π_{p}^{s}((EG ×_{C_{G}K} X^{K})_{+}) ⊗_{ℤ} H_{q}^{K}(∗) ⟵^{id ⊗H_{q}^{K}(p_{1})∘ind_{c(g):H→K}} π_{p}^{s}((EG ×_{C_{G}K} X^{K})_{+}) ⊗_{ℤ} H_{q}^{H}(∗)
↓^{D_{p,q}^{K}}  ^{D_{p,q}^{H}} ↓
H_{p+q}^{K}(EG ×_{C_{G}K} X^{K})  ⟵^{H_{q}^{K}(p_{2})∘ind_{c(g):H→K}}  H_{p+q}^{H}(EG ×_{C_{G}K} X^{K})
≅↑^{ind}^{pr:C_{G}K×K→K}  ^{ind}^{pr:C_{G}K×H→H} ↑≅
H_{p+q}^{C_{G}K×K}(EG × X^{K})  ⟵^{H_{q}^{K}(p_{3})∘ind_{id ×c(g)}}  H_{p+q}^{C_{G}K×H}(EG × X^{K})
↓^{ind}^{m_{K}}  ^{ind}^{m_{K} ∘id ×c(g)}↓
H_{p+q}^{G}(ind_{m_{K}} EG × X^{K})  ⟵^{H_{p+q}^{G}(ind_{m_{K}} p_{3})∘H_{p+q}^{G}(f_{1})} H_{p+q}^{G}(ind_{m_{K}∘id ×c(g)} EG × X^{K})
↓^{H_{p+q}^{G}(ind_{m_{K}} pr_{2})}  ^{H_{p+q}^{G}(ind_{m_{K} ∘id ×c(g)} pr_{2})}↓
H_{p+q}^{G}(ind_{m_{K}} X^{K})  ⟵^{H_{p+q}^{G}(ind_{m_{K}} p_{4})∘H_{p+q}^{G}(f_{2})}  H_{p+q}^{G}(ind_{m_{K}∘id ×c(g)} X^{K})

and

π_{p}^{s}((EG ×_{C_{G}K} X^{K})_{+}) ⊗_{R} H_{q}^{H}(∗)  ⟶^{π_{p}^{s}(\overline{R_{g}×L_{g^{−1}}} )⊗id}  π_{p}^{s}((EG ×_{C_{G}K} X^{H})_{+}) ⊗_{R} H_{q}^{H}(∗)
↓^{D_{p,q}^{H}}  ^{D_{p,q}^{H}} ↓
H_{p+q}^{H}(EG ×_{C_{G}K} X^{K})  ⟶^{H_{p+q}^{G}(\overline{R_{g}×L_{g^{−1}}} )}  H_{p+q}^{H}(EG ×_{C_{G}K} X^{H})
≅↑^{ind}^{pr:C_{G}K×H→H}  ^{ind}^{pr:C_{G}H×H→H} ↑≅
H_{p+q}^{C_{G}K×H}(EG × X^{K})  ⟶^{H_{p+q}^{G}({ \atop R_{g}×L_{g^{−1}}} )∘ind_{c(g^{−1})×id}}  H_{p+q}^{C_{G}H×H}(EG × X^{H})
↓^{ind}^{m_{K} ∘id ×c(g)}  ^{ind}^{m_{H}} ↓
H_{p+q}^{G}(ind_{m_{K}∘id ×c(g)} EG × X^{K}) ⟶^{H_{p+q}^{G}(ind_{m_{H}} { \atop R_{g}×L_{g^{−1}}} )∘H_{p+q}^{G}(f_{3})∘ind_{c(g^{−1})}}  H_{p+q}^{G}(ind_{m_{H}} EG × X^{H})
↓^{H_{p+q}^{G}(ind_{m_{K} ∘id ×c(g)} pr_{2})}  ^{H_{p+q}^{G}(ind_{m_{H}} pr_{2})}↓
H_{p+q}^{G}(ind_{m_{K}∘id ×c(g)} X^{K})  ⟶^{H_{p+q}^{G}(ind_{m_{H}} { \atop L_{g^{−1}}})∘H_{p+q}^{G}(f_{4})∘ind_{c(g^{−1})}}  H_{p+q}^{G}(ind_{m_{H}} X^{H})
↓^{H_{p+q}^{G}(ind_{m_{K}} p_{4})∘H_{p+q}^{G}(f_{2})}  ^{H_{p+q}^{G}(v_{H})}↓
H_{p+q}^{G}(ind_{m_{K}} X^{K})  ⟶^{H_{p+q}^{G}(v_{K})}  H_{p+q}^{G}(X)

Now assertion (b) follows from an easy diagram chase in the three commutative diagrams above and Lemma 1.2.

(c) Its proof is similar to the one of (b) but much easier and hence left to the reader. This finishes the proof of Lemma 4.3.

Theorem 4.4 Let R be a commutative ring with ℚ ⊂ R holds. Let H_{∗}^{?} be a proper equivariant homology theory with values in R-modules. Suppose for every group G that the RSub(G, F)-module H_{q}^{G}(G/?) is flat for all q ∈ ℤ. Then there is an isomorphism, called equivariant Chern character, of proper equivariant homology theories

ch_{∗}^{?} : BH_{∗}^{?} ⟶^{≅} H_{∗}^{?},

i.e. for every group G and any proper G-CW -pair (X, A) there is an in (X, A) natural isomorphism

ch_{n}^{G}(X, A) : ⨁_{p+q=n} H_{p}^{Or (G,F)}(X, A; H_{q}^{G}(G/?)) ⟶^{≅} H_{n}^{G}(X, A)

such that the obvious compatibility conditions for the boundary homomorphisms of pairs and the induction structures hold.

\underline{\underline{Proof}} : We get for a pair of proper G-CW -complexes (X, A) from the collection of the homomorphisms of (4.2), the identification (3.6), Lemma 3.7 and Lemma 4.3 (a) and (b) (which holds for pairs (X, A) also) a natural R-homomorphism

ch_{p,q}^{G}(X, A) : H_{p}^{Or (G,F)}(X, A; H_{q}^{G}(G/?)) ≅ H_{p}(C_{∗}^{Sub (G,F)}(X, A)) ⊗_{RSub (G,F)} H_{q}^{G}(G/?) → H_{p+q}^{G}(X).

Taking their direct sum for p + q = n yields an in (X, A)-natural homomorphism

ch_{n}^{G}(X, A) : BH_{n}^{G}(X, A) → H_{n}^{G}(X).  (4.5)

One easily checks that ch_{∗}^{G} : BH_{∗}^{G} → H_{∗}^{G} is a transformation of G-homology theories. Essentially one has to check that it is compatible with the boundary maps in the long exact sequences of pairs.

Next we show that ch_{∗}^{G} is a natural equivalence, i.e. ch_{n}^{G}(X, A) is bijective for all n ∈ ℤ and all proper G-CW -pairs (X, A). The disjoint union axiom implies that both G-homology theories are compatible with colimits over directed systems indexed by the natural numbers such as the system given by the skeletal filtration X_{0} ⊂ X_{1} ⊂ X_{2} … ∪_{n≥0} X_{n} = X. The argument for this claim is analogous to the one in [24, 7.53]. Hence it suffices to prove the bijectivity of ch_{n}^{G}(X, A) for finite-dimensional pairs. By excision, the exact sequence of pairs, the disjoint union axiom and the five-lemma one reduces the proof of the bijectivity of ch_{n}^{G}(X, A) to the special case (X, A) = (G/H, ∅) for finite H ⊂ G. In this case the bijectivity follows from the consequence of Lemma 4.3 (c) that ch_{n}^{G}(G/H) is the identity under the obvious identification of its source with H_{n}^{G}(G/H) coming from (3.2).

Example 4.6 Given a homology theory K with values in R-modules for ℚ ⊂ R, we can associate to it an equivariant homology theory H_{∗}^{?} in two ways as explained in Example 1.3. There is an obvious equivariant Chern character coming from the non-equivariant one of Remark 4.1. Our general construction reduces to it by the following elementary observation. For any finite group H the natural map K_{q}(BH) → K_{q}(∗) is an isomorphism by the Atiyah-Hirzebruch spectral sequence since H_{p}(BH; ℚ) → H_{q}(∗; ℚ) is bijective. Hence in both cases the RSub(G, F)-module H_{q}^{G}(G/?) = H_{q}^{?}(∗) is constant with value K_{q}(∗). Therefore it is isomorphic to ℚ mor_{Sub (G,F)}(1,?) ⊗_{ℚ} K_{q}(∗) which is obviously a projective RSub(G, F)-module. By (2.12) the source of our equivariant Chern character reduces in this special case to

⨁_{p+q=n} H_{p}^{Or (G,F)}(X, A; H_{q}^{G}(G/?)) ≅ ⨁_{p+q=n} H_{p}(G\(X, A); K_{q}(∗)).

Remark 4.7 Let H_{G}^{∗} be an equivariant proper cohomology theory with values in F -modules for a field F of characteristic zero. It is defined axiomatically in the obvious way analogous to the definition of an equivariant homology theory. Suppose that H_{H}^{n}(∗) is a finite-dimensional F -vector space for all finite groups H and n ∈ ℤ. Put H_{n}^{G}(X, A) := hom_{F} (H_{G}^{n}(X, A), F). This defines an equivariant homology theory for proper finite G-CW -pairs (X, A). We can rediscover H_{G}^{n}(X, A) by hom_{F} (H_{n}^{G}(X, A), F) for proper finite G-CW -pairs (X, A). If one obtains a bijective Chern character for H_{∗}^{G} for proper finite G-CW -pairs, dualizing yields a bijective Chern character from H_{G}^{∗} to the associated equivariant Bredon cohomology for proper finite G-CW -pairs.

This applies for instance to equivariant K-cohomology after tensoring with ℚ over ℤ. Equivariant Chern characters for equivariant K-cohomology have been constructed for K_{G}^{∗}(X) ⊗_{ℤ} ℂ in [2] and for K_{G}^{∗}(X) ⊗_{ℤ} ℚ in [17]. Our construction of an equivariant Chern character for proper equivariant homology theories is motivated by [17].

5. Mackey functors

In order to apply Theorem 4.4, we have to check the flatness condition about the RSub(G, F)-module H_{q}^{G}(G/?). We will see that the existence of a Mackey structure will guarantee that it is projective and hence flat. This would not work if we would consider H_{q}^{G}(G/?) over the orbit category. Recall that we can consider it over Sub(G, F) because of Lemma 1.2 which is a consequence of the induction structure. The desired Mackey structures do exist in all relevant examples.

Let R be an associative commutative ring with unit. Let FGINJbe the category of finite groups with injective group homomorphisms as morphisms. Let M : FGINJ → R − MOD be a bifunctor, i.e. a pair (M_{∗}, M^{∗}) consisting of a covariant functor M_{∗} and a contravariant functor M^{∗} from FGINJto R − MOD which agree on objects. We will often denote for an injective group homomorphism f : H → G the map M_{∗}(f) : M(H) → M(G) by ind_{f} and the map M^{∗}(f) : M(G) → M(H) by res_{f} and write ind_{H}^{G} = ind_{f} and res_{G}^{H} = res_{f} if f is an inclusion of groups. We call such a bifunctor M a Mackey functor with values in R-modules if

(a) For an inner automorphism c(g) : G → G we have M_{∗}(c(g)) = id : M(G) → M(G);

(b) For an isomorphism of groups f : G ⟶^{≅} H the composites res_{f} ∘ ind_{f} and ind_{f} ∘ res_{f} are the identity;

(c) Double coset formula We have for two subgroups H, K ⊂ G

res_{G}^{K} ∘ ind_{H}^{G} = ∑_{KgH∈K\G/H} ind_{c(g):H∩g^{−1}Kg→K} ∘ res_{H}^{H∩g^{−1}Kg},

where c(g) is conjugation with g, i.e. c(g)(h) = ghg^{−1}.

Our main examples of Mackey functors will be R_{ℚ}(H), K_{q}(RH), L_{q}(RH) and K_{q}^{top}(C_{∗}^{r}(H, F)). Recall that for a subgroup H ⊂ G we denote by N_{G}H and C_{G}H the normalizer and the centralizer of H in G and by W_{G}H the quotient N_{G}H/H · C_{G}H. In the sequel we will use the identification W_{G}H ≅ aut_{Sub (G,F)}(H) which sends the class of n ∈ N_{G}H to the class of c(n) : H → H. We have introduced S_{H}P = P (H)/P (H)_{s} for a covariant Sub(G, F)-module P in (2.9). Notice for the sequel that

P (H)_{s} = im ( ⨁_{\begin{matrix}K⊂H \\ K≠H \end{matrix}} ind_{K}^{H} : ⨁_{\begin{matrix}K⊂H \\ K≠H \end{matrix}} P (K) → P (H)) .  (5.1)

Given a left R[W_{G}H]-module Q, we have defined the covariant RSub(G, F)-module E_{H}Q in (2.7). Recall that (H) has two meanings, namely, the set of subgroups of G which are conjugate to H and the isomorphism class of objects in Sub(G, F). One easily checks that these two interpretations give the same.

Theorem 5.2 Let R be a commutative ring with ℚ ⊂ R. Let M be a Mackey functor with values in R-modules. It induces a covariant RSub(G, F)-module denoted in the same way

M : Sub(G, F) → R − MOD, (f : H → K) ↦ (M_{∗}(f) : M(H) → M(K)) .

Each R[W_{G}H]-module S_{H}M is projective. For any finite subgroup H ⊂ G choose a section σ_{H} : S_{H}M → M(H) of the canonical projection M(H) → S_{H}M. Put I = Is(Sub(G, F)). Then the homomorphism defined in (2.10)

T : ⨁_{(H)∈I} E_{H} ∘ S_{H}M → M

is an isomorphism and the RSub(G, F)-module M is projective and hence flat.

\underline{\underline{Proof}} : Since W_{G}H is finite, any R[W_{G}H]-module is projective. Because of Theorem 2.11 it suffices to show for any finite subgroup K ⊂ G that T (K) is injective. Consider an element u in the kernel of T (K). Put J(H) = mor_{Sub (G,F)}(H, K)/(W_{G}H). Choose for any (H) ∈ I a representative H ∈ (H). Then fix for any element \overline{f} ∈ J(H) a representative f : H → K in mor_{Sub (G,F)}(H, K). We can find elements x_{H,f} ∈ S_{H}M for (H) ∈ I and \overline{f} ∈ J(H) such that only finitely many are different from zero and u can be written as

u = ∑_{(H)∈I} ∑_{\overline{f}∈J(H)}(f : H → K) ⊗_{R[W_{G}H]} x_{H,f}.

We want to show that all elements x_{H,f} are zero. Suppose that this is not the case. Let (H_{0}) be maximal among those elements (H) ∈ I for which there is \overline{f} ∈ J(H) with x_{H,f} ≠ 0, i.e. if for (H) ∈ I the element x_{H,f} is different from zero for some morphism f : H → K in Sub(G, F) and there is a morphism H_{0} → H in Sub(G, F), then (H_{0}) = (H). In the sequel we choose for any of the morphisms f : H → K in Sub(G, F) a group homomorphism denoted in the same way f : H → K representing it. Recall that f : H → K is given by conjugation with an appropriate element g ∈ G. Fix f_{0} : H_{0} → K with x_{H_{0}.f_{0}} ≠ 0. We claim that the composite

A : ⨁_{(H)∈I} E_{H} ∘ S_{H}M(K) ⟶^{T (K)} M(K) ⟶^{res_{K}^{im(f_{0})}} M(im(f_{0})) ⟶^{ind_{f{−1 \atop 0}:im(f_{0})→H_{0}}} M(H_{0}) ⟶^{pr_{H_{0}}} S_{H_{0}}M

maps u to m · x_{H_{0},f_{0}} for some integer m > 0. This would lead to a contradiction because of T (K)(u) = 0 and x_{H_{0},f_{0}} ≠ 0.

Consider (H) ∈ I and \overline{f} ∈ J(H). It suffices to show that A ((f : H → K) ⊗_{R[W_{G}H]} x_{H,f}) is [K ∩ N_{G} im(f_{0}) : im(f_{0})] · x_{H,f} if (H) = (H_{0}) and \overline{f} = \overline{f_{0}}, and is zero otherwise. One easily checks that A((f : H → K) ⊗_{R[W_{G}H]} x_{H,f}) is the image of x_{H,f} under the composite

a(H, f) : S_{H}M ⟶^{σ_{H}} M(H) ⟶^{ind_{f:H→im(f)}} M(im(f)) ⟶^{ind_{im(f)}^{K}} M(K) ⟶^{res_{K}^{im(f_{0})}} M(im(f_{0}))

⟶^{ind_{f{−1 \atop 0}:im(f_{0})→H_{0}}} M(H_{0}) ⟶^{pr_{H_{0}}} S_{H_{0}}M.

The Double Coset formula implies

res_{K}^{im(f_{0})} ∘ ind_{im(f)}^{K} = ∑_{k∈im(f_{0})\K/ im(f)} ind_{c(k):im(f)∩k^{−1} im(f_{0})k→im(f_{0})} ∘ res_{im(f)}^{im(f)∩k^{−1} im(f_{0})k} .

The composites pr_{H_{0}} ∘ ind_{f_{0}^{−1}:im(f_{0})→H_{0}} ∘ ind_{c(k):im(f)∩k^{−1} im(f_{0})k→im(f_{0})} is trivial, if c(k) : im(f) ∩ k^{−1} im(f_{0})k → im(f_{0}) is not an isomorphism. Suppose that c(k) : im(f) ∩ k^{−1} im(f_{0})k → im(f_{0}) is an isomorphism. Then k^{−1} im(f_{0})k ⊂ im(f). Since H_{0} has been choosen maximal among the H for which x_{H,f} ≠ 0 for some morphism f : H → K, this implies x_{H,f} = 0 or that k^{−1} im(f_{0})k = im(f). Suppose k^{−1} im(f_{0})k = im(f). Then (H) = (H_{0}) which implies H = H_{0}. Moreover, the homomorphisms in Sub(G, F) represented by f_{0} and f agree. Hence the group homomorphisms f_{0} and f agree themselves and we get k ∈ N_{G} im(f_{0}) ∩ K. This implies that a(H, f) = [K ∩ N_{G} im(f_{0}) : im(f_{0})] · id if (H) = (H_{0}) and \overline{f} = \overline{f_{0}}, and that otherwise a(H, f) = 0 or x_{H,f} = 0 holds. Hence the map T is injective. This finishes the proof of Theorem 5.2.

Now Theorem 0.1 and Theorem 0.2 follow from Theorem 4.4 and Theorem 5.2 using (2.12).

6. Restriction structures and multiplicative structures

Before we simplify the source of the equivariant Chern character further in the presence of a module structure over the Green functor ℚ ⊗_{ℤ} R_{ℚ}(?) on H_{q}^{?}(∗) in Section 7, we introduce an additional structure on an equivariant homology theory called restriction structure. It will guarantee that the Mackey structure appearing in Theorem 0.1 and and Theorem 0.2 exists. This restriction structure is canonically given in all relevant examples. We also briefly deal with multiplicative structures. The material of this section is not needed for the following sections.

A restriction structure on an equivariant homology theory H_{∗}^{?} consists of the following data. For any injective group homomorphism α : H → G, whose image has finite index in G, we require in (X, A) natural homomorphisms

res_{α} : H_{n}^{G}(X, A) → H_{n}^{H}(res_{α}(X, A)),

where (X, A) is a pair of G-CW -complexes and res_{α}(X, A) is the H-CW -pair obtained from (X, A) by restriction with α. If α is an inclusion of a subgroup H ⊂ G, we also write res_{G}^{H} instead of res_{α}. We require

(a) Compatibility with the boundary homomorphisms The restriction homomorphism res_{α} is compatible with the boundary homomorphism δ_{n}^{G} and δ_{n}^{H};

(b) Functoriality If β : G → K is another injective group homomorphism whose image has finite index in K, then res_{β∘α} = res_{α} ∘ res_{β};

(c) Compatibility of induction and restriction for isomorphisms

If α : H ⟶^{≅} G is an isomorphism of groups, then the composite

H_{n}^{G}(X) ⟶^{res_{α}} H_{n}^{H}(res_{α} X) ⟶^{ind_{α}} H_{n}^{G}(ind_{α} res_{α} X) ⟶^{T (X)} H_{n}^{G}(X)

is the identity, where T (X) : ind_{α} res_{α} X → X is the canonical G-homeomorphism;

(d) Double Coset formula Let H, K ⊂ G be subgroups such that K has finite index in G. Let (X, A) be a H-CW -pair. (Notice for the sequel that K\G/H is finite.) Denote by

f : ∐_{KgH∈K\G/H} ind_{c(g):H∩g^{−1}Kg→K} res_{H}^{H∩g^{−1}Kg}(X, A) ⟶^{≅} res_{G}^{K} ind_{H}^{G}(X, A)

the canonical K-homeomorphism.

Then the following two composites agree for all q ∈ ℤ

H_{q}^{H}(X) \underrightarrow{∏_{KgH∈K\G/H} ind_{c(g):H∩g^{−1}Kg→K} ∘ res_{H}^{H∩g^{−1}Kg}}

∏_{KgH∈K\G/H} H_{q}^{K} (ind_{c(g):H∩g^{−1}Kg→K} res_{H}^{H∩g^{−1}Kg}(X, A))

⟶^{≅} H_{q}^{K} ( ∐_{KgH∈K\G/H} ind_{c(g):H∩g^{−1}Kg→K} res_{H}^{H∩g^{−1}Kg}(X, A)) ⟶^{H_{q}^{K}(f)} H_{q}^{K}(res_{G}^{K} ind_{H}^{G}(X, A))

and

res_{G}^{K} ∘ ind_{H}^{G} : H_{q}^{H}(X, A) → H_{q}^{K}(res_{G}^{K} ind_{H}^{G}(X, A)).

If H_{∗}^{?} is an equivariant homology theory with a restriction structure, BH_{∗}^{?} inherits a restriction structure as follows. For K ⊂ H we get a natural map H/K → res_{α} ind_{α} H/K as the adjoint of the identity on ind_{α} H/K. It induces H_{q}^{H}(H/K) → H_{q}^{H}(res_{α} ind_{α} H/K). We get a ROr(G, F)-module H_{q}^{H}(res_{α} G/?) which assigns to G/K the R-module H_{q}^{H}(res_{α} G/K). Thus we obtain a transformation of covariant ROr(H, F)- modules H_{q}^{H}(H/?) → res_{α} H_{q}^{H}(res_{α} G/?). Its adjoint is a map of ROr(G, F)-modules

i_{q} : ind_{α} H_{q}^{H}(H/?) ⟶^{≅} H_{q}^{H}(res_{α} G/?),

which turns out to be bijective. This can be seen from its more explicit description as the composites of isomorphisms

ind_{α} H_{q}^{H}(H/?) = R mor_{Or (G,F)}(ind_{α} H/?, G/??) ⊗_{ROr (H,F)} H_{q}^{H}(H/?)

⟶^{μ} R mor_{Or (H,F)}(H/?, res_{α} G/??) ⊗_{ROr (H,F)} H_{q}^{H}(H/?) ⟶^{ν} H_{q}^{G}(G/??),

where μ comes from the adjunction of ind_{α} and res_{α} and ν sends (f : H/? → res_{α} G/??) ⊗ x to H_{q}^{H}(f)(x). The restriction structure on H_{∗}^{?} induces a map of Or(G, F)-modules

H_{q}^{G}(G/?) → H_{q}^{H}(res_{α} G/?).

There is a natural isomorphism of ROr(H, F)-chain complexes

C_{∗}^{Or (H,F)}(res_{α}(X, A)) ⟶^{≅} res_{α} C_{∗}^{Or (G,F)}(X, A).

There is a natural isomorphism of R-modules (compare (2.5))

(res_{α} C_{∗}^{Or (G,F)}(X, A))) ⊗_{ROr (H,F)} H_{q}^{H}(H/?) ⟶^{≅} C_{∗}^{Or (G,F)}(X, A) ⊗_{ROr (G,F)} (ind_{α} H_{q}^{H}(H/?)) .

The last four maps together can be combined to a map of R-chain complexes

C_{∗}^{Or (G,F)}(X, A) ⊗_{ROr (G,F)} H_{q}^{G}(G/?) → C_{∗}^{Or (H,F)}(res_{α}(X, A)) ⊗_{ROr (H,F)} H_{q}^{H}(H/?).

It induces on homology homomorphisms

res_{α} : H_{p}^{Or (G,F)}(X, A; H_{q}^{G}(X, A)) → H_{p}^{Or (H,F)}(res_{α}(X, A); H_{q}^{H}(H/?)).

Their direct sum yields the desired natural homomorphism

res_{α} : BH^{G}(X, A) → BH^{H}(res_{α}(X, A)).

We leave it to the reader to check that the axioms of a restriction structure are fullfilled.

Next we introduce multiplicative structures. An external product on H_{∗}^{G} assigns to any two groups G and G′ and pairs of (proper) G-CW -complexes (X, A) and G′-CW -complexes (X′, A′) an in (X, A) and (X′, A′) natural homomorphism

× : H_{n}^{G}(X, A) ⊗_{R} H_{n′}^{G′}(X′, A′) → H_{n+n′}^{G×G′}((X, A) × (X′, A′)),  (6.1)

where (X, A)×(X′, A′) is the pair of (proper) G×G′-CW -complexes (X ×X′, X ×A′ ∪A×X′). We mention that we work in the category of compactly generated spaces (see [25], [26, I.4]) so that (X, A) × (X′, A′) is indeed a (proper) G × G′-CW -pair. These parings are required to be compatible with the boundary homomorphisms, namely, for u ∈ H_{p}^{G}(X, A) and v ∈ H_{q}^{G}(Y, B) we have

∂(u × v) = ∂(u) × v + (−1)^{p} · u × ∂(v).

We also assume that these pairings are compatible with induction, i.e. for group homomorphisms α : H → G and α′ : H′ → G′ and u ∈ H_{p}^{H}(X, A) and u′ ∈ H_{q}^{H′}(Y, B) we require

H_{p+p′}^{G×G′}(f) (ind_{α}(u) × ind_{α′}(u′)) = ind_{α×α′}(u × v)

for f : ind_{α}(X, A) × ind_{α′}(X′, A′) ⟶^{≅} ind_{α×α′} ((X, A) × (X′, A′)) the canonical G × G′-homeomorphism. Furthermore we require that the external product × is associative, graded commutative and has a unit element 1 in H_{0}^{{1}}(∗).

If H_{∗}^{?} comes with an external product, we call it a multiplicative (proper) equivariant homology theory with values in R-modules. If H_{∗}^{?} comes with a restriction structure, we will require that the multiplicative structure and restriction structure are compatible. Namely, for injective group homomorphisms α : H → G and α′ : H′ → G′, whose images have finite index, and u ∈ H_{p}^{G}(X, A) and u′ ∈ H_{q}^{G′}(Y, B) we require

res_{α}(u) × res_{α′}(u′) = res_{α×α′}(u × u′).

Next we explain how a multiplicative structure on H_{∗}^{?} induces a multiplicative structure on the associated Bredon homology BH_{∗}^{?}. Let (X, A) be a proper G-CW -pair and let (X′, A′) be a proper G′-CW -pair. Let C_{∗}(X, A) ⊗_{R} C_{∗}(X′, A′) be the obvious ROr(G, F) × Or(G′, F)-chain complex. Denote by I : Or(G, F) × Or(G′, F) → Or(G × G′, F) the functor sending (G/H, G′/H′) to G × G′/H × H′. There is a natural isomorphism of Or(G × H, F)-chain complexes

ind_{I} (C_{∗}^{Or (G,F)}(X, A) ⊗_{R} C_{∗}^{Or (G′,F)}(X′, A′)) ⟶^{≅} C_{∗}^{Or (G×G′,F)}((X, A) × (X′, A′)),

which comes from the adjunction (2.4) and the natural isomorphism of the cellular chain complex of a product of two (non-equivariant) CW -complexes with the tensor product of the individual cellular chain complexes. The multiplicative structure on H_{∗}^{?} induces a natural transformation of ROr(G, F) × Or(H, F)- modules

H_{p}^{G}(G/?) ⊗_{R} H_{q}^{G′}(G′/?′) → res_{I} H_{p+q}^{G×G′}(G × G′/??).

There are natural isomorphisms of R-chain complexes

(C_{∗}^{Or (G,F)}(X, A) ⊗_{ROr (G,F)} H_{p}^{G}(G/?)) ⊗_{R} (C_{∗}^{Or (G′,F)}(X′, A′) ⊗_{ROr (G′,F)} H_{p}^{G′}(G′/?′))

⟶^{≅} (C_{∗}^{Or (G,F)}(X, A) ⊗_{R} C_{∗}^{Or (G′,F)}(X′, A′)) ⊗_{ROr (G,F)×Or (G′,F)} (H_{p}^{G}(G/?) ⊗_{R} H_{p}^{G′}(G′/?′))

and (see (2.5))

ind_{I} (C_{∗}^{Or (G,F)}(X, A) ⊗_{R} C_{∗}^{Or (G′,F)}(X′, A′)) ⊗_{ROr (G×G′,F)} H_{p+q}^{G×G′}(G × G′/??)

⟶^{≅} (C_{∗}^{Or (G,F)}(X, A) ⊗_{R} C_{∗}^{Or (G′,F)}(X′, A′))) ⊗_{ROr (G,F)×Or (G′,F)} (res_{I} H_{p+q}^{G×G′}(G × G′/??)) .

Combining the last four maps yields a chain map

(C_{∗}^{Or (G,F)}(X, A) ⊗_{ROr (G,F)} H_{p}^{G}(G/?)) ⊗_{R} (C_{∗}^{Or (G′,F)}(X′, A′) ⊗_{ROr (G′,F)} H_{p}^{G′}(G′/?′))
→ C_{∗}^{Or (G×G′,F)}((X, A) × (X′, A′)) ⊗_{ROr (G×G′,F)} H_{p+q}^{G×G′}(G × G′/??).

It induces the required multiplicative structure

BH_{m}^{G}(X, A) ⊗_{R} BH_{n}^{G′}(X′, A′) → BH_{m+n}^{G×G′}((X, A) × (X′, A′)).  (6.2)

We leave it to the reader to verify the axioms of a multiplicative proper equivariant homology theory for BH_{∗}^{?}.

Theorem 6.3 Let R be a commutative ring with ℚ ⊂ R. Let H_{∗}^{?} be a proper equivariant homology theory with values in R-modules. Suppose that H_{∗}^{?} possesses a restriction structure. Let I be the set of conjugacy classes (H) of finite subgroups H of G. Then there is an isomorphism of proper homology theories

ch_{∗}^{?} : BH_{∗}^{?} ⟶^{≅} H_{∗}^{?}

such that

BH_{n}^{G}(X, A) = ⨁_{p+q=n} ⨁_{(H)∈I} H_{p}(C_{G}H\(X^{H}, A^{H}); R) ⊗_{R[W_{G}H]} S_{H}H_{q}^{G}(G/?).

The isomorphism ch_{∗}^{?} is compatible with the given restriction structure on H_{∗}^{?} and the induced restriction structure on BH_{∗}^{?}. If H^{∗} comes with a multiplicative structure and we equip BH_{∗}^{?} with the associated multiplicative structure, ch_{∗}^{?} is also compatible with the multiplicative structures.

\underline{\underline{Proof}} : Given a proper equivariant homology theory H_{∗}^{?} with values in R-modules together with restriction structure, then H_{q}^{?}(∗) inherits a Mackey structure in the obvious way. Given an injective group homomor-

phism f : H → K of finite groups, induction is given by the composite H^{H}(∗) ⟶^{ind_{f}} H_{q}^{K}(ind_{f} ∗) ⟶^{H_{q}^{K}(pr)} H_{q}^{K}(∗) and restriction by res_{f} : H_{q}^{K}(∗) → H_{q}^{H}(∗). Now apply Theorem 0.1 and Theorem 0.2. We leave the lengthy but straighforward verification that the equivariant Chern character is compatible with the restriction structures and multiplicative structures to the reader.

Example 6.4 Equivariant bordism as introduced in Example 1.4 has an obvious restriction structure coming from restriction of spaces and an obvious multiplicative structure coming from the cartesian product. Hence Theorem 6.3 applies to it and yields an isomorphism of multiplicative proper equivariant homology theories with restriction structure

ch_{n}^{G}(X, A) : ⨁_{p+q=n} ⨁_{(H)∈I} H_{p}(C_{G}H\(X^{H}, A^{H}); ℚ) ⊗_{ℚ[W_{G}H]} S_{H}(ℚ ⊗_{ℤ} Ω_{q}^{G}(G/?)) ⟶^{≅} ℚ ⊗_{ℤ} Ω_{n}^{G}(X, A),

where S_{H}(ℚ ⊗_{ℤ} Ω_{q}^{G}(G/?)) = coker (⨁_{K⊂H,K≠H} ℚ ⊗_{ℤ} Ω_{q}^{K}(∗) → ℚ ⊗_{ℤ} Ω_{q}^{H}(∗)) .

7. Green functors

Next we simplify the source of the equivariant Chern character further in the presence of a module structure over the Green functor ℚ ⊗_{ℤ} R_{ℚ}(?) on H_{q}^{?}(∗). Such an additional structure is given in the situation of our main Example 1.5.

Let ϕ : R → S be a homomorphism of associative commutative rings with unit. Let M be a Mackey functor with values in R-modules and let N and P be Mackey functors with values in S-modules. A pairing with respect to ϕ is a family of maps

m(H) : M(H) × N(H) → P (H), (x, y) ↦ m(H)(x, y) =: x · y,

where H runs through the finite groups and we require the following properties for all injective group homomorphisms f : H → K of finite groups:

(x_{1} + x_{2}) · y = x_{1} · y + x_{2} · y for x_{1}, x_{2} ∈ M(H), y ∈ N(H);
x · (y_{1} + y_{2}) = x · y_{1} + x · y_{2} for x ∈ M(H), y_{1}, y_{2} ∈ N(H);
(rx) · y = ϕ(r)(x · y)  for r ∈ R, x ∈ M(H), y ∈ N(H);
x · sy = s(x · y)  for s ∈ S, x ∈ M(H), y ∈ N(H);
res_{f}(x · y) = res_{f}(x) · res_{f}(y) for x ∈ M(K), y ∈ N(K);
ind_{f}(x) · y = ind_{f} (x · res_{f}(y)) for x ∈ M(H), y ∈ N(K);
x · ind_{f}(y) = ind_{f} (res_{f}(x) · y) for x ∈ M(K), y ∈ N(H).

A Green functor with values in R-modules is a Mackey functor U together with a pairing with respect to id : R → R and elements 1_{H} ∈ U(H) for each finite group H such that for each finite group H the pairing U(H) × U(H) → U(H) induces the structure of an R-algebra on U(H) with unit 1_{H} and for any morphism f : H → K in FGINJthe map U^{∗}(f) : U(K) → U(H) is a homomorphism of R-algebras with unit. Let U be a Green functor with values in R-modules and M be a Mackey functor with values in S-modules. A (left) U-module structure on M with respect to the ring homomorphism ϕ : R → S is a pairing such that any of the maps U(H) × M(H) → M(H) induces the structure of a (left) module over the R-algebra U(H) on the R-module ϕ^{∗}M(H) which is obtained from the S-module M(H) by rx := ϕ(r)x for r ∈ R and x ∈ M(H).

Lemma 7.1 Let ϕ : R → S be a homomorphism of associative commutative rings with unit. Let U be a Green functor with values in R-modules and let M be a Mackey functor with values in S-modules such that M comes with a U-module structure with respect to ϕ. Let S be a set of subgroups of the finite group H. Suppose that the map

⨁_{K∈S} ind_{K}^{H} : ⨁_{K∈S} U(K) → U(H)

is surjective. Then the map

⨁_{K∈S} ind_{K}^{H} : ⨁_{K∈S} M(K) → M(H)

is surjective.

\underline{\underline{Proof}} : By hypothesis there are elements u_{K} ∈ U(K) for K ∈ S satisfying 1_{H} = ∑_{K∈S} ind_{K}^{H} u_{K} in U(H). This implies for x ∈ M(H).

x = 1_{H} · x = ( ∑_{K∈S} ind_{K}^{H} u_{K}) · x = ∑_{K∈S} ind_{K}^{H} (u_{K} · res_{H}^{K} x) .

Our main example of a Green functor with values in ℚ-modules ℚ ⊗_{ℤ} R_{ℚ}(?) assigns to a finite group H the ℚ-module ℚ ⊗_{ℤ} R_{ℚ}(H), where R_{ℚ}(H) denotes the rational representation ring. Notice that R_{ℚ}(H) is the same as the projective class group K_{0}(ℚH). The Mackey structure comes from induction and restriction of representations. The pairing ℚ ⊗_{ℤ} R_{ℚ}(H) × ℚ ⊗_{ℤ} R_{ℚ}(H) → ℚ ⊗_{ℤ} R_{ℚ}(H) comes from the tensor product P ⊗_{ℚ} Q of two ℚH-modules P and Q equipped with the diagonal H-action. The unit element is the class of ℚ equipped with the trivial H-action.

Let class_{ℚ}(H) be the ℚ-vector space of functions H → ℚ which are invariant under ℚ-conjugation, i.e. we have f(h_{1}) = f(h_{2}) for two elements h_{1}, h_{2} ∈ H if the cyclic subgroups ⟨h_{1}⟩ and ⟨h_{2}⟩ generated by h_{1} and h_{2} are conjugate in H. Elementwise multiplication defines the structure of a ℚ-algebra on class_{ℚ} with the function which is constant 1 as unit element. Taking the character of a rational representation yields an isomorphism of ℚ-algebras [23, Theorem 29 on page 102]

χ^{H} : ℚ ⊗_{ℤ} R_{ℚ}(H) ⟶^{≅} class_{ℚ}(H).  (7.2)

We define a Mackey structure on class_{ℚ}(?) as follows. Let f : H → K be an injective group homomorphism. For a character χ ∈ class_{ℚ}(H) define its induction with f to be the character ind_{f}(χ) ∈ class_{ℚ}(K) given by

ind_{f}(χ)(k) = \frac{1}{|H|} · ∑_{\begin{matrix}l∈K,h∈H \\ f(h)=l^{−1}kl \end{matrix}} χ(h).

For a character χ ∈ class_{ℚ}(H) define its restriction with f to be the character res_{f}(χ) ∈ class_{ℚ}(H) given by

res_{f}(χ)(h) := χ(f(h)).

One easily checks that this yields the structure of a Green functor on class_{ℚ}(?) and that the family of isomorphisms χ^{H} defined in (7.2) yields an isomorphism of Green functors from ℚ ⊗_{ℤ} R_{ℚ}(?) to class_{ℚ}(?).

For a finite group H and any cyclic subgroup C ⊂ H, define

θ_{C}^{H} ∈ class_{ℚ}(H)  (7.3)

to be the function which sends h ∈ H to 1 if ⟨h⟩ and C are conjugate in H and to 0 otherwise.

Lemma 7.4 Let ϕ : ℚ → R be a homomorphism of associative commutative rings with unit. Let M be a Mackey functor with values in R-modules which is a module over the Green functor ℚ ⊗_{ℤ} R_{ℚ}(H) with respect to ϕ. Then

(a) For a finite group H the map

⨁_{\begin{matrix}C⊂H \\ Ccyclic \end{matrix}} ind_{C}^{H} : ⨁_{\begin{matrix}C⊂H \\ Ccyclic \end{matrix}} M(C) → M(H)

is surjective;

(b) Let C be a finite cyclic group. Let

θ_{C}^{C} : M(C) → M(C)

be the map induced by the ℚ ⊗_{ℤ} R_{ℚ}(C)-module structure and multiplication with the preimage of the element θ_{C}^{C} ∈ class_{ℚ}(C) under the isomorphism χ^{C} : ℚ ⊗_{ℤ} R_{ℚ}(C) ≅ class_{ℚ}(C) of (7.2). Then the cokernel

⨁_{\begin{matrix}D⊂C \\ D≠C \end{matrix}} ind_{D}^{C} : ⨁_{\begin{matrix}D⊂C \\ D≠C \end{matrix}} M(D) → M(C)

is equal to the image of the map θ_{C}^{C} : M(C) → M(C).

\underline{\underline{Proof}} : Let C ⊂ H be a cyclic subgroup of the finite group H. Then we get for h ∈ H

\frac{1}{[H : C]} · ind_{C}^{H} θ_{C}^{C}(h) = \frac{1}{[H : C]} · \frac{1}{|C|} · ∑_{\begin{matrix}l∈H \\ l^{−1}hl∈C \end{matrix}} θ_{C}^{C}(l^{−1}hl) = \frac{1}{|H|} · ∑_{\begin{matrix}l∈H \\ ⟨l^{−1}hl⟩=C \end{matrix}} 1.

This implies in ℚ ⊗_{ℤ} R_{ℚ}(H) ≅ class_{ℚ}(H)

1_{H} = ∑_{\begin{matrix}C⊂H \\ Ccyclic \end{matrix}} \frac{1}{[H : C]} · ind_{C}^{H} θ_{C}^{C}  (7.5)

since for any l ∈ H and h ∈ H there is precisely one cyclic subgroup C ⊂ H with C = ⟨l^{−1}hl⟩. Now assertion (a) follows from the following calculation for x ∈ M(H)

x = 1_{H} · x = ( ∑_{\begin{matrix}C⊂H \\ Ccyclic \end{matrix}} \frac{1}{[H : C]} · ind_{C}^{H} θ_{C}^{C} ) · x = ∑_{\begin{matrix}C⊂H \\ Ccyclic \end{matrix}} \frac{1}{[H : C]} · ind_{C}^{H}(θ_{C}^{C} · res_{H}^{C} x).

It remains to prove assertion (b). Obviously θ_{C}^{C} is an idempotent for any cyclic group C. We get for x ∈ M(C) from (7.5)

(1_{C} − θ_{C}^{C}) · x = ( ∑_{\begin{matrix}D⊂C \\ D≠C \end{matrix}} \frac{1}{[C : D]} · ind_{D}^{C} θ_{D}^{D}) · x = ∑_{\begin{matrix}D⊂C \\ D≠C \end{matrix}} \frac{1}{[C : D]} · ind_{D}^{C}(θ_{D}^{D} · res_{C}^{D} x)

and for D ⊂ C, D ≠ C and y ∈ M(D)

θ_{C}^{C} · ind_{D}^{C} y = ind_{D}^{C}(res_{C}^{D} θ_{C}^{C} · y) = ind_{D}^{C}(0 · y) = 0.

This finishes the proof of Lemma 7.4.

Now Theorem 0.3 follows from Theorem 0.1, Theorem 0.2 and Lemma 7.4. For more information about Mackey and Green functors and induction theorems we refer for instance to [6, Section 6] and [8].

8. Applications to K- and L-theory

In this section we apply Theorem 0.3 to the equivariant homology theories of Example 1.5. Thus we obtain explicit computations of the rationalized source of the assembly map (1.6). These give explicit computations of the rationalized algebraic K- and L-groups of RG and of the topological K-groups of the real and complex reduced group C^{∗}-algebras of G, provided that the Farrell-Jones Conjecture with respect to the family F of finite subgroups and the Baum Connes Conjecture are true for G. Before we carry out this program, we mention the following facts. Notice for the sequel that the different versions of L-groups, symmetric, quadratic or decorated L-groups, differ only by 2-torsion and hence agree after inverting 2.

Theorem 8.1 There are natural isomorphisms

L_{n}(ℤG)[1/2] ⟶^{≅} L_{n}(ℚG)[1/2];
K_{n}(C_{r}^{∗}(G, ℝ))[1/2] ⟶^{≅} L_{n}(C_{r}^{∗}(G, ℝ))[1/2];
K_{n}(C_{r}^{∗}(G, ℂ))[1/2] ⟶^{≅} L_{n}(C_{r}^{∗}(G, ℂ))[1/2].

\underline{\underline{Proof}} : The proof of the first isomorphism can be found in [20, page 376]. The other two isomorphism are explained in [22, Theorem 1.8 and 1.11], where they are attributed to Karoubi, Miller and Mishchenko.

Next we introduce a Mackey structure and then a module structure over the Green functor ℚ ⊗_{ℤ} R_{ℚ}(?) on the various K- and L-groups. Let R be an associative commutative ring with unit satisfying ℚ ⊂ R and let F be ℝ, ℂ. Induction and restriction yield obvious Mackey functors

ℚ ⊗_{ℤ} K_{q}(R?) : FGINJ → ℚ − MOD,  H ↦ ℚ ⊗_{ℤ} K_{q}(RH);
ℚ ⊗_{ℤ} L_{q}(R?) : FGINJ → ℚ − MOD,  H ↦ ℚ ⊗_{ℤ} L_{q}(RH);
ℚ ⊗_{ℤ} K_{q}^{top}(C_{r}^{∗}(?, F)) : FGINJ → ℚ − MOD,  H ↦ ℚ ⊗_{ℤ} K_{q}^{top}(C_{r}^{∗}(H, F)).

The tensor product over R or F with the diagonal action induces on ℚ ⊗_{ℤ} K_{0}(R?), ℚ ⊗_{ℤ} L_{0}(R?) and ℚ ⊗_{ℤ} K_{0}^{top}(C^{∗}(?, F)) the structure of a Green functor with values in ℚ-modules and the structure of a module over these Green functors on ℚ ⊗_{ℤ} K_{q}(R?), ℚ ⊗_{ℤ} L_{q}(R?) and ℚ ⊗_{ℤ} K_{q}^{top}(C^{∗}(?, F)) for all q ∈ ℤ. The change of ring maps

ℚ ⊗_{ℤ} K_{0}(ℚ?) → ℚ ⊗_{ℤ} K_{0}(R?);
ℚ ⊗_{ℤ} L_{0}(ℚ?) → ℚ ⊗_{ℤ} L_{0}(R?);
ℚ ⊗_{ℤ} K_{0}^{top}(C_{r}^{∗}(?, ℝ)) → ℚ ⊗_{ℤ} K_{0}^{top}(C_{r}^{∗}(?, ℂ))

induce maps of Green functors. Since ℚ ⊗_{ℤ} K_{0}(ℚ?) ≅ ℚ ⊗_{ℤ} R_{ℚ}(?), we get a module structure over the Green functor ℚ ⊗_{ℤ} R_{ℚ}(?) on each Mackey functor ℚ ⊗_{ℤ} K_{q}(R?). The change of rings map

ℚ ⊗_{ℤ} L_{0}(ℚ?) ⟶^{≅} ℚ ⊗_{ℤ} L_{0}(ℝ?)

is known to be an isomorphism (see [21, Proposition 22.19 on page 237]. There is an isomorphism of Green functors (see Theorem 8.1 or [21, Proposition 22.33 on page 252])

ℚ ⊗_{ℤ} K_{0}(ℝ?) ⟶^{≅} ℚ ⊗_{ℤ} L_{0}(ℝ?).

Thus we get a morphism of Green functors

ℚ ⊗_{ℤ} R_{ℚ}(?) ⟶^{≅} ℚ ⊗_{ℤ} L_{0}(ℚ?).

Hence we obtain a module structure over the Green functor ℚ⊗_{ℤ} R_{ℚ}(?) on the Mackey functor ℚ⊗_{ℤ} L_{q}(R?). Since K_{0}(ℝ?) = K_{0}^{top}(C_{r}^{∗}(?, ℝ)), we finally obtain also a module structure over the Green functor ℚ ⊗_{ℤ} R_{ℚ}(?) on the Mackey functor ℚ ⊗_{ℤ} K_{q}^{top}(C_{r}^{∗}(?, F)). If ℚ ⊂ R, then the cellular R[C_{G}H]-chain complex C_{∗}(E(G, F)^{H}) is a projective resolution of the trivial R[C_{G}H]-module R and we obtain for any finite group H ⊂ G an identification

H_{p}(C_{G}H\E(G, F)^{H}; R) ≅ H_{p}(C_{G}H; R).  (8.2)

Notice that now Theorem 0.4 follows from Theorem 0.3 and Example 1.5. The homomorphisms appearing in Theorem 0.4 are compatible with the various change of rings or of K-theory maps since these maps are compatible with the relevant module structures over the Green functor ℚ ⊗_{ℤ} R_{ℚ}(?).

If the ring R is a field of characteristic zero and we are willing to extend ℚ to a larger field, then we can simplify the right side of the various maps appearing in Theorem 0.4 as follows. Let F be a field of characteristic zero. Fix an integer m ≥ 1. Let F (ζ_{m}) ⊃ F be the Galois extension given by adjoining the primitive m-th root of unity ζ_{m} to F. Denote by Γ(m, F) the Galois group of this extension of fields, i.e. the group of automorphisms σ : F (ζ_{m}) → F (ζ_{m}) which induce the identity on F. It can be identified with a subgroup of ℤ/m^{∗} by sending σ to the unique element u(σ) ∈ ℤ/m^{∗} for which σ(ζ_{m}) = ζ_{m}^{u(σ)} holds. Given a finite cyclic group C of order |C|, the Galois group Γ(|C|, F) acts on C by sending c to c^{u(σ)}, and thus on the set Gen(C) of generators of C. Let V be a F (ζ_{|C|})-module. Denote by res_{σ} V for σ ∈ Γ(|C|, F) the F (ζ_{|C|})- module obtained from V by restriction with σ, i.e. the underlying abelian groups of res_{σ} V and V agree and multiplication with x ∈ F (ζ_{m}) on res_{σ} V is given by multiplication with σ(x) on V. Thus we obtain an action of Γ(|C|, F) on K_{q}(F (ζ_{|C|})) by sending σ ∈ Γ(|C|, F) to the automorphism res_{σ} : K_{q}(F (ζ_{|C|})) → K_{q}(F (ζ_{|C|})) coming from the functor V ↦ res_{σ} V. This action extends to an action of the Galois group Γ(|C|, F) on F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})) by σ · (v ⊗ w) := v ⊗ res_{σ}(w). Equip map (Gen(C), F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})))^{Γ(|C|,F)} and F (ζ_{|C|}) ⊗_{ℤ} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}(F [C])) with the obvious F (ζ_{|C|})-module structures.

Lemma 8.3 Let F be a field of characteristic zero. Let C be a finite cyclic group. Then there is an isomorphism of F (ζ_{|C|})-modules

map (Gen(C), F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})))^{Γ(|C|,F)} ⟶^{≅} F (ζ_{|C|}) ⊗_{ℚ} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}(F [C])) ,

which is natural with respect to automorphisms of C.

Its proof needs some preparation. Let G be a group. Given a positive integer m and a F (ζ_{m})[G]-module V, we define an in V natural isomorphism of F (ζ_{m})[G]-modules

Φ : ind_{F}^{F (ζ_{m})} res_{F (ζ_{m})}^{F} V = F (ζ_{m}) ⊗_{F} V ⟶^{≅} ⨁_{σ∈Γ(m,F)} res_{σ} V,  x ⊗ v ↦ (σ(x)v)_{σ∈Γ(m,F)}.

Obviously Φ is natural in V and F (ζ_{m})[G]-linear. We claim that an inverse of Φ is given by

Φ^{−1} : ⨁_{σ∈G} res_{σ} V → ind_{F}^{F (ζ_{m})} res_{F (ζ_{m})}^{F} V = F (ζ_{m}) ⊗_{F} V,
(v_{σ})_{σ∈Γ(m,F)} ↦ \frac{1}{m} · ∑_{i=1}^{m} ∑_{σ∈Γ(m,F)} ζ_{m}^{−i} ⊗_{F} σ(ζ_{m})^{i}v_{σ}.

This follows from an easy calculation using the facts that for a m-th root of unity ζ the sum ∑_{i=1}^{m} ζ^{i} is zero if ζ ≠ 1, and is m if ζ = 1, and that an element x ∈ F (ζ_{m}) belongs to F if and only if σ(x) = x for all σ ∈ Γ(m, F) holds. Fix a F -basis {b_{σ} | σ ∈ Γ(m, F)} for F (ζ_{m}). Given a F G-module W, we obtain an in W natural F G-isomorphism

Ψ : ⨁_{Γ(m,F)} W ⟶^{≅} res_{F (ζ_{m})}^{F} ind_{F}^{F (ζ_{m})} W = F (ζ_{m}) ⊗_{F} W, (w_{σ})_{σ∈Γ(m,F)} ↦ ∑_{σ∈Γ(m,F)} b_{σ} ⊗_{F} w_{σ}

and an in W natural F (ζ_{m})[G]-isomorphism for σ ∈ Γ(m, F)

Λ : ind_{F}^{F (ζ_{m})} W = F (ζ_{m}) ⊗_{F} W → res_{σ} ind_{F}^{F (ζ_{m})} W, x ⊗_{F} w ↦ σ(x) ⊗_{F} w.

From the existence of the natural isomorphisms Φ,Ψ and Λ above we conclude for the homomorphisms

ind_{F}^{F (ζ_{m})} : K_{q}(F G) → K_{q}(F (ζ_{m})[G]);
res_{F (ζ_{m})}^{F} : K_{q}(F (ζ_{m})[G]) → K_{q}(F G);
res_{σ} : K_{q}(F (ζ_{m})[G]) → K_{q}(F (ζ_{m})[G]),

that res_{F (ζ_{m})}^{F} ∘ ind_{F}^{F (ζ_{m})} = |Γ(m, F)| · id, ind_{F}^{F (ζ_{m})} ∘ res_{F (ζ_{m})}^{F} = ∑_{σ∈Γ(m,F)} res_{σ} and res_{σ} ∘ ind_{F}^{F (ζ_{m})} =

ind_{F}^{F (ζ_{m})} holds for σ ∈ Γ(m, F). The various maps res_{σ} induce a Γ(m, F)-action on K_{q}(F (ζ_{m})[G]). We conclude

Lemma 8.4 Induction induces an isomorphism

ℚ ⊗_{ℤ} ind_{F}^{F (ζ_{m})} : ℚ ⊗_{ℤ} K_{q}(F G) ⟶^{≅} ℚ ⊗_{ℤ} K_{q}(F (ζ)[G])^{Γ(m,F)}.

Let C be a finite cyclic group of order |C|. Then all irreducible F (ζ_{|C|})-representations of C are 1dimensional. The number of isomorphism classes of irreducible F (ζ_{|C|})-representations is equal to |C|. Given a finite-dimensional F (ζ_{|C|})-representation V of C, we obtain a functor from the category of finitely generated projective F (ζ_{|C|})-modules to the category of finitely generated projective F (ζ_{|C|})[C]-modules by tensoring with V over F (ζ_{m}) and thus a map K_{q}(F (ζ_{|C|})) → K_{q}(F (ζ_{|C|})[C]). This yields a homomorphism

α : K_{0}(F (ζ_{|C|})[C]) ⊗_{ℤ} K_{q}(F (ζ_{|C|})) ⟶^{≅} K_{q}(F (ζ_{|C|})[C]),  (8.5)

which is an isomorphism by the following elementary facts. Given a F (ζ_{|C|})[C]-module U and an irreducible F (ζ_{|C|})[C]-module V, denote by U_{V} the V -isotypical summand. This is the F (ζ_{|C|})[C]-submodule of U generated by all element u ∈ U for which there exists a F (ζ_{|C|})[C]-submodule U′ ⊂ U which contains u and is F (ζ_{|C|})[C]-isomorphic to V. For any homomorphism f : U → W of finitely generated projective F (ζ_{|C|})[C]-modules there are natural splittings U = ⨁_{V} U_{V} and W = ⨁_{V} W_{V}, where V runs over the irreducible representations, f maps U_{V} to W_{V} and aut_{F (ζ_{|C|})[C]}(V) = {x · id_{V} | x ∈ F (ζ_{|C|})}.

An element σ ∈ Γ(|C|, F) induces automorphisms res_{σ} of K_{q}(F (ζ_{|C|})) and of K_{q}(F (ζ_{|C|})[C]) by restriction with σ : F (ζ_{|C|}) → F (ζ_{|C|}) and σ : F (ζ_{|C|})[C] → F (ζ_{|C|})[C], ∑_{c∈C} x_{c} · c ↦ ∑_{c∈C} σ(x_{c}) · c. We get for σ ∈ Γ(|C|, F)

res_{σ} ∘α = α ∘ (res_{σ} ⊗_{ℤ} res_{σ}).

Taking the character of a representation yields an isomorphism

χ : F (ζ_{|C|}) ⊗_{ℤ} K_{0}(F (ζ_{|C|})[C]) ⟶^{≅} map(C, F (ζ_{|C|})),  x ⊗ [V] ↦ x · χ_{V} .  (8.6)

The operation of Γ(|C|, F) on K_{0}(F (ζ_{|C|})[C]) extends to an operation on F (ζ_{|C|})⊗_{ℤ}K_{0}(F (ζ_{|C|})[C]) by taking the tensor product id ⊗_{ℤ}?. We define a Γ(|C|, F)-operation on map(C, F (ζ_{|C|})) by assigning to σ ∈ Γ(|C|, F) and χ ∈ map(C, F (ζ_{|C|})) the element σ · χ which sends c ∈ C to χ(c^{u(σ)}). The map χ is compatible with these Γ(|C|, F)-actions. It suffices to check this for 1 ⊗_{ℤ} [V] if V is an irreducible F (ζ_{|C|})[C]-representation. Its character is a homomorphism χ_{V} : C → F (ζ_{|C|}) whose values are multiples of ζ_{|C|} and c ∈ C acts on V by multiplication with χ_{V} (c). Hence c ∈ C acts on res_{σ} V by multiplication with σ(χ_{V} (c)) on V. This implies χ_{res_{σ} V} (c) = σ(χ_{V} (c)) = χ_{V} (c)^{u(σ)} = χ_{V} (c^{u(σ)}). We have the obvious isomorphism

β : map (C, F (ζ_{|C|})) ⊗_{ℤ} K_{q}(F (ζ_{|C|})) ⟶^{≅} map (C, F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|}))) .  (8.7)

Now the maps α, χ and β defined in (8.5), (8.6) and (8.7) can be combined to an isomorphism of F (ζ_{|C|})- modules

γ = (id ⊗α) ∘ (χ ⊗ id)^{−1} ∘ β^{−1} : map (C, F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})) ⟶^{≅} F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})[C]). (8.8)

It is Γ(|C|, F)-equivariant, where we use on the source the action given by (σ · χ)(c) := (id ⊗σ)(χ(c^{u(σ)})) and on the target by res_{σ} ⊗ id.

Next we treat the various ℚ ⊗_{ℚ} R_{ℚ}(C)-module structures. The source of α and the source of χ inherit a module structure over ℚ ⊗_{ℚ} R_{ℚ}(C) by the obvious ring homomorphism ind_{ℚ}^{F (ζ_{|C|})} : R_{ℚ}(C) = K_{0}(ℚ[C]) → K_{0}(F (ζ_{|C|})[C]). We equip the target of α with the ℚ ⊗_{ℚ} R_{ℚ}(C)-module structure for which α becomes a

ℚ ⊗_{ℚ} R_{ℚ}(C)-homomorphism. We have introduced the isomorphism of ℚ-algebras χ^{C} : ℚ ⊗_{ℤ} R_{ℚ}(C) ⟶^{≅} class_{ℚ}(C) in (7.2). The target of the isomorphism χ^{C} is a module over class_{ℚ}(C) by the obvious inclusion of rings class_{ℚ}(C) → map(C, F (ζ_{|C|})). Then χ is a ℚ ⊗_{ℚ} R_{ℚ}(C)-homomorphism. Equip the source of the isomorphism β with the ℚ ⊗_{ℤ} R_{ℚ}(C)-module structure given by the one on the target of χ and the trivial one on K_{q}(F (ζ_{|C|})). Equip the target of β with the ℚ ⊗_{ℚ} R_{ℚ}(C)-structure for which β becomes a ℚ ⊗_{ℚ} R_{ℚ}(C)-homomorphism. Then the isomorphism γ is a ℚ ⊗_{ℚ} R_{ℚ}(C)-homomorphism. Therefore we obtain a commutative diagram of F (ζ_{|C|})-modules where all maps are Γ(|C|, F)-equivariant

map (C, F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|}))) ⟶^{γ} K_{q}(F (ζ_{|C|})[C]) ⊗_{ℤ} F (ζ_{|C|})
θ_{C}^{C} ↓  ↓^{θ_{C}^{C}}
map (C, F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|}))) ⟶_{γ} K_{q}(F (ζ_{|C|})[C]) ⊗_{ℤ} F (ζ_{|C|})

By taking the fixed point sets, we obtain a commutative diagram of F (ζ_{|C|})-modules

map (C, F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})))^{Γ(|C|,F)} ⟶^{γ} K_{q}(F (ζ_{|C|})[C])^{Γ(|C|,F)} ⊗_{ℤ} F (ζ_{|C|})
θ_{C}^{C} ↓  ↓^{θ_{C}^{C}}
map (C, F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})))^{Γ(|C|,F)} ⟶_{γ} K_{q}(F (ζ_{|C|})[C])^{Γ(|C|,F)} ⊗_{ℤ} F (ζ_{|C|})

Thus we obtain an isomorphism from the image of the left vertical arrow in the diagram above to the the image of the right vertical arrow. Recall that θ_{C}^{C} is the character which sends a generator of C to 1 and all other elements to 0. Hence the image of the left vertical arrow is canonically isomorphic to map(Gen(C), F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|}))^{Γ(|C|,F)}. The image of the right vertical arrow is by Lemma 8.4 canonically isomorphic to the image of θ_{C}^{C} : K_{q}(F [C]) ⊗_{ℤ} F (ζ_{|C|}) → K_{q}(F [C]) ⊗_{ℤ} F (ζ_{|C|}). This finishes the proof of Lemma 8.3.

We conclude from Theorem 0.4 and Lemma 8.3

Theorem 8.9 Let G be a group. Let F be a field of characteristic zero. Let F ⊂ \overline{F} be a field extension such that for any finite cyclic subgroup C ⊂ G the primitive |C|-th root of unity belongs to \overline{F}. Let J be the set of conjugacy classes (C) of finite cyclic subgroups of G. Then the assembly map (1.6) in the Farrell-Jones Conjecture with respect to F for the algebraic K-groups K_{n}(F G) can be identified after applying \overline{F} ⊗_{ℤ}? with

⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; \overline{F}) ⊗_{\overline{F} [W_{G}C]} map (Gen(C), \overline{F} ⊗_{ℤ} K_{q}(F (ζ_{|C|})))^{Γ(|C|,F)} → \overline{F} ⊗_{ℤ} K_{n}(F G).

If the Farrell-Jones Conjecture with respect to F is true, then this map is an isomorphism.

Remark 8.10 The following remark was pointed out by the referee. Notice that the fixed point set of the operation of Γ(|C|, F) on F (ζ_{|C|}) is F itself. There is a second Γ(|C|, F)-operation on map (Gen(C), F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|}))) which comes from the obvious operations on Gen(C) and F (ζ_{|C|}) and the trivial operation on K_{q}(F (ζ_{|C|})). Define a Γ(|C|, F)-operation on F (ζ_{|C|}) ⊗_{ℚ} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}(F [C])) by the obvious operation on F (ζ_{|C|}) and the trivial operation on θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}(F [C])). Then the isomorphism appearing in Lemma 8.3 is Γ(|C|, F)-equivariant with respect to these operations and we obtain an isomorphism of F -modules

map (Gen(C), F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})))^{Γ(|C|,F)×Γ(|C|,F)} ⟶^{≅} F ⊗_{ℚ} θ_{C}^{C} · (ℚ ⊗_{ℤ} K_{q}(F [C])) ,

which is natural with respect to automorphisms of C. Moreover, we get an improvement of the isomorphism appearing in Theorem 8.9 to an isomorphism of F -modules

⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; F)⊗_{F [W_{G}C]}map (Gen(C), F (ζ_{|C|}) ⊗_{ℤ} K_{q}(F (ζ_{|C|})))^{Γ(|C|,F)×Γ(|C|,F)} ⟶^{≅} F ⊗_{ℤ}K_{n}(F G).

Example 8.11 If F = ℂ, then F (ζ_{|C|}) = ℂ and Γ(|C|, ℂ) = 1. Let T be the set of conjugacy classes (g) of elements g ∈ G of finite order. The action of W_{G}C on Gen(C) is free. Then the assembly maps (1.6) in the Farrell-Jones Conjecture with respect to F and in the Baum-Connes conjecture can be identified after applying ℂ⊗_{ℤ}? with

⨁_{p+q=n} ⨁_{(g)∈T} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}(ℂ) → ℂ ⊗_{ℤ} K_{n}(ℂG);
⨁_{p+q=n} ⨁_{(g)∈T} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} L_{q}(ℂ) → ℂ ⊗_{ℤ} L_{n}(ℂG);
⨁_{p+q=n} ⨁_{(g)∈T} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}^{top}(ℂ) → ℂ ⊗_{ℤ} K_{n}^{top}(C_{r}^{∗}(G, ℂ)),

where we use in the definition of L_{q}(ℂ) and L_{n}(ℂG) the involutions coming from complex conjugation. We get the first one from Theorem 8.9. The proof for the third is completely analogous to the one of the first. The proof of the second can be reduced to the one of the third by Theorem 8.1. In particular this proves Theorem 0.5. We mention that the restriction of the upper horizontal arrow in Theorem 0.5 to the part for q = 0 has been shown to be split injective for all groups G using the Dennis trace map but not the Farell-Jones Conjecture in [19].

If we use the trivial involution on ℂ in the definition of L_{n}(ℂG), then the Farrell-Jones Conjecture with respect to F implies L_{n}(ℂG)[1/2] = 0 since L_{n}(ℂH)[1/2]) = 0 is known for all finite groups H with respect to the trivial involution on ℂ [21, Proposition 22.21 on page 239]. Notice that the Farrell-Jones Conjecture with respect to F and the Baum Connes Conjecture together with Theorem 8.1 imply that the change of ring maps L_{n}(ℂG) → L_{n}(C_{r}^{∗}(G, ℂ)) becomes a bijection after inverting 2.

Example 8.12 Next we consider the case F = ℝ. Put \overline{F} = ℂ. We call g_{1} and g_{2} in G ℝ-conjugate if (g_{1}) = (g_{2}) or (g_{1}) = (g_{2}^{−1}). Denote by (g)_{ℝ} the ℝ-conjugacy class of g ∈ G. Denote by T_{ℝ} the set of ℝ-conjugacy classes of elements of finite order in G. This splits as the disjoint union T′_{ℝ} ∐ T′′_{ℝ}, where T′_{ℝ} resp. T′′_{ℝ} consists of classes (g)_{ℝ} with (g) ≠ (g^{−1}) resp. (g) = (g^{−1}). For a class (g)_{ℝ} ∈ T′′_{ℝ} we can find an element g′ ∈ G such that the homomorphism c(g′) : G → G given by conjugation with g′ maps g to g^{−1}. Then c(g′) induces also an automorphism C_{G}⟨g⟩ → C_{G}⟨g⟩. The induced automorphism of H_{p}(C_{G}⟨g⟩; ℂ) does not depend on the choice of g′ and is of order two. Thus we obtain a ℤ/2-action on H_{p}(C_{G}⟨g⟩; ℂ). The Galois group of the field extension ℝ ⊂ ℂ is ℤ/2 with complex conjugation as generator. Complex conjugation induces a ℤ[ℤ/2]-structure on K_{q}(ℂ) and K_{q}^{top}(ℂ). We obtain analogously to Example 8.11 an identification of the assembly maps (1.6) in the Farrell-Jones Conjecture with respect to F and in the Baum-Connes conjecture after applying ℂ⊗_{ℤ}? with

⨁_{p+q=n} (( ⨁_{(g)_{ℝ}∈T′_{ℝ}} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}(ℂ)) ⨁ ( ⨁_{(g)_{ℝ}∈T′′_{ℝ}} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ[ℤ/2]} K_{q}(ℂ)))
→ ℂ ⊗_{ℤ} K_{n}(ℝG),
⨁_{p+q=n} (( ⨁_{(g)_{ℝ}∈T′_{ℝ}} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} L_{q}(ℂ)) ⨁ ( ⨁_{(g)_{ℝ}∈T′′_{ℝ}} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ[ℤ/2]} L_{q}(ℂ)))
→ ℂ ⊗_{ℤ} L_{n}(ℝG),
⨁_{p+q=n} (( ⨁_{(g)_{ℝ}∈T′_{ℝ}} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ} K_{q}^{top}(ℂ)) ⨁ ( ⨁_{(g)_{ℝ}∈T′′_{ℝ}} H_{p}(C_{G}⟨g⟩; ℂ) ⊗_{ℤ[ℤ/2]} K_{q}^{top}(ℂ)))
→ ℂ ⊗_{ℤ} K_{n}^{top}(C_{r}^{∗}(G, ℝ)),

where we use in the definition of L_{q}(ℂ) the involution coming from complex conjugation. Notice that the Farrell-Jones Conjecture with respect to F and the Baum Connes Conjecture together with Theorem 8.1 imply that the change of ring maps L_{n}(ℚG) → L_{n}(ℝG) and L_{n}(ℝG) → L_{n}(C_{r}^{∗}(G, ℝ)) become bijections after inverting 2 since L_{n}(ℚH) → L_{n}(ℝH) is known to be bijective after inverting 2 for finite groups H [21, Proposition 22.33 on page 252].

Example 8.13 If F = ℚ, then Γ(|C|, ℚ) = ℤ/|C|^{∗} = aut(C). Since Γ(|C|, ℚ) acts freely and transitively on Gen(C), we obtain after the choice of a generator c ∈ C an isomorphism

map (Gen(C), ℚ(ζ_{|C|}) ⊗_{ℤ} K_{q}(ℚ(ζ_{|C|})))^{Γ(|C|,ℚ)} ≅ ℚ(ζ_{|C|}) ⊗_{ℤ} K_{q}(ℚ(ζ_{|C|})).

It is natural with respect to automorphisms of C, if f ∈ aut(C) acts on ℚ(ζ_{|C|}) ⊗_{ℤ} K_{q}(ℚ(ζ_{|C|})) by id ⊗ res_{σ} for the element σ in the Galois group Γ(|C|, ℚ) for which σ(ζ) = ζ^{u} and f(c) = c^{u} holds. Let J be the set of conjugacy classes (C) of finite cyclic subgroups of G. We conclude from Theorem 8.9 that the assembly map (1.6) in the Farrell-Jones Conjecture with respect to F can be identified with

⨁_{p+q=n} ⨁_{(C)∈J} H_{p}(C_{G}C; \overline{ℚ}) ⊗_{\overline{ℚ}[W_{G}C]} \overline{ℚ} ⊗_{ℤ} K_{q}(ℚ(ζ_{|C|})) → \overline{ℚ} ⊗_{ℤ} K_{n}(ℚG).

Example 8.14 Let F be a field of characteristic zero and let G be a group. Let g_{1} and g_{2} be two elements of G of finite order. We call them F -conjugate if for some (and hence all) positive integers m with g_{1}^{m} = g_{2}^{m} = 1 there exists an element σ in the Galois group Γ(m, F) with the property (g_{1}^{u(σ)}) = (g_{2}). Denote by con_{F} (G) the set of F -conjugacy classes (g)_{F} of elements g ∈ G of finite order. Let class_{F} (G) be the F -vector space with the set con_{F} (G) as basis, or, equivalently, the F -vector space of functions con_{F} (G) → F with finite support. Recall that for a finite group H taking characters yields an isomorphism [23, Corollary 1 on page 96]

χ : F ⊗_{ℤ} R_{F} (H) = F ⊗_{ℤ} K_{0}(F H) ⟶^{≅} class_{F} (H).  (8.15)

By Theorem 0.4 and (8.15) the assembly map (1.6) of the Farrell-Jones Conjecture with respect to F for K_{0}(F G) can be identified with a map

class_{F} (G) → F ⊗_{ℤ} K_{0}(F G).

If the Farrell-Jones Conjecture with respect to F for K_{0}(F G) is true, this map is an isomorphism. This generalizes (8.15) for finite groups to infinite groups. This example is related to the Hattori-Stalling rank and the Bass Conjecture [1].

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