Isomorphism Conjecture for homotopy K-theory and groups acting on trees

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

July 14, 2005

Abstract

We discuss an analogon to the Farrell-Jones Conjecture for homotopy algebraic K-theory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the Farrell-Jones Conjecture in algebraic K-theory. Key words: K-theory and homotopy K-theory of group rings, Isomorphism Conjectures, Actions on trees. Mathematics subject classification 2000: 19D35, 19D55.

0 Introduction

The Farrell-Jones Conjecture [12] in algebraic K-theory is concerned with the K-theory K_{n}(RG) of group rings RG for a group G and a ring R. The conjecture states that the assembly map

H_{n}^{G}(E_{VCYC}(G); K_{R}) → K_{n}(RG)  (0.1)

is an isomorphism. (This map is constructed by applying a certain Ghomology theory H_{n}^{G}(−; K_{R}) to the projection E_{VCYC}(G) → pt, see Definition 1.1 and Remark 6.6.) There seem to occur two quite different phenomena in the algebraic K-theory of such group rings. Firstly, K_{n}(RG) contains elements coming from the K-theory of RF for finite subgroups F of G. Secondly, it contains nilgroup information. This is already illuminated in the simple case G = ℤ, then R[ℤ] = R[t, t^{−1}] and by the Bass-Heller-Swan formula [7], [14]

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

K_{n}(R[ℤ]) ≅ K_{n}(R) ⊕ K_{n−1}(R) ⊕ NK_{n}(R) ⊕ NK_{n}(R). (0.2)

Here NK_{n}(R) are the Nil-groups of R, which can be defined as the kernel of the projection K_{n}(R[t]) → K_{n}(R) induced from t ↦ 0. In general it is known [1] that the domain of the assembly map (0.1) splits as

H_{n}^{G}(E_{FIN} (G); K_{R}) ⊕ H_{n}^{G}(E_{VCYC}(G), E_{FIN} (G); K_{R}).  (0.3)

Thus, the Farrell-Jones Conjecture predicts a similar splitting for K_{n}(RG).

In this paper we will formulate a (Fibered) Isomorphism Conjecture for homotopy algebraic K-theory, see Conjecture 7.3. This variant of K-theory was defined by Weibel [31], building on the definition of Karoubi-Villamayor K-theory. The homotopy algebraic K-theory groups of a ring R are denoted by KH_{n}(R). Their crucial property is homotopy invariance: KH_{n}(R) ≅ KH_{n}(R[t]). In particular, homotopy algebraic K-theory does not contain Nilgroups. We think about this KH -Isomorphism Conjecture as an Isomorphism Conjecture for algebraic K-theory modulo Nil-groups. For a more precise formulation of the relation of the Farrell-Jones Conjecture in algebraic Ktheory to the KH -Isomorphism Conjecture see Section 8.

Our main results concerning the KH -Isomorphism Conjecture are inheritance properties. A group G acts on a tree T, if T is a 1-dimensional G-CW - complex which is contractible (after forgetting the group action).

Definition 0.4 (The class of groups C_{0}). We define the following properties a class C of groups may or may not have.

(FIN) All finite groups belong to C;

(TREE) Suppose that G acts on a tree T. Assume that for each x ∈ T the isotropy group G_{x} belongs to C. Then G belongs to C;

(COL) Let G be a group with a directed system of subgroups {G_{i} | i ∈ I}, which is directed by inclusion and satisfies ⋃_{i∈I} G_{i} = G. If each G_{i} belongs to C, then G ∈ C;

(SUB) If G ∈ C and H ⊆ G is a subgroup, then H ∈ C.

We define C_{0} to be the smallest class of groups satisfying (FIN), (TREE) and (COL).

It is not hard to check that the class C_{0} is closed under taking subgroups. For instance if H is a subgroup of a group G acting on a tree, then H acts also on this tree and the isotropy groups satisfy H_{x} ⊆ G_{x}. By induction we may assume that the G_{x} are closed under taking subgroups and therefore H ∈ C_{0}.

Theorem 0.5. (Inheritance properties of the KH -Isomorphism Conjecture) The class of groups satisfying the Fibered KH -Isomorphism Conjecture for a fixed coefficient ring R has the properties (FIN), (TREE), (COL) and (SUB). The class of groups satisfying the KH -Isomorphism Conjecture for a fixed coefficient ring R has the properties (FIN), (TREE) and (COL). In particular, all groups in C_{0} satisfy the (Fibered) KH -Isomorphism Conjecture.

Remark 0.6. The class of groups satisfying the KH -Isomorphism Conjecture is stricly bigger than C_{0} since it contains all fundamental groups of closed Riemannian manifolds with negative sectional curvature by [5] and Theorem 8.4 (i).

This result has the following applications.

Theorem 0.7 (Extensions of groups and actions on trees). Let 1 → K → G → Q → 1 be an extension of groups. Suppose that K acts on a tree with finite stabilizers and that Q satisfies the Fibered KH -Isomorphism Conjecture 7.3 for the ring R. Then G satisfies the Fibered KH -Isomorphism Conjecture 7.3 for the ring R.

A ring R is called regular if it is Noetherian and every finitely generated R-module possesses a finite-dimensional resolution by finitely generated projective modules.

Theorem 0.8. (Conclusions for the K-theoretic Farrell-Jones Conjecture for groups in C). Let G be a group in the class C_{0} defined above in (0.4). Then

(i) Let R be a regular ring with ℚ ⊆ R. Then the assembly map

H_{n}^{G}(E_{FIN} (G); K_{R}) → K_{n}(RG)

is injective, or, equivalently, the injectivity part of the Farrell-Jones Isomorphism Conjecture for algebraic K-theory is true for (G, R).

(ii) Let R be the ring ℤ of integers. Then the assembly map

H_{n}^{G}(E_{FIN} (G); K_{ℤ}) → K_{n}(ℤG)

is rationally injective, or, equivalently, the rational injectivity part of the Farrell-Jones Isomorphism Conjecture for algebraic K-theory is true for (G, ℤ).

Proposition 0.9. The following classes of groups belong to C_{0}:

(i) One relator groups; (ii) G is poly-free, i.e. there is a filtration

{1} = G_{0} ⊆ G_{1} ⊆ G_{2} ⊆ … ⊆ G_{n} = G

such that G_{i} is normal in G_{i+1} with a free group as quotient G_{i+1}/G_{i} The pure braid group is an example;

(iii) Let M be a compact orientable 3-manifold with prime decomposition M = M_{1}♯M_{2}♯ … ♯M_{n}. Suppose that each M_{i}, which has infinite fundamental group and is aspherical, has a boundary or is a Haken manifold. Then π_{1}(M) ∈ C_{0};

(iv) If M is a compact 2-dimensional manifold, then π_{1}(M) ∈ C_{0};

(v) If M is a submanifold of S^{3}, then π_{1}(M) ∈ C_{0}.

Next we discuss similar inheritance properties for the Farrell-Jones Conjecture in algebraic K-theory. A ring R is called regular coherent if every finitely presented R-module possesses a finite-dimensional resolution by finitely generated projective R-modules. A ring R is regular if and only if it is regular coherent and Noetherian. A group G is called regular or regular coherent respectively if for any regular ring R the group ring RG is regular respectively regular coherent. For more information about these notions we refer to [29, Theorem 19.1].

Definition 0.10 (The classes of groups CL and CL′). Consider the following further properties a class C of groups may have.

(TRI) The trivial group belongs to C;

(VCYC) All virtually cyclic groups belong to C;

(TREE_{R}) Suppose that G acts on a tree T. Assume that for each x ∈ T the isotropy group G_{x} belongs to C. For each edge e of T, assume that the isotropy group G_{e} is regular coherent. Then G belongs to C;

The class CL is defined as the smallest class of groups satisfying (TRI), (TREE_{R}) and (COL). The class CL′ is defined as the smallest class of groups satisfying (VCYC), (TREE_{R}) and (COL).

All groups appearing in CL are torsionfree. Similar to the class C_{0}, the classes CL and CL′ are closed under taking subgroups [29, Proposition 19.3]. We conclude from Waldhausen [29, Theorem 17.5 on page 250] that CL contains a group G appearing in Proposition 0.9 under (i), (iii), (iv), and (v) provided that G is torsionfree. One of the main results in Waldhausen’s article [29] is that for a regular ring R the K-theoretic assembly map

H_{n}(BG; K_{R}) → K_{n}(RG)

is an isomorphism. Actually, Waldhausen states this only for n ≥ 0, but the embedding of K_{n−1}(R) into K_{n}(R[ℤ]) allows the extension to all n, see for example Remark 10.3. Furthermore, Waldhausen considers HNN-extensions and amalgamated products rather than action on trees, but this does not change the class CL, compare Remark 4.7 and Lemma 5.1.

Theorem 0.11. Let R be a regular ring. The class of groups satisfying the Farrell-Jones Conjecture in algebraic K-theory for the ring R has the properties (VCYC), (TREE_{R}) and (COL). In particular, all groups in CL′ satisfy the Farrell-Jones Conjecture in algebraic K-theory for the ring R.

Related results can be found in [15] and [24]. It is an interesting question, whether the class of groups satisfying the Farrell-Jones Conjecture in algebraic K-theory has the property (TREE).

Theorem 0.5 and Waldhausen’s result imply

Corollary 0.12. Let R be a regular ring. Let G be a group in the class CL. Then the canonical map

K_{n}(RG) → KH_{n}(RG)

is bijective for n ∈ ℤ.

Isomorphism Conjectures can be formulated in the quite general context of equivariant homology theories, see Definition 1.1. We show in Theorem 4.2 that the property (TREE) holds for the class of groups satisfying the Isomorphism Conjecture for such an equivariant homology theory whenever the equivariant homology theory satisfies the tree property, see Definitions 4.1. The weaker property (TREE_{R}) is related to the regular tree property (see Definition 4.1), which is a weakening of the tree property. The above Theorem 0.7 has also an analogon in this setting, see Corollary 4.4. The tree property means essentially that there are Mayer-Vietoris sequences for amalgamated products and HNN extensions of groups in the equivariant homology theory (see Lemma 4.5). On spectrum level this means that there are certain homotopy cartesian diagrams of spectra (see Lemma 5.1 and Remark 5.6). In Section 7 we define the equivariant homology theory H_{∗}^{?}(−, KH_{R}) that is relevant for the KH -Isomorphism Conjecture. We prove in Theorem 11.1 that this theory satisfies the tree property. In the case of algebraic K-theory amalgamated free products and HNN extensions have been analyzed by Waldhausen [29]. In both cases there are long exact sequences, but they involve as an additional term Waldhausen’s Nil-groups. Their nontriviality obstructs the equivariant homology theory H_{∗}^{?}(−; K_{R}) relevant for the Farrell-Jones Conjecture in algebraic K-theory from having the tree property. Our proof of the tree property for H_{∗}^{?}(−, KH_{R}), consists essentially of showing that Waldhausen’s Nil-groups are killed under the transition from K to KH . On the other hand vanishing results for Waldhausen’s Nil-groups can be used to show (see Theorem 11.4) that H_{∗}^{?}(−; K_{R}) has the regular tree property (see Definition 4.1). This implies then, that for a regular ring R the class of groups satisfying the Farrell-Jones Conjecture has the property (TREE_{R}), see Theorem 4.2 (iii). It is an interesting question for which rings R the equivariant homology theory H_{∗}^{?}(−; K_{R}) ⊗ ℚ has the tree property. It is worthwhile to consider also H_{∗}^{?}(−; L_{R}^{−∞}), the equivariant homology theory relevant for the Farrell-Jones Conjecture in L-theory. In this case amalgamated free products and HNN extensions have been analyzed by Cappell [9]. Again additional terms appear in the long exact sequences, the UNil-groups and non-triviality of those obstructs this theory from having the tree property. On the other hand these UNil-groups are known to be 2-torsion [9], thus H_{n}^{?}(−; L_{R}^{−∞}) ⊗ ℤ[\frac{1}{2}] does have the tree property. Thus we obtain the following result.

Theorem 0.13. (Conclusions for the L-theoretic Farrell-Jones Conjecture for groups in C). The class of groups for which the assembly map

H_{n}^{G}(E_{FIN} (G); L_{R}^{−∞}) → L_{n}^{−∞}(RG)

becomes an isomorphism after tensoring with ℤ[\frac{1}{2}], has the properties (FIN), (TREE) and (COL). In particular, this class contains the class in C_{0} from Definition 0.4.

In the context of topological K-theory, i.e. for the Baum-Connes Conjecture one can apply our results to the equivariant K-theory H_{n}^{G}(−; K^{top}) = K_{n}^{G}(−). Then one obtains the analogon of our Theorem 0.5. In this case amalgamated products and HNN extensions have been analyzed Pimsner-Voiculescu [20] and Pimsner [21]. Here the situation is much better, since no Nil-groups appear. This analogon has already been proved by Oyono-Oyono [19] for the Baum-Connes Conjecture (with coefficients).

We are indebted to Holger Reich for pointing out the reference [30] to us.

The papers is organized as follows:

1.  Isomorphism Conjectures for equivariant homology theories
2.  Homological aspects
3  Continuous equivariant homology theories
4.  The tree property
5.  Equivariant homology theories constructed from spectra
6.  Isomorphism Conjectures for spectra
7.  The KH -Isomorphism Conjecture
8.  The Relation between the K- and the KH -Isomorphism Conjecture
9  Non-connective Waldhausen Nil
10.  Waldhausen’s cartesian squares
11.  The tree property for KH

References

1 Isomorphism Conjectures for equivariant homology theories

We will use the notion of an equivariant homology theory H_{∗}^{?} with values in Λ-modules for a commutative associative ring Λ with unit from [16, 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)).

We will later discuss examples, the most important ones will be given by those equivariant homology theories which appear in the Baum-Connes Conjecture and the Farrell-Jones Conjecture. These conjectures are special cases of the following more general formulation of a (Fibered) Isomorphism Conjecture (see Section 6).

A family F of subgroups of G is a set of subgroups which is closed under conjugation and taking subgroups. If C is a class of groups that is closed under taking subgroups and isomorphisms, then the collections of subgroups of G that are in C forms a family C(G) of subgroups of G. Abusing notation, we will denote this family often by C. Examples are the families FIN of finite subgroups and VCYC 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 i: H → G is the inclusion of a subgroup, then we write often F|_{H} for i^{∗}F. Associated to such a family 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. For F = ALL the family of all subgroups, we can take the one-point-space pt as a model for E_{ALL}(G). For more information about the spaces we refer for instance to [17].

Definition 1.1 ((Fibered) Isomorphism Conjecture for H_{∗}^{?}). Let H_{∗}^{?} be an equivariant homology theory with values in Λ-modules. 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 to the one-point-space 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 known to be true in general in the nonfibered case.

Lemma 1.2. Let H_{∗}^{?} be an equivariant homology theory, let ϕ: K → G be a group homomorphism and let F be a family of subgroups. If (G, F) satisfies the Fibered Isomorphism Conjecture 1.1 (in the range ≤ N), then (K, ϕ^{∗}F) satisfies the Fibered Isomorphism Conjecture 1.1 (in the range ≤ N).

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

In particular, if for a given class of groups C, which is closed under isomorphism and taking subgroups, the Fibered Isomorphism Conjecture 1.1 is true for (G, C(G)) and if H ⊆ G is a subgroup, then the Fibered Isomorphism Conjecture 1.1 is true for (H, C(H)).

2 Homological aspects

The disjoint union axiom ensures that a G-homology is compatible with directed colimits.

Lemma 2.1. Let H_{∗}^{G} be a G-homology theory. Let X be a G-CW -complex and {X_{i} | i ∈ I} be a directed system of G-CW -subcomplexes directed by inclusion such that X = ⋃_{i∈I} X_{i}. Then for all n ∈ ℤ the natural map

colim_{i∈I} H_{n}^{G}(X_{i}) ⟶^{≅} H_{n}^{G}(X)

is bijective.

Proof. Compare for example with [27, Proposition 7.53 on page 121], where the non-equivariant case for I = ℕ is treated. The main point is that the functor colimit over a directed system of R-modules is exact.

Lemma 2.2. Let H_{∗}^{?} be an equivariant homology theory with values in Λmodules in the sense of [16, Section 1]. Let G be a group and let F be a family of subgroups of G. Let Z be a G-CW -complex. Consider N ∈ ℤ ∪ {∞}. Suppose for each H ⊆ G which occurs as isotropy group in Z that the G-map induced by the projection pr: E_{F|_{H}} (H) → pt

H_{n}^{H}(pr): H_{n}^{H}(E_{F|_{H}} (H)) → H_{n}^{H}(pt)

is bijective for all n ∈ ℤ, n ≤ N.

Then the map induced by the projection pr_{1} : E_{F}(G) × Z → Z

H_{n}^{G}(pr_{1}): H_{n}^{G}(E_{F}(G) × Z) → H_{n}^{G}(Z)

is bijective for n ∈ ℤ, n ≤ N.

Proof. We first prove the claim for finite-dimensional G-CW -complexes by induction over d = dim(Z). The induction beginning dim(Z) = −1, i.e. Z = ∅, is trivial. In the induction step from (d − 1) to d we choose a Gpushout

∐_{i∈I_{d}} G/H_{i} × S^{d−1} ⟶ Z_{d−1}
↓  ↓
∐_{i∈I_{d}} G/H_{i} × D^{d} ⟶ Z_{d}

If we cross it with E_{F}(G), we obtain another G-pushout of G-CW -complexes. The various projections induce a map from the Mayer-Vietoris sequence of the latter G-pushout to the Mayer-Vietoris sequence of the first G-pushout. By the Five-Lemma it suffices to prove that the following maps

H_{n}^{G}(pr_{2}): H_{n}^{G} (E_{F}(G) × ∐_{i∈I_{d}} G/H_{i} × S^{d−1}) → H_{n}^{G} ( ∐_{i∈I_{d}} G/H_{i} × S^{d−1});
H_{n}^{G}(pr_{3}): H_{n}^{G}(E_{F}(G) × Z_{d−1}) → H_{n}^{G}(Z_{d−1});
H_{n}^{G}(pr_{4}): H_{n}^{G} (E_{F}(G) × ∐_{i∈I_{d}} G/H_{i} × D^{n}) → H_{n}^{G} ( ∐_{i∈I_{d}} G/H_{i} × D^{n}) ,

are bijective for n ∈ ℤ, n ≤ N. This follows from the induction hypothesis for the first two maps. Because of the disjoint union axiom and G-homotopy invariance of H_{∗}^{?} the claim follows for the third map if we can show for any H ⊆ G which occurs as isotropy group in Z that the map

H_{n}^{G}(pr_{1}): H_{n}^{G}(E_{F}(G) × G/H) → H_{n}^{G}(G/H)  (2.3)

is bijective for n ∈ ℤ, n ≤ N. The G-map

G ×_{H} res_{G}^{H} E_{F}(G) → G/H × E_{F}(G) (g, x) ↦ (gH, gx)

is a G-homeomorphism where res_{G}^{H} denotes the restriction of the G-action to an H-action. Obviously res_{G}^{H} E_{F}(G) is a model for E_{F|_{H}} (H). Since for any

H-CW -complex Y there is a natural isomorphism H_{n}^{H}(Y) ⟶^{≅} H_{n}^{G}(G ×_{H} Y), the map (2.3) can be identified with the map

H_{n}^{G}(pr): H_{n}^{H}(E_{F|_{H}} (H)) → H_{n}^{H}(pt)

which is bijective for all n ∈ ℤ, n ≤ N by assumption. This finishes the proof in the case that Z is finite-dimensional.

Finally we consider an arbitrary G-CW -complex Z. It can be written as the colimit colim_{d→∞} Z_{d}. The natural maps

colim_{d→∞} H_{n}^{G}(E_{F}(G) × Z_{d}) ⟶^{≅} H_{n}^{G}(E_{F}(G) × Z);
colim_{d→∞} H_{n}^{G}(Z_{d}) ⟶^{≅} H_{n}^{G}(Z),

are bijective by Lemma 2.1. Since the colimit of isomorphisms is an isomorphism again, Lemma 2.2 follows.

Theorem 2.4 (Reducing the family). Let H_{∗}^{?} be an equivariant homology theory with values in Λ-modules. Let G be a group and let F ⊆ G be families of subgroups of G. Consider N ∈ ℤ ∪ {∞}. Suppose for each H ∈ G, or, more generally, suppose for each isotropy group appearing in a specific model for E_{G}(G) that (H, F|_{H}) satisfies the (Fibered) Isomorphism Conjecture 1.1 (in the range ≤ N).

Then (G, G) satisfies the (Fibered) Isomorphism Conjecture 1.1 (in the range ≤ N) if and only if (G, F) satisfies the (Fibered) Isomorphism Conjecture 1.1 (in the range ≤ N).

Proof. For the Isomorphism Conjecture this follows from Lemma 2.2 applied to the case Z = E_{G}(G) and the fact that E_{F}(G) × E_{G}(G) is a model for E_{F}(G). The case of the Fibered Isomorphism Conjecture is easily reduced to the former case.

Lemma 2.5. Let H_{∗}^{?} be an equivariant homology theory with values in Λmodules. Let C be a class of groups that is closed under isomorphisms, subgroups and quotients. Let 1 → L → G ⟶^{p} Q → 1 be an extension of groups. Suppose that (Q; C(Q)) satisfies the Fibered Isomorphism Conjecture 1.1 (in the range ≤ N) and that for H ∈ p^{∗}C(Q) the pair (H, C(H)) satisfies the Fibered Isomorphism Conjecture 1.1 (in the range ≤ N).

Then (G, C(G)) satisfies the Fibered Isomorphism Conjecture 1.1 (in the range ≤ N).

Proof. By Lemma 1.2 the pair (G, p^{∗}C(Q)) satisfies the Fibered Isomorphism Conjecture 1.1 (in the range ≤ N). Since C is closed under quotients we have C(G) ⊆ p^{∗}C(Q). Now the assumption on the subgroups H ∈ p^{∗}C(Q) and Theorem 2.4 imply the result.

3 Continuous equivariant homology theories

In this section we explain a criterion for an equivariant homology theory ensuring that for a class C of groups closed under subgroups and isomorphisms the (Fibered) Isomorphism Conjecture 1.1 is true for (G, C(G)) provided that G is a directed union G = ⋃_{i∈I} G_{i} of groups G_{i} and the (Fibered) C-Isomorphism Conjecture 1.1 is true for (G_{i}, C(G_{i})) for all i ∈ I.

Definition 3.1 (Continuous equivariant homology theory). An equivariant homology theory H_{∗}^{?} is called continuous if for each group G and directed system of subgroups {G_{i} | i ∈ I}, which is directed by inclusion and satisfies ⋃_{i∈I} G_{i} = G, and each n ∈ ℤ the map

colim_{i∈I} j_{i}: colim_{i∈I} H_{n}^{G_{i}}(pt) → H_{n}^{G}(pt)

is an isomorphism, where j_{i}: H_{n}^{G_{i}}(pt) → H_{n}^{G}(pt) is the composition of the induction isomorphism H_{n}^{G_{i}}(pt) ⟶^{≅} H_{n}^{G}(G/G_{i}) with the map induced by the projection G/G_{i} → pt.

Lemma 3.2. Let H_{∗}^{?} be a continuous equivariant homology theory. Let G be a group with a directed system of subgroups {G_{i} | i ∈ I}, which is directed by inclusion and satisfies ⋃_{i∈I} G_{i} = G.

Then for each G-CW -complex X and each n ∈ ℤ the map

colim_{i∈I} j_{i}: colim_{i∈I} H_{n}^{G_{i}}(res_{G}^{G_{i}} X) → H_{n}^{G}(X)

is an isomorphism, where j_{i}: H_{n}^{G_{i}}(res_{G}^{G_{i}} X) → H_{n}^{G}(X) is the composition of the induction isomorphism H_{n}^{G_{i}}(res_{G}^{G_{i}} X) ⟶^{≅} H_{n}^{G}(G ×_{G_{i}} res_{G}^{G_{i}} X) with the homomorphism induced by the G-map G ×_{G_{i}} res_{G}^{G_{i}} X → X that sends (g, x) to gx.

Proof. Since colim_{i∈I} is an exact functor, colim_{i∈I} H_{n}^{G_{i}}(res_{G}^{G_{i}} X) is a Ghomology theory in X. The map colim_{i∈I} j_{i} is a transformation of G-homology theories. Therefore it suffices to prove that

colim_{i∈I} j_{i}: colim_{i∈I} H_{n}^{G_{i}}(res_{G}^{G_{i}} G/H) → H_{n}^{G}(G/H)

is an isomorphism for every subgroup H ⊆ G and n ∈ ℤ.

For i ∈ I let k_{i}: G_{i}/G_{i} ∩ H → res_{G}^{G_{i}} G/G_{i} be the obvious injective G_{i} map. Then the following diagram commutes

\xymatrix{colim_{i∈I} H_{n}^{G_{i}∩H}(pt) \ar[r]^{≅} \ar[dd]_{colim_{i∈I} j_{i}}^{≅} & colim_{i∈I} H_{n}^{G_{i}}(G_{i}/G_{i} ∩ H) \ar[d]^{colim_{i∈I} k_{i}} \\  & colim_{i∈I} H_{n}^{G_{i}}(res_{G}^{G_{i}} G/H) \ar[d]^{colim_{i∈I} j_{i}} \\ H_{n}^{H}(pt) \ar[r]^{≅} & H_{n}^{G}(G/H)}

where the horizontal maps are the isomorphism given by induction. The left vertical arrow is bijective since H_{∗}^{?} is continuous by assumption. Hence it remains to show that the map

colim_{i∈I} k_{i}: colim_{i∈I} H_{n}^{G_{i}}(G_{i}/G_{i} ∩ H) → colim_{i∈I} H_{n}^{G_{i}}(res_{G}^{G_{i}} G/H) (3.3)

is surjective.

We get an obvious decomposition of G_{i}-sets

res_{G}^{G_{i}} G/H = ∐_{G_{i}gH∈G_{i}\G/H} G_{i}/G_{i} ∩ gHg^{−1}.

It induces an identification

H_{n}^{G_{i}}(res_{G}^{G_{i}} G/H) = ⨁_{G_{i}gH∈G_{i}\G/H} H_{n}^{G_{i}}(G_{i}/G_{i} ∩ gHg^{−1}).

The summand corresponding to G_{i}1H is precisely the image of

H_{n}^{G_{i}}(k_{i}): H_{n}^{G_{i}}(G_{i}/G_{i} ∩ H) → H_{n}^{G_{i}}(res_{G}^{G_{i}} G/H).

Consider an element G_{i}gH ∈ G_{i}\G/H. Choose an index j with j ≥ i and g ∈ G_{j}. Then the structure map for i ≤ j is a map H_{n}^{G_{i}}(res_{G}^{G_{i}} G/H) → H_{n}^{G_{j}}(res_{G}^{G_{j}} G/H) which sends the summand corresponding to G_{i}gH ∈ G_{i}\G/H to the image of

H_{n}^{G_{j}}(k_{j}): H_{n}^{G_{j}}(G_{j}/G_{j} ∩ H) → H_{n}^{G_{j}}(res_{G}^{G_{j}} G/H).

This implies that the map (3.3) is surjective. This finishes the proof of Lemma 3.2.

Proposition 3.4. Let H_{∗}^{?} be an equivariant homology theory which is continuous. Let C be a class of groups that is closed under isomorphism and taking subgroups. Let G be the directed union G = ⋃_{i∈I} G_{i} of subgroups G_{i} such that the (Fibered) Isomorphism Conjecture 1.1 (in the range ≤ N) is true for (G_{i}, C(G_{i})) for all i ∈ I.

Then the (Fibered) Isomorphism Conjecture 1.1 (in the range ≤ N) is true for (G, C(G)).

Proof. Since H_{∗}^{?} is continuous by assumption, we get the isomorphism

colim_{i∈I} H_{∗}^{G_{i}}(pt) = H_{∗}^{G}(pt)

and from Lemma 3.2 the isomorphism

H_{∗}^{G}(E_{C(G)}(G)) = colim_{i∈I} H_{∗}^{G_{i}}(res_{G}^{G_{i}} E_{C(G)}(G)).

The result follows for the Isomorphism Conjecture since the colimit of an isomorphism is an isomorphism and since res_{G}^{G_{i}} E_{C(G)}(G) is a model for E_{C(G_{i})}(G_{i}). If ϕ: K → G is a group homomorphism then the same argument can be applied to the triple (K, ϕ^{∗}C(G), {ϕ^{−1}(G_{i}) | i ∈ I}) in place of (G, C(G), {G_{i} | i ∈ I}) and this implies the statement for the Fibered Isomorphism Conjecture.

4 The tree property

In this section we study criteria for an equivariant homology theory that ensure that the class of groups G for which (G, FIN) satisfies the (Fibered) Isomorphism Conjecture 1.1 has property (TREE) from Definition 0.4 or that the class of groups G for which (G, VCYC) satisfies the Isomorphism Conjecture 1.1 has property (TREE_{R}) from Definition 0.10.

Definition 4.1 (Tree Property). An equivariant homology theory H_{∗}^{?} has the tree property if for any group G that acts on a tree T, the projection pr: T → pt induces for all n ∈ ℤ (with n ≤ N) isomorphisms

H_{n}^{G}(pr): H_{n}^{G}(T) ⟶ H_{n}^{G}(pt).

It has the regular tree property if for any group G that acts on a tree T, such that for each edge e of T the isotropy group G_{e} is regular coherent, the projection pr: T → pt induces for all n ∈ ℤ (with n ≤ N) isomorphisms

H_{n}^{G}(pr): H_{n}^{G}(T) ⟶ H_{n}^{G}(pt).

Theorem 4.2. (The tree property and inheritance properties of Isomorphism Conjectures) Let H_{∗}^{?} be an equivariant homology theory. Let C be a class of groups closed under subgroups and isomorphisms. Let D_{fib}(C) be the class of groups G for which Fibered Isomorphism Conjecture 1.1 (in the range ≤ N) is true for (G, C(G)) and let D(C) be the class of groups G for which Isomorphism Conjecture 1.1 (in the range ≤ N) is true for (G, C(G)).

(i) Suppose that H_{∗}^{?} has the tree property (4.1). Then the class D_{fib}(C) has the property (TREE) from Definition 0.4.

(ii) Suppose H_{∗}^{?} has the tree property (4.1) and C ⊆ FIN. Then the class D(C) has the property (TREE) from Definition 0.4.

(iii) Suppose that H_{∗}^{?} has the regular tree property (4.1) and C ⊆ VCYC. Then the class D(C) has the property (TREE_{R}) from Definition 0.10.

Proof. Let G act on a tree T. Denote by V the set of vertices of T and by E the set of edges. For x ∈ V ∪ E denote by G_{x} the isotropy group of x and by ϕ_{x}: G_{x} → G the inclusion. Let I_{T} = {H ≤ G | T^{H} ≠ ∅}. Since in a tree there is a unique geodesic between any two points, the fixed set T^{H} is contractible for H ∈ I_{T}. Thus T is a model for E_{I_{T}} (G).

Next we prove (i). In this case we assume that for each x ∈ V ∪ E the pair (G_{x}, C(G_{x})) satisfies the Fibered Isomorphism Conjecture 1.1. Let ϕ: K → G be a group homomorphism. Then K acts via ϕ on T. Equipped with this action T is also a model for E_{ϕ^{∗}I_{T}} (K). The tree property implies that (K, ϕ^{∗}I_{T}) satisfies the Isomorphism Conjecture 1.1. Thus (G, I_{T}) satisfies the Fibered Isomorphism Conjecture 1.1. Since the isotropy groups of T satisfy the Fibered Isomorphism Conjecture 1.1 with respect to C, we can deduce from Theorem 2.4 that (G, C(G)∩I_{T}) satisfies the Fibered Isomorphism Conjecture 1.1. Finally we use the fact that for the Fibered Isomorphism Conjecture 1.1 we can always enlarge the family (see [4, Lemma 1.6]) to conclude that the pair (G, C(G)) satisfies the Fibered Isomorphism Conjecture 1.1.

Next we prove (ii). In this case we assume that for each x ∈ V ∪ E the pair (G_{x}, C(G_{x})) satisfies the Isomorphism Conjecture 1.1. Arguing as above we conclude that (G, I_{T} ∩ C(G)) satisfies the Isomorphism Conjecture 1.1. Finite groups cannot act without fixed points on trees [26, Theorem 15 in 6.1 on page 58 and 6.3.1 on page 60]. Therefore I_{T} ∩ C(G) = C(G).

Finally, we prove (iii). In this case we assume that for each x ∈ V ∪ E the pair (G_{x}, C(G_{x})) satisfies the Isomorphism Conjecture 1.1 and that G_{e} is regular coherent for each e ∈ E. Arguing as before we conclude that (G, I_{T} ∩ C(G)) satisfies the Isomorphism Conjecture 1.1. We have to show that (G, C(G)) satisfies the Isomorphism Conjecture 1.1. Because of Theorem 2.4 it suffices to show for any virtually cyclic group V ∈ C(G) that the Isomorphism Conjecture 1.1 holds for (V, I_{T} ∩ C(G)|_{V}) = (V, I_{T}).

We first consider the case, where V contains a non-trivial normal finite subgroup F. We saw above that T^{F} is not empty and contractible. By Lemma 4.3 regular coherent groups are torsionfree. Thus isotropy groups of edges are torsionfree, therefore T^{F} is just a single vertex of T. Since F is normal in V, the action of V leaves the fixed points of F invariant. Therefore the vertex T^{F} is a fixed point for V. Hence we have V ∈ I_{T} so that I_{T} consists of all subgroups of V.

If V does not contain a non-trivial normal finite subgroup F, it is either ℤ or the infinite dihedral group. In both cases V acts on the tree ℝ with finite stabilizers such that the stabilizers of the edges are trivial and every finite subgroup of V occurs as stabilizer. The tree ℝ is a model for E_{FIN} (V). Since H_{∗}^{?} has the regular tree property (4.1) the map H_{n}^{V} (ℝ) → H_{n}^{V} (pt) is bijective for all n ∈ ℤ. This shows that V satisfies the Isomorphism Conjecture 1.1 for (V, FIN (V)). If V = ℤ, then every subgroup H ⊆ V is trivial or isomorphic to ℤ. If V is the infinite dihedral group, then any subgroup H of V is finite, infinite cyclic or infinite dihedral. We conclude from Theorem 2.4 that V satisfies the Isomorphism Conjecture 1.1 for every family which contains FIN, in particular for I_{T}.

Lemma 4.3. Regular coherent groups are torsionfree.

Proof. Assume that F is a finite subgroup of a regular coherent group G. Then the ℤG-module ℤ[G/F] is finitely presented and has a finite-dimensional resolution by finitely generated projective ℤG-modules since G is regular coherent and the ring ℤ is regular. Thus the restriction of ℤ[G/F] to a ℤF - module has a finite-dimensional resolution by projective (but no longer finite generated) ℤF -modules. As an ℤF -module ℤ[G/F] contains ℤ (with the constant F -action) as a direct summand. Therefore ℤ has a finite-dimensional resolution by projective ℤF -modules. This is only possible if F is the trivial group.

Corollary 4.4. Let H_{∗}^{?} be an equivariant homology theory which has the tree property (see Definition 4.1). Let 1 → K → G → Q → 1 be an extension of groups. Suppose that K acts on a tree with finite stabilizers and that (Q, FIN) satisfies the Fibered Isomorphism Conjecture 1.1 (in the range ≤ N).

Then (G, FIN) satisfies the Fibered Isomorphism Conjecture 1.1 (in the range ≤ N).

Proof. We first treat the case Q = {1}. Then the claim follows from Theorem 4.2 (i) because for a finite group F the pair (F, FIN) obviously satisfies the Fibered Isomorphism Conjecture 1.1.

Next we treat the case, where Q is finite. By a result of Dunwoody [11, Theorem 1.1] a group K acts on a tree with finite stabilizers if and only if H^{p}(K; ℚ) = 0 for each p ≥ 2. Since K acts on a tree with finite stabilizers, the trivial ℚK-module ℚ has a 1-dimensional projective resolution. Hence the trivial ℚG-module ℚ has a 1-dimensional projective resolution since [G : K] is finite and invertible in ℚ. This implies H^{p}(G; ℚ) = 0 for each p ≥ 2. Hence also G acts on a tree with finite stabilizers if Q is finite. This proves the claim for finite Q.

Now the general case follows from Lemma 2.5.

Lemma 4.5. Let H_{∗}^{?} be an equivariant homology theory which is continuous. Then the following assertions are equivalent.

(i) For each 1-dimensional G-CW -complex T for which each component is contractible (after forgetting the group action), the projection pr_{T} : T → π_{0}(T) induces isomorphisms

H_{n}^{G}(pr_{T}): H_{n}^{G}(T) ⟶^{≅} H_{n}^{G}(π_{0}(T)),

for each n ∈ ℤ, where we consider π_{0}(T) as a G-space using the discrete topology;

(ii) H_{∗}^{?} has the tree property, i.e. for each 1-dimensional G-CW -complex T, which is contractible (after forgetting the group action), and each n ∈ ℤ we obtain isomorphisms

H_{n}^{G}(pr_{T}): H_{n}^{G}(T) ⟶^{≅} H_{n}^{G}(pt);

(iii) For each 1-dimensional G-CW -complex X, which is contractible (after forgetting the group action) and has only one equivariant 1-cell, and each n ∈ ℤ we obtain isomorphisms

H_{n}^{G}(pr_{X}): H_{n}^{G}(X) ⟶^{≅} H_{n}^{G}(pt).

These three assertions remain equivalent if we add the requirement that the isotropy groups of edges are regular coherent to each assertion. (Thus (ii) becomes the assertion that H_{∗}^{?} has the regular tree property.)

Proof. (i) ⇒ (ii) ⇒ (iii) is obvious.

(iii) ⇒ (i) We prove the claim first under the assumption that G\T has finitely many 1-cells.

We use induction over the number of 1-cells in G\T. In the induction beginning, where G\T has no 1-cell, T is the disjoint union of homogeneous spaces and the claim follows from the fact that H_{∗}^{?} satisfies the disjoint union axiom.

In the induction step we can write T as a G-pushout

G/H × S^{0} ⟶^{q} T_{0}
↓  ↓
G/H × D^{1} ⟶ T

for a G-CW -subcomplex T_{0} ⊆ T such that G\T_{0} has one 1-cell less than G\T. Here H is the isotropy group of the 1-cell of T that is not contained in T_{0}. Since a connected subgraph of a tree is again a tree, each component of T_{0} is contractible. The induction hypothesis applies to T_{0}, G/H × S^{0} and G/H × D^{1}. Define X to be the G-pushout

G/H × S^{0} ⟶^{pr_{T_{0}} ∘q} π_{0}(T_{0})
↓  ↓
G/H × D^{1} ⟶ X

The G-maps pr_{T_{0}} : T_{0} → π_{0}(T_{0}), id_{G/H×S^{0}} and id_{G/H×D^{1}} are non-equivariant homotopy equivalences and induce a G-map f: T → X which is a nonequivariant homotopy equivalence since G/H × S^{0} → G/H × D^{1} is a cofibration. In particular X is a 1-dimensional G-CW -complex whose components are contractible. By a Mayer-Vietoris argument and the Five-Lemma the map

H_{n}^{G}(f): H_{n}^{G}(T) ⟶^{≅} H_{n}^{G}(X)

is bijective for all n ∈ ℤ. The following diagram commutes

T ⟶^{pr_{T}} π_{0}(T)
f↓  ↓^{π_{0}(f)}
X ⟶_{pr_{X}} π_{0}(X)

Since the map π_{0}(f) is bijective and hence a G-homeomorphism, H_{n}^{G}(π_{0}(f)) is bijective for all n ∈ ℤ. Recall that we have to show that H_{n}^{G}(pr_{T}) is bijective for all n ∈ ℤ. Hence it suffices to show that H_{n}^{G}(pr_{X}) is bijective for all n ∈ ℤ. This follows from the fact that we can write X as a disjoint union of a G-CW -complex Y, for which the assumption (iii) applies, and a 0-dimensional G-CW -complex Z, for which the induction beginning applies, and that H_{∗}^{?} satisfies the disjoint union axiom.

Next we treat the general case. Because H_{∗}^{?} satisfies the disjoint union axiom, we can assume without loss of generality that G\T is connected. Since we can write T = G ×_{H} T′ for a path component T′ and we have natural isomorphisms H^{H}(T′) ⟶^{≅} H_{n}^{G}(T) and H^{H}(pt) ⟶^{≅} H_{n}^{G}(G/H), we can assume without loss of generality that T is contractible.

Fix a 0-cell e ∈ G\T. Let I be the set of finite connected CW -subcomplexes Z ⊆ G\T with e ∈ Z. It can be directed by inclusion and satisfies G\T = ⋃_{Z∈I} Z. Let p: T → G\T be the projection. Then T is the directed union of the G-CW -subcomplexes p^{−1}(Z). Because of Lemma 2.1 the canonical map

colim_{Z∈I} H_{n}^{G}(p^{−1}(Z)) ⟶^{≅} H_{n}^{G}(T)

is bijective. Since each G-CW -complex p^{−1}(Z) has only finitely many equivariant 1-cells and hence satisfies the claim, and a colimit of a system of isomorphisms is again an isomorphism, it suffices to show that

colim_{Z∈I} H_{n}^{G}(π_{0}(p^{−1}(Z)) → H_{n}^{G}(pt)  (4.6)

is bijective. Fix ẽ ∈ T with p(ẽ) = e. Let G_{Z} be the isotropy group of the path component of p^{−1}(Z) containing ẽ in the G-set π_{0}(p^{−1}(Z)). Since each Z is connected, π_{0}(p^{−1}Z) is G/G_{Z}. We have for any inclusion Z_{1} ⊆ Z_{2} for elements Z_{1}, Z_{2} ∈ I, that G_{Z_{1}} is a subgroup of G_{Z_{2}}. We have G = ⋃_{Z∈I} G_{Z}. Since H_{∗}^{?} is continuous, we get an isomorphism

colim_{Z∈I} H_{n}^{G}(G/G_{Z}) ⟶^{≅} H_{n}^{G}(pt).

But this isomorphism can easily be identified with the map (4.6). This finishes the proof of Lemma 4.5.

Remark 4.7. Let G act on a tree T, such that G\T has only finitely many 1-cells. The proof of Lemma 4.5 shows that then G acts on tree X with the following properties: The quotient G\X has only one 1-cell. For each edge e of X the isotropy group G_{e} is also the isotropy group of an edge e′ of T. For each vertex v of X there is a subtree T_{v} of T that is invariant under the isotropy group G_{v} and for which G_{v}\T_{v} has one less 1-cell than G\T. In combination with the colimit argument from the proof of Lemma 4.5 this means that a class of groups C that has property (COL) from Definition 0.4 has property (TREE) from Definition 0.4 if and only if it has the following property

(TREE’) Suppose that G acts on a tree T where T has only one equivariant 1-cell. Assume that for each x ∈ T the isotropy group G_{x} belongs to C. Then G belongs to C;

and has property (TREE_{R}) if and only if it has the property (TREE’_{R}) Suppose that G acts on a tree T. Assume that for each x ∈ T

the isotropy group G_{x} belongs to C. For each edge e of T, assume that the isotropy group G_{e} is regular coherent. Then G belongs to C.

Note on the other hand, that the statement that the Fibered Isomorphism Conjecture 1.1 has property (TREE’) is really a statement about arbitrary actions on trees: If G acts on a tree T where T has only one equivariant 1-cell and ϕ: K → G is a group homomorphism, then the induced action of K on T may have more equivariant 1-cells and may even be no longer cocompact. Therefore we have to consider general trees in the formulation of the tree property in Definition 4.1.

5 Equivariant homology theories constructed from spectra

In this section we want to give a criterion when an equivariant homology theory has the tree property provided that it arises from a covariant functor E: GROUPOIDS → SPECTRAwhich sends equivalences of groupoids to weak equivalences of spectra. This will be the main example for us.

Fix a group G. The transport groupoid G^{G}(S) of a G-set S has S as set of objects and the set of morphism from s_{1} to s_{2} consists of those element g ∈ G with gs_{1} = s_{2}. Composition of morphisms comes from the group structure on G. The orbit category Or(G) has as objects homogeneous spaces G/H and as morphisms G-maps. We obtain a covariant functor G^{G}: Or(G) → GROUPOIDS, G/H ↦ G^{G}(G/H). Define the covariant functor E^{G}: Or(G) → SPECTRAby E ∘ G^{G}. Let H_{∗}^{G}(−; E) be the G-homology theory associated to E^{G} in [10, Section 4 and Section 7]. It is not hard to construct the relevant induction structure to get an equivariant homology theory H_{∗}^{?}(−; E). It has the property that for each group G with subgroup H ⊆ G and each n ∈ ℤ we have canonical isomorphisms

H_{n}^{G}(G/H; E) ≅ H_{n}^{H}(pt; E) ≅ π_{n}(E(H)).

In the expression E(H) we think of the group H as a groupoid with one object. More details of the construction of H_{∗}^{?}(−; E) can be found in [18] and [25].

Lemma 5.1. The equivariant homology theory H_{∗}^{?}(−, E) is continuous and has the tree property if and only if the following conditions are satisfied

(i) For each group G and directed system of subgroups {G_{i} | i ∈ I}, which is directed by inclusion and satisfies ⋃_{i∈I} G_{i} = G, and each n ∈ ℤ the map

colim_{i∈I} j_{i}: colim_{i∈I} π_{n}(E(G_{i})) → π_{n}(E(G))

is an isomorphism, where j_{i} is the homomorphism induced by the inclusion G_{i} → G; (ii) Consider a pushout of groups

\begin{matrix}H_{0} & ⟶^{i_{1}} & H_{1} \\ ^{i_{2}}↓ & & ↓^{j_{1}} \\ H_{2} & ⟶_{j_{2}} & G \end{matrix}  (5.2)

such that i_{1} and i_{2} are injective. In other words, G is the amalgamated product of H_{1} and H_{2} over H_{0} with respect to the injections i_{1} and i_{2}. Then for each such pushout (5.2) the following square of spectra is homotopy cocartesian

E^{G}(G/H_{0}) ∨ E^{G}(G/H_{0}) ⟶^{E^{G}(pr_{1})∨E^{G}(pr_{2})} E^{G}(G/H_{1}) ∨ E^{G}(G/H_{2})
id∨id↓  ^{E^{G}(pr_{3})∨E^{G}(pr_{4})}↓
E^{G}(G/H_{0})  ⟶_{E^{G}(pr_{5})}  E^{G}(G/G)

(5.3)

where the maps labeled pr_{i} denote canonical projections.

(iii) Let i_{0}, i_{1} : H → K be injective group homomorphisms. Let G be the HNN-extension associated to i_{0} and i_{1}. The HNN-extension comes with an inclusion j: K → G and t ∈ G such that j ∘i_{0} = c_{t}∘j ∘i_{1}, where c_{t} is conjugation by t. (This is the defining property of the HNN-extension.) We will use i_{0} to consider H as a subgroup of G. Then the following square of spectra is homotopy cocartesian

E^{G}(G/H) ∨ E^{G}(G/H) ⟶^{E^{G}(pr_{0})∨E^{G}(β)} E^{G}(G/K)
id∨id↓  ↓^{E^{G}(pr_{1})}
E^{G}(G/H)  ⟶_{E^{G}(pr_{2})} E^{G}(G/G)

(5.4)

where the maps labeled pr_{i} are canonical projections while β is defined by β(gH) = gtK.

For conditions (ii) and (iii) one can also consider the regular versions where (5.3) is only required to be homotopy cartesian if in addition H_{0} is regular coherent and torsionfree and (5.4) is only required to be homotopy cartesian if in addition H is regular coherent and torsionfree. Then the equivariant homology theory H_{∗}^{?}(−, E) is continuous and has the regular tree property if and only the condition (i) and the regular versions of conditions (ii) and (iii) hold.

Proof. Obviously condition (i) is equivalent to the condition that H_{∗}^{?} is continuous. From now on we assume that H_{∗}^{?} is continuous.

Suppose that the two conditions (ii) and (iii) are satisfied. Because of Lemma 4.5 it suffices to prove the tree property only for 1-dimensional contractible G-CW -complexes T such that there is precisely one equivariant 1-cell. Such a G-CW -complex will have precisely one or precisely two equivariant 0-cells. We only treat the case, where there are two equivariant 0-cells, the proof of the other case is analogous using condition (iii) instead of condition (ii).

We can write T as a G-pushout

G/H_{0} × S^{0} \underrightarrow{pr_{1} ∐ pr_{2}} G/H_{1} ∐ G/H_{2}
↓  ↓
G/H_{0} × D^{1} ⟶  T

where H_{0} is a subgroup of both H_{1} and H_{2} and pr_{1} and pr_{2} are the canonical projections. Recall that a G-space Z defines a contravariant functor Or(G) → SPACES, G/H ↦ map_{G}(G/H, Z) and that we get a spectrum map_{G}(G/?, Z) ∧_{Or (G)} E^{G} by the tensor product over the orbit category (see [10, Section 1]). If we apply map_{G}(G/?, −) ∧_{Or (G)} E^{G} to the G-pushout above, we obtain a homotopy cocartesian diagram of spectra

E^{G}(G/H_{0}) ∨ E^{G}(G/H_{0}) ⟶^{E^{G}(pr_{1})∨E(pr_{2})} E^{G}(G/H_{1}) ∨ E^{G}(G/H_{2})
id∨id↓  ↓
E^{G}(G/H_{0})  ⟶  map_{G}(G/?, T) ∧_{Or (G)} E^{G}

(5.5)

The following diagram is a pushout of groups

H_{0} ⟶^{i_{1}} H_{1}
i_{2}↓  ↓^{j_{1}}
H_{2} ⟶_{j_{2}} G

where i_{k}: H_{0} → H_{k}, j_{k}: H_{k} → G are inclusion (see [26, Example 1 on page 43]). Hence by condition (ii) we have the homotopy cocartesian square (5.3). The projection pr: T → G/G induces a map from the right lower corner of the diagram (5.5) to the right lower corner of the diagram (5.3), if we identify G/G ∧_{Or (G)} E^{G} = E^{G}(G/G). If we take the identity on the other three corners, we get a map between homotopy cocartesian squares of spectra. Since the three identity maps are obviously weak equivalences, the fourth map induced by the projection is a weak equivalence. But this map induces on homotopy groups the map H_{n}^{G}(pr; E): H_{n}^{G}(T; E) → H_{n}^{G}(pt) which is hence bijective for each n ∈ ℤ.

This shows that H_{∗}^{?}(−; E) has the tree property if conditions (ii) and (iii) are satisfied. It is now also obvious that conditions (ii) and (iii) hold if H_{∗}^{?}(−; E) has the tree property.

Remark 5.6.

(i) In the situation of Lemma 5.1 (ii) diagram (5.3) is homotopy cocartesian if and only if the commutative diagram

E(H) ∨ E(H) ⟶^{E(i_{1})∨E(i_{2})} E(G_{1}) ∨ E(G_{2})
id∨id↓  ↓^{E(j_{1})∨E(j_{2})}
E(H)  ⟶_{E(j_{0})}  E(G)

where j_{0} : H → G is defined to be j_{1} ∘ i_{1} = j_{2} ∘ i_{2}, is homotopy cocartesian since there is a canonical weak equivalences from each corner of this square to the corresponding corner of (5.3).

(ii) The situation in Lemma 5.1 (iii) is a bit more complicated. The natural diagram to consider is

E(H) ∨ E(H) ⟶^{E(i_{0})∨E(i_{1})} E(K)
id∨id↓  ↓^{E(j)}
E(H)  ⟶_{E(j∘i_{0})} E(G)

(5.7)

However, (5.7) is not commutative, while (5.4) is commutative. There is a canonical weak equivalence from each corner of (5.4) to the corresponding corner of (5.7), but those maps do not make the square

E(H) ∨ E(H)  ⟶^{E(i_{0})∨E(i_{1})}  E(K)
↓  ↓
E^{G}(G/H) ∨ E^{G}(G/H) ⟶_{E^{G}(pr_{0})∨E^{G}(β)} E^{G}(G/K)

commutative. The failure of the commutativity of (5.7) stems from the fact, that the underlying diagram of groups commutes only up to conjugation, i.e. j ∘ i_{0} ≠ j ∘ i_{1} = c_{t} ∘ j ∘ i_{0}. It is a consequence of the definitions that E(c_{t}) is weakly homotopic to id_{E(G)}, but in general there is no preferred homotopy. On the other hand E: GROUPOIDS→ SPECTRAis often slightly better than required in the discussion before Lemma 5.1, namely E is a 2-functor. This means that if τ is a natural transformation between functors f, g between groupoids, then there is a (preferred) homotopy E(τ) from E(f) to E(g). Under this stronger assumption on E there is a canonical homotopy that makes (5.7) homotopy commutative and then condition (iii) in Lemma 5.1 is equivalent to requiring that (5.7) is homotopy cocartesian with respect to the canonical homotopy.

6 Isomorphism Conjectures for spectra

In this section we relate the (Fibered) Isomorphism Conjecture 1.1 for an equivariant homology theory H_{∗}^{?} to the versions appearing in Farrell-Jones [12] for algebraic K- and L-theory.

Consider a group homomorphism ϕ: K → G, a K-CW -complex Z and a covariant functor E: SPACES→ SPECTRA , which sends weak equivalences to weak equivalences and is compatible with disjoint unions, i.e. for a family {Y_{i} | i ∈ I} of spaces the map induced by the inclusions j_{i}: Y_{i} → ∐_{i∈I} Y_{i}

⋁_{i∈I} E(j_{i}): ⋁_{i∈I} E(Y_{i}) → E (∐_{i∈I} Y_{i})

is a weak equivalence. We obtain a covariant functor

E_{Z}^{K}: Or(K) → SPECTRA , K/H ↦ E(Z ×_{K} K/H).

Recall that for each covariant functor F: Or(K) → SPECTRAthere is a Khomology theory H_{∗}^{K}(−; F) defined for K-CW -complexes with the property that H_{n}^{K}(K/H; F) ≅ π_{n}(F(K/H)) holds for H ⊆ K and n ∈ ℤ. [10, Section 4 and Section 7]. We denote by ϕ_{∗}Z the G-space G×_{ϕ} Z obtained by induction with ϕ from the K-space Z. For a G-space X let ϕ^{∗}X be the K-space obtained by restricting the G-action to a K-action using ϕ.

Lemma 6.1. For any G-CW -complex X there is an isomorphism, natural in X, Z and E,

ϕ_{∗}: H_{n}^{K}(ϕ^{∗}X; E_{Z}^{K}) ⟶^{≅} H_{n}^{G}(X; E_{ϕ_{∗}Z}^{G}).

Proof. Let ϕ: Or(K) → Or(G), K/H → G/ϕ(H) be the functor induced by ϕ. Given a contravariant (pointed) Or(G)-space A and a covariant (pointed) Or(K)-space B there is an adjunction

res_{ϕ} A ⊗_{Or (K)} B ⟶^{≅} A ⊗_{Or (G)} ind_{ϕ} B,  (6.2)

where res_{ϕ} is restriction and ind_{ϕ} denotes induction with the functor ϕ: Or(K) → Or(G) (see [10, Lemma 1.9]). It induces a natural isomorphism

H_{n}^{K}(ϕ^{∗}X; E_{Z}^{K}) ⟶^{≅} H_{n}^{G}(X; ind_{ϕ} E_{Z}^{K})

There is a weak equivalence of covariant Or(G)-spectra

ind_{ϕ} E_{Z}^{K} ⟶^{≅} E_{ϕ_{∗}Z}^{G}

coming from

map_{G}(G/ϕ(?), G/??)) ⊗_{Or (K)} (Z ×_{K} K/?) ⟶^{≅} Z ×_{ϕ} G/??,

(f, (z, k?)) ↦ (z, f(ϕ(k)ϕ(?)))

and the fact that E is compatible with disjoint unions.

Lemma 6.3. Let F be a family of subgroups of G. Let N ∈ ℤ. Then the following assertions are equivalent:

(i) For any free G-CW -complex Z and n ∈ ℤ (with n ≤ N) the assembly

map

H_{n}^{G}(E_{F}(G); E_{Z}^{G}) ⟶^{≅} H_{n}^{G}(pt; E_{Z}^{G})

is bijective;

(ii) For each injective group homomorphism ϕ: K → G and any free connected K-CW -complex Z and n ∈ ℤ (with n ≤ N) the assembly map

H_{n}^{K}(E_{ϕ^{∗}(F)}(K); E_{Z}^{K}) ⟶^{≅} H_{n}^{K}(pt; E_{Z}^{K})

is bijective;

(iii) For each group homomorphism ϕ: K → G and any free simply connected K-CW -complex Z and n ∈ ℤ (with n ≤ N) the assembly map

H_{n}^{K}(E_{ϕ^{∗}(F)}(K); E_{Z}^{K}) ⟶^{≅} H_{n}^{K}(pt; E_{Z}^{K})

is bijective.

Proof. (i) ⇒ (ii) and (i) ⇒ (iii) These implications follow from Lemma 6.1 since for any group homomorphism ϕ: K → G we have ϕ^{∗}(E_{F}(G)) = E_{ϕ^{∗}(F)}(K). (ii) ⇒ (i) We can write a G-CW -complex Z as Z = ∐_{i∈I} G ×_{G_{i}} Z_{i} for subgroups G_{i} ⊆ G and connected free G_{i}-CW -complexes Z_{i}. Since E is compatible with disjoint unions, we conclude from [10, Lemma 4.6] that we can assume without loss of generality that I consists of one element 0, i.e. Z = G ×_{G_{0}} Z_{0}. Now the claim follows from Lemma 6.1 applied to the inclusion ϕ: G_{0} → G and the free connected G_{0}-CW -complex Z_{0}.

(iii) ⇒ (ii) There is an extension of groups 1 → π_{1}(Z) → K̃ ⟶^{p} K → 1 and a K̃-action on the universal covering Z̃ which extends the π_{1}(Z)-action on Z̃ and covers the K-action on Z. Moreover, Z̃ inherits the structure of a free K̃-CW -complex. Now the claim follows from Lemma 6.1 applied to the group homomorphism p: K̃ → K and the simply connected free K̃-CW -complex Z̃ since p_{∗}Z̃ = Z and p^{∗}(ϕ^{∗}F) = (ϕ ∘ p)^{∗}F.

Lemma 6.4. Suppose that for any two-connected map f: X → Y the induced map E(f): E(X) → E(Y) is a weak equivalence. Let Z be a simply-connected G-CW -complex and f: Z → EG be the classifying map.

Then it induces a weak equivalence of Or(G)-spectra

f : E_{Z}^{G} → E_{EG}^{G}

and in particular for each G-CW -complex and each n ∈ ℤ a natural isomorphism

H_{n}^{G}(X; E_{Z}^{G}) ⟶^{≅} H_{n}^{G}(X; E_{EG}^{G}).

Proof. The map f: Z → EG is 2-connected. Hence the induced map f ×_{G} id_{G/H}: Z ×_{G} G/H → EG ×_{G} G/H is 2-connected for all subgroups H ⊆ G. Now apply [10, Lemma 4.6].

Definition 6.5 (Fibered Isomorphism Conjecture for spectra). We say that a group G satisfies the Isomorphism Conjecture for F and E (in the range ≤ N) if the assembly map induced by the projection pr: E_{F}(G) → pt

asmb: H_{n}^{G}(E_{F}(G); E_{EG}^{G})) → H_{n}^{G}(pt; E_{EG}^{G}))

is bijective for all n ∈ ℤ (with n ≤ N).

We say that a group G satisfies the Fibered Isomorphism Conjecture for F and E (in the range ≤ N) if for any free G-CW -complex Z the assembly map induced by the projection pr: E_{F}(G) → pt

asmb: H_{n}^{G}(E_{F}(G); E_{Z}^{G})) → H_{n}^{G}(pt; E_{Z}^{G}))

is bijective for all n ∈ ℤ (with n ≤ N).

Remark 6.6. The (Fibered) Isomorphism Conjecture of Farrell-Jones [12] for algebraic K-theory or L-theory respectively is equivalent to the (Fibered) Isomorphisms Conjecture 6.5 for (G, VCYC, E) if E: SPACES → SPECTRA sends X to the non-connective algebraic K-theory spectrum or L^{⟨−∞⟩}-theory spectrum of the fundamental groupoid of X respectively. Since a 2-connected map f: X → Y induces an equivalence on the fundamental groupoids, Lemma 6.4 applies. Let R be a ring. Consider the covariant functors

K_{R}: GROUPOIDS → SPECTRA
L_{R}: GROUPOIDS → SPECTRA

defined in [10, Section 2] satisfying π_{n}(K_{R}(G)) = K_{n}(RG) and π_{n}(L_{R}(G)) = L_{n}^{⟨−∞⟩}(RG) for each group G and n ∈ ℤ. Let H_{∗}^{?}(−, K_{R}) and H_{∗}^{?}(−, L_{R}) be the associated equivariant homology theories. Then the (Fibered) Isomorphism Conjecture for algebraic K-theory or algebraic L-theory respectively for the group G in the sense of Farrell-Jones [12] is equivalent to the (Fibered) Isomorphisms Conjecture 1.1 for H_{∗}^{?}(−, K_{R}) and H_{∗}^{?}(−, L_{R}) for the pair (G, VCYC). This follows from Lemma 6.3 and Lemma 6.4 .

For more information about the various conjectures such as the version for pseudoisotopy or the Baum-Connes Conjecture we refer for instance to [18].

7 The KH-Isomorphism Conjecture

In this section we will formulate the KH -Isomorphism Conjecture. The construction of homotopy algebraic K-Theory is a simplicial construction, so we will quickly fix the notation. The category Δ has as objects finite ordered sets of the form \underline{n} = \{0 < 1 < ⋯ < n\} and order preserving maps as morphisms. The n-simplex Δ_{•}^{n} is the simplicial set \underline{m} ↦ Map_{Δ}(\underline{m}, \underline{n}). Let R be a ring. The simplicial ring R[•] is defined by

R[\underline{n}] = R[t_{0}, …, t_{n}]/(t_{0} + ⋯ + t_{n} = 1).

Here the structure maps acts as follows: if f: \underline{n} → \underline{m} is order preserving then f^{∗}: R[\underline{m}] → R[\underline{n}] is defined by

f^{∗}(t_{k}) = ∑_{j∈f^{−1}(k)} t_{j}.

In [31] the homotopy algebraic K-theory KH_{∗}(R) of R is defined as the homotopy groups of the realization KHR of the simplicial spectrum K^{−∞}R[•]. Here K^{−∞} denotes the non-connected K-theory spectrum; a construction is reviewed before Definition 9.4. To illustrate the construction of homotopy algebraic K-theory we give a proof of the following fundamental property of homotopy algebraic K-theory, cf. [31, 1.2.(i)].

Proposition 7.1. The inclusion R ↪ R[X] gives an isomorphism KH_{n}(R) ≅ KH_{n}(R[X]) for all n ∈ ℤ.

Proof. It suffices to show that R[•] → R[X][•] is a homotopy equivalence of simplicial rings (cf. Remark 7.2). To see this we need to show that X ↦ 0 is homotopic to the identity of R[X][•]. Such a homotopy R[X][•] × Δ_{•}^{1} → R[X][•] is given by

(X, f) ↦ ( ∑_{j∈f^{−1}(0)} t_{j}) X,

where f: \underline{n} ↦ \underline{1}.

Remark 7.2. If S is a ring and F is a finite set, then S × F has a ring structure (S × F ≅ ∏_{f∈F} S). Therefore we may view R[X][•] × Δ_{•}^{1} as a simplicial ring and the above homotopy as a map of simplicial rings. Therefore we get a map |K^{−∞}(R[X][•] × Δ_{•}^{1})| → |K^{−∞}(R[X][•])|. On the other hand, there is a map of simplicial spectra K^{−∞}(R[X][•]) × Δ_{•}^{1} → K^{−∞}(R[X][•] × Δ_{•}^{1}) defined as follows. For f ∈ Δ1 ; \underline{n} there is an obvious map of rings ι_{f}: R[X][\underline{n}] → R[X][\underline{n}] × \{f\} ↪ R[X][\underline{n}] × Δ_{n}^{1}. Thus we can map (x, f) ∈ K^{−∞}(R[X][\underline{n}]) × Δ_{n}^{1} to K^{−∞}(ι_{f})(x) ∈ K^{−∞}(R[X][\underline{n}] × Δ_{n}^{1}).

In order to define an equivariant homology theory we define the functor

KH_{R}: GROUPOIDS→ SPECTRA

as the realization of the simplicial functor

K_{R[•]} : GROUPOIDS→ SPECTRA .

Thus π_{n}(KH_{R}(G)) = KH_{n}(RG). Since the realization of a weak equivalence is again a weak equivalence, KH_{R} sends equivalences of groupoids to weak equivalences of spectra.

Conjecture 7.3 ((Fibered) KH -Isomorphism Conjecture). A group G is said to satisfy the (Fibered) KH -Isomorphism Conjecture (for a ring R) if the pair (G, FIN) satisfies the (Fibered) Isomorphism Conjecture 1.1 for the equivariant homology theory H_{∗}^{?}(−; KH_{R}).

Remark 7.4. All virtually cyclic groups act on trees with finite stabilizers. For a finite group F the (Fibered) KH -Isomorphism Conjecture holds (since E_{FIN} (F) = F/F). Thus by Theorem 0.5 the (Fibered) KH -Isomorphism Conjecture holds for virtually cyclic groups. Therefore Theorem 2.4 implies, that it makes no difference if we replace the family of finite groups with the family of virtually cyclic groups in the formulation of the (Fibered) KH - Isomorphism Conjecture.

8 The Relation between the K- and the KH-Isomorphism Conjecture

There is a natural map K^{−∞}R → KHR induced from the inclusion of the constant simplicial ring R into R[•]. Similarly we obtain a natural transformation K_{R} → KH_{R} of functors from GROUPOIDS to SPECTRA . Thus we obtain a natural transformation of equivariant homology theories H_{∗}^{?}(−; K_{R}) → H_{∗}^{?}(−; KH_{R}) and a commutative diagram between assembly maps

\begin{matrix}H_{n}^{G}(E_{FIN} (G); K_{R}) & ⟶ & K_{n}(RG) \\ ↓ & & ↓ \\ H_{n}^{G}(E_{FIN} (G); KH_{R}) & ⟶ & KH_{n}(RG) \end{matrix}  (8.1)

We first explain what the KH -Isomorphism Conjecture 7.3 implies for the K-Isomorphism Conjecture, i.e. the Farrell-Jones Conjecture for algebraic K-theory (see Remark 6.6). In order to state the connection we need to recall the groups N^{p}K_{n}(R) [6, XIII]. They can be defined by N^{0}K_{n}(R) = K_{n}(R) and

N^{p}K_{n}(R) = ker(N^{p−1}(q): N^{p−1}K_{n}(R[t]) → N^{p−1}K_{n}(R)),

where q(t) = 0. For regular rings N^{p}K_{n}(R) = 0 for p ≥ 1, see [13].

Proposition 8.2. Let G be a group that satisfies the KH -Isomorphism Conjecture 7.3 for the ring R.

(i) Suppose that N^{p}K_{n}(RF) = 0 for all finite subgroups F of G and all n ∈ ℤ, p ≥ 1. Then the assembly map with respect to the family FIN in algebraic K-theory, i.e. the top row in (8.1), is split injective.

(ii) Suppose that N^{p}K_{n}(RF)⊗ℚ = 0 for all finite subgroups F of G and all n ∈ ℤ, p ≥ 1. Then assembly map with respect to the family FIN in algebraic K-theory, i.e. the top row in (8.1), is rationally split injective.

Proof. By the spectral sequence from [31, 1.3] the canonical map K_{∗}(A) → KH_{∗}(A) is an isomorphism if N^{p}K_{n}(A) = 0 for all n ∈ ℤ and p ≥ 1 and a rational isomorphism if N^{p}K_{n}(A) ⊗ ℚ = 0 for all n ∈ ℤ and p ≥ 1. Therefore these assumptions imply by a spectral sequence argument that the left vertical map in (8.1) is an isomorphism or a rational isomorphism respectively.

Remark 8.3. The assumptions of Proposition 8.2 (i) and (ii) are satisfied in the following cases.

(i) If R is a regular ring containing ℚ then RF is regular for all finite groups F. Thus the assumption in 8.2 (i) is satisfied.

(ii) If R = ℤ then the assumption in 8.2 (ii) is satisfied. This follows from [30, 6.4], which implies

NK_{∗}(ℤ[t_{1}, …, t_{n}]F) ⊗ ℚ ≅ NK_{∗}(ℚ[t_{1}, …, t_{n}]F).

Thus for a finite group F it follows that, NK_{∗}(ℤ[t_{1}, …, t_{n}]F) vanishes rationally, since ℚ[t_{1}, …, t_{n}]F is regular. A straight forward induction shows that this implies N^{p}K_{n}(ℤF) ⊗ ℚ = 0 for p ≥ 1.

(iii) If G is torsionfree and R is regular, then the assumption in 8.2 (i) is satisfied. In this case the Farrell-Jones Conjecture in algebraic K-theory asserts, that the top vertical map in (8.1) is an isomorphism. Thus, in this situation (R regular, G torsionfree) the Farrell-Jones Conjecture in algebraic K-theory holds if N_{n}^{p}(RG) = 0 for all n ∈ ℤ and p ≥ 1 and if G satisfies the KH -Isomorphism Conjecture.

Next we explain the reverse connection.

Theorem 8.4 (The K-theory version implies the KH -version). Let G be a group and let R be a ring.

(i) Suppose that the (Fibered) Farrell-Jones Conjecture in algebraic Ktheory is true for (G, R[x_{1}, x_{2}, … x_{n}]) for all n ≥ 1 then the (Fibered) KH -Isomorphism Conjecture 7.3 is true for (G, R).

(ii) Suppose that the Fibered Farrell-Jones Conjecture in algebraic K-theory is true for (G×ℤ^{n}, R) for all n ≥ 1. Then the Fibered KH -Isomorphism Conjecture 7.3 is true for (G, R).

Proof. Let ϕ: K → G be a group homomorphism. The assembly map H_{n}^{K}(E_{ϕ^{∗}VCYC}(K); K_{R[\underline{n}]}) → K_{n}(R[\underline{n}][K]) is on the level of spectra given by the map

E_{ϕ^{∗}VCYC}(K) ⊗_{Or (K)} K_{R[\underline{n}]} → K/K ⊗_{Or K} K_{R[\underline{n}]} ≃ K_{R[\underline{n}]}(K) induced by E_{ϕ^{∗}VCYC}(K) → K/K. The assumption in (i) is that this map of spectra is a weak equivalence. Using the fact that the realization of a map of simplicial spectra that is levelwise a weak equivalences is a weak equivalence and the identification

|E_{ϕ^{∗}VCYC}(K) ⊗_{Or (K)} K_{R[•]}| ≅ E_{VCYC}(K) ⊗_{Or (K)} |K_{R[•]}|

we conclude that the (Fibered) Farrell-Jones Conjecture for (G, R[x_{1}, …, x_{n}]) for all n implies the (Fibered) KH -Isomorphism Conjecture for (G, R) with the family of finite subgroups replaced by the family of virtually cyclic subgroups. By Remark 7.4 this is equivalent to the (Fibered) KH -Isomorphism Conjecture 7.3 for (G, R).

Next we prove (ii) by reducing it to (i). For a group K we denote by p_{K}: K ×ℤ → K the canonical projection. We observe first that the (Fibered) Isomorphism Conjecture for (G, VCYC, R[ℤ]) is equivalent to the (Fibered) Isomorphism Conjecture (G × ℤ, (p_{G})^{∗}VCYC, R) because for every group K and every K-space X there is a natural isomorphism

H_{n}^{K}(X; K_{R[ℤ]}) ≅ H_{n}^{K×ℤ}(p_{K}^{∗}X; K_{R})

where p_{K}^{∗}X denotes the K × ℤ-space obtained by restriction of X along p_{K} and because p_{K}^{∗}E_{F}(K) = E_{(p_{K})^{∗}F}(K × ℤ) for every family of subgroups of K. If the Fibered Isomorphism Conjecture holds for a family F, then it will also hold for every family G that contains F [4, Lemma 1.6]. Because the family of virtually cyclic subgroups of G × ℤ is contained in (p_{G})^{∗}VCYC the Fibered Farrell-Jones Conjecture for (G × ℤ, R) implies the Fibered Farrell-Jones Conjecture for (G, R[ℤ]). By the Bass-Heller-Swan splittings [14] the latter is equivalent to the Fibered Farrell-Jones Conjecture for (G, R[x]). By induction on n this means that the assumption in (ii) implies the assumption of (i).

Remark 8.5. It is not unreasonable to expect that the non-fibered version of Theorem 8.4 (ii) is also valid. Our argument would also prove the nonfibered version if we were to know that for every virtually cyclic group V the product V × ℤ satisfies the Farrell-Jones Conjecture. This seems very likely, but we could not find such a statement in the literature.

Remark 8.6. Let us briefly list some consequences of the Farrell-Jones Conjecture for algebraic K-theory. Suppose that the Farrell-Jones Conjecture for algebraic K-theory holds for the group G and every regular ring R. Now consider a group G and a regular ring R with the property that either ℚ ⊆ R holds or G is torsionfree. The proof of Proposition 8.2, Remark 8.3 and Theorem 8.4 imply:

(i) The KH -Isomorphism Conjecture 7.3 is true for G and R;

(ii) The canonical map K_{n}(RG) → KH_{n}(RG) is bijective for n ∈ ℤ; (iii) N^{p}K_{n}(RG) = 0 for p ≥ 1 and n ∈ ℤ, see [3, Proposition 7.4].

Remark 8.7 (Injectivitiy of the KH -assembly map). In many cases injectivitiy of the assembly map

H_{n}^{G}(E_{FIN} (G); K_{R}) → K_{n}(RG)

is proven by construction of a spectrum T(R, G) and a map of spectra K^{−∞}RG → T(R, G) such that for many groups the composition of the assembly map on the level of spectra with this map is a weak equivalence. The construction of K^{−∞}RG → T(R, G) is always natural in the coefficient ring R. Therefore applying the arguments of the proof of Theorem 8.4 (i) we can use T(R[•], G) to split the KH -assembly map in this cases. This proves that the KH -assembly map is split injective for groups G of finite asymptotic dimension that admit a finite model for BG [2] and for groups G for which E_{FIN} (G) has a compactification with certain properties [23].

9 Non-connective Waldhausen Nil

Before we can show that H_{∗}^{?}(−; KH_{R}) has the tree property, we will need to recall Waldhausen’s work on K-theory of generalized free products [29]. We start with Waldhausen’s Nil-groups.

Definition 9.1 (Nil-categories). Let R be a ring and X, Y, Z, W be Rbimodules.

(i) The category NIL(R; X, Y) has as objects quadruples (P, Q, p, q), where P and Q are finitely generated projective R-modules and p: P → Q⊗_{R} X, q : Q → P ⊗_{R}Y are R-linear maps subject to the following nilpotence condition: Let P_{0} = Q_{0} = 0, P_{n+1} = p^{−1}(Q_{n} ⊗_{R} X) and Q_{n+1} = q^{−1}(P_{n} ⊗_{R} Y). It is required that for sufficient large N, P = P_{N} and Q = Q_{N}.

(ii) The category NIL(R; X, Y, Z, W) has as objects quadruples (P, Q, p, q), where P and Q are finitely generated projective R-modules and p: P → Q ⊗_{R} X ⊕ P ⊗_{R} Z, q : Q → P ⊗_{R} Y ⊕ Q ⊗_{R} W are R-linear maps subject to the following nilpotence condition: Let P_{0} = Q_{0} = 0, P_{n+1} = p^{−1}(Q_{n} ⊗_{R} X ⊕ P_{n} ⊗_{R} Z) and Q_{n+1} = q^{−1}(P_{n} ⊗_{R} Y ⊕ Q_{n} ⊗_{R} W). It is required that for sufficient large N, P = P_{N} and Q = Q_{N}.

Morphisms are in both cases R-linear maps P → P′, Q → Q′ that are compatible with p, p′, q and q′. Both categories are exact categories, where sequences are exact whenever they map to exact sequences of modules under (P, Q, p, q) ↦ P and (P, Q, p, q) ↦ Q.

Remark 9.2. Let f_{R}: R → S be a map of rings and f_{X}: X → X′, f_{Y} : Y → Y′ be maps over f_{R}, i.e. X′ and Y′ are S-bimodules, f_{X}(rxr′) = f_{R}(r)f_{X}(x)f_{R}(r′) and similar for f_{Y}. Then (f_{R}, f_{X}, f_{Y}) induce an exact functor NIL(R; X, Y) → NIL(S; X′, Y′) sending (P, Q, p, q) to (P ⊗_{R} S, Q ⊗_{R} S, p_{S}, q_{S}) where p_{S} and q_{S} are the canonical maps. For example, p_{S} is the composition

P ⊗_{R} S → Q ⊗_{R} X ⊗_{R} S → Q ⊗_{R} X′ ≅ Q ⊗_{R} S ⊗_{S} X′,

where the first map uses p and the second uses f_{X} and left multiplication of S. In particular, we get a functor S ⊗ −: NIL(R; X, Y) → NIL(S ⊗ R; S ⊗X, S ⊗Y). If f: S → S′ is a map of rings, then we get another functor f_{∗}: NIL(S⊗R; S⊗X, S⊗Y) → NIL(S′⊗R; S′⊗X, S′⊗Y). The functoriality of NIL(R; X, Y, Z, W) is similar.

We review next [28, 2.5] in a slightly more modern language and discuss applications to Waldhausen Nil-categories.

A sum ring is a ring S together with elements v, v̄, u and ū of S such that uū = 1, vv̄ = 1 and v̄v + ūu = 1. This implies that uv̄ = 0 and vū = 0. Moreover, the map f_{⊕}: S ⊕ S → S defined by (r, s) ↦ ūru + v̄sv is a ring homomorphism. Let M, N be S-modules. Denote by (M, N) the direct sum M ⊕ N considered as an S ⊕ S-module. The S-modules (M, N) ⊗_{f_{⊕}} S and M ⊕ N (considered as an S-module as usual) are naturally isomorphic. Such an isomorphism and its inverse are given by

\begin{aligned}M ⊕ N & ∋ & m ⊕ n & ↦ & (m, n) ⊗ (ū + v̄) & ∈ & (M, N) ⊗_{f_{⊕}} S \\ (M, N) ⊗_{f_{⊕}} S & ∋ & (m, n) ⊗ r & ↦ & mur ⊕ nvr & ∈ & M ⊕ N. \end{aligned}

An infinite sum ring is a sum ring together with a ring endomorphism f_{∞} such that f_{⊕}(r, f_{∞}(r)) = f_{∞}(r).

Remark 9.3. The functor M ↦ M ⊗_{f_{∞}} S is an Eilenberg swindle on the category P_{S} of finitely generated projective modules over such an infinite sum ring. Indeed,

M ⊗_{f_{∞}} S = M ⊗_{f_{⊕}∘(id_{S},f_{∞})} S
≅ M ⊗_{(id_{S},f_{∞})} (S ⊕ S) ⊗_{f_{⊕}} S
≅ (M, M ⊗_{f_{∞}} S) ⊗_{f_{⊕}} S
≅ M ⊕ (M ⊗_{f_{∞}} S).

The same swindle applies to Waldhausen’s Nil-categories: Fix an infinite sum ring S. Let X and Y be bimodules over another ring R. Then the endofunctor (f_{∞})_{∗} is equal to the composition

\xymatrix{NIL(S ⊗ R; S ⊗ X, S ⊗ Y) \ar[d]_{(id,f_{∞})_{∗}} \\ NIL((S ⊕ S) ⊗ R; (S ⊕ S) ⊗ X, (S ⊕ S) ⊗ Y) \ar[d]_{(f_{⊕})_{∗}} \\ NIL(S ⊗ R; S ⊗ X, S ⊗ Y)}

Using the natural isomorphism from above there is a natural transformation from this composition to id ⊕(f_{∞})_{∗}. Thus, (f_{∞})_{∗} is an Eilenberg swindle. This swindle is compatible with the two forgetful functors NIL(S ⊗ R; S ⊗ X, S ⊗Y) → P_{S⊗R}. Analogous considerations apply to NIL(S ⊗R; S ⊗X, S ⊗ Y, S ⊗ Z, S ⊗ Z, S ⊗ W).

The cone ring Λℤ of ℤ is the ring of column and row finite ℕ×ℕ-matrices over ℤ, i.e. matrices such that every column and every row contains only finitely many non-zero entries. The suspension ring Σℤ is the quotient of Λℤ by the ideal of finite matrices. For an arbitrary ring R we define ΛR = Λℤ⊗R and ΣR = Σℤ ⊗ R. We will view Λ and Σ as functors. Every bijection ℕ → ℕ × ℕ induces the structure of an infinite sum ring on the cone ring ΛR, cf. [28, p.355]. We can consider ΛR as a subring of the ring of column and row finite ℕ × ℕ-matrices over R. However, this inclusion is not always an equality.

Next we want to define a non-connective spectrum associated to Waldhausen’s Nil-categories. First we recall the construction for K-theory. Denote by 𝕂R the K-theory space of a ring (obtained for example by applying Waldhausen’s S_{•}-construction to the category P_{R} of finitely generated projective R-modules). The n-th space of the spectrum K^{−∞}R is by definition 𝕂Σ^{n}R. The composition 𝕂R → 𝕂ΛR → 𝕂ΣR is constant. The choice of an bijection ℕ → ℕ×ℕ gives an Eilenberg swindle on ΛR, cf. Remark 9.3. If we fix such a bijection we get a functorial way of contracting 𝕂ΛR to the basepoint. This induces the structure maps Σ(𝕂Σ^{n}R) → 𝕂Σ^{n+1}R.

Definition 9.4 (Non-connective Nil-spectra). Let R be a ring and X and Y be R-bimodules. The (non-connective) spectrum NiL^{−∞}(R; X, Y) has 𝕂NIL(Σ^{n}R; Σ^{n}X, Σ^{n}Y) as its nth space. Here Σ^{n}X = Σ^{n}ℤ ⊗ X and we define similarly Σ^{n}Y, ΛX and ΛY. The structure maps are defined in an analogous ways as for the non-connective K-theory spectrum: The functoriality discussed in Remark 9.2 allows us to consider the (constant) composition

NIL(C; A′, B′) → NIL(ΛC; ΛA′, ΛB′) → NIL(ΣC; ΣA′, ΣB′).

The structure maps for NiL^{−∞}(R; X, Y) are now defined using the Eilenberg swindle on the second category discussed in Remark 9.3. Similarly, we define a (non-connective) spectrum NiL^{−∞}(R; X, Y, Z, W) with

𝕂NIL(Σ^{n}R; Σ^{n}X, Σ^{n}Y, Σ^{n}Z, Σ^{n}W)

as its nth space.

An inclusion α: C → A of rings is called pure if A = α(C) ⊕ A′ as Cbimodules. It is called pure and free if in addition A′ is free as a left C-module. If H → G is an inclusion of groups, then the inclusion RH → RG of rings is pure and free. The following observation is straight forward.

Lemma 9.5. If α is pure (and free) then Σα and Λα are also pure (and free).

Let α: C → A and β: C → B be both pure. The ring R = A ∗_{C} B, the free product of A and B, amalgamated at C (w.r.t. α, β), is defined by the push-out

\xymatrix{C \ar[r]^{α} \ar[d]_{β} & A \ar[d] \\ B \ar[r] & R.}

For group rings this corresponds to amalgamated products of groups.

Lemma 9.6. The cone respectively suspension ring of A ∗_{C} B is naturally isomorphic to ΛA ∗_{ΛC} ΛB respectively ΣA ∗_{ΣC} ΣB.

Proof. This follows from the universal property.

Let α, β: C → A be pure and free. The Laurent extension w.r.t. α and β is the universal ring R = A_{α,β}{t^{±1}} that contains A and an invertible element t and satisfies

α(c)t = tβ(c) for c ∈ C.

Existence is explained in [29, p.149]. For group rings this corresponds to HNN-extensions.

Lemma 9.7. The cone respectively suspension ring of A_{α,β}{t^{±1}} is naturally isomorphic to ΛA_{Λα,Λβ}{T^{±1}} respectively ΣA_{Σα,Σβ}{T^{±1}}.

Proof. This follows from the universal property.

10 Waldhausen’s cartesian squares

Let α: C → A and β: C → B be pure and free. Write A = α(C) ⊕ A′ and B = β(C) ⊕ B′ as C-bimodules. Let R = A ∗_{C} B. Consider the square

\begin{matrix}NIL(C; A′, B′) & ⟶ & P_{A} × P_{B} \\ ↓ & & ↓ \\ P_{C} & ⟶ & P_{R}^{∗} \end{matrix}  (10.1)

The two functors starting at the upper left hand corner are defined by sending (P, Q, p, q) to (P ⊕Q) ∈ P_{C} respectively to (P ⊗_{α}A, Q⊗_{β}B). The category P_{R}^{∗} is defined in [29, p.205]. It is a cofinal full subcategory of P_{R} and contains all finitely generated free modules. There is an obvious natural transformation between the two ways to go through the diagram. However, there is also a not quite so obvious more complicated natural transformation that makes use of p and q, cf. [29, 1.4,11.3]: Let i_{P} : P ⊗_{C} A ⊗_{A} R → P ⊗_{C} R and i_{Q}: Q ⊗_{C} B ⊗_{B} R → Q ⊗_{C} R be the natural isomorphisms. Define N by the commutative diagram

P  ⟶^{p} Q ⊗_{C} A
↓  ↓
P ⊗_{C} A ⊗_{A} R ⟶_{N} Q ⊗_{C} R

and similarly M: Q ⊗_{C} B ⊗_{B} R → P ⊗_{C} R. Then

( \begin{matrix}i_{P} & M \\ N & i_{Q} \end{matrix} )

is an isomorphism and defines the more complicated natural transformation. It is a result of Waldhausen [29, 11.3], that applying 𝕂 to (10.1) yields a homotopy cartesian square (w.r.t. the homotopy induced by the more complicated natural transformation). We will need a non-connective version of Waldhausen’s result.

Theorem 10.2. (Non-connective versions of Waldhausen’s homotopy cartesian squares for amalgamation) We have the following diagram of spectra

NiL^{−∞}(C; A′, B′) ⟶ K^{−∞}A ∧ K^{−∞}B
↓  ↓
K^{−∞}C  ⟶  K^{−∞}R

The more complicated natural transformations combine to a homotopy between the two ways to go through this diagram. The diagram is homotopy cartesian w.r.t. this homotopy.

Proof. The diagram of spectra is obtained from (10.1) by tensoring everything in sight by Σ^{n}ℤ (and applying 𝕂). We need to check compatibility with the structure maps. Those come from an intermediate diagram where we apply Λℤ ⊗ − and use an Eilenberg swindle. This Eilenberg swindle happens on the left of this tensor product, while everything else happens on the right. This proves compatibility with the structure maps. If we use P_{R}^{∗} rather then P_{R} then the diagram is homotopy cartesian by Waldhausen’s result and Lemma 9.6. However, since the former category contains all finitely generated free modules and we use non-connective K-theory we can also use P_{R}.

Remark 10.3. (Waldhausen Nil for amalgamations vanishes for regular coherent rings) Waldhausen proved that for a regular coherent ring C, the functor P_{C} × P_{C} → NIL(C; A′, B′) defined by (P, Q) ↦ (P, Q, 0, 0) induces an isomorphism in connective K-theory, [29, 12.2]. A priori, this does not immediately imply that the induced map α : K^{−∞}C ∨ K^{−∞}C → NiL^{−∞}(C; A′, B′) is a weak equivalence because it is not clear whether ΣC is again regular coherent. However, if C is regular or more generally, if C is a group ring with a regular coefficient ring over a regular coherent group, then α is a weak equivalence. This can be seen as follows. The functor (P, Q, p, q) ↦ (P, Q) splits α, thus α will injective on homotopy groups. To prove surjectivity we use the fact that there is an in C natural map of rings C[ℤ^{n}] → Σ^{n}C, that is naturally split surjective in connective K-theory, [28, Section 6]. We get the following commutative diagram

\begin{matrix}P_{C[ℤ^{n}]} × P_{C[ℤ^{n}]} & ⟶ & NIL(C[ℤ^{n}]; A′[ℤ^{n}], B′[ℤ^{n}]) \\ ↓ & & ↓ \\ P_{Σ^{n}C} × P_{Σ^{n}C} & ⟶ & NIL(Σ^{n}C; Σ^{n}A′, Σ^{n}B′) \end{matrix}  (10.4)

Apply C[ℤ^{n}] → Σ^{n}C map to the long exact sequence obtained from (10.1) by [29, 11.3]. A little diagram chase in the resulting ladder diagram shows that the right vertical map in (10.4) is surjective in connective K-theory. The assumptions on C imply that C[ℤ^{n}] is regular coherent. Therefore the top horizontal map in (10.4) is an isomorphism in connective K-theory. Therefore the bottom horizontal map in (10.4) is also surjective in connective K-theory. This implies that α is surjective on homotopy groups.

Next we discuss the analogous cartesian square for Laurent extensions. Let α, β: C → A be pure and free and R = A_{α,β}{t^{±1}}. We denote by ι: A → R the inclusion. Write A = α(C) ⊕ A′ and A = β(C) ⊕ A′′ as C-bimodules. Consider the square

\begin{matrix}NIL(C;_{α} A′_{α},_{β} A′′_{β},_{β} A_{α},_{α} A_{β}) & ⟶ & P_{A} \\ ↓ & & ↓^{ι_{∗}} \\ P_{C} & ⟶_{(ι∘α)_{∗}} & P_{R}^{∗} \end{matrix}  (10.5)

Here we we use α and β to indicate the C-bimodule structures. The two functors starting at the upper right hand corner are defined by sending (P, Q, p, q) to (P ⊕ Q) ∈ P_{C} respectively to (P ⊗_{α} A ⊕ Q ⊗_{β} A). The category P_{R}^{∗} is defined in [29, p.205]. It is a cofinal full subcategory of P_{R} and contains all finitely generated free modules. As before there is an obvious and a more complicated natural transformation between the two ways to go through the diagram [29, 2.4,12.3]: Let i_{P} : P ⊗_{α} A ⊗_{A} ⊗R → P ⊗_{α} R and i_{Q}: Q⊗_{β} A⊗R → Q⊗_{α} R denote the canonical isomorphisms. (Here i_{Q} uses an extra t, i.e. i_{Q}(y ⊗ a ⊗ r) = y ⊗ tar.) These isomorphisms give the obvious natural transformation. The more complicated natural transformation is obtained by adding a nilpotent term which we review next. Write p = p_{0} + p_{1}, where p_{0} : P → P ⊗_{β} A_{α} and p_{1} : P → Q ⊗_{α} A′_{α}. Define N_{0} and N_{1} by the commutative diagrams

\begin{matrix}P & ⟶^{p_{0}} & P ⊗_{β} A_{α} \\ ↓ & & ↓ \\ P ⊗_{α} A ⊗_{A} R & ⟶_{N_{0}} & P ⊗_{α} R \end{matrix}  \begin{matrix}P & ⟶^{p_{1}} & Q ⊗_{β} A′_{α} \\ ↓ & & ↓ \\ P ⊗_{α} A ⊗_{A} R & ⟶_{N_{1}} & Q ⊗_{α} R \end{matrix}

(The second vertical arrow is x ⊗ a ↦ x ⊗ ta.) Write q = q_{0} + q_{1}, where q_{0} : Q → Q ⊗_{α} A_{β} and q_{1} : Q → P ⊗_{β} A′′_{β}. Define M_{0} and M_{1} by the commutative diagrams

\begin{matrix}Q & ⟶^{q_{0}} & Q ⊗_{α} A_{β} \\ ↓ & & ↓ \\ Q ⊗_{β} A ⊗_{A} R & ⟶_{M_{0}} & Q ⊗_{α} R \end{matrix}  \begin{matrix}Q & ⟶^{q_{1}} & P ⊗_{α} A′′_{β} \\ ↓ & & ↓ \\ Q ⊗_{β} A ⊗_{A} R & ⟶_{M_{1}} & P ⊗_{α} R \end{matrix}

(The forth vertical arrow is x ⊗ a ↦ x ⊗ ta.) The more complicated natural transformation is then given by the isomorphism

( \begin{matrix}i_{P} & 0 \\ 0 & i_{Q} \end{matrix} ) + ( \begin{matrix}N_{0} & M_{1} \\ N_{1} & M_{0} \end{matrix} ) .

It is a result of Waldhausen [29, 12.3], that applying 𝕂 to (10.5) yields a homotopy cartesian square (w.r.t. the homotopy induced by the more complicated natural transformation). The arguments used to prove Theorem 10.2 can be used to prove a non-connective version of this result.

Theorem 10.6. (Non-connective versions of Waldhausen’s homotopy cartesian squares for Laurent extensions) We have the following diagram of spectra

NiL^{−∞}(C;_{α} A′_{α},_{β} A′′_{β},_{β} A_{α},_{α} A_{β}) ⟶ K^{−∞}(A)
↓  ↓^{K^{−∞}(ι)}
K^{−∞}(C)  ⟶_{K(ι∘α)} K^{−∞}(R)

The more complicated natural transformations combine to a homotopy between the two ways to go through this diagram. The diagram is homotopy cartesian w.r.t. this homotopy.

Remark 10.7. (Waldhausen Nil for Laurent extensions vanishes for regular coherent rings) The reasoning in Remark 10.3 also applies to Nil(C;_{α} A′_{α},_{β} A′′_{β},_{β} A_{α},_{α} A_{β}). If C is a group ring with a regular coefficient ring over a regular coherent group, then the functor (P, Q) ↦ (P, Q, 0, 0) induces a weak equivalence K^{−∞}C∨K^{−∞}C → NiL^{−∞}(C;_{α} A′_{α},_{β} A′′_{β},_{β} A_{α},_{α} A_{β}).

11 The tree property for Homotopy K-Theory

This section contains the proof of the following result.

Theorem 11.1 (Continuity and tree-property for H_{∗}^{?}(−; KH_{R})). The equivariant homology theory H_{∗}^{?}(−; KH_{R}) is continuous and has the tree property.

Let X, Y,Z and W be bimodules over R. Consider the simplicial spectra \underline{n} ↦ NiL(ℤ[\underline{n}]⊗R; ℤ[\underline{n}]⊗X, ℤ[\underline{n}]⊗Y) and \underline{n} ↦ NiL(ℤ[\underline{n}]⊗R; ℤ[\underline{n}]⊗X, ℤ[\underline{n}]⊗ Y, ℤ[\underline{n}] ⊗ Z, ℤ[\underline{n}] ⊗ W). We will denote the realization of these simplicial spectra by NH(R; X, Y) or NH(R; X, Y, Z, W) respectively. However, the point here is that this process kills the additional information in Waldhausen’s Nil-groups.

Proposition 11.2.

(i) The functor P_{R} × P_{R} → NIL(R; X, Y) defined by (P, Q) ↦ (P, Q, 0, 0) induces an equivalence KH(R) ∨ KH(R) → NH(R; X, Y).

(ii) The functor P_{R}×P_{R} → NIL(R; X, Y, Z, W) defined by (P, Q) ↦ (P, Q, 0, 0) induces an equivalence KH(R) ∨ KH(R) → NH(R; X, Y, Z, W).

Proof. We prove only (i), the proof of (ii) is similar. It suffices to show that the functor (P, Q, p, q) ↦ (P, Q, 0, 0) mapping the simplicial category N(•) = NIL(ℤ[•] ⊗ R; ℤ[•] ⊗ X, ℤ[•] ⊗ Y) to itself is simplicially homotopic to the identity. Such a homotopy N(•) × Δ_{•}^{1} → N(•) is given by

(P, Q, p, q) ↦ (P, Q, ∑_{j∈f^{−1}(0)} t_{j} ⊗ p, ∑_{j∈f^{−1}(0)} t_{j} ⊗ q)

where f: \underline{n} ↦ \underline{1}.

The homotopy algebraic K-theory of a free product or a Laurent extension does therefore not involve Nil-groups.

Theorem 11.3. (Homotopy cartesian squares for Homotopy K-theory)

(i) Consider the free product R = A∗_{C}B (w.r.t. pure and free maps α: C → A and β: C → B). Then the commutative diagram

KH(C) ∨ KH(C) ⟶^{KH(α)∨KH(β)} KH(A) ∨ KH(B)
id∨id↓  ↓^{KH(ι_{A})∨KH(ι_{B})}
KH(C)  ⟶_{KH(α∘ι_{A})}  KH(R)

is homotopy cartesian. Here ι_{A} and ι_{B} are the obvious inclusions of rings.

(ii) Consider the Laurent extension R = A_{α,β}{t^{±1}} (w.r.t. pure and free maps α: C → A and β: C → A). Then the diagram

KH(C) ∨ KH(C) ⟶^{KH(α)∨KH(β)} KH(A)
id∨id↓  ↓^{KH(ι)}
KH(C)  ⟶_{KH(ι∘α)} KH(R)

is homotopy cartesian w.r.t. the obvious homotopy between the two ways to go through this diagram, cf. Section 10. Here ι is the obvious inclusions of rings.

Proof. The realization of a simplicial diagram of spectra that is degreewise homotopy cartesian is again homotopy cartesian. Thus the result follows by combining Theorems 10.2 or 10.6 respectively and Proposition 11.2. There is no longer a difference between the obvious and the more complicated natural transformation, since we got rid of the nil-categories.

Proof of Theorem 11.1. We use Lemma 5.1. We discuss first continuity, i.e. condition 5.1 (i). For K_{R} this follows from the compatibility of K-theory with directed colimits [22]. Since realizations of simplicial spectra commutes with directed colimits, this implies condition 5.1 (i).

Next we discuss the tree property, i.e. conditions 5.1 (ii) and (iii). Note that KH_{R} is a 2-functor as discussed in Remark 5.6. For K_{R} this holds since natural equivalences between functors F and G induce naturally a homotopy from K^{−∞}(F) to K^{−∞}(G). This homotopy is preserved under realization. Now observe that the obvious homotopy in Theorem 11.3 (ii) is in the case of an HNN-extension the homotopy coming from conjugation as discussed in Remark 5.6 (ii). Thus conditions 5.1 (ii) and (iii) are satisfied for KH_{R} by Theorem 11.3 and Remark 5.6.

Using Theorems 10.2 and 10.6 and Remarks 10.3 and 10.7 the above arguments also prove a version of Theorem 11.1 for algebraic K-theory.

Theorem 11.4 (Continuity and tree-property for H_{∗}^{?}(−; K_{R})). The equivariant homology theory H_{∗}^{?}(−; K_{R}) is continuous and if R is regular then it has the regular tree property.

Now we can finish the proof of the various results stated in the Introduction. We start with Theorem 0.5 and Theorem 0.11. The property (FIN) respectively (VCYC) hold for trivial reasons. Similar (SUB) in the Fibered case is a formal consequence of the Definitions, compare Lemma 1.2. The property (COL) is a consequence of Proposition 3.4, Theorem 11.1 and Theorem 11.4. The properties (TREE) respectively (TREE_{R}) follow from Theorem 4.2 and Theorem 11.1 respectively Theorem 11.4. Theorem 0.7 is a consequence of Corollary 4.4 and Theorem 11.1. Theorem 0.8 is a consequence of Theorem 0.5, Proposition 8.2 and Remark 8.3. It remains to prove Proposition 0.9.

Proof. (i) This result is stated in [19, page 133]. We give an outline of the proof. We consider first a group G which possesses a finite presentation with one relation. Let r be the number of generators appearing in the word describing the relation. If r ≤ 1, then G is the amalgamated product of a free group and a finite cyclic group. Obviously any finite group and ℤ belong to C_{0} and C_{0} is closed under free products. Hence G belongs to C_{0}. It remains to treat r ≥ 2. Here we use induction over the length l of the word describing the relation. In our case l ≥ 2. Then G acts on a tree with stabilizers which are subgroups of one-relator groups whose relation have length ≤ (l − 1), [8, Theorem 7.7] and hence belong to C_{0} by the induction hypothesis. Therefore G belongs to C_{0}. For a general one-relator group G there are finitely generated subgroups G_{i} which are free or one-relator groups, such that G is is the directed colimit over the G_{i}.

(ii) Let {1} = G_{0} ⊆ G_{1} ⊆ … ⊆ G_{n} = G be a sequence of subgroups such that G_{i−1} is normal in G_{i} and the quotient G_{i}/G_{i−1} is free for i = 1, 2 …, n. We prove by induction over n that G belongs to C_{0}. The induction beginning n = 0 is trivial because of property (FIN), the induction step done as follows. We can write G_{n}/G_{n−1} as a directed union of its finitely generated subgroups. Hence G_{n} is the directed union of the preimages of the finitely generated subgroups of G_{n}/G_{n−1}. Since any finitely generated subgroup of a free group F is a finitely generated free group, it suffices to treat the case, where G_{n}/G_{n−1} is finitely generated free by property (COL). Since G_{n}/G_{n−1} acts on a tree with trivial stabilizers, G_{n} acts on a tree with stabilizers which are all isomorphic to G_{n−1} and hence belong to C_{0}. Hence G belongs to C_{0} by property (TREE).

(iii), (iv), and (v) These follow from [29, Theorem 17.5 on page 250].

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