Induction Theorems and Isomorphism Conjectures for K- and L-Theory

Arthur Bartels^{∗}and Wolfgang Lück^{†} Fachbereich Mathematik Universität Münster Einsteinstr. 62 48149 Münster Germany

August 29, 2005

Abstract

The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C^{∗}-algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with ℤ as kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups. Key words: K- and L-groups of group rings and group C^{∗}-algebras, induction theorems. Mathematics subject classification 2000: 19A31, 19B28, 19DXX, 46L80.

0. Introduction

The goal of this paper is to reduce the family of virtually cyclic subgroups to a smaller subfamily in the statements of the (Fibered) Farrell-Jones Conjecture for algebraic K- and L-theory of group rings. The strategy is to extend the classical induction results for finite groups to infinite groups.

∗email: bartelsa@math.uni-muenster.de †email: lueck@math.uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ fax: +49 251 8338370

Let FIN and VCY respectively be the class of finite groups and virtually cyclic groups respectively. Let F ⊆ FIN be a subclass of the class of finite groups which is closed under isomorphism of groups and taking subgroups. Define

F′ ⊆ VCY

to be the class of groups V for which either

(a) there exists an extension 1 → ℤ → V → F → 1 for a group F ∈ F or

(b) V ∈ F holds.

With this notion we get VCY = FIN′.

Let p be a prime. A finite group G is called p-elementary if it is isomorphic to C ×P for a cyclic group C and a p-group P such that the order |C| is prime to p. A finite group G is called p-hyperelementary if it can be written as an extension 1 → C → G → P → 1 for a cyclic group C and a p-group P such that the order |C| is prime to p. A finite group G is called elementary or hyperelementary respectively if it is p-elementary or p-hyperelementary respectively for some prime p. Let FCY be the class of finite cyclic groups. Let E_{p} and H_{p} respectively be the class of groups which are p-elementary groups and p-hyperelementary respectively for a prime p. Let E and H respectively be the class of groups which are elementary and hyperelementary respectively.

A family of subgroups of G is a set of subgroups which is closed under conjugation and taking subgroups. For a class F of groups which is closed under isomorphism of groups and taking subgroups we denote by F(G) the family of subgroups of G whose members are in F. Thus FIN (G) is the family of finite subgroups of G. If G is clear from the context we will often abuse notation and write simply F for F(G).

The Isomorphism Conjecture of Farrell and Jones appeared in [10, 1.6]. For a survey of this conjecture see [22].

Theorem 0.1 (Induction theorem for algebraic K-theory). Let G be a group and let N be an integer. Then the following hold.

(a) The group G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family VCY if and only if G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family H′.

(b) Let p be a prime. Then G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family VCY after applying ℤ_{(p)} ⊗_{ℤ} − if and only if G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family H′_{p} after applying ℤ_{(p)} ⊗_{ℤ} −.

(c) Suppose that R is regular and ℚ ⊆ R. Then the group G satisfies the Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family VCY if and only if G satisfies the Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family H.

If we assume that R is regular and ℂ ⊆ R, then we can replace H by E.

(d) Suppose that R is regular and ℚ ⊆ R. Let p be a prime. Then G satisfies the Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family VCY after applying ℤ_{(p)} ⊗_{ℤ} − if and only if G satisfies the Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family H_{p} after applying ℤ_{(p)} ⊗_{ℤ} −.

If we assume that R is regular and ℂ ⊆ R, then we can replace H_{p} by E_{p}.

Note that in (c) and (d) above there are no claims about the Fibered Isomorphism Conjecture. The problem is that we do not know whether the relevant Nil-groups for amalgamated products and HNN-extensions with rational coefficients vanish for all groups. If the group ℤ satisfies the Fibered Isomorphism Conjecture for algebraic K-theory with coefficients in ℚ for the trivial family, then the Nil groups NK_{∗}(ℚΓ) vanishes for all groups Γ. It seems to be not known whether these Nil groups vanish. (Of course for the group ℤ the families H, H_{p} and FIN are all trivial.)

Remark 0.2 (Relative assembly maps). We have stated the main theorems in terms of the (Fibered) Isomorphism Conjectures. The main new result we prove is that the relevant relative assembly maps are bijective. This statement is true in general and is independent of the question whether the Isomorphism Conjecture is true or not but implies our main results.

For instance, the assembly map for the family H′ factorizes as

H_{n}(E_{H′}G) → H_{n}(E_{FCY}G) → H_{n}^{G}({pt .}) = K_{n}(RG)

where H_{∗}^{G} is a certain G-homology theory related to algebraic K-theory of group rings with coefficients in a ring R, the first map is a relative assembly map which we will prove is bijective, and the second map is the assembly map for VCY. The Isomorphism Conjecture for algebraic K-theory for H′ or VCY respectively says that the assembly map for H′ or VCY respectively is bijective for n ∈ ℤ. The same remark applies to the L-theory version below.

Theorem 0.3 (Induction theorem for algebraic L-theory). Let G be a group. Then the following hold.

(a) The group G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) for algebraic L-theory with coefficients in R for the family VCY if and only if G satisfies the (Fibered) Isomorphism Conjecture for algebraic

L-theory with coefficients in R for the family (H_{2} ∪ ⋃_{p prime,p≠2} E_{p})′.

(b) The group G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) for algebraic L-theory with coefficients in R for the family VCY after applying ℤ[1/2] ⊗_{ℤ} − if and only if G satisfies the (Fibered) Isomorphism Conjecture for algebraic L-theory with coefficients in R for the family ⋃_{p prime,p≠2} E_{p} after applying ℤ[1/2] ⊗_{ℤ} −.

Remark 0.4 (Rationalized versions). We omit the discussion of the rationalized versions, i.e. the versions obtained after applying ℚ ⊗_{ℤ} −. In this case one gets more precise information as discussed in detail in [22, Section 8]. The results presented there are based on [1], [13], [17] and [19].

The next result is due to Mislin and Matthey [23] for the complex case. We will give a proof for both the complex and the real case in our framework. It is not clear to us whether it is possible to extend the methods of [23] to the real case.

Theorem 0.5 (Induction theorem for topological K-theory). Let G be a group.

Then the relative assembly map

K_{n}^{G}(E_{FCY}G) → K_{n}^{G}(E_{FIN} G)

is bijective for all n ∈ ℤ.

In particular G satisfies the Baum-Connes Conjecture if and only if G satisfies the Baum-Connes Conjecture for the family FCY.

The corresponding statements are also true if one replaces complex equivariant K-homology K_{∗}^{G} by real equivariant K-homology KO_{∗}^{G} and uses the real reduced group C^{∗}-algebra in the Baum-Connes Conjecture.

The rational version of the Induction theorem for topological real K-theory has been used by Stolz [29, p.695]. Using Artin Induction [28, Theorem 26 on page 97] the methods we use to proof the Induction Theorem for algebraic K-theory can also be used to give a simpler proof of the rational version of Theorem 0.5.

The paper is organized as follows

1. Transitivity Principles
2. General Induction Theorems
3. The Swan Group as a Functor on Groupoids
4. Proof of the Main Result for Algebraic K-Theory
5. Outline of the Proof of the Main Result for Algebraic L-Theory
6. Proof of the Main Result for Topological K-Theory
7. Versions in terms of colimits
8. On Quinn’s Hyperelementary Induction Conjecture

References

1. Transitivity Principles

In this section we fix an equivariant homology theory H_{∗}^{?} with values in Λmodules for a commutative associative ring Λ with unit in the sense of [19, Section 1]. This essentially means that we get for each group G a G-homology theory H_{∗}^{G} which assigns to a (not necessarily proper or cocompact) pair of G-CW -complexes (X, A) a ℤ-graded Λ-module H_{n}^{G}(X, A), such that there exists natural long exact sequences of pairs and G-homotopy invariance, excision, and the disjoint union axiom are satisfied. Moreover, an induction structure is required which in particular implies for a subgroup H ⊆ G and a H-CW -pair (X, A) that there is a natural isomorphism H_{n}^{H}(X, A) ⟶^{≅} H_{n}^{G}(G ×_{H} (X, A)).

Recall that a family F of subgroups of G is a set of subgroups which is closed under conjugation and taking subgroups. Examples are the families FIN (G) of finite subgroups and VCY(G) of virtually cyclic subgroups. Given a group homomorphism ϕ: K → G and a family F of subgroups of G, define the family ϕ^{∗}F of subgroups of K by

ϕ^{∗}F := {H ⊆ K | ϕ(H) ∈ F}.

If ϕ is an inclusion of subgroups, we also write

ϕ^{∗}F = K ∩ F = {H ⊆ K | H ∈ F} = {L ∩ K | L ∈ F}.

If ψ: H → K is another group homomorphism, then

ψ^{∗}(ϕ^{∗}F) = (ϕ ∘ ψ)^{∗}F.  (1.1)

Associated to a family F there is a G-CW -complex E_{F}G (unique up to G-homotopy equivalence) with the property that the fixpoint sets (E_{F}G)^{H} are contractible for H ∈ F and empty for H ∉ F. It is called the classifying space of the family F. For more information about these spaces we refer for instance to [20], [33, pp.46].

Definition 1.2 ((Fibered) Isomorphism Conjecture). Fix an equivariant homology theory H_{∗}^{?} with values in Λ-modules for a commutative associative ring Λ. A group G together with a family of subgroups F satisfies the Isomorphism Conjecture (in the range ≤ N) if the projection pr: E_{F}G → {pt .} induces an isomorphism

H_{n}^{G}(pr): H_{n}^{G}(E_{F}G) ⟶^{≅} H_{n}^{G}({pt .})

for n ∈ ℤ (with n ≤ N).

The pair (G, F) satisfies the Fibered Isomorphism Conjecture (in the range ≤ N) if for each group homomorphism ϕ: K → G the pair (K, ϕ^{∗}F) satisfies the Isomorphism Conjecture (in the range ≤ N).

Built in into the Fibered Isomorphism Conjecture is the following obvious inheritance property which is not true in general in the non-fibered case.

Lemma 1.3. Let ϕ: K → G be a group homomorphism and let F a family of subgroups. If (G, F) satisfies the Fibered Isomorphism Conjecture 1.2, then (K, ϕ^{∗}F) satisfies the Fibered Isomorphism Conjecture 1.2.

Proof. If ψ: L → K is a group homomorphism, then ψ^{∗}(ϕ^{∗}F) = (ϕ∘ψ)^{∗}F from (1.1).

We will use several times the following result proved in [22, Theorem 2.9]

Theorem 1.4 (Transitivity Principle for equivariant homology). Suppose F ⊂ G are two families of subgroups of the group G. Let N be an integer. If for every H ∈ G and every n ≤ N the map induced by the projection

H_{n}^{H}(E_{F∩H}H) → H_{n}^{H}({pt .})

is an isomorphism, then for every n ≤ N the map induced by the up to Ghomotopy unique G-map E_{F}G → E_{G}G

H_{n}^{G}(E_{F}G) → H_{n}^{G}(E_{G}G)

is an isomorphism.

This implies the following transitivity principle for the (Fibered) Isomorphism Conjecture. At the level of spectra this transitivity principle is due to Farrell and Jones [10, TheoremA.10].

Theorem 1.5 (Transitivity Principle). Suppose F ⊆ G are two families of subgroups of G. Assume that for every element H ∈ G the group H satisfies the (Fibered) Isomorphism Conjecture for F ∩ H (in the range ≤ N).

Then (G, G) satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) if and only if (G, F) satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N).

Proof. We first prove the claim for the Fibered Isomorphism Conjecture. Consider a group homomorphism ϕ: K → G. Then for every subgroup H of K we conclude

(ϕ|_{H})^{∗}(F ∩ ϕ(H)) = (ϕ^{∗}F) ∩ H

from (1.1), where ϕ|_{H}: H → ϕ(H) is the group homomorphism induced by ϕ. For every element H ∈ ϕ^{∗}G the map

H_{p}^{H}(E_{(ϕ|_{H})^{∗}(F∩ϕ(H))}H) = H_{p}^{H}(E_{ϕ^{∗}F∩H}H) → H_{p}^{H}({pt .})

is bijective for all p ∈ ℤ ( with p ≤ N) by the assumption that that the element ϕ(H) ∈ G satisfies the Fibered Isomorphism Conjecture for F ∩ ϕ(H) (in the range ≤ N). Hence by Theorem 1.4 applied to the inclusion ϕ^{∗}F ⊆ ϕ^{∗}G of families of subgroups of K we get an isomorphism

H_{p}^{K}(E_{ϕ^{∗}F}K) ⟶^{≅} H_{p}^{K}(E_{ϕ^{∗}G}K)

(for p ≤ N). Therefore the map H_{p}^{K}(E_{ϕ^{∗}F}K) → H_{p}^{K}({pt .}) is bijective for all p ∈ ℤ (with p ≤ N) if and only if the map H_{p}^{K}(E_{ϕ^{∗}G}K) → H_{p}^{K}({pt .}) is bijective for all p ∈ ℤ (with p ≤ N).

If we specialize this argument to ϕ = id_{G} then only the Isomorphism Conjecture for F ∩ H (in the range ≤ N) is needed and this proves the claim for the Isomorphism Conjecture.

As an easy application we see that we can always enlarge the family in the Fibered Isomorphism Conjecture. Note that this argument is not valid for the Isomorphism Conjecture.

Lemma 1.6. Let G be a group and let F ⊂ G be families of subgroups of G. Suppose that G satisfies the Fibered Isomorphism Conjecture 1.2 (in the range n ≤ N) for the family F.

Then G satisfies the Fibered Isomorphism Conjecture 1.2 (in the range n ≤ N) for the family G.

Proof. We want to use Theorem 1.5. Therefore we have to show for each subgroup K in G that it satisfies the Fibered Isomorphism Conjecture for F ∩ K. If i: K → G is the inclusion, then i^{∗}F = F ∩ K. Now apply Lemma 1.3.

We have defined in the introduction for every class F of finite groups the class F′ consisting of those virtually cyclic groups V for which there exists an extension 1 → ℤ → V → F → 1 for a group F ∈ F or for which V ∈ F holds.

Theorem 1.7. Let F be a class of finite groups closed under isomorphism and taking subgroups. Suppose that every finite group F satisfies the Fibered Isomorphism Conjecture (in the range ≤ N) with respect to the family F(F) = {H ⊆ F | H ∈ F}. Let G be a group.

Then G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) with respect to the family F′(G) if and only if G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) with respect to the family VCY(G) = {H ⊆ G | H ∈ VCY}.

Proof. Consider V ∈ VCY. Because of Theorem 1.5 we have to show that V satisfies the Fibered Isomorphism Conjecture (in the range ≤ N) for the family F′(V) = F′(G) ∩ V. If V is finite, then the claim follows from the assumptions. It remains to treat the case, where V can be written as an extension 1 → ℤ → V ⟶^{p} F → 1 for a finite group F.

Since F satisfies the Fibered Isomorphism Conjecture (in the range ≤ N) for F by assumption, V satisfies the Fibered Isomorphism Conjecture (in the range ≤ N) for p^{∗}F by Lemma 1.3. Obviously p^{∗}F(F) ⊆ F′(V) but in general these two families of subgroups of V do not agree. Because of Lemma 1.6 V satisfies the Fibered Isomorphism Conjecture (in the range ≤ N) for F′(V).

The Fibered Isomorphism Conjecture is also well behaved with respect to finite intersections of families of subgroups.

Lemma 1.8. Let G be a groups and F and G families of subgroups. Suppose that G satisfies the Fibered Isomorphism Conjecture (in the range n ≤ N) for both F and G.

Then G satisfies the Fibered Isomorphism Conjecture (in the range n ≤ N) for F ∪ G and the Fibered Isomorphism Conjecture (in the range n ≤ N − 1) for F ∩ G.

Proof. We conclude from Lemma 1.6 that G satisfies the Fibered Isomorphism Conjecture (in the range n ≤ N) for F ∪ G. Next we prove the claim for F ∩ G.

Consider a group homomorphism ϕ: K → G. Choose G-CW -models E_{F∩G}G, E_{F}G and E_{G}G such that E_{F∩G}G is a G-CW -subcomplex of both E_{F}G and E_{G}G. This can be arranged by mapping cylinder constructions. Define a G-CW -complex

X = E_{F}G ∪_{E_{F∩G}G} E_{G}G.

For any subgroup H ⊆ G we get

X^{H} = (E_{F}G)^{H} ∪_{(E_{F∩G}G)^{H}} (E_{G}G)^{H}.

If (E_{F}G)^{H} and (E_{G}G)^{H} are empty, the same is true for X^{H}. If (E_{F}G)^{H} is empty, then (E_{G}G)^{H} = X^{H}. If (E_{G}G)^{H} is empty, then (E_{F}G)^{H} = X^{H}. If (E_{F}G)^{H}, (E_{G}G)^{H} and (E_{F∩G}G)^{H} are contractible, the same is true for X^{H}. Hence X is a model for E_{F∪G}G. If we apply restriction with ϕ, we get a decomposition of E_{ϕ^{∗}F∪ϕ^{∗}G}K = ϕ^{∗}E_{F∪G}G as the union of E_{ϕ^{∗}F}K = ϕ^{∗}E_{F}G and E_{ϕ^{∗}G}K = ϕ^{∗}E_{G}G such that the intersection of E_{ϕ^{∗}F}K and E_{ϕ^{∗}G}K is E_{ϕ^{∗}(F∩G)}K = ϕ^{∗}E_{F∩G}G. Now the claim follows from the Mayer-Vietoris sequence for E_{ϕ^{∗}F∪ϕ^{∗}G}K and the Five-Lemma.

2. General Induction Theorems

Let G be a finite group. Let G-FSETSbe the category of finite G-sets. Morphisms are G-maps. Let Λ be an associative commutative ring with unit. Denote by Λ-MOD the abelian category of Λ-modules. A bi-functor M from G-FSETS to Λ-MOD consists of a covariant functor

M_{∗}: G-FSETS→ Λ-MOD

and a contravariant functor

M^{∗}: G-FSETS→ Λ-MOD

which agree on objects.

Definition 2.1 (Mackey functor). A Mackey functor M for G with values in Λ-modules is a bifunctor from G-FSETSto Λ-MOD such that

• Double Coset formula For any cartesian square of finite G-sets

S ⟶^{\overline{f_{1}}} S_{1}
\overline{f_{2}}↓  ↓^{f_{1}}
S_{2} ⟶_{f_{2}} S_{0}

the diagram

M(S) ⟶^{M_{∗}(\overline{f_{1}})} M(S_{1})
M^{∗}(\overline{f_{2}})↑  ^{M^{∗}(f_{1})}↑
M(S_{2}) ⟶^{M_{∗}(f_{2})} M(S_{0})

commutes.

• Additivity Consider two finite G-sets S_{0} and S_{1}. Let i_{k}: S_{k} → S_{0} ⨿ S_{1} for k = 0, 1 be the inclusion. Then the map

M^{∗}(i_{0}) × M^{∗}(i_{1}): M(S_{0} ⨿ S_{1}) → M(S_{0}) × M(S_{1})

is bijective.

One easily checks that the condition Additivity is equivalent to the requirement that

M_{∗}(i_{0}) ⊕ M_{∗}(i_{1}): M(S_{0}) ⊕ M(S_{1}) → M(S_{0} ⨿ S_{1})

is bijective since the double coset formula implies that (M^{∗}(i_{0}) × M^{∗}(i_{1})) ∘ (M_{∗}(i_{0}) ⊕ M_{∗}(i_{1})) is the identity.

Let M be a Mackey functor. Let S be a finite non-empty G-set. Then we get another Mackey functor M_{S} by (M_{S})_{∗}(T) = M_{∗}(S × T) and (M_{S})^{∗}(T) = M^{∗}(S × T). The projection pr: S × T → T defines natural transformations of bifunctors from from G-FSETSto Λ-MOD

θ_{S}: M_{S} → M,  θ_{S}(T) = M_{∗}(pr);
θ^{S}: M → M_{S},  θ^{S}(T) = M^{∗}(pr).

We call M S-projective if θ_{S} is split surjective as a natural transformation of bifunctors. We call M S-injective if θ^{S} is split injective as a natural transformation of bifunctors.

Given a Mackey functor M, we can associate to it the Λ-chain complex

… ⟶^{c_{4}} M(S^{3}) ⟶^{c_{3}} M(S^{2}) ⟶^{c_{2}} M(S^{1}) ⟶^{c_{1}} M(S^{0}) → 0 → 0 → …  (2.2)

and the Λ-cochain complex

… ⟵^{c^{3}} M(S^{3}) ⟵^{c^{2}} M(S^{2}) ⟵^{c^{1}} M(S^{1}) ⟵^{c^{0}} M(S^{0}) ← 0 ← 0 ← ….  (2.3)

Here S^{n} = ∏_{i=1}^{n} S for n ≥ 1 and S^{0} = G/G. If pr_{i}^{n}: S^{n} → S^{n−1} for i = 1, 2, … n is the projection omitting the i-th coordinate, then c_{n} = ∑_{i=1}^{n}(−1)^{i} · M_{∗}(pr_{i}^{n}) and c^{n} = ∑_{i=1}^{n+1}(−1)^{i} · M^{∗}(pr_{i}^{n+1}).

The elementary proof of the following result can be found for instance in [32, Proposition 6.1.3 and Proposition 6.1.6]. The original sources are the papers of Dress [7] and [8].

Lemma 2.4. Let M be a Mackey functor for the finite group G with values in Λ-modules and let S be a finite G-set. Then

(a) The following statements are equivalent:

(i) M is S-projective;

(ii) M is S-injective;

(iii) M is a direct summand of M_{S} as a bifunctor from G-FSETSto Λ-MOD;

(b) M_{S} is always S-injective and S-projective;

(c) If M is S-projective, then the Λ-chain complex (2.2) is contractible.

If M is S-injective, then the Λ-cochain complex (2.3) is contractible.

Let Λ be an associative commutative ring with unit. Let Λ-MOD be the abelian category of Λ-modules. For a small category C let ΛC-MOD be the abelian category of contravariant ΛC-modules, i.e. objects are contravariant functors C → Λ-MOD and morphisms are natural transformations. In the sequel we use the notation of [18, Sections 9 and 17], where more information about the abelian category ΛC-MOD and its homological algebra can be found. If T is a set, then ΛT or Λ(T) denote the free Λ-module with the set T as basis. The orbit category Or(G) is the small category whose objects are homogeneous G-spaces G/H and whose morphisms are G-maps.

Lemma 2.5. Let M be a Mackey functor for the finite group G with values in Λ-modules and let S be a finite non-empty G-set. Suppose that M is S-injective, or, equivalently, that M is S-projective. Let F(S) be the family of subgroups of H ⊆ G with S^{H} ≠ ∅. Let \underline{Λ}_{F(S)} be the contravariant ΛOr(G)-module which sends G/H to Λ if H belongs to F(S) and to zero otherwise and which sends a morphism G/H → G/K to id: Λ → Λ if both H and K belong to F(S) and to the zero map otherwise.

Then there are natural Λ-isomorphisms

Tor_{p}^{ΛOr _{F} (G)}(\underline{Λ}_{F(S)}, M) ⟶^{≅} \{ \begin{matrix}M(G/G) & p = 0; \\ 0 & p ≥ 1. \end{matrix}
Ext_{ΛOr _{F} (G)}^{p}(\underline{Λ}_{F(S)}, M) ⟶^{≅} \{ \begin{matrix}M(G/G) & p = 0; \\ 0 & p ≥ 1. \end{matrix}

Proof. Let T be a set, possibly empty. We can assign to it a Λ-chain complex C_{∗}(T) which has as n-th Λ-chain module Λ(T^{n+1}) and whose n-th differential c_{n}: Λ(T^{n+1}) → Λ(T^{n}) is ∑_{i=1}^{n+1}(−1)^{i} · Λ(pr_{i}^{n+1}) for pr_{i}^{n+1} : T^{n+1} → T^{n} the projection given by leaving out the i-th coordinate. There are in T natural isomorphisms

H_{p}(C_{∗}(T)) ⟶^{≅} \{ \begin{matrix}Λ & p = 0 and T ≠ ∅; \\ 0 & p ≥ 1 or T = ∅. \end{matrix}

Of course the isomorphism H_{0}(C_{∗}(T)) ⟶^{≅} Λ is induced by the augmentation Λ-homomorphism Λ[T] → Λ for T ≠ ∅. Fix an element t_{0} ∈ T. The maps

Λ(T^{n}) → Λ(T^{n+1}), (t_{1}, t_{2}, … t_{n}) ↦ − (t_{0}, t_{1}, t_{2}, … t_{n})

yields the necessary chain homotopies.

Actually, this chain complex belongs to the fat realization of the nerve of the category which has T as set of objects and for which the set of morphisms between two objects consists of exactly one element. Notice that every object t_{0} is both initial and terminal.

Next we define a ΛOr(G)-chain complex P_{∗} by assigning to G/H the Λ-chain complex C_{∗}(map_{G}(G/H, S)) for the given finite G-set S. We obtain a natural identification of ΛOr(G)-modules

H_{0}(C_{∗}) ⟶^{≅} \underline{Λ}_{F(S)}.

Obviously each ΛOr(G)-chain module P_{n} is finitely generated free and hence in particular projective. Hence P_{∗} is a projective ΛOr(G)-resolution of \underline{Λ}_{F(S)}. Thus we get by definition

Tor_{p}^{ΛOr _{F} (G)}(\underline{Λ}_{F(S)}, M) = H_{p} (P_{∗} ⊗_{ΛOr (G)} M_{∗});
Ext_{ΛOr _{F} (G)}^{p}(\underline{Λ}_{F(S)}, M) = H^{p} (Hom_{ΛOr (G)}(P_{∗}, M^{∗})) .

There is an obvious identification of Λ-chain complexes P_{∗} ⊗_{ΛOr (G)} M_{∗} with the Λ-chain complex (2.2) if we replace M(S^{0}) by zero and put M(S^{n}) as (n−1)-th Λ-chain module for n ≥ 1. Analogously there is an obvious identification of Λ-chain complexes Hom_{ΛOr (G)}(P_{∗}, M^{∗}) with the Λ-cochain complex (2.3) if we replace M(S^{0}) by zero and put M(S^{n}) as (n−1)-th Λ-cochain module for n ≥ 1. Now Lemma 2.5 follows from Lemma 2.4 (c)

Next we deal with the question whether a given Mackey functor is S-projective or, equivalently, S-injective.

Let M, N and L be bi-functors for the finite group G with values in Λmodules. A pairing

M × N → L

is a family of Λ-bilinear maps

μ(S): M(S) × N(S) → L(S), (m, n) ↦ μ(m, n) = m · n

indexed by the objects S of G-FSETSsuch that for every morphism f: S → T in G-FSETSwe have

\begin{aligned}L^{∗}(f)(x · y) & = & M^{∗}(f)(x) · N^{∗}(f)(y), & x ∈ M(T), y ∈ N(T); \\ x · N_{∗}(f)(y) & = & L_{∗}(f)(M^{∗}(f)(x) · y), & x ∈ M(T), y ∈ N(S); \\ M_{∗}(f)(x) · y & = & L_{∗}(f)(x · N^{∗}(f)(y)), & x ∈ M(S), y ∈ N(T). \end{aligned} (2.6)

Definition 2.7 (Green functor). A Green functor for the finite group G with values in Λ-modules is a Mackey functor U together with a pairing

μ: U × U → U

and a choice of elements 1_{S} ∈ U(S) for each finite G-set S such that for each finite G-set S the pairing μ(S): U(S) × U(S) → U(S) and the element 1_{S} determine the structure of an associative Λ-algebra with unit on U(S). Moreover, it is required that U^{∗}(f)(1_{T}) = 1_{S} for every morphism f: S → T in G-FSETS.

A (left) U-module M is a Mackey functor for the finite group G with values in Λ-modules together with a pairing

ν: U × M → M

such that for every finite G-set S the pairing ν(S): U(S) × M(S) → M(S) defines the structure of a U(S)-module on M(S), where 1_{S} acts as id_{M(S)}.

The proof of the next result can be found for instance in [32, Theorem 6.2.2].

Theorem 2.8 (Criterion for S-projectivity). Let G be a finite group and let Λ be an associative commutative ring with unit. Let S be a finite G-set. Let U be a Green functor. Then the following statements are equivalent:

(a) The projection pr: S → G/G induces an epimorphism U_{∗}(pr): U(S) → U(G/G);

(b) U is S-injective;

(c) Every U-module is S-injective.

The results of this section have the following application to equivariant homology theories.

Theorem 2.9 (Criterion for induction for G-homology theories). Let G be a finite group and F be a family of subgroups of G. Let H_{∗}^{G} be a G-homology theory with values in Λ-modules for an associative commutative ring Λ with unit. Suppose that the following conditions are satisfied:

• There exists a Green functor U for the finite group G with values in Λmodules such that the Λ-homomorphism

⨁_{H∈F} U(pr_{H}): ⨁_{H∈F} U(G/H) → U(G/G)

is surjective, where pr_{H}: G/H → G/G is the projection.

• For every n ∈ ℤ there is a (left) U-module M such that the covariant functor M_{∗}: G-FSETS→ Λ-MOD is naturally equivalent to the covariant functor

H_{n}^{G}: G-FSETS→ Λ-MOD , S ↦ H_{n}^{G}(S).

Then the projection pr: E_{F}G → G/G induces for all n ∈ ℤ a Λ-isomorphism

H_{n}^{G}(pr): H_{n}^{G}(E_{F}G) ⟶^{≅} H_{n}^{G}(G/G).  (2.10)

and the canonical map

colim_{Or _{F} (G)} H_{n}^{G}(G/?) ⟶^{≅} H_{n}^{G}(G/G)  (2.11)

is bijective, where Or_{F}(G) ⊂ Or(G) is the full subcategory of the orbit category whose objects are homogeneous spaces G/H with H ∈ F.

Proof. Let S be the G-set ⨿_{H∈F}G/H. The first condition together with Lemma 2.4 (a) and Theorem 2.8 implies that M is S-projective. We conclude from the second condition and Lemma 2.5 that there is a canonical Λ-isomorphism

Tor_{p}^{ΛOr _{F} (G)}(\underline{Λ}_{F(S)}, H_{q}^{G}(G/?)) ⟶^{≅} \{ \begin{matrix}H_{q}^{G}(G/G) & p = 0; \\ 0 & p ≥ 1, \end{matrix}

for all q ∈ ℤ. The cellular ΛOr(G)-chain complex of E_{F}G is a projective ΛOr(G)-resolution of \underline{Λ}_{F(S)}. Hence Tor_{p}^{ΛOr _{F} (G)}(\underline{Λ}_{F(S)}, H_{q}^{G}(G/?) agrees with

the Bredon homology H_{p}^{ΛOr (G)}(E_{F}G; H_{q}^{G}(G/?)). But this is exactly the E^{2}- term in the equivariant Atiyah-Hirzebruch spectral sequence which converges to H_{p+q}^{G}(E_{F}G). This implies that the E^{2}-term is concentrated on the y-axis and the spectral sequence collapses. Hence the edge homomorphism yields an isomorphism

H_{n}^{G}(E_{F}G) ⟶^{≅} H_{n}^{G}(G/G)

for all n ∈ ℤ which can easily be identified with the Λ-map H_{n}^{G}(pr). There is a natural identification

Tor_{0}^{ΛOr _{F} (G)}(\underline{Λ}_{F(S)}, H_{q}^{G}(G/?)) ⟶^{≅} colim_{Or _{F} (G)} H_{q}^{G}(G/?).

This finishes the proof of Theorem 2.9.

Remark 2.12. The bijectivity of the map (2.11) is a consequence of the exactness of the complex (2.2) at M(S^{0}) and M(S^{1}). It is not hard to see that this map factors as

colim_{Or _{F} (G)} H_{n}^{G}(G/?) → H_{n}^{G}(E_{F}G) → H_{n}^{G}(G/G)

and therefore (2.10) is onto. In order to get our applications to assembly maps it is important that (2.10) is in fact an isomorphism. (Surjectivity alone is not helpful, since the Transitivity Principle 1.5 does not apply to surjections.) The proof of the bijectivity of (2.10) is based on the result due to Dress that the complex (2.2) is not only exact at M(S^{0}) and M(S^{1}) but is contractible.

The dual version of the map (2.11) leads to the following induction result in [9] that for a group Γ (in their setting a Bieberbach group) which can be written as an extension 1 → Γ_{0} → Γ ⟶^{p} G → 1 for a finite group G the map

L_{n}(ℤΓ) ⊗_{ℤ} ℚ ⟶^{≅} lim L_{n}(ℤ[p^{−1}(C)]) ⊗_{ℤ} ℚ

is bijective, where the inverse limit runs over the cyclic subgroups of G. We expect that there is a dual version of (2.10) which leads to kind of dual version of induction theorems for groups mapping surjectively to finite groups.

3. The Swan Group as a Functor on Groupoids

In this section we will construct Green functors and modules over them that will be used in the application of Theorem 2.9 in the proof of Theorem 0.1 in Section 4.

Let G be a small groupoid. Our main example is the transport groupoid G^{G}(S) of a G-set S. The set of objects is given by the set S itself. The set of morphisms from s_{1} to s_{2} consists of those elements g ∈ G which satisfy gs_{1} = s_{2}. Composition comes from the multiplication in G. A G-map f: S → T induces a functor G^{G}(f): G^{G}(S) → G^{G}(T). Let R be an associative ring with unit. Denote by RG-FGPMOD the category of contravariant functors from G to finitely generated projective R-modules and by K_{n}(RG) its K-theory. Let func(G, ℤ-FGMOD) and func(G, ℤ-FGFMOD) respectively be the category of contravariant functors from G to the category ℤ-FGMOD of finitely generated ℤ-modules and to the category ℤ-FGFMOD of finitely generated free ℤ-modules respectively. Let Sw(G) and Sw^{f}(G) respectively be the K_{0}-group of func(G, ℤ-FGMOD) and func(G, ℤ-FGFMOD) respectively. The forgetful map

Sw^{f}(G) ⟶^{≅} Sw(G)

is a bijection. This is proved for groups in [25, page 890] and carries easily over to groupoids. We will concentrate our discussion mostly on Sw^{f}(G).

Given a contravariant ℤG-module M and a contravariant RG-module N, let M ⊗_{ℤ} N be the contravariant RG-module which assigns to an object c the Rmodule M(c) ⊗_{ℤ} N(c). If M belongs to func(G, ℤ-FGFMOD), then the functor M ⊗_{ℤ} − is exact. If M and N belong to func(C, ℤ-FGFMOD), then M ⊗_{ℤ} N belongs to func(C, ℤ-FGFMOD). Hence ⊗_{ℤ} induce a pairing

Sw^{f}(G) × Sw^{f}(G) → Sw^{f}(G).  (3.1)

With this pairing Sw^{f}(G) is a commutative associative ring with the class of the constant contravariant ℤC-module with value ℤ as unit.

If M belongs to func(G, ℤ-FGFMOD), then M ⊗_{ℤ} − sends finitely generated projective RG-modules to finitely generated projective RG-modules. This is well-known if the groupoid C has only one object, i.e. in case of group rings, and hence holds also for a groupoid G. Hence we get for a groupoid a pairing

Sw^{f}(G) × K_{n}(RG) → K_{n}(RG)  (3.2)

which turns K_{n}(RG) into a Sw^{f}(G)-module.

Given a functor F: C → D, induction defines an functor

RF_{∗}: RC-MOD → RD-MOD

and restriction defines a functor

RF^{∗}: RD-MOD → RC-MOD .

The induction functor is given by − ⊗_{RC} R mor_{D}(??, F (?)) and restriction by − ⊗_{RD} R mor_{D}(F (?),??) for the RC-RD-bimodule R mor_{D}(??, F (?)) and the RD-RC-bimodule R mor_{D}(F (?),??), where ? runs through the objects in C and ?? through the objects in D (see [18, 9.15 on page 166]). We have the obvious equalities (F_{2} ∘ F_{1})_{∗} = (F_{2})_{∗} ∘ (F_{1})_{∗} and (F_{2} ∘ F_{1})^{∗} = (F_{1})^{∗} ∘ (F_{2})^{∗} for functors F_{1} : C_{1} → C_{2} and F_{2} : C_{2} → C_{3}.

Let F: C → D be a functor of groupoids. Induction with F sends finitely generated projective RC-modules to finitely generated projective RD-modules and respects direct sums. Hence induction induces for n ∈ ℤ homomorphisms of abelian groups

F_{∗}: K_{n}(RC) → K_{n}(RD).

Restriction with F yield exact functors func(D, ℤ-FGFMOD) → func(C, ℤ-FGFMOD) and func(D, ℤ-FGMOD) → func(C, ℤ-FGMOD) and thus ring homomorphisms

F^{∗}: Sw^{f}(D) → Sw^{f}(C);
F^{∗}: Sw(D) → Sw(C).

We call F admissible if for each object c ∈ C the group homomorphism aut_{C}(c) → aut_{D}(F (c)) induced by F is injective and its image has finite index and the map π_{0}(F): π_{0}(C) → π_{0}(D) has the property that the preimage of any element in π_{0}(D) is finite. Note that if G is a (not necessary finite) group and f : S → T is a map of finite G-sets then G^{G}(f) is admissible. For admissible F induction and restriction do also induce homomorphisms of abelian groups

F^{∗}: K_{n}(RD) → K_{n}(RC);
F_{∗}: Sw^{f}(C) → Sw^{f}(D);
F_{∗}: Sw(C) → Sw(D).

The various claims above are well-known for groups, i.e. groupoids with one object and therefore carry easily over to groupoids.

Let E, F: C → D be functors which are naturally equivalent. Then we get the following equalities of homomorphisms:

E^{∗} = F^{∗}: Sw^{f}(D) → Sw^{f}(C);
E^{∗} = F^{∗}: Sw(D) → Sw(C);
E_{∗} = F_{∗}: K_{n}(RC) → K_{n}(RD)

and E is admissible if and only if F is and in this case also the following homomorphisms agree

E_{∗} = F_{∗}: Sw^{f}(C) → Sw^{f}(D);
E^{∗} = F^{∗}: Sw(C) → Sw(D);
E^{∗} = F^{∗}: K_{n}(RD) → K_{n}(RC).

Let G be a groupoid with a finite set π_{0}(G) of components. For a component C ∈ π_{0}(G) let i_{C}: C → G be the inclusion. Then we obtain to one another inverse isomorphisms

⨁_{C∈π_{0}(G)}(i_{C})_{∗}: ⨁_{C∈π_{0}(G)} K_{n}(RC) ⟶^{≅} K_{n}(RG);

∏_{C∈π_{0}(G)}(i_{C})^{∗}: K_{n}(RG) ⟶^{≅} ∏_{C∈π_{0}(G)} K_{n}(RC)

and to one another inverse isomorphisms

⨁_{C∈π_{0}(G)}(i_{C})_{∗}: ⨁_{C∈π_{0}(G)} Sw^{f}(C) ⟶^{≅} Sw^{f}(G);

∏_{C∈π_{0}(G)}(i_{C})^{∗}: Sw^{f}(G) ⟶^{≅} ∏_{C∈π_{0}(G)} Sw^{f}(C)

and similarly for Sw. If G is a connected groupoid and x an object in G, then the inclusion i: aut_{G}(x) → G induces to another inverse isomorphisms

i_{∗}: K_{n}(R[aut_{G}(x)]) ⟶^{≅} K_{n}(RG);
i^{∗}: K_{n}(RG) ⟶^{≅} K_{n}(R[aut_{G}(x)]),

and analogously for Sw^{f} and Sw.

Lemma 3.3. Consider a cartesian square of G-sets

S ⟶^{\overline{f_{1}}} S_{1}
\overline{f_{2}}↓  ↓^{f_{1}}
S_{2} ⟶_{f_{2}} S_{0}

Then the following diagram of functors commutes up to natural equivalence

RG^{G}(S)-MOD ⟶^{RG^{g}(\overline{f_{1}})_{∗}} RG^{G}(S_{1})-MOD
RG^{G}(\overline{f_{2}})^{∗}↑  ^{RG^{G}(f_{1})^{∗}}↑
RG^{G}(S_{2})-MOD ⟶^{RG^{g}(f_{2})_{∗}} RG^{G}(S_{0})-MOD

Proof. The composition

RG^{G}(S_{2})-MOD ⟶^{RG^{g}(f_{2})_{∗}} RG^{G}(S_{0})-MOD ⟶^{RG^{G}(f_{1})^{∗}} RG^{G}(S_{1})-MOD

is given by

− ⊗_{RG^{G}(S_{2})} R mor_{G^{G}(S_{0})}(!, G^{G}(f_{2})(??)) ⊗_{RG^{G}(S_{0})} R mor_{G^{G}(S_{0})}(G^{G}(f_{1})(?),!), where ?, ?? and ! run through objects in G^{G}(S_{1}), G^{G}(S_{2}) and G^{G}(S_{0}). The composition

RG^{G}(S_{2})-MOD ⟶^{RG^{g}(\overline{f_{2}})^{∗}} RG^{G}(S)-MOD ⟶^{RG^{G}(\overline{f_{1}})_{∗}} RG^{G}(S_{1})-MOD

is given by

− ⊗_{RG^{G}(S_{2})} R mor_{G^{G}(S_{2})}(G^{G}(\overline{f_{2}})(!!),??) ⊗_{RG^{G}(S)} R mor_{G^{G}(S_{1})}(?, G^{G}(\overline{f_{1}})(!!)),

where ?, ?? and !! run through objects in G^{G}(S_{1}), G^{G}(S_{2}) and G^{G}(S). Hence it suffices to construct an isomorphism of RG^{G}(S_{2})-RG^{G}(S_{1})-bimodules

R mor_{G^{G}(S_{0})}(!, G^{G}(f_{2})(??)) ⊗_{RG^{G}(S_{0})} R mor_{G^{G}(S_{0})}(G^{G}(f_{1})(?),!)

⟶^{≅} R mor_{G^{G}(S_{2})}(G^{G}(\overline{f_{2}})(!!),??) ⊗_{RG^{G}(S)} R mor_{G^{G}(S_{1})}(?, G^{G}(\overline{f_{1}})(!!)).

For this purpose it suffices to construct an isomorphism of G^{G}(S_{2})-G^{G}(S_{1})-bisets

mor_{G^{G}(S_{0})}(!, G^{G}(f_{2})(??)) ⊗_{G^{G}(S_{0})} mor_{G^{G}(S_{0})}(G^{G}(f_{1})(?),!)

⟶^{≅} mor_{G^{G}(S_{2})}(G^{G}(\overline{f_{2}})(!!),??) ⊗_{G^{G}(S)} mor_{G^{G}(S_{1})}(?, G^{G}(\overline{f_{1}})(!!)) (3.4)

where ⊗_{G^{G}(S)} is now to be understood with respect to the category of sets. There is an obvious bijection of G^{G}(S_{2})-G^{G}(S_{1})-bisets

mor_{G^{G}(S_{0})}(!, G^{G}(f_{2})(??)) ⊗_{G^{G}(S_{0})} mor_{G^{G}(S_{0})}(G^{G}(f_{1})(?),!)

⟶^{≅} mor_{G^{G}(S_{0})}(G^{G}(f_{1})(?), G^{G}(f_{2})(??))

which sends u ⊗ v to u ∘ v. Its inverse sends w: G^{G}(f_{1})(?) → G^{G}(\overline{f_{2}})(??) to id_{G^{G}(\overline{f_{2}})(??)} ⊗w. There is a map of G^{G}(S_{2})-G^{G}(S_{1})-bisets

mor_{G^{G}(S_{2})}(G^{G}(\overline{f_{2}})(!!),??) ⊗_{G^{G}(S)} mor_{G^{G}(S_{1})}(?, G^{G}(\overline{f_{1}})(!!))

⟶^{≅} mor_{G^{G}(S_{0})}(G^{G}(f_{1})(?), G^{G}(f_{2})(??))

which sends u ⊗ v to G^{G}(f_{2})(u) ∘ G^{G}(f_{1})(v). This makes sense, since G^{G}(f_{1}) ∘ G^{G}(\overline{f_{1}}) and G^{G}(f_{2}) ∘ G^{G}(\overline{f_{2}}) coincide. In order to show that this map is a bijection of bisets, we need the assumption that the square of G-sets appearing in Lemma 3.3 is cartesian. Namely, we construct the inverse

mor_{G^{G}(S_{0})}(G^{G}(f_{1})(?), G^{G}(f_{2})(??))

⟶^{≅} mor_{G^{G}(S_{2})}(G^{G}(\overline{f_{2}})(!!),??) ⊗_{G^{G}(S)} mor_{G^{G}(S_{1})}(?, G^{G}(\overline{f_{1}})(!!))

as follows. Consider a morphism w: G^{G}(f_{1})(?) → G^{G}(\overline{f_{2}})(??). It is given by an element g ∈ G satisfying g · f_{1}(?) = f_{2}(??). The element (g·?,??) ∈ S_{1} × S_{2} defines a unique element !! ∈ S since f_{1}(g·?) = f_{2}(??). We have \overline{f_{1}}(!!) = g·? and \overline{f_{2}}(!!) =??. The element g defines a morphism u: ? → G^{G}(\overline{f_{1}})(!!). Now define the image of w by id_{??} ⊗u. One easily checks that these two maps of bisets are inverse to one another. The desired map of bisets (3.4) is given by the two bijections of bisets above. This finishes the proof of Lemma 3.3.

We can now define the Green functors that we will need. Let ϕ : K → G be a group homomorphism whose target is a finite group G. Then S ↦ Sw^{f}(G^{G}(S)) and S ↦ Sw^{f}(G^{K}(ϕ^{∗}S)) are Green functors. (Here ϕ^{∗}S denotes the finite Kset obtained by restricting the finite G-set with ϕ.) The covariant functorial structure comes from induction and the contravariant functorial structure from restriction with G^{G}(f) and G^{K}(ϕ^{∗}f). The double coset formula for Sw^{f}(G^{G}(−)) follows directly from Lemma 3.3 and for Sw^{f}(G^{K}(ϕ^{∗}−) from Lemma 3.3 using the fact that ϕ^{∗} sends a cartesian square of finite G-sets to a cartesian square of finite K-sets. The required pairings come from (3.1). We leave the verification of the conditions (2.6) to the reader, they follow from certain natural equivalences of functors on the level of RG-modules.

The group homomorphism ϕ induces a functor G^{ϕ}(S): G^{K}(ϕ^{∗}S) → G^{G}(S) for every finite F -set which is natural in S. Restriction with it induces a morphisms of Green functors for G

ϕ^{∗}: Sw^{f}(G^{G}(−)) → Sw^{f}(G^{K}(ϕ^{∗}−).

Analogously one constructs a left Sw^{f}(G^{K}(ϕ^{∗}−)-module K_{n}(G^{K}(ϕ^{∗}−). The Mackey structure on K_{n}(G^{K}(ϕ^{∗}−) comes from induction and restriction and the desired pairing from (3.2). Using ϕ^{∗}: Sw^{f}(G^{G}(−)) → Sw^{f}(G^{K}(ϕ^{∗}−) we obtain a left Sw^{f}(G^{G}(−))-module structure on K_{n}(G^{K}(ϕ^{∗}−).

If ℂ ⊆ R, we can replace Sw^{f}(G) by the version Sw(ℂG), where one uses modules over ℂ instead of ℤ. If H is a finite group, then Sw(ℂH) agrees with the complex representation ring of H.

4. Proof of the Main Result for Algebraic K-Theory

Let R be an associative ring with unit. Let H_{∗}^{?}(−; K_{R}) be the equivariant homology theory associated to the covariant functor K_{R} from the category GROUPOIDS of small groupoids to the category of Ω-SPECTRAwhich sends a groupoid G to the (non-connective) algebraic K-theory spectrum of the category of finitely generated projective contravariant RG-modules. Notice that

K_{n}(RG) = π_{n}(K_{R}(G))

and

H_{n}^{G}(G/H; K_{R}) = H_{n}^{H}({pt .}; K_{R}) = K_{n}(RH)

holds for all n ∈ ℤ. (See [22, Chapter 6] for more details). Recall that the Farrell-Jones Conjecture for algebraic K-theory and a group G says that the projection pr: E_{VCY}G → G/G induces for all n ∈ ℤ an isomorphism

H_{n}^{G}(pr; K_{R}): H_{n}^{G}(E_{VCY}G; K_{R}) ⟶^{≅} H_{n}^{G}(G/G; K_{R}) = K_{n}(RG).

Let Λ be a commutative associative ring with unit such that Λ is flat as a ℤmodule, or, equivalently, Λ is torsionfree as an abelian group. Let ϕ: K → G be a group homomorphism. Then X ↦ Λ ⊗_{ℤ} H_{∗}^{K}(ϕ^{∗}X; K_{R}) defines a G-homology theory, where for a G-CW -complex X we denote by ϕ^{∗}X the K-CW -complex obtained from X by restriction with ϕ.

Lemma 4.1. Let F be a class of finite groups closed under isomorphism and taking subgroups. Let ϕ: K → G be a group homomorphism with a finite group G as target.

Then the G-homology theory Λ ⊗_{ℤ} H_{∗}^{K}(ϕ^{∗}−; K_{R}) satisfies the assumptions appearing in Theorem 2.9 for the family F(G) = {H ⊆ G | H ∈ F} in the following cases:

(a) F is the class H of hyperelementary groups and Λ = ℤ;

(b) F is the class E of elementary groups and Λ = ℤ, provided ℂ ⊆ R;

(c) For a given prime p the family F is the class H_{p} of p-hyperelementary groups and Λ = ℤ_{(p)};

(d) For a given prime p the family F is the class E_{p} of p-elementary groups and Λ = ℤ_{(p)}, provided ℂ ⊆ R;

(e) F is the class FCY of finite cyclic groups and Λ = ℚ.

Proof. In Section 3 we have constructed the Green functor Λ⊗Sw^{f}(G^{G}(−)) and the Λ ⊗ Sw^{f}(G^{G}(−))-module Λ ⊗ K_{n}(RG^{K}(ϕ^{∗}−)). There is a natural equivalence of covariant functors G-FSETS→ ℤ − MOD from K_{n}(RG^{K}(ϕ^{∗}−)) to H_{n}^{G}(ϕ^{∗}−; K_{R}). Hence it remains to check the following

(a) The map coming from induction with respect to the various inclusions

H ⊆ G

⨁_{H∈H} Sw^{f}(H) → Sw^{f}(G)

is surjective;

(b) The map coming from induction with respect to the various inclusions

H ⊆ G

⨁_{H∈E} Sw(ℂH) → Sw(ℂG)

is surjective;

(c) For a given prime p the map coming from induction with respect to the various inclusions H ⊆ G

⨁_{H∈H_{p}} Sw^{f}(H)_{(p)} → Sw^{f}(G)_{(p)}

is surjective; (d) For a given prime p the map coming from induction with respect to the

various inclusions H ⊆ G

⨁_{H∈E_{p}} Sw(ℂH)_{(p)} → Sw(ℂG)_{(p)}

is surjective;

(e) The map coming from induction with respect to the various inclusions

H ⊆ G

⨁_{H∈FCY} ℚ ⊗_{ℤ} Sw^{f}(H) → ℚ ⊗_{ℤ} Sw^{f}(G)

is surjective.

(a) and (e) are proved by Swan [30, Corollary 4.2] for Sw(G^{G}(−)) which is isomorphic to Sw^{f}(G^{G}(−)) by [30, Proposition 1.1]. We conclude from [30, Lemma 4.1] and [31, Section 12] that the torsion elements of Sw(G) are all nilpotent. Together with (e) and Theorem 2.8 this show that the assumptions of [32, 6.3.3] are satisfied. Thus (c) follows from [32, 6.3.3]. We get (b) and (d) from [28, Theorem 27 and Theorem 28 on page 98], since Sw(ℂH) for a finite group H is the same as the complex representation ring of H.

Next we give the proof of Theorem 0.1.

Proof. If we combine Theorem 2.9 and Lemma 4.1, then we get for any group homomorphism ϕ: K → G whose target G is a finite group that the projection pr: ϕ^{∗}E_{F(G)}G → {pt .} induces for all n ∈ ℤ bijections

id_{Λ} ⊗_{ℤ}H_{n}^{K}(pr; K_{R}) : Λ ⊗_{ℤ} H_{n}^{K}(ϕ^{∗}E_{F(G)}G; K_{R}) → Λ ⊗_{ℤ} H_{n}^{K}({pt .}; K_{R})

in the following cases

(a) F is the class H of hyperelementary groups and Λ = ℤ;

(b) F is the class E of elementary groups and Λ = ℤ, provided that ℂ ⊆ R;

(c) For a given prime p the family F is the class H_{p} of p-hyperelementary groups and Λ = ℤ_{(p)};

(d) For a given prime p the family F is the class E_{p} of p-elementary groups and Λ = ℤ_{(p)}, provided that ℂ ⊆ R;

(e) F is the class FCY of finite cyclic groups and Λ = ℚ.

Since ϕ^{∗}E_{F(G)}G = E_{ϕ^{∗}F(G)}K holds, this shows that for the finite group G and the equivariant homology theory Λ⊗_{ℤ} H_{∗}^{?}(−; K_{R}) the Fibered Isomorphism Conjecture 1.2 for the family F(G) holds. Now apply Theorem 1.7 and use the fact that the (Fibered) Isomorphism Conjecture in the sense of Definition 1.2 for Λ ⊗_{ℤ} H_{∗}^{?}(−; K_{R}) is the same as the original (Fibered) Farrell-Jones Conjecture for algebraic K-theory for group rings with coefficients in R. For the Farrell-Jones Conjecture this is indicated in [6, p.239], a careful proof can be found in [14, Corollary 9.2]. For the Fibered Farrell-Jones Conjecture compare [2, Remark 6.6] and [22, Remark 4.14]. This finishes the proof of assertions (a) and (b) of Theorem 0.1.

For the proof of assertions (c) and (d) use the fact that for regular R with ℚ ⊆ R the group G satisfies the (Fibered) Isomorphism Conjecture (in the range ≤ N) for algebraic K-theory with coefficients in R for the family VCY if and only if G satisfies the Isomorphism Conjecture for algebraic K-theory with coefficients in R for the family FIN. This is proved in [22, Proposition 2.7]. Now we can reduce from FIN to H or to E, provided that ℂ ⊆ R, or after localization at p to H_{p} or to E_{p}, provided that ℂ ⊆ R, by the Transitivity Principle 1.4.

5. Outline of the Proof of the Main Result for Algebraic L-Theory

We mention that for L-theory we always use the decoration −∞, the Isomorphism Conjecture is not true for the other decorations such as s, h and p (see [12]).

The proof of Theorem 0.3, which is the L-theory version of Theorem 0.1, is analogous. The only difference is that one has to take the involutions into account and replace the functor Sw^{f} and the results about it due to Swan [30, Corollary 4.2] by its L-theoretic version denoted by GW (H; R) and studied by Dress in [8, Theorem 2].

The reduction from FIN to VCY explained in [22, Proposition 2.18] works also in the Fibered case since the relevant UNil-terms for amalgamated products and HNN-extensions vanish for all groups (see [4]).

6. Proof of the Main Result for Topological K-Theory

The proof of Theorem 0.5 will require two ingredients: a Completion Theorem and a Universal Coefficient Theorem for topological K-theory.

In [21, Theorem 6.5] a family version of the Atiyah-Segal Completion Theorem is proved. In the special case, where G is finite, X = {pt .} and the family is the family FCY of finite cyclic subgroups it yields an isomorphism of pro-groups

{KF_{G}^{∗}(pr^{(n)})}_{n≥0} : {KF_{G}^{∗}({pt .})}_{n≥0} ⟶^{≅} {KF_{G}^{∗}(E_{FCY}G^{(n)})}_{n≥0}, (6.1)

where the first pro-group is given by the constant system with KF_{G}^{∗}({pt .}) as value, E_{FCY}G^{(n)} is the n-skeleton of the G-CW -complex E_{FCY}G and the Gmap pr^{(n)} : E_{FCY}G^{(n)} → G/G is the projection. This means that for every n there is m > n and f_{m,n} : KF_{G}^{∗}({pt .}) → KF_{G}^{∗}(E_{FCY}G^{(m)}) such that

\xymatrix{KF_{G}^{∗}(E_{FCY}G^{(n)}) \ar[d]_{_{pr_{(n)}}} & KF_{G}^{∗}(E_{FCY}G^{(m)}) \ar[l] \ar[d]^{_{pr_{(n)}}} \\ KF_{G}^{∗}(\{pt .\}) & KF_{G}^{∗}(\{pt .\}) \ar[l]^{=} \ar[ul]^{f_{m,n}}}

commutes. The point is that no I-adic completion occurs since the map

R_{ℂ}(G) = KF_{G}^{0}({pt .}) → ∏_{(C),C cyclic} KF_{H}^{0}({pt .}) = R_{ℂ}(C)

is injective and hence I is the zero ideal. Here KF is either complex K-theory (KF = K) or real K-theory (KF = KO). The complex case occurs already in [15, Theorem 5.1].

Let G be a finite group. In [3] Bökstedt proves Universal Coefficient Theorems that express equivariant K-cohomology in terms of equivariant K-homology. Using S-duality [24, Chapter XVI.7] his results provide also Universal Coefficient Theorems that express equivariant K-homology in terms of equivariant K-cohomology. Let X be a finite G-CW -complex. For complex K-theory Bökstedt’s result asserts that there are natural short exact sequences

0 → Ext_{K_{G}^{∗}({pt .})}^{1}(K_{∗}^{G}(X), K_{∗}^{G}({pt .})) → K_{G}^{∗}(X)

→ Hom_{K_{G}^{∗}({pt .})}(K_{∗}^{G}(X), K_{∗}^{G}({pt .}) → 0, (6.2)

0 → Ext_{K_{G}^{∗}({pt .})}^{1}(K_{G}^{∗}(X), K_{G}^{∗}({pt .})) → K_{∗}^{G}(X)

→ Hom_{K_{G}^{∗}({pt .})}(K_{G}^{∗}(X), K_{G}^{∗}({pt .}) → 0. (6.3)

For KO-theory his results provide spectral sequences

E_{2}^{p,∗} = Ext_{KO_{G}^{∗}({pt .})}^{p}(KO_{∗}^{G}(X), KO_{∗}^{G}({pt .})) ⟹ KO_{G}^{∗}(X), (6.4)
E_{p,∗}^{2} = Ext_{KO_{G}^{∗}({pt .})}^{p}(KO_{G}^{∗}(X), KO_{G}^{∗}({pt .})) ⟹ KO_{∗}^{G}(X) (6.5)

such that E_{∞}^{p,∗} = 0 and E_{p,∗}^{∞} = 0 for p ≥ 2. In [16] the Universal Coefficient Theorem for complex K-theory is generalized to infinite groups and proper G-CW -complexes.

We can now give the proof of Theorem 0.5.

Proof. Because of the Transitivity Principle 1.5 it suffices to show that

KF_{n}^{G}(pr): KF_{n}^{G}(E_{FCY}G) ⟶^{≅} KF_{n}^{G}({pt .})  (6.6)

is bijective for all finite groups G and n ∈ ℤ. There exists a G-CW -model for E_{FCY}G whose skeleta are all finite G-CW -complexes. This follows for instance from the functorial construction in [6, Section 3 and Lemma 7.6] using the fact that Or_{F}(G) is a category with finitely many morphisms.

If we apply (6.3) to pr^{(n)} : E_{FCY}G^{(n)} → G/G we obtain a map between two short exact sequences. Since colim_{n→∞} is an exact functor, these sequences stay exact if we apply colim_{n→∞}. Because of the isomorphism of pro-groups (6.1), we get isomorphism in the first and third term of (6.3) (see for instance [16]). By the Five-Lemma this implies that

colim_{n→∞} K_{∗}^{G}(E_{FCY}G^{(n−1)}) → colim_{n→∞} K_{∗}^{G}({pt .})

is bijective. Since K-homology is compatible with colimits, this map can be identified with the map (6.6) for complex K-theory.

For KO-theory we can proceed similarly. The isomorphism of pro-groups (6.1) yields isomorphisms on the colimit of the E^{2}-Term of the spectral sequence (6.5). Since colim_{n→∞} is an exact functor this yields also isomorphisms on the colimit of the E^{∞} term. Because this E^{∞} term has only a finite number of lines a simple diagram chase shows that the induced map

colim_{n→∞} KO_{∗}^{G}(E_{FCY}G^{(n)}) → colim_{n→∞} KO_{∗}^{G}({pt .})

on the right hand side of (6.5) is an isomorphisms. Since KO-homology is compatible with colimits, this map can be identified with the map (6.6) for KO-theory.

7. Versions in Terms of Colimits

The classical induction theorems can be stated in terms of colimits. If we combine Theorem 2.9 and Lemma 4.1 and (its L-theory version) we get for an extension 1 → K → G ⟶^{p} F → 1 for a finite group F isomorphisms

colim_{F/L∈Or _{H}(F)} K_{n}(R[p^{−1}(L)]) ⟶^{≅} K_{n}(RG);
colim_{F/L∈Or _{Hp} (F)} K_{n}(R[p^{−1}(L)])_{(p)} ⟶^{≅} K_{n}(RG)_{(p)};
colim_{F/L∈Or _{H_{2}∪⋃_{p prime,p≠2} Ep} (F)} L_{n}^{⟨−∞⟩}(R[p^{−1}(L)]) ⟶^{≅} L_{n}^{⟨−∞⟩}(RG);
colim_{F/L∈Or _{⋃_{p prime,p≠2} Ep} (F)} L_{n}(R[p^{−1}(L)])[1/2] ⟶^{≅} L_{n}(RG)[1/2].

But in general our results cannot be stated in such an elementary way. For instance, Theorem 0.5 says for a finite group G that there is an isomorphism

K_{0}^{G}(E_{FCY}G) ⟶^{≅} K_{0}^{G}(G/G)

In general K_{0}^{G}(G/H) is the complex representation ring R_{ℂ}(H) of the finite group H. It is not true in general that the canonical map

colim_{G/H∈Or _{FCY} (G)} R_{ℂ}(H) → R_{ℂ}(G)

is bijective or, equivalently, that the canonical map

colim_{G/H∈Or _{FCY} (G)} K_{0}^{G}(G/H) → K_{0}^{G}(G/G)

is bijective.

In some special cases one can get formulations of our results in terms of colimits.

Theorem 7.1. (a) The group G satisfies the Isomorphism Conjecture in the range n ≤ −1 for algebraic K-theory with coefficients in R = ℤ for the family VCY if and only if

K_{n}(ℤG) = 0 for n ≤ −2

and the canonical map

colim_{G/H∈Or _{H}(G)} K_{−1}(ℤH) ⟶^{≅} K_{−1}(ℤG)

is bijective;

(b) Suppose that R is regular and ℚ ⊆ R. Then the group G satisfies the Isomorphism Conjecture in the range n ≤ 0 for algebraic K-theory with coefficients in R for the family VCY if and only if

K_{n}(RG) = 0 for n ≤ −1

and the canonical map

colim_{G/H∈Or _{H}(G)} K_{0}(RH) ⟶^{≅} K_{0}(RG)

is bijective.

If we assume that R is regular and ℂ ⊆ R, then we can replace H by E.

Proof. (a) It follows from [11] that for a virtually cyclic group V the assembly map

H_{n}^{G}(E_{FIN} V; K_{ℤ}) ⟶ K_{n}(ℤV)

is surjective in the range n ≤ −1. Moreover, this map is known to be injective in all degrees [1], [27]. Theorem 2.9 and Lemma 4.1 imply that for every finite group F the assembly map

H_{n}^{F} (E_{H}F; K_{ℤ}) ⟶^{≅} K_{n}(ℤF)

is bijective. Applying the Transitivity Principle for equivariant homology 1.4 twice we conclude that a group G satisfies the Isomorphism Conjecture for algebraic K-theory with coefficients in R = ℤ for the family VCY in the range n ≤ −1 if and only if the assembly map

H_{n}^{G}(E_{H}G; K_{ℤ}) ⟶^{≅} K_{n}(ℤG)

is bijective for n ≤ −1. It is proven in [5] that for finite groups F and n ≤ −2 K_{n}(ℤF) = 0. Now a spectral sequence argument shows that H_{n}^{G}(E_{H}G; K_{ℤ}) vanishes for n ≤ −2 and can be identified with colim_{G/H∈Or _{H}(F)} K_{−1}(ℤH) for n = −1.

(b) By Theorem 0.1 (c) the group G satisfies the Isomorphism Conjecture in the range n ≤ 0 for algebraic K-theory with coefficients in R for the family VCY in the range n ≤ 0 if and only if the canonical map

H_{n}^{G}(E_{H}G; K_{R}) ⟶^{≅} K_{n}(RG)

is bijective for n ≤ 0. Since RH is regular for a finite group H, K_{q}(RH) = 0 for q ≤ −1. Now a spectral sequence argument shows that H_{n}^{G}(E_{H}G; K_{R}) vanishes for n ≤ −1 and can be identified with colim_{G/H∈Or _{H}(G)} K_{0}(RH) for n = 0.

8. On Quinn’s Hyperelementary Induction Conjecture

Quinn [26] states the following conjecture

Conjecture 8.1 (Hyperelementary Induction Conjecture). All groups satisfy induction for the class \overline{H} of (possibly infinite) hyperelementary groups.

The phrase that a group G satisfies induction for a family F of subgroups of G means that the Fibered Farrell-Jones Conjecture is true for the family F and algebraic K-theory with coefficients in a given associative ring with unit R. The Fibered Isomorphisms Conjecture is due to Farrell-Jones [10] (see also [22, Section 4.2.2]). Quinn calls a group H p-hyperelementary if there exists a prime p, a finite p-group P and a (possibly infinite) cyclic group C such that H can be written as an extension 1 → C → H → P → 1, and hyperelementary if it is p-hyperelementary for some prime p. The class \overline{H} appearing in the Hyperelementary Induction Conjecture 8.1 is to be understood with respect to Quinn’s definition of hyperelementary group as above. Of course \overline{H} ∩ FIN is the class H of finite hyperelementary groups. There is an inclusion \overline{H} ⊆ H′ but the classes \overline{H} and H′ are different. One would like to prove that Hyperelementary Induction Conjecture 8.1 follows from the Fibered Isomorphism Conjecture for algebraic K-theory. This is in view of Theorem 0.1 (a) and Theorem 1.5 equivalent to a positive answer to the following question:

Question 8.2. Is for every group H ∈ H′ the Hyperelementary Induction Conjecture 8.1 true?

We have no evidence for a positive answer from general machinery, a proof of a positive answer will need some special information about the Nil-groups NK_{n}(RH).

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