K-and L-theory of the semi-direct product of the discrete three-dimensional Heisenberg group by

ℤ/4

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

December 8, 2004

Abstract

We compute the group homology, the topological K-theory of the reduced C^{∗}-algebra and the algebraic K- and L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by ℤ/4. These computations will follow from the more general treatment of a certain class of groups G which occur as extensions 1 → K → G → Q → 1 of a torsionfree group K by a group Q which satisfies certain assumptions. The key ingredients are the Baum-Connes and Farrell-Jones Conjectures and methods from equivariant algebraic topology. Key words: K- and L-groups of group rings and group C^{∗}-algebras, threedimensional Heisenberg group. Mathematics subject classification 2000: 19A31, 19B28, 19D50, 19G24, 19K99, 55N99.

0 Introduction

The original motivation for this paper was the question of Chris Phillips how the topological K-theory of the reduced (complex) C^{∗}-algebra of the semi-direct product Hei ⋊ℤ/4 looks like. Here Hei is the three-dimensional discrete Heisenberg group which is the subgroup of GL_{3}(ℤ) consisting of upper triangular matrices with 1 on the diagonals. The ℤ/4-action is given by

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

( \begin{matrix}1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} ) ↦ ( \begin{matrix}1 & −z & y − xz \\ 0 & 1 & x \\ 0 & 0 & 1 \end{matrix} ) .

The answer, which is proved in Theorem 2.6, consists of an explicit isomorphism

j_{0} ⨁ c[0]′_{0} ⨁ c[2]′_{0} : K_{0}({∗}) ⨁ R̃_{ℂ}(ℤ/4) ⨁ R̃_{ℂ}(ℤ/2) ⟶^{≅} K_{0}(C_{r}^{∗}(Hei ⋊ℤ/4))

and a short exact sequence

0 → R̃_{ℂ}(ℤ/4) ⨁ R̃_{ℂ}(ℤ/2) \underrightarrow{c[0]′_{1} ⨁ c[2]′_{1}} K_{1}(C_{r}^{∗}(Hei ⋊ℤ/4)) ⟶^{c_{1}} K̃_{1}(S^{3}) → 0,

which splits since K̃_{1}(S^{3}) ≅ ℤ. Here R̃_{ℂ}(ℤ/m) is the kernel of the split surjective map R_{ℂ}(ℤ/m) → R_{ℂ}({1}) ≅ ℤ which sends the class of a complex ℤ/m-representation to the class of ℂ ⊗_{ℂ[ℤ/m]} V. As abelian group we get for n ∈ ℤ

K_{n}(C_{r}^{∗}(Hei ⋊ℤ/4)) ≅ ℤ^{5}.

This computation will pay a role in the paper by Lück, Phillips and Walters [20], where certain C^{∗}-algebras given by semi-direct products of rotation algebras with finite cyclic groups are classified.

Although the group Hei ⋊ℤ/4 is very explicit, this computation is highly nontrivial and requires besides the Baum-Connes Conjecture a lot of machinery from equivariant algebraic topology. Even harder is the computation of the middle and lower K-theory. The result is (see Corollary 3.9)

Wh_{n}(Hei ⋊ℤ/4) ≅ \{ \begin{matrix}NK_{n}(ℤ[ℤ/4]) ⨁ NK_{n}(ℤ[ℤ/4]) & for n = 0, 1; \\ 0 & for n ≤ −1, \end{matrix}

where NK_{n}(ℤ[ℤ/4]) denotes the n-th Nil-group of ℤ[ℤ/4] which appears in the Bass-Heller-Swan decomposition of ℤ[ℤ/4 × ℤ]. So the lower K-theory is trivial and the middle K-theory is completely made up of Nil-groups.

We also treat the L-groups. The answer and calculation is rather messy due to the appearance of UNil-terms and the structure of the family of infinite virtually cyclic subgroups (see Theorem 4.11). If one is willing to invert 2, these UNil-terms and questions about decorations disappear and the answer is given by the short split exact sequence

0 → L_{n}(ℤ) [\frac{1}{2}] ⨁ L̃_{n}(ℤ[ℤ/2]) [\frac{1}{2}] ⨁ L̃_{n−1}(ℤ[ℤ/2]) [\frac{1}{2}]

⨁ L̃_{n}(ℤ[ℤ/4]) [\frac{1}{2}] ⨁ L̃_{n−1}(ℤ[ℤ/4]) [\frac{1}{2}]

⟶^{j} L_{n}(ℤ[Hei ⋊ℤ/4]) [\frac{1}{2}] → L_{n−3}(ℤ) [\frac{1}{2}] → 0.

Finally we will also compute the group homology (see Theorem 5.6)

H_{n}(G) = ℤ/2 × ℤ/4 for n ≥ 1, n ≠ 2, 3;
H_{2}(G) = ℤ/2;
H_{3}(G) = ℤ × ℤ/2 × ℤ/4.

In turns out that we can handle a much more general setting. Namely, we will consider an extension of (discrete) groups

1 → K ⟶^{i} G ⟶^{p} Q → 1  (0.1)

which satisfies the following conditions:

(M) Each non-trivial finite subgroup of Q is contained in a unique maximal finite subgroup;

(NM) Let M be a maximal finite subgroup of Q. Then N_{Q}M = M unless G is torsionfree;

(T) K is torsionfree.

The special case, where K is trivial, is treated in [11, Theorem 5.1]. In [11, page 101] it is explained using [22, Lemma 4.5]), [22, Lemma 6.3] and [23, Propositions 5.17, 5.18 and 5.19 in II.5 on pages 107 and 108] why the following groups satisfy conditions (M) and (NM):

• Extensions 1 → ℤ^{n} → Q → F → 1 for finite F such that the conjugation action of F on ℤ^{n} is free outside 0 ∈ ℤ^{n};

• Fuchsian groups;

• One-relator groups.

Of course Hei ⋊ℤ/4 is an example for G. For such groups G we will establish certain exact Mayer-Vietoris sequences relating the K- or L-theory of G to the K- and L-theory of p^{−1}(M) for maximal finite subgroups M ⊆ Q and terms involving the quotients G\\underline{E}G and p^{−1}(M)\\underline{E}p^{−1}(M). The classifying space \underline{E}G for proper G-actions plays an important role and often there are nice small geometric models for them. One key ingredient in the computations for Hei ⋊ℤ/4 will be to figure out that G\\underline{E}G is in this case S^{3}. For instance the computation of the group homology illustrates that it is often very convenient to work with these spaces G\\underline{E}G although one wants information about BG.

1 Topological K-theory

For a G-CW -complex X let K_{∗}^{G}(X) be its equivariant K-homology theory. If G is trivial, we abbreviate K_{∗}(X). For a C^{∗}-algebra A let K_{∗}(A) be its topological K-theory. Recall that a model \underline{E}G for the classifying space for proper G-actions is a G-CW -complex with finite isotropy groups such that (\underline{E}G)^{H} is contractible for each finite subgroup H ⊆ G. It has the property that for any G-CW -complex X with finite isotropy groups there is up to G-homotopy precisely one G-map from X to \underline{E}G. In particular two models for \underline{E}G are G-homotopy equivalent. For more information about these spaces \underline{E}G we refer for instance to [6], [19], [24]. [30]. Recall that the Baum-Connes Conjecture (see [6, Conjecture 3.15 on page 254]) says that the assembly map

asmb: K_{n}^{G}(\underline{E}G) ⟶^{≅} K_{n}(C_{r}^{∗}(G))

is an isomorphism for each n ∈ ℤ, where C_{r}^{∗}(G) is the reduced group C^{∗}-algebra associated to G. Let EG be a model for the classifying space for free G-actions, i.e. a free G-CW -complex which is contractible (after forgetting the group action). Up to G-homotopy there is precisely one G-map s: EG → \underline{E}G. The classical assembly map a is defined as the composition

a: K_{n}(BG) = K_{n}^{G}(EG) ⟶^{K_{p}^{G}(s)} K_{n}^{G}(\underline{E}G) ⟶^{asmb} K_{n}(C_{r}^{∗}(G).

For more information about the Baum-Connes Conjecture we refer for instance to [6], [21], [24], [31].

From now on consider a group G as described in (0.1). We want to compute K_{n}^{G}(\underline{E}G) which is by the Baum-Connes Conjecture the same as K_{n}(C_{r}^{∗}(G)).

First we construct a nice model for \underline{E}Q. Let \{(M_{i}) | i ∈ I\} be the set of conjugacy classes of maximal finite subgroups of M_{i} ⊆ Q. By attaching free Q-cells we get an inclusion of Q-CW -complexes j_{1} : ∐_{i∈I} Q ×_{M_{i}} EM_{i} → EQ. Define \underline{E}Q as the Q-pushout

\begin{matrix}∐_{i∈I} Q ×_{M_{i}} EM_{i} & ⟶^{j_{1}} & EQ \\ ^{u_{1}}↓ & & ↓^{f_{1}} \\ ∐_{i∈I} Q/M_{i} & ⟶_{k_{1}} & \underline{E}Q \end{matrix}  (1.1)

where u_{1} is the obvious Q-map obtained by collapsing each EM_{i} to a point.

We have to explain why \underline{E}Q is a model for the classifying space for proper actions of Q. Obviously it is a Q-CW -complex. Its isotropy groups are all finite. We have to show for H ⊆ Q finite that (\underline{E}Q)^{H} contractible. We begin with the case H ≠ {1}. Because of conditions (M) and (NM) there is precisely one index i_{0} ∈ I such that H is subconjugated to M_{i_{0}} and is not subconjugated to M_{i} for i ≠ i_{0} and we get

(∐_{i∈I} Q/M_{i})^{H} = (Q/M_{i_{0}})^{H} = {∗}.

It remains to treat H = {1}. Since u_{1} is a non-equivariant homotopy equivalence and j_{1} is a cofibration, f_{1} is a non-equivariant homotopy equivalence and hence \underline{E}Q is contractible (after forgetting the group action).

Let X be a Q-CW -complex and Y be a G-CW -complex. Then X × Y with the G-action given by g · (x, y) = (p(g)x, gy) is a G-CW -complex and the G-isotropy group G_{(x,y)} of (x, y) is p^{−1}(H_{x}) ∩ G_{y}. Hence \underline{E}Q × \underline{E}G is a G-CW - model for \underline{E}G and EQ × \underline{E}G is a G-CW -model for EG, since ker(p: G → Q) is torsionfree by assumption. Let Z be a M_{i}-CW -complex. Then there is a G-homeomorphism

G×_{p^{−1}(M_{i})} (Z × res_{G}^{p^{−1}(M_{i})} Y ) ⟶^{≅} (Q×_{M_{i}} Z)×Y (g, (z, y)) ↦ ((p(g), z), gy).

The inverse sends ((q, z), y) to (g, (z, g^{−1}y) for any choice of g ∈ G with p(g) = q. If we cross the Q-pushout (1.1) with \underline{E}G, then we obtain the following G-pushout

\begin{matrix}∐_{i∈I} G ×_{p^{−1}(M_{i})} Ep^{−1}(M_{i}) & ⟶^{j_{2}} & EG \\ ^{u_{2}}↓ & & ↓^{f_{2}} \\ ∐_{i∈I} G ×_{p^{−1}(M_{i})} \underline{E}p^{−1}(M_{i}) & ⟶_{k_{2}} & \underline{E}G \end{matrix}  (1.2)

If we divide out the G-action in the pushout (1.2) above we obtain the pushout

\begin{matrix}∐_{i∈I} Bp^{−1}(M_{i}) & ⟶^{j_{3}} & BG \\ ^{u_{3}}↓ & & ↓^{f_{3}} \\ ∐_{i∈I} p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}) & ⟶_{k_{3}} & G\\underline{E}G \end{matrix}  (1.3)

If we divide out the Q-action in the pushout (1.1) we obtain the pushout

\begin{matrix}∐_{i∈I} BM_{i} & ⟶^{j_{4}} & BQ \\ ^{u_{4}}↓ & & ↓^{f_{4}} \\ ∐_{i∈I}\{∗\} & ⟶_{k_{4}} & Q\\underline{E}Q \end{matrix}  (1.4)

Theorem 1.5. Let G be the group appearing in (0.1) and assume that conditions (M), (NM) and (T) hold. Assume that G and all groups p^{−1}(M_{i}) satisfy the Baum-Connes Conjecture. Then the Mayer-Vietoris sequence associated to (1.2) yields the long exact sequence of abelian groups

… ⟶^{∂_{n+1}} ⨁_{i∈I} K_{n}(Bp^{−1}(M_{i}))

\underrightarrow{(⨁_{i∈I} K_{n}(Bl_{i})) ⨁(⨁_{i∈I} a[i]_{n})} K_{n}(BG) ⨁ (⨁_{i∈I} K_{n}(C_{r}^{∗}(p^{−1}(M_{i})))

\underrightarrow{a_{n} ⨁(⨁_{i∈I} K_{n}(C_{r}^{∗}(l_{i})))} K_{n}(C_{r}^{∗}(G)) ⟶^{∂_{n}} ⨁_{i∈I} K_{n−1}(Bp^{−1}(M_{i}))

\underrightarrow{(⨁_{i∈I} K_{n−1}(Bl_{i})) ⨁(⨁_{i∈I} a[i]_{n−1})} K_{n−1}(BG) ⨁ (⨁_{i∈I} K_{n−1}(C_{r}^{∗}(p^{−1}(M_{i})))

\underrightarrow{a_{n−1} ⨁(⨁_{i∈I} K_{n−1}(C_{r}^{∗}(l_{i})))} …

Here the maps a[i]_{n} and a are classical assembly maps and l_{i}: p^{−1}(M_{i}) → G is the inclusion.

Let Λ be a ring with ℤ ⊆ Λ ⊆ ℚ such that the order of each finite subgroup of G is invertible in Λ. Then the composition

Λ ⊗_{ℤ} K_{n}(Bp^{−1}(M_{i})) ⟶^{id_{Λ} ⊗_{ℤ}a[i]_{n}} K_{n}(C_{r}^{∗}(p^{−1}(M_{i})) = K_{n}^{p^{−1}(M_{i})}(\underline{E}p^{−1}(M_{i}))

⟶^{id_{Λ} ⊗_{ℤ} ind_{p^{−1}(M_{i})→\{1\}}} Λ ⊗_{ℤ} K_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}))

is an isomorphism, where ind denotes the induction map. In particular the long exact sequence above reduces after applying Λ⊗_{ℤ} to split exact short exact sequences of Λ-modules

0 → ⨁_{i∈I} Λ ⊗_{ℤ} K_{n}(Bp^{−1}(M_{i})) \underrightarrow{(⨁_{i∈I} id_{Λ} ⊗_{ℤ}K_{n}(Bl_{i})) ⨁(⨁_{i∈I} id_{Λ} ⊗_{ℤ}a[i]_{n})}

Λ ⊗_{ℤ} K_{n}(BG) ⨁ (⨁_{i∈I} Λ ⊗_{ℤ} K_{n}(C_{r}^{∗}(p^{−1}(M_{i}))))

\underrightarrow{⨁_{i∈I} id_{Λ} ⊗_{ℤ}K_{n}(C_{r}^{∗}(l_{i})) ⨁ id_{Λ} ⊗_{ℤ}a_{n}} Λ ⊗_{ℤ} K_{n}(C_{r}^{∗}(G)) → 0.

Proof. The Mayer Vietoris sequence is obvious using the fact that for a free G-CW -complex X there is a canonical isomorphism K_{n}^{G}(X) ⟶^{≅} K_{n}(G\X). The composition

Λ ⊗_{ℤ} K_{n}(Bp^{−1}(M_{i})) ⟶^{id_{Λ} ⊗_{ℤ}a[i]_{n}} K_{n}(C_{r}^{∗}(p^{−1}(M_{i})) = K_{n}^{p^{−1}(M_{i})}(\underline{E}p^{−1}(M_{i}))

⟶^{id_{Λ} ⊗_{ℤ} ind_{p^{−1}(M_{i})→\{1\}}} Λ ⊗_{ℤ} K_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}))

is bijective by [22, Lemma 2.8 (a)].

The advantage of the following version is that it involves instead of the spaces BG the spaces G\\underline{E}G which often have rather small geometric models. In the case G = Hei ⋊ℤ/4 we will see that G\\underline{E}G is the three-dimensional sphere S^{3} (see Lemma 2.4).

Theorem 1.6. Let G be the group appearing in (0.1) and assume conditions (M), (NM) and (T) hold. Assume that G and all groups p^{−1}(M_{i}) satisfy the Baum-Connes Conjecture. Then there is a long exact sequence of abelian groups

… \underrightarrow{c_{n+1} ⨁_{i∈I} d[i]_{n+1}} K_{n+1}(G\\underline{E}G) ⟶^{∂_{n+1}} ⨁_{i∈I} K_{n}(C_{r}^{∗}(p^{−1}(M_{i})))

\underrightarrow{(⨁_{i∈I} K_{n}(C_{r}^{∗}(l_{i}))) ⨁(⨁_{i∈I} c[i]_{n})} K_{n}(C_{r}^{∗}(G)) ⨁ (⨁_{i∈I} K_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i})))

\underrightarrow{c_{n} ⨁_{i∈I} d[i]_{n}} K_{n}(G\\underline{E}G) ⟶^{∂_{n}} ⨁_{i∈I} K_{n−1}(C_{r}^{∗}(p^{−1}(M_{i})))

\underrightarrow{(⨁_{i∈I} K_{n−1}(C_{r}^{∗}(l_{i}))) ⨁(⨁_{i∈I} c[i]_{n−1})} …

Here the homomorphisms d[i]_{n} come from the up to equivariant homotopy unique (p^{−1}(M_{i}) → G)-equivariant maps \underline{E}p^{−1}(M_{i}) → \underline{E}G. The maps c_{n} and (analogously for c[i]_{n}) are the compositions

K_{n}(C_{r}^{∗}(G)) ⟶^{asmb^{−1}} K_{n}^{G}(\underline{E}G) ⟶^{ind_{G→\{1\}}} K_{n}(G\\underline{E}G).

Let Λ be a ring with ℤ ⊆ Λ ⊆ ℚ such that the order of each finite subgroup of G is invertible in Λ. Then the composition

Λ ⊗_{ℤ} K_{n}(BG) ⟶^{id_{Λ} ⊗_{ℤ}a_{n}} Λ ⊗_{ℤ} K_{n}(C_{r}^{∗}(G)) ⟶^{id_{Λ} ⊗_{ℤ}c_{n}} Λ ⊗_{ℤ} K_{n}(G\\underline{E}G)

is an isomorphism of Λ-modules. In particular the long exact sequence above reduces after applying Λ ⊗_{ℤ} − to split exact short sequences of Λ-modules

0 → ⨁_{i∈I} Λ ⊗_{ℤ} K_{n}(C_{r}^{∗}(p^{−1}(M_{i}))) \underrightarrow{(⨁_{i∈I} id_{Λ} ⊗_{ℤ}K_{n}(C_{r}^{∗}(l_{i}))) ⨁(⨁_{i∈I} id_{Λ} ⊗_{ℤ}c[i]_{n})}

Λ ⊗_{ℤ} K_{n}(C_{r}^{∗}(G)) ⨁ (⨁_{i∈I} Λ ⊗_{ℤ} K_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i})))

\underrightarrow{id_{Λ} ⊗_{ℤ}c_{n} ⨁_{i∈I} id_{Λ} ⊗_{ℤ}d[i]_{n}} Λ ⊗_{ℤ} K_{n}(G\\underline{E}G) → 0.

Proof. We get from the pushout (1.3) the long exact Mayer Vietoris sequence for (non-equivariant) topological K-theory

… ⟶^{∂_{n+1}} ⨁_{i∈I} K_{n}(Bp^{−1}(M_{i})) \underrightarrow{(⨁_{i∈I} K_{n}(Bl_{i})) ⨁(⨁_{i∈I} K_{n}((p^{−1}(M_{i})\s_{i}))}

K_{n}(BG) ⨁ (⨁_{i∈I} K_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}))) \underrightarrow{H_{n}(G\s) ⨁(⨁_{i∈I} d[i]_{n})} K_{n}(G\\underline{E}G)

⟶^{∂_{n}} ⨁_{i∈I} K_{n−1}(Bp^{−1}(M_{i})) \underrightarrow{(⨁_{i∈I} K_{n−1}(Bl_{i})) ⨁(⨁_{i∈I} K_{n−1}(p^{−1}(M_{i})\s_{i}))}

K_{n−1}(BG) ⨁ (⨁_{i∈I} K_{n−1}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}))) \underrightarrow{K_{n−1}(G\s) ⨁(⨁_{i∈I} d[i]_{n−1})} …

where s_{i}: Ep^{−1}(M_{i}) → \underline{E}p^{−1}(M_{i}) and s: EG → \underline{E}G are the (up to equivariant homotopy unique) equivariant maps. Now one splices the long exact Mayer-Vietoris sequences from above and from Theorem 1.5 together.

2 The semi-direct product of the Heisenberg group and a cyclic group of order four

We want to study the following example. Let Hei be the discrete Heisenberg group. We will use the presentation

Hei = ⟨u, v, z | [u, z] = 1, [v, z] = 1, [u, v] = z⟩.  (2.1)

Throughout this section let G be the semi-direct product

G = Hei ⋊ℤ/4

with respect to the homomorphism ℤ/4 → aut(Hei) which sends the generator t of ℤ/4 to the automorphism of Hei given on generators by z ↦ z, u ↦ v and v ↦ u^{−1}. Let Q be the semi-direct product ℤ^{2} ⋊ ℤ/4 with respect to the automorphism ℤ^{2} → ℤ^{2} which comes from multiplication with the complex number i and the inclusion ℤ^{2} ⊆ ℂ. Since the action of ℤ/4 on ℤ^{2} is free outside 0, the group Q satisfies (M) and (NM) (see [22, Lemma 6.3]). The group G has the presentation

G = ⟨u, v, z, t | [u, z] = [v, z] = [t, z] = t^{4} = 1, [u, v] = z, tut^{−1} = v, tvt^{−1} = u^{−1}⟩.

Let i: ℤ → G be the inclusion sending the generator of ℤ to z. Let p: G → Q be the group homomorphism, which sends z to the unit element, u to (1, 0) in ℤ^{2} ⊆ Q, v to (0, 1) in ℤ^{2} ⊆ Q and t to the generator of ℤ/4 ⊆ Q. Then 1 → ℤ → G → Q → 1 is a central extension which satisfies the conditions (M), (NM) and (T) appearing in (0.1). Moreover, G is amenable and hence G and all its subgroups satisfy the Baum-Connes Conjecture [15].

In order to apply the general results above we have to figure out the conjugacy classes of finite subgroups of Q = ℤ^{2} ⋊ ℤ/4 and among them the maximal ones. An element of order 2 in Q must have the form xt^{2} for x ∈ ℤ^{2}. In the sequel we write the group multiplication in Q and G multiplicatively and in ℤ^{2} additively. We compute (xt^{2})^{2} = xt^{2}xt^{2} = (x − x) = 0. Hence the set of elements of order two in Q is {xt^{2} | x ∈ ℤ^{2}}. Consider e_{1} = (1, 0) and e_{2} = (0, 1) in ℤ^{2}. We claim that up to conjugacy there are the following subgroups of order two: ⟨e_{1}t^{2}⟩, ⟨e_{1}e_{2}t^{2}⟩, ⟨t^{2}⟩. This follows from the computations for x, y ∈ ℤ^{2}

y(xt^{2})y^{−1} = yxyt^{2} = (x + 2y)t^{2};
t(xt^{2})t^{−1} = txt^{−1}t^{2} = (ix)t^{2}.

An element of order 4 must have the form xt for x ∈ ℤ^{2}. We compute

(xt)^{4} = xtxt^{−1}t^{2}xt^{−2}t^{3}xt^{−3} = (x + ix + i^{2}x + i^{3}x) = (1 + i + i^{2} + i^{3})x = 0x = 0.

Hence the set of elements of order four in Q is {xt | x ∈ ℤ^{2}}. We claim that up to conjugacy there are the following subgroups of order four: ⟨e_{1}t⟩, ⟨t⟩. This follows from the computations for x, y ∈ ℤ^{2}

y(xt)y^{−1} = (x + y − iy)t;
t(xt)t^{−1} = (ix)t.

We have (e_{1}t)^{2} = e_{1}te_{1}t = e_{1}ie_{1}t^{2} = e_{1}e_{2}t^{2}. The considerations above imply

Lemma 2.2. Up to conjugacy Q has the following non-trivial finite subgroups

⟨e_{1}t^{2}⟩, ⟨e_{1}e_{2}t^{2}⟩, ⟨t^{2}⟩, ⟨e_{1}t⟩, ⟨t⟩.

The maximal finite subgroups are up to conjugacy

M_{0} = ⟨t⟩, M_{1} = ⟨e_{1}t⟩, M_{2} = ⟨e_{1}t^{2}⟩.

Since t^{4} = 1, (ut^{2})^{2} = ut^{2}ut^{−2} = uu^{−1} = 1 and (ut)^{4} = utut^{−1}t^{2}ut^{−2}t^{3}ut^{−3} = uvu^{−1}v^{−1} = z hold in G, the preimages of these groups under p: G → Q are given by

p^{−1}(M_{0}) = ⟨t, z⟩ ≅ ℤ/4 × ℤ;
p^{−1}(M_{1}) = ⟨ut, z⟩ = ⟨ut⟩ ≅ ℤ;
p^{−1}(M_{2}) = ⟨ut^{2}, z⟩ ≅ ℤ/2 × ℤ.

On easily checks

Lemma 2.3. Up to conjugacy the finite subgroups of G are ⟨t⟩, ⟨t^{2}⟩ and ⟨ut^{2}⟩.

Next we construct nice geometric models for \underline{E}G and its orbit space \underline{E}G. Let Hei(ℝ) be the real Heisenberg group, i.e. the Lie group of real (3, 3)-matrices of the special form

( \begin{matrix}1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix} ) .

In the sequel we identify such a matrix with the element (x, y, z) ∈ ℝ^{3}. Thus Hei(ℝ) can be identified with the Lie group whose underlying manifold is ℝ^{3} and whose group multiplication is given by

(a, b, c) • (x, y, z) = (a + x, b + y + az, c + z).

The discrete Heisenberg group is given by the subgroup where all the entries x, y, z are integers. In the presentation of the discrete Heisenberg group (2.1) the elements u, v and z correspond to (1, 0, 0), (0, 0, 1) and (0, 1, 0). Obviously Hei is a torsionfree discrete subgroup of the contractible Lie group Hei(ℝ). Hence Hei(ℝ) is a model for EHei and Hei \ Hei(ℝ) for BHei. We have the following ℤ/4-action on Hei(ℝ), where the generator t acts by (x, y, z) ↦ (−z, y − xz, x). This is an action by automorphisms of Lie groups and induces on Hei the homomorphism ℤ/4 → aut(Hei) which we have used above to define G = Hei ⋊ℤ/4. The Hei-action and ℤ/4-action on Hei(ℝ) above fit together to a G = Hei ⋊ℤ/4action. The next result is the main geometric input for the desired computations.

Lemma 2.4. The manifold Hei(ℝ) with the G-action above is a model for \underline{E}G. The quotient space G\\underline{E}G is homeomorphic to S^{3}.

Proof. Let ℝ ⊆ Hei(ℝ) be the subgroup of elements {(0, y, 0) | y ∈ ℝ}. This is the center of Hei(ℝ). The intersection ℝ ∩ Hei is ℤ ⊆ ℝ. Thus we get a ℝ/ℤ = S^{1}-action on Hei \ Hei(ℝ). One easily checks that this S^{1}-action and the ℤ/4-action above commute so that we see a S^{1} × ℤ/4-action on Hei \ Hei(ℝ). The S^{1}-action is free, but the S^{1} × ℤ/4-action is not. Next we figure out its fixed points.

Obviously t^{2} sends (x, y, z) to (−x, y, −z). We compute for (a, b, c) ∈ Hei, u ∈ ℝ and (x, y, z) ∈ Hei(ℝ)

(a, b, c) · (0, u, 0) · t · (x, y, z) = (a − z, u + b + y − xz − ax, c + x);
(a, b, c) · (0, u, 0) · t^{2} · (x, y, z) = (a − x, u + b + y − az, c − z);
(a, b, c) · (0, u, 0) · (x, y, z) = (a + x, u + b + y + az, c + z).

Hence the isotropy group of Hei ·(x, y, z) ∈ Hei \ Hei(ℝ) under the S^{1} × ℤ/4action contains (exp(2πiu), t) in its isotropy group under the S^{1} × ℤ/4-action if and only if (a − z, u + b + y − xz − ax, c + x) = (x, y, z) holds for some integers a, b, c. The last statement is equivalent to the condition that 2x and x + z are integers, y is an arbitrary real number and u − 3x^{2} ∈ ℤ.

The isotropy group of Hei ·(x, y, z) ∈ Hei \ Hei(ℝ) contains (exp(2πiu), t^{2}) in its isotropy group under the S^{1} × ℤ/4-action if and only if (a − x, u + b + y − az, c − z) = (x, y, z) holds for some integers a, b, c. Obviously the last statement is equivalent to the condition that 2x, 2z and u − 2xz are integers and y is an arbitrary real number.

The isotropy group of Hei ·(x, y, z) ∈ Hei \ Hei(ℝ) contains (exp(2πiu), 1) in its isotropy group under the S^{1}×ℤ/4-action if and only if (a+x, u+b+y+az, c+ z) = (x, y, z) holds for some integers a, b, c. The last statement is equivalent to the condition that x = 0, z = 0, u is an integer and y is an arbitrary real number.

This implies that the orbits under the S^{1} × ℤ/4-action on Hei \ Hei(ℝ^{3}) are free except the orbits through Hei ·(1/2, 0, 1/2), whose isotropy group is the cyclic subgroup of order four generated by (exp(3πi/4), t), and the orbits though Hei ·(0, 0, 0), whose isotropy group is the cyclic subgroup of order four generated by (exp(0), t), and the orbits though Hei ·(1/2, 0, 0) and Hei ·(0, 0, 1/2), whose isotropy groups are the cyclic subgroup of order two generated by (exp(0), t^{2}). By the slice theorem any point p ∈ Hei \ Hei(ℝ) has a neighborhood of the form S^{1} × ℤ/4 ×_{H_{p}} U_{p}, where H_{p} is its isotropy group and U_{p} a 2-dimensional real H_{p}-representation, namely the tangent space of Hei \ Hei(ℝ) at p. Since there are only finitely S^{1} × ℤ/4-orbits which are nonfree, the H_{p}-action on U_{p} is free outside the origin for each p ∈ Hei \ Hei(ℝ). In particular H_{p}\U_{p} is a manifold without boundary. If the isotropy group H_{p} is mapped under the projection pr: S^{1} × ℤ/4 → S^{1} to the trivial group, then ℤ/4\ (S^{1} × ℤ/4 ×_{H_{p}} U_{p}) is S^{1}-homeomorphic to S^{1} × H_{p}\U_{p} and hence a free S^{1}-manifold without boundary. If the projection pr: S^{1} × ℤ/4 → S^{1} is injective on H_{p}, then ℤ/4\ (S^{1} × ℤ/4 ×_{H_{p}} U_{p}) is the S^{1}-manifold S^{1} ×_{H_{p}} U_{p} with respect to the free H-action on S^{1} induced by p which has no boundary and precisely one non-free S^{1}-orbit. This shows that the quotient of Hei \ Hei(ℝ^{3}) under the ℤ/4-action is a closed S^{1}-manifold with precisely one non-free orbit.

The fixed point set of any finite subgroup of G of the G-space Hei(ℝ) = ℝ^{3} is a non-empty affine real subspace of Hei(ℝ) = ℝ^{3} and hence contractible. This shows that Hei(ℝ) with its G-action is a model for \underline{E}G. Hence G\\underline{E}G is a closed S^{1}-manifold with precisely one non-free orbit, whose quotient space under the S^{1}-action is the orbit space of T^{2} under the ℤ/4-action. One easily checks for the rational homology

H_{n} ((ℤ/4)\T^{2}; ℚ) ≅_{ℚ} H_{n}(T^{2}) ⊗_{ℤ[ℤ/4]} ℚ ≅ H_{n}(S^{2}; ℚ).

This implies that the S^{1}-space G\\underline{E}G is a Seifert bundle over (ℤ/4)\T^{2} ≅ S^{2} with precisely one singular fiber. Since the orbifold fundamental group of this orbifold S^{2} with precisely one cone point vanishes, the map e: π_{1}(S^{1}) → π_{1}(G\\underline{E}G) given by evaluating the S^{1}-action at some base point is surjective by [28, Lemma 3.2]. The Hurewicz map h: π_{1}(G\\underline{E}G) → H_{1}(G\\underline{E}G) is bijective since π_{1}(G\\underline{E}G) is a quotient of π_{1}(S^{1}) and hence is abelian. The composition

π_{1}(S^{1}) ⟶^{e} π_{1}(G\\underline{E}G) ⟶^{h} H_{1}(G\\underline{E}G)

agrees with the composition

π_{1}(S^{1}) ⟶^{h′} H_{1}(S^{1}) = H_{1}(ℤ\ℝ) ⟶^{e′} H_{1}(Hei \ Hei(ℝ)) ⟶^{H_{1}(pr)} H_{1}(G\\underline{E}G),

where h′ is the Hurewicz map, e′ given by evaluating the S^{1}-operation and pr is the obvious projection. The map H_{1}(ℤ\ℝ) → H_{1}(Hei \ Hei(ℝ)) is trivial since the element z ∈ Hei is a commutator, namely [u, v]. Hence G\\underline{E}G is a simply connected closed Seifert fibered 3-manifold. We conclude from [28, Lemma 3.1] that G\\underline{E}G is homeomorphic to S^{3}.

Next we investigate what information Theorem 1.6 gives in combination with Lemma 2.4.

We have to analyze the maps

c[i]_{n}: K_{n}(C_{r}^{∗}(p^{−1}(M_{i}))) → K_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i})),

which are defined as the compositions

K_{n}(C_{r}^{∗}(p^{−1}(M_{i}))) ⟶^{asmb^{−1}} K_{n}^{p^{−1}(M_{i})}(\underline{E}p^{−1}(M_{i}))

⟶^{ind_{p^{−1}(M_{i})→\{1\}}} K_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i})).

For i = 1 the group p^{−1}(M_{i}) is isomorphic to ℤ and hence the maps c[1]_{n} are all isomorphisms. In the case i = 0, 2 the group p^{−1}(M_{i}) looks like H_{i} × ℤ for H_{0} = ⟨t⟩ ≅ ℤ/4 and H_{2} = ⟨ut^{2}⟩ ≅ ℤ/2. The following diagram commutes

K_{n}(C_{r}^{∗}(H_{i} × ℤ)) ⟵_{≅}^{asmb} K_{n}^{H_{i}×ℤ}(\underline{E}H_{i} × ℤ) ⟵^{≅} K_{n}^{H_{i}}(\{∗\}) ⨁ K_{n−1}^{H_{i}}(\{∗\})
K_{n}(C_{r}^{∗}(pr_{i}))↓  ^{ind}^{H_{i}×ℤ→ℤ}↓  ^{ind}^{H_{i}→{1}} ⨁^{ind}^{H_{i}→{1}}↓
K_{n}(C_{r}^{∗}(ℤ)) ⟵_{≅}^{asmb}  K_{n}^{ℤ}(Eℤ)  ⟵^{≅} K_{n}({∗}) ⨁ K_{n−1}({∗})

The map ind_{H_{i}→{1}}: K_{n}^{H_{i}}({∗}) → K_{n}({∗}) is the map id: 0 → 0 for n odd. For n even it can be identified with the homomorphism ϵ: R_{ℂ}(H_{i}) → ℤ which sends the class of a complex H_{i}-representation V to the complex dimension of ℂ ⊗_{ℂH_{i}} V. This map is split surjective. The kernel of ϵ is denoted by R̃_{ℂ}(H_{i}). Define for i = 0, 2 maps

c[i]′_{n}: R̃_{ℂ}(H_{i}) → K_{n}(C_{r}^{∗}(G))  (2.5)

as follows. For n even it is the composition

R̃_{ℂ}(H_{i}) ⊆ R_{ℂ}(H_{i}) = K_{n}(C_{r}^{∗}(H_{i})) ⟶^{K_{n}(C_{r}^{∗}(l′_{i}))} K_{n}(C_{r}^{∗}(G)),

where l′_{i}: H_{i} → G is the inclusion. For n odd it is the composition

R̃_{ℂ}(H_{i}) ⊆ R_{ℂ}(H_{i}) = K_{n−1}(C_{r}^{∗}(H_{i})) ⟶^{x_{i}} K_{n}(C_{r}^{∗}(H_{i}×ℤ)) ⟶^{K_{n}(C_{r}^{∗}(l_{i}))} K_{n}(C_{r}^{∗}(G)),

where l_{i}: H_{i} × ℤ = p^{−1}(M_{i}) → G is the inclusion and

x_{i} ⨁ K_{n}(y_{i}): K_{n−1}(C_{r}^{∗}(H_{i})) ⨁ K_{n}(C_{r}^{∗}(H_{i})) ⟶^{≅} K_{n}(C_{r}^{∗}(H_{i} × ℤ))

is the canonical isomorphism for y_{i}: H_{i} → H_{i} × ℤ the inclusion. The map

∂_{n}: K_{n}(G\\underline{E}G) → ⨁_{i∈I} K_{n−1}(C_{r}^{∗}(p^{−1}(M_{i})))

appearing in Theorem 1.6 vanishes after applying ℚ ⊗_{ℤ} −. Since the target is a finitely generated torsionfree abelian group, the map itself is trivial. Hence we obtain from Theorem 1.6 short exact sequences for n ∈ ℤ

0 → R̃_{ℂ}(ℤ/4) ⨁ R̃_{ℂ}(ℤ/2) \underrightarrow{c[0]′_{n} ⨁ c[2]′_{n}} K_{n}(C_{r}^{∗}(G)) ⟶^{c_{n}} K_{n}(S^{3}) → 0,

where we identify H_{0} = ⟨t⟩ = ℤ/4 and H_{2} = ⟨ut^{2}⟩ = ℤ/2 and G\\underline{E}G = S^{3} using Lemma 2.4. If j_{n}: K_{n}({∗}) = K_{n}(C_{r}^{∗}({1})) → K_{n}(C_{r}^{∗}(G)) is induced by the inclusion of the trivial subgroup, we can rewrite the sequence above as the short exact sequence

0 → K_{n}(\{∗\}) ⨁ R̃_{ℂ}(ℤ/4) ⨁ R̃_{ℂ}(ℤ/2) \underrightarrow{j_{n} ⨁ c[0]′_{n} ⨁ c[2]′_{n}} K_{n}(C_{r}^{∗}(G)) ⟶^{c_{n}}

K̃_{n}(S^{3}) → 0,

where K̃_{n}(Y) is for a path connected space Y the cokernel of the obvious map K_{n}({∗}) → K_{n}(Y). We have K̃_{0}(S^{3}) = 0 and K̃_{1}(S^{3}) ≅ ℤ. Thus we get

Theorem 2.6. We have the isomorphism

j_{0} ⨁ c[0]′_{0} ⨁ c[2]′_{0} : K_{0}({∗}) ⨁ R̃_{ℂ}(ℤ/4) ⨁ R̃_{ℂ}(ℤ/2) ⟶^{≅} K_{0}(C_{r}^{∗}(Hei ⋊ℤ/4))

and the short exact sequence

0 → R̃_{ℂ}(ℤ/4) ⨁ R̃_{ℂ}(ℤ/2) \underrightarrow{c[0]′_{1} ⨁ c[2]′_{1}} K_{1}(C_{r}^{∗}(G)) ⟶^{c_{1}} K̃_{1}(S^{3}) → 0,

where the maps c[i]′_{n} have been defined in (2.5). In particular K_{n}(C_{r}^{∗}(G)) is a free abelian group of rank five for all n.

Remark 2.7. These computations are consistent with the computation of K_{n}(C_{r}^{∗}(G)) [\frac{1}{2}] coming from the Chern character constructed in [18].

Remark 2.8. One can also use these methods to compute the topological Ktheory of the real reduced group C^{∗}-algebra C_{r}^{∗}(Hei ⋊ℤ/4; ℝ). One obtains the short exact sequence

0 → KO_{n}({∗}) ⨁ K̃_{n}(C_{r}^{∗}(ℤ/2; ℝ)) ⨁ K̃_{n−1}(C_{r}^{∗}(ℤ/2; ℝ))

⨁ K̃_{n}(C_{r}^{∗}(ℤ/4; ℝ)) ⨁ K̃_{n−1}(C_{r}^{∗}(ℤ/4; ℝ))
→ K_{n}(C_{r}^{∗}(Hei ⋊ℤ/4)) → K̃O_{n}(S^{3}) → 0,

which splits after inverting 2.

3 Algebraic K-theory

In this section we want to describe what the methods above yield for the algebraic K-theory provided that instead of the Baum-Connes Conjecture the relevant version of the Farrell-Jones Conjecture for algebraic K-theory (see [12]) is true. The L-theory will be treated in the next section. We want to prove

Theorem 3.1. Let R be a regular ring, for instance R = ℤ. Let G be the group appearing in (0.1) and assume that conditions (M), (NM), and (T) are satisfied. Suppose that G and all subgroups p^{−1}(M_{i}) satisfy the Farrell-Jones Conjecture for algebraic K-theory with coefficients in R. Then we get for n ∈ ℤ the isomorphism

⨁_{i∈I} Wh_{n}(Rl_{i}): Wh_{n}(R[p^{−1}(M_{i})]) ⟶^{≅} Wh_{n}(RG),

where l_{i}: p^{−1}(M_{i}) → G is the inclusion.

Notice that in the context of the Farrell-Jones Conjecture one has to consider the family of virtually cyclic subgroups VCYC and only under special assumptions it suffices to consider the family FIN of finite subgroups. Recall that a family F of subgroups is a set of subgroups closed under conjugation and taking subgroups and that a model for the classifying space E_{F}(G) for the family F is a G-CW - complex whose isotropy groups belong to F and whose H-fixed point set is contractible for each H ∈ F. It is characterized up to G-homotopy by the property that any G-CW -complex, whose isotropy groups belong to F, possesses up to G-homotopy precisely one G-map to E_{F}(G). In particular two models for E_{F}(G) are G-homotopy equivalent and for an inclusion of families F ⊂ G there is up to G-homotopy precisely one G-map E_{F}(G) → E_{G}(G). The space \underline{E}G is the same as E_{FIN} (G).

Let H_{∗}^{G}(X; K(R?)) and H_{∗}^{G}(X; L^{⟨−∞⟩}(R?)) be the G-homology theories associated to the algebraic K and L-theory spectra over the orbit category K(R?) and L^{⟨−∞⟩}(R?) (see [10]). They satisfy for each subgroup H ⊆ G

H_{n}^{G}(G/H; K(R?)) ≅ K_{n}(RH);
H_{n}^{G}(G/H; L^{⟨−∞⟩}(R?)) ≅ L_{n}^{⟨−∞⟩}(RH).

The Farrell-Jones Conjecture (see [12, 1.6 on page 257]) says that the projection E_{VCYC}(G) → G/G induces isomorphisms

H_{n}^{G}(E_{VCYC}(G); K(R?)) ⟶^{≅} H_{n}^{G}(G/G; K(R?)) = K_{n}(RG);
H_{n}^{G}(E_{VCYC}(G); L^{⟨−∞⟩}(R?)) ⟶^{≅} H_{n}^{G}(G/G; L^{⟨−∞⟩}(R?)) = L_{n}^{⟨−∞⟩}(RG).

In the L-theory case one must use L^{⟨−∞⟩}. There are counterexamples to the Farrell-Jones Conjecture for the other decorations p, h and s (see [14]).

In the sequel we denote for a G-map f: X → Y by H_{n}^{G}(f: X → Y; K(R?)) the value of H_{n}^{G} on the pair given by the mapping cylinder of f and Y viewed as a G-subspace. We will often use the long exact sequence associated to this pair

… → H_{n}^{G}(X; K(R?)) → H_{n}^{G}(Y; K(R?)) → H_{n}^{G}(f: X → Y; K(R?))

→ H_{n−1}^{G}(X; K(R?)) → H_{n−1}^{G}(Y; K(R?)) → …

The following result is taken from [3].

Theorem 3.2. There are isomorphisms

H_{n}^{G}(\underline{E}G; K(ℝ?)) ⨁ H_{n}^{G} (\underline{E}G → E_{VCYC}(G); K(R?))

⟶^{≅} H_{n}^{G}(E_{VCYC}(G); K(R?));

H_{n}^{G}(\underline{E}G; L^{⟨−∞⟩}(ℝ?)) ⨁ H_{n}^{G} (\underline{E}G → E_{VCYC}(G); L^{⟨−∞⟩}(R?))

⟶^{≅} H_{n}^{G}(E_{VCYC}(G); L^{⟨−∞⟩}(R?)),

where in the K-theory context G and R are arbitrary and in the L-theory context G is arbitrary and we assume for any virtually cyclic subgroup V ⊆ G that K_{−i}(RV) = 0 for sufficiently large i.

For a virtually cyclic group V we have K_{−i}(ℤV) = 0 for n ≥ 2 (see [13]). The terms H_{n}^{G} (\underline{E}G → E_{VCYC}(G); K(R?)) vanish for instance if R is a regular

ring containing ℚ. The terms H_{n}^{G} (\underline{E}G → E_{VCYC}(G); L^{⟨−∞⟩}(R?)) vanish after inverting 2 (see Lemma 4.2). Recall that the Whitehead group Wh_{n}(RG) is by definition H_{n}^{G}(EG → G/G; K(R?)). This implies that Wh_{n}(RG) = H_{n}^{G}(EG → E_{VCYC}(G); K(R?)) if the Farrell-Jones Isomorphism Conjecture for algebraic Ktheory holds for RG. The group Wh_{1}(ℤG) is the classical Whitehead group Wh(G). If R is a principal ideal domain, then Wh_{0}(RG) is K̃_{0}(RG) and Wh_{n}(RG) = K_{n}(RG) for n ≤ −1.

If we cross the Q-pushout (1.1) with E_{VCYC}(G) we obtain the G-pushout

\begin{matrix}∐_{i∈I} G ×_{p^{−1}(M_{i})} E_{VCYC(K∩p^{−1}(M_{i}))}(p^{−1}(M_{i})) & ⟶ & E_{VCYC(K)}(G) \\ ↓ & & ↓ \\ ∐_{i∈I} G ×_{p^{−1}(M_{i})} E_{VCYC}(p^{−1}(M_{i})) & ⟶ & E_{VCYC_{f}} (G) \end{matrix} (3.3)

where VCYC(K ∩p^{−1}(M_{i})) is the family of virtually cyclic subgroups of p^{−1}(M_{i}), which are contained in K∩p^{−1}(M_{i}), and VCYC(K) is the family of virtually cyclic subgroups of G, which are contained in K, and VCYC_{f} is the family of virtually cyclic subgroups of G, whose image under p: G → Q is finite. Since K is torsionfree, elements in VCYC(K ∩ p^{−1}(M_{i})) and VCYC(K) are trivial or infinite cyclic groups. The following result is taken from [22, Theorem 2.3].

Theorem 3.4. Let F ⊂ G be families of subgroups of the group Γ. Let Λ be a ring with ℤ ⊆ Λ ⊆ ℚ and N be an integer. Suppose for every H ∈ G that the assembly map induces for n ≤ N an isomorphism

Λ ⊗_{ℤ} H_{n}^{H}(E_{H∩F}(H); K(R?)) → Λ ⊗_{ℤ} H_{n}^{H}(H/H; K(R?)),

where H ∩ F is the family of subgroups K ⊆ H with K ∈ F. Then the map

Λ ⊗_{ℤ} H_{n}^{Γ}(E_{F}(Γ); K(R?)) → Λ ⊗_{ℤ} H_{n}^{Γ}(E_{G}(Γ); K(R?))

is an isomorphism for n ≤ N. The analogous result is true for L^{⟨−∞⟩}(R?) instead of K(R?).

In the sequel we will apply Theorem 3.4 using the fact that for a infinite cyclic group or an infinite dihedral group H the map

asmb: H_{n}^{H}(\underline{E}H; K(R?)) → H_{n}^{K}(H/H; K(R?)) = K_{n}(RH)

is bijective for n ∈ ℤ. This follows for the infinite cyclic group from the Bass-Heller-decomposition and for the infinite dihedral group from Waldhausen [32, Corollary 11.5 and the following Remark] (see also [4] and [21, Section 2.2]).

The Farrell-Jones Conjecture for algebraic K-theory for the trivial family TR consisting of the trivial subgroup only is true for infinite cyclic groups and regular rings R as coefficients. We conclude from Theorem 3.4 that for a regular ring R the maps

H_{n}^{p^{−1}(M_{i})}(E(p^{−1}(M_{i})); K(R?))

⟶^{≅} H_{n}^{p^{−1}(M_{i})}(E_{VCYC(K∩p^{−1}(M_{i}))}(p^{−1}(M_{i})); K(R?))

and

H_{n}^{G}(EG; K(R?)) ⟶^{≅} H_{n}^{G}(E_{VCYC(K)}(G); K(R?))

are bijective for all n ∈ ℤ. Hence we obtain for a regular ring R from the G-pushout (3.3) an isomorphism

⨁_{i∈I} H_{n}^{p^{−1}(M_{i})} (E(p^{−1}(M_{i})) → E_{VCYC}(p^{−1}(M_{i})); K(R?))

⟶^{≅} H_{n}^{G} (EG → E_{VCYC_{f}} (G); K(R?)) . (3.5)

Let VCYC_{1} be the family of virtually cyclic subgroups of G whose intersection with K is trivial. Since VCYC is the union VCYC_{f} ∪ VCYC_{1} and the intersection VCYC_{f} ∩ VCYC_{1} is FIN, we obtain a G-pushout

\begin{matrix}\underline{E}G & ⟶ & E_{VCYC_{1}}(G) \\ ↓ & & ↓ \\ E_{VCYC_{f}} (G) & ⟶ & E_{VCYC}(G) \end{matrix}  (3.6)

The following conditions are equivalent for a virtually cyclic group V: i.) V admits an epimorphism to ℤ with finite kernel, ii.) H_{1}(V; ℤ) is infinite, iii.) The center of V is infinite. A virtually cyclic subgroup does not satisfy these three equivalent conditions if and only if it admits an epimorphism onto D_{∞} with finite kernel.

Lemma 3.7. Any virtually cyclic subgroup of Q is finite, infinite cyclic or isomorphic to D_{∞}.

Proof. Suppose that V ⊆ Q is an infinite virtually cyclic subgroup. Choose a finite normal subgroup F ⊆ V such that V/F is ℤ or D_{∞}. We have to show that F is trivial. Suppose F is not trivial. By assumption there is a unique maximal finite subgroup M ⊆ Q with F ⊆ M. Consider q ∈ N_{G}F. Then F ⊆ q^{−1}Mq ∩ M. This implies q ∈ N_{G}M = M. Hence N_{G}F is contained in the finite group M what contradicts V ⊆ N_{G}F. Hence F must be trivial.

Now we can prove Theorem 3.1.

Proof. Lemma 3.7 implies that any infinite subgroup appearing in VCYC_{1} is an infinite cyclic group or an infinite dihedral group. Hence Theorem 3.4 implies that H_{n}^{G} (\underline{E}G → E_{VCYC_{1}}(G); K(R?)) vanishes for n ∈ ℤ. We conclude from the G-pushout (3.6) that H_{n}^{G} (E_{VCYC_{f}} (G) → E_{VCYC}(G); K(R?)) vanishes for n ∈ ℤ. Now Theorem 3.1 follows from (3.5).

Now let us investigate what the results above imply for the middle and lower algebraic K-theory with integral coefficients of the group G = Hei ⋊ℤ/4 introduced in Section 2 and R = ℤ. The Farrell-Jones Conjecture for algebraic K-theory is true for G and R = ℤ in the range n ≤ 1 since G is a discrete cocompact subgroup of the virtually connected Lie group Hei(ℝ) ⋊ ℤ/4 (see [12]). Each group p^{−1}(M_{i}) is virtually cyclic and satisfies the Farrell-Jones Conjecture for algebraic K-theory for trivial reasons. From Theorem 3.1 we get for n ≤ 1 an isomorphism

Wh_{n}(p^{−1}(M_{0})) ⨁ Wh_{n}(p^{−1}(M_{1})) ⨁ Wh_{n}(p^{−1}(M_{2})) ⟶^{≅} Wh_{n}(G),

which comes from the various inclusions of subgroups and the subgroups M_{0}, M_{1} and M_{2} of Q have been introduced in Lemma 2.2. The Bass-Heller-Swan decomposition yields for any group H an isomorphism

Wh_{n}(H × ℤ) ≅ Wh_{n−1}(H) ⨁ Wh_{n}(H) ⨁ NK_{n}(ℤH) ⨁NK_{n}(ℤH). (3.8)

The groups Wh_{n}(ℤ^{k}) and Wh_{n}(ℤ/2 × ℤ^{k}) vanish for n ≤ 1 and k ≥ 0. The groups Wh_{n}(ℤ/4) are trivial for n ≤ 1. References for these claims are given in the proof of [22, Theorem 3.2]. The groups Wh_{n}(ℤ/4 × ℤ^{k}) vanish for n ≤ −1 and k ≥ 0. This follows from [13]. Thus we get Corollary 3.9. Let G be the group Hei ⋊ℤ/4 introduced in Section 2. Then

Wh_{n}(G) ≅ \{ \begin{matrix}NK_{n}(ℤ[ℤ/4]) ⨁ NK_{n}(ℤ[ℤ/4]) & for n = 0, 1; \\ 0 & for n ≤ −1. \end{matrix} (3.10)

where the isomorphism for n = 0, 1 comes from the inclusions of the subgroup p^{−1}(M_{0}) = ⟨t, z⟩ = ℤ × ℤ/4 into G and the Bass-Heller-Swan decomposition (3.8).

Some information about NK_{n}(ℤ[ℤ/4]) is given in [5, Theorem 10.6 on page 695]. Their exponent divides 4^{d} for some natural number d.

4 L-theory

In this section we want to describe what the methods above yield for the algebraic L-theory provided that instead of the Baum-Connes Conjecture the relevant version of the Farrell-Jones Conjecture for algebraic L-theory (see [12]) is true.

Theorem 4.1. Let G be the group appearing in (0.1) and assume that conditions (M), (NM), and (T) are satisfied. Suppose that G and all the groups p^{−1}(M) for M ⊆ Q maximal finite satisfy the Farrell-Jones Conjecture for L-theory with coefficients in R.

(i) Then there is a long exact sequence of abelian groups

… → H_{n+1}(G\\underline{E}G; L^{⟨−∞⟩}(R)) → ⨁_{i∈I} L_{n}^{⟨−∞⟩}(R[p^{−1}(M_{i})])

→ H_{n}^{G}(\underline{E}G; L^{⟨−∞⟩}(R?)) ⨁ (⨁_{i∈I} H_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}); L^{⟨−∞⟩}(R?)))

→ H_{n}(G\\underline{E}G; L^{⟨−∞⟩}(R)) → ⨁_{i∈I} L_{n−1}^{⟨−∞⟩}(R[p^{−1}(M_{i})]) → ….

Let Λ be a ring with ℤ ⊆ Λ ⊆ ℚ such that the order of each finite subgroup of G is invertible in Λ. Then the long exact sequence above reduces after applying Λ ⊗_{ℤ} − to short split exact sequences of Λ-modules

0 → ⨁_{i∈I} Λ ⊗_{ℤ} L_{n}^{⟨−∞⟩}(R[p^{−1}(M_{i})]) → Λ ⊗_{ℤ} H_{n}^{G}(\underline{E}G; L^{⟨−∞⟩}(R?))

⨁ (⨁_{i∈I} Λ ⊗_{ℤ} H_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}); L^{⟨−∞⟩}(R)))

→ Λ ⊗_{ℤ} H_{n}(G\\underline{E}G; L^{⟨−∞⟩}(R)) → 0; (ii) Suppose for any virtually cyclic subgroup V ⊆ G that K_{−i}(RV) = 0 for

sufficiently large i. Then there is a canonical isomorphism

H_{n}^{G}(\underline{E}G; L^{⟨−∞⟩}(ℝ?)) ⨁ H_{n}^{G} (\underline{E}G → E_{VCYC}(G); L^{⟨−∞⟩}(R?))

⟶^{≅} L_{n}^{⟨−∞⟩}(RG);

(iii) We have

H_{n}^{G} (\underline{E}G → E_{VCYC}(G); L^{⟨−∞⟩}(R?)) [\frac{1}{2}] = 0.

Proof. (i) is proven completely analogous to Theorem 1.6. (ii) follows from Theorem 3.2. (iii) follows from the next Lemma 4.2.

Lemma 4.2. Let Γ be a group. Let VCYC be the family of virtually cyclic subgroups of Γ and VCYC_{ℤ} be the subfamily of VCYC consisting of subgroups of Γ which admit an epimorphism to ℤ with finite kernel. Let F and G be families of subgroups of Γ. If FIN ⊆ F ⊆ G ⊆ VCYC_{ℤ} holds, then

H_{n}^{Γ} (E_{F}(Γ) → E_{G}(Γ); L^{⟨−∞⟩}(R?)) = 0.

If FIN ⊆ F ⊆ G ⊆ VCYC holds, then

H_{n}^{Γ} (E_{F}(Γ) → E_{G}(Γ); L^{⟨−∞⟩}(R?)) [\frac{1}{2}] = 0.

Proof. From Theorem 3.4 we conclude that it suffices to show for a virtually cyclic group V, which admits an epimorphism to ℤ, that the map

H_{n}^{V} (\underline{E}V; L^{⟨−∞⟩}(R?)) → L_{n}^{⟨−∞⟩}(RV)

is bijective and for a virtually cyclic group V, which admits an epimorphism to D_{∞}, that the map above is bijective after inverting two.

We begin with the case, where V = F ⋊_{ϕ} ℤ for an automorphism ϕ: F → F of a finite group F. There is a long exact sequence which can be derived from [25] and [26]

… → L_{n}^{⟨−∞⟩}(RF) ⟶^{id −L_{n}^{⟨−∞⟩}(Rϕ)} L_{n}^{⟨−∞⟩}(RF) → L_{n}^{⟨−∞⟩}(RV) →

→ L_{n−1}^{⟨−∞⟩}(RF) ⟶^{id −L_{n−1}^{⟨−∞⟩}(Rϕ)} L_{n−1}^{⟨−∞⟩}(RF) → …

Since ℝ with the action of V coming from the epimorphism to ℤ and the action of ℤ by translation is a model for \underline{E}V, we also obtain a long exact Mayer-Vietoris sequence

… → L_{n}^{⟨−∞⟩}(RF) ⟶^{id −L_{n}^{⟨−∞⟩}(Rϕ)} L_{n}^{⟨−∞⟩}(RF) → H_{n}^{V} (\underline{E}V; L^{⟨−∞⟩}(R?)) →

→ L_{n−1}^{⟨−∞⟩}(RF) ⟶^{id −L_{n−1}^{⟨−∞⟩}(Rϕ)} L_{n−1}^{⟨−∞⟩}(RF) → ….

These two sequences are compatible with the assembly map

asmb: H_{n}^{V} (\underline{E}V; L^{⟨−∞⟩}(R?)) → H_{n}^{V} (V/V; L^{⟨−∞⟩}(R?)) = L_{n}^{⟨−∞⟩}(RV),

which must be an isomorphism by the Five-Lemma.

Suppose that V admits an epimorphism onto D_{∞} = ℤ/2 ∗ ℤ/2. Then we can write V as an amalgamated product F_{1} ∗_{F_{0}} F_{2} for finite groups F_{1} and F_{2} and a common subgroup F_{0}. We can think of V as a graph of groups associated to a segment and obtain an action without inversions on a tree which yields a 1-dimensional model for \underline{E}V with two equivariant 0-cells V/F_{1} and V/F_{2} and one equivariant one-cell V/F_{0} × D^{1} (see [29, §5]). The associated long Mayer-Vietoris sequence looks like

… → L_{n}^{⟨−∞⟩}(RF_{0}) → L_{n}^{⟨−∞⟩}(RF_{1}) ⨁ L_{n}^{⟨−∞⟩}(RF_{2}) → H_{n}^{V} (\underline{E}V; L^{⟨−∞⟩}(R?))

→ L_{n−1}^{⟨−∞⟩}(RF_{0}) → L_{n−1}^{⟨−∞⟩}(RF_{1}) ⨁ L_{n−1}^{⟨−∞⟩}(RF_{2}) → ….

There is a corresponding exact sequence, where H_{n}^{V} (\underline{E}V; L^{⟨−∞⟩}(R?)) is replaced by L_{n}^{⟨−∞⟩}(RV) and additional UNil-terms occur which vanish after inverting two (see for ℤ ⊆ R ⊆ ℚ [7, Corollary 6] or see [27, Remark 8.7] and [25]). Now a Five-Lemma argument proves the claim.

Theorem 4.3. Let G be the group appearing in (0.1) and assume that conditions (M), (NM), and (T) are satisfied. Suppose that Q contains no element of order 2. Suppose that G and all the groups p^{−1}(M) for M ⊆ Q maximal finite satisfy the Farrell-Jones Conjecture for L-theory with coefficients in R. Then there is a long exact sequence of abelian groups

… → H_{n+1}(G\\underline{E}G; L^{⟨−∞⟩}(R?)) → ⨁_{i∈I} L_{n}^{⟨−∞⟩}(R[p^{−1}(M_{i})])

→ L_{n}^{⟨−∞⟩}(RG) ⨁ (⨁_{i∈I} H_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}); L^{⟨−∞⟩}(R?)))

→ H_{n}(G\\underline{E}G; L^{⟨−∞⟩}(R?)) → ….

Let Λ be a ring with ℤ ⊆ Λ ⊆ ℚ such that the order of each finite subgroup of G is invertible in Λ. Then the long exact sequence above reduces after applying Λ ⊗_{ℤ} − to short split exact sequences of Λ-modules

0 → ⨁_{i∈I} Λ ⊗_{ℤ} L_{n}^{⟨−∞⟩}(R[p^{−1}(M_{i})]) →

Λ ⊗_{ℤ} L_{n}^{⟨−∞⟩}(RG) ⨁ (⨁_{i∈I} Λ ⊗_{ℤ} H_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}); L^{⟨−∞⟩}(R)))

→ Λ ⊗_{ℤ} H_{n}(G\\underline{E}G; L^{⟨−∞⟩}(R)) → 0.

Proof. Because of Theorem 4.1 and Lemma 4.2 it suffices to prove for an infinite virtually cyclic subgroup V ⊂ G that V admits an epimorphism to ℤ. If V ∩ K is trivial, then V is an infinite virtually cyclic subgroup of Q and hence is isomorphic to ℤ by Lemma 3.7. Suppose that V ∩ K is non-trivial. Then V can be written as an extension 1 → K ∩ V → V → p(V) → 1 for a finite subgroup p(V) ⊆ Q. The group K ∩ V is infinite cyclic and p(V) must have odd order. Hence V contains a central infinite cyclic subgroup. This implies that V admits an epimorphism to ℤ.

From now on we assume that Q is an extension 1 → ℤ^{n} → Q → F → 1 for a finite group F such that the conjugation action of F on ℤ^{n} is free outside the origin.

Let V ⊆ Q be infinite virtually cyclic. Either F is of odd order or F has a unique element f_{2} of order two [22, Lemma 6.2]. Because of Lemma 3.7 either V is infinite cyclic or V is isomorphic to D_{∞} and each element v_{2} ∈ V of order two is mapped under Q → F to the unique element f_{2} ∈ F of order two in F.

Suppose that V ⊆ Q is isomorphic to D_{∞}. Then V contains at least one element v_{2} ∈ V of order two. Any other element of order two is of the shape v_{2}u for u ∈ V ∩ ℤ^{n}. Hence V is the subgroup ⟨v_{2}, V ∩ ℤ^{n}⟩ generated by v_{2} and the infinite cyclic group V ∩ ℤ^{n} regardless which element v_{2} ∈ V of order two we choose. For any infinite cyclic subgroup C ⊆ ℤ^{n} let C_{max} be the kernel of the projection ℤ^{n} → (ℤ^{n}/C)/ tors(ℤ^{n}/C). This is the maximal infinite cyclic subgroup of ℤ^{n} which contains C. Define V_{max} ⊆ Q to be

V_{max} := ⟨v_{2}, (V ∩ ℤ^{n})_{max}⟩.

The subgroup V_{max} is isomorphic to D_{∞} and satisfies V ⊆ V_{max}. Moreover, it is a maximal virtually cyclic subgroup, i.e. V_{max} ⊆ W for a virtually cyclic subgroup W ⊆ Q implies V_{max} = W. Let V ⊆ W be virtually cyclic subgroups of Q such that V ≅ D_{∞}. Then W ≅ D_{∞} and V_{max} = W_{max}. Hence each virtually cyclic subgroup V with V ≅ D_{∞} is contained in a unique maximal virtually cyclic subgroup of Q isomorphic to D_{∞}, namely V_{max}.

Next we show N_{G}V = V if V is a subgroup of Q with V ≅ D_{∞} and V = V_{max}. Let v_{2} ∈ V be an element of order two. Consider an element q ∈ N_{G}V. Then the conjugation action of q on ℤ^{n} sends V ∩ℤ^{n} to itself. Hence the conjugation action of q^{2} on ℤ^{n} induces the identity on V ∩ℤ^{n}. This implies that q is mapped under Q → F to the unit element or the unique element of order two f_{2}. Hence q is of the shape u or v_{2}u for some u ∈ ℤ^{n}. Since (v_{2}u)v_{2}(v_{2}u)^{−1} = v_{2}uv_{2}u^{−1}v_{2} = v_{2}u^{2} and uv_{2}u^{−1} = v_{2}u^{2}, we conclude v_{2}u^{2} ∈ V. This implies that u^{2} ∈ V ∩ ℤ^{n}. Since V = V_{max}, we get u ∈ V and hence q ∈ V.

Let J be a complete system of representatives V for the set of conjugacy classes (V) of subgroups V ⊆ Q with V ≅ D_{∞} and V = V_{max}. In the sequel let ICOF be the set of subgroups H with are infinite cyclic or finite. By attaching equivariant cells we construct a model for E_{ICOF}(Q) which contains ∐_{V ∈J} Q×_{V} E_{ICOF}(V) as Q-CW -subcomplex. Define a Q-CW -complex E_{VCYC}(Q) by the Qpushout

\begin{matrix}∐_{V ∈J} Q ×_{V} E_{ICOF}(V) & ⟶ & E_{ICOF}(Q) \\ ^{u_{5}}↓ & & ↓^{f_{5}} \\ ∐_{V ∈J} Q/V & ⟶ & E_{VCYC}(Q) \end{matrix}  (4.4)

where the map u_{5} is the obvious projection and the upper horizontal arrow is the inclusion.

We have to show that E_{VCYC}(Q) is a model for the classifying space for the family VCYC of virtually cyclic subgroups of Q. Obviously all its isotropy groups belong to VCYC. Let H ⊆ Q be a virtually cyclic group with H ∈ ICOF. Choose a map of sets s: Q/V → Q such that its composition with the projection Q → Q/V is the identity. For any V -space X, there is a homeomorphism

(Q ×_{V} X)^{H} ⟶^{≅} ∐_{\begin{matrix}qV ∈Q/V \\ s(qV)^{−1}Hs(qV)⊆V \end{matrix}} X^{s(qV)^{−1}Hs(qV)}, (q, x) ↦ s(qV)^{−1}qx,

whose inverse sends x of the summand X^{s(qV)^{−1}Hs(qV)} belonging to qV ∈ Q/V with s(qV)^{−1}Hs(qV) ⊆ V to (s(qV), x). Since E_{ICOF}(Q)^{s(qV)^{−1}Hs(qV)} and (V/V)^{s(qV)^{−1}Hs(qV)} are contractible for each qV ∈ Q/V with s(qV)^{−1}Hs(qV) ⊆ V, the map u_{5}^{H} is a homotopy equivalence. Hence f_{5}^{H} is a homotopy equivalence. The space E_{ICOF}(Q)^{H} is contractible. Therefore E_{VCYC}(Q)^{H} is contractible. Let H ⊆ Q be a virtually cyclic group with H ∉ ICOF. Then

( ∐_{V ∈J} Q/V )^{H} = {∗};
(Q ×_{V} E_{ICOF}(V))^{H} = ∅;
E_{ICOF}(Q)^{H} = ∅.

This implies that E_{VCYC}(Q)^{H} = {∗} is contractible.

Recall that VCYC_{f} is the family of virtually cyclic subgroups of G whose image under p: G → Q is finite and VCYC_{1} is the family of virtually cyclic subgroups of G whose intersection with K = ker(p) is trivial. Let VCYC_{icof} be the family of subgroups of G whose image under p: G → Q is contained in ICOF. If we cross the Q-pushout (4.4) with E_{VCYC}(G), we obtain the G-pushout

∐_{V ∈J} G ×_{p^{−1}(V)} E_{ICOF}(p^{−1}(V)) ⟶ E_{VCYC_{icof}} (G)
u_{6}↓  ↓^{f_{6}}
∐_{V ∈J} G ×_{p^{−1}(V)} E_{VCYC}(p^{−1}(V)) ⟶ E_{VCYC}(G)

Because of Lemma 4.2 this G-pushout induces for n ∈ ℤ isomorphisms

⨁_{V ∈J} H_{n}^{p^{−1}(V)} (\underline{E}p^{−1}(V) → E_{VCYC}(p^{−1}(V)); L^{⟨−∞⟩}(R?))

⟶^{≅} H_{n}^{G} (E_{VCYC_{f}} (G) → E_{VCYC}(G); L^{⟨−∞⟩}(R?)) .

Now we conclude from Lemma 4.2

Lemma 4.5. Let 1 → ℤ^{n} → Q → F → 1 be an extension such that the conjugation action of F on ℤ^{n} is free outside the origin. Let 1 → K ⟶^{i} G ⟶^{p} Q → 1 be an extension. Suppose that for any virtually cyclic group V ⊆ G with p(V) finite there exists an epimorphism V → ℤ. (This condition is satisfied if K is abelian and contained in the center of G.) Then there is an isomorphism

⨁_{V ∈J} H_{n}^{p^{−1}(V)} (\underline{E}p^{−1}(V) → E_{VCYC}(p^{−1}(V)); L^{⟨−∞⟩}(R?))

⟶^{≅} H_{n}^{G} (\underline{E}G → E_{VCYC}(G); L^{⟨−∞⟩}(R?)) .

Next we apply Theorem 4.1 and Lemma 4.5 to the special example G = Hei ⋊ℤ/4 introduced in Section 2. We begin with constructing an explicit choice for J and determining the preimages p^{−1}(V) for V ∈ J. Recall that J is a complete system of representatives of the conjugacy classes (V) of subgroups V ⊆ Q with V ≅ D_{∞} and V = V_{max}. Let IC(ℤ^{2}) be the set of infinite cyclic subgroups L of ℤ^{2}. Any subgroup V ⊆ Q with V ≅ D_{∞} can be written as V = ⟨v_{2}, V ∩ ℤ^{2}⟩ for v_{2} ∈ V any element of order two. Hence we can write V = ⟨t^{2}a, L⟩ for L ∈ IC(ℤ^{2}) and a ∈ ℤ^{2}. We have ⟨t^{2}a, L⟩ = ⟨t^{2}a′, L′⟩ if and only if L = L′ and a − a′ ∈ L = L′. We have V = V_{max} for V = ⟨t^{2}a, L⟩ if and only if L ⊂ ℤ^{2} is maximal.

Let IC^{+}(ℤ) be the subset for which L ∈ IC(ℤ^{2}) meets {(n_{1}, n_{2}) ∈ ℤ^{2} | n_{1} ≥ 0, n_{2} > 0}. The ℤ/4-action on ℤ^{2} induces a ℤ/2-action on L_{1}(ℤ^{2}) by sending L to i · L. Notice that IC^{+}(ℤ^{2}) is a fundamental domain for this action, i.e. IC(ℤ^{2}) is the disjoint union of IC^{+}(ℤ^{2}) and its image under this involution. We claim that a complete system of representatives of conjugacy classes (V) of subgroups V of Q = ℤ^{2} ⋊ ℤ/4 with V ≅ D_{∞} is

⟨t^{2}, (n_{1}, n_{2})⟩  n_{1} even;
⟨t^{2}(0, 1), (n_{1}, n_{2})⟩  n_{1} even;
⟨t^{2}, (n_{1}, n_{2})⟩  n_{2} even;
⟨t^{2}(1, 0), (n_{1}, n_{2})⟩  n_{2} even;
⟨t^{2}, (n_{1}, n_{2})⟩  n_{1} and n_{2} odd;
⟨t^{2}(1, 0), (n_{1}, n_{2})⟩  n_{1} and n_{2} odd,

(4.6)

where (n_{1}, n_{2}) runs through IC^{+} = {(n_{1}, n_{2}) ∈ ℤ^{2} | n_{1} > 0, n_{2} ≥ 0, (n_{1}, n_{2}) = 1}. This follows from the computations

(m_{1}, m_{2})^{−1}(t^{2}(n_{1}, n_{2}))(m_{1}, m_{2}) = t^{2}(n_{1} + 2m_{1}, n_{2} + 2m_{2});
t(t^{2}(n_{1}, n_{2}))t^{−1} = t^{2}(−n_{2}, n_{1});
t^{2}(t^{2}(n_{1}, n_{2}))(t^{2})^{−1} = t^{2}(−n_{1}, −n_{2}).

Now we list the preimages p^{−1}(V) of these subgroups above and determine their isomorphism type. We claim that they can be described by the following generators

⟨t^{2}, t^{2}(n_{1}, n_{1}n_{2}/2, n_{2}), (0, 1, 0)⟩  n_{1} even;
⟨t^{2}(0, 0, 1), t^{2}(n_{1}, n_{1}n_{2}/2, n_{2}), (0, 1, 0)⟩  n_{1} even;
⟨t^{2}, t^{2}(n_{1}, n_{1}n_{2}/2, n_{2}), (0, 1, 0)⟩  n_{2} even;
⟨t^{2}(1, 0, 0), t^{2}(n_{1}, n_{1}n_{2}/2, n_{2}), (0, 1, 0)⟩  n_{2} even;
⟨t^{2}, t^{2}(2n_{1}, 2n_{1}n_{2}, 2n_{2}), (n_{1}, \frac{n_{1}n_{2}+1}{2}, n_{2})⟩  n_{1} and n_{2} odd;
⟨t^{2}(1, 0, 0), t^{2}(2n_{1} + 1, 2n_{1}n_{2} + n_{2}, 2n_{2}), (n_{1}, \frac{n_{1}n_{2}+1}{2}, n_{2})⟩  n_{1} and n_{2} odd,

(4.7)

where (n_{1}, n_{2}) runs through IC^{+} = {(n_{1}, n_{2}) ∈ ℤ^{2} | n_{1} > 0, n_{2} ≥ 0, (n_{1}, n_{2}) = 1}. This is obvious for the first four groups and follows for the last two groups from the computation

(n_{1}, \frac{n_{1}n_{2} + 1}{2}, n_{2})^{2} = (2n_{1}, 2n_{1}n_{2}, 2n_{2}) · (0, 1, 0);
(2n_{1} + 1, 2n_{1}n_{2} + n_{2}, 2n_{2}) = (1, 0, 0) · (2n_{1}, 2n_{1}n_{2}, 2n_{2}).

The first four groups are isomorphic to D_{∞} × ℤ and the last two are isomorphic to the semi-direct product D_{∞} ⋊_{a} ℤ with respect to the automorphism a of D_{∞} = ℤ/2∗ℤ/2 = ⟨s_{1}, s_{2} | s_{2}^{2} = s_{2}^{2} = 1⟩ which send s_{1} to s_{2} and s_{2} to s_{1}. For the first four groups there are explicit isomorphisms from D_{∞} ×ℤ = ⟨s_{1}, s_{2}, z | s_{1}^{2} = s_{2}^{2} = [s_{1}, z] = [s_{2}, z] = 1⟩ which send s_{1}, s_{2}, z to the three generators appearing in the presentation above. Similarly for the last two groups there are explicit isomorphisms from D_{∞} ⋊_{a} ℤ = ⟨s_{1}, s_{2}, z | s_{1}^{2} = s_{2}^{2} = 1, z^{−1}s_{1}z = s_{2}⟩ which send s_{1}, s_{2}, z to the three generators appearing in the presentation above. We leave it to the reader to check that these generators appearing in the presentation above do satisfy the required relations.

Next we compute the groups H_{n}^{D_{∞}×ℤ} (\underline{E}D_{∞} × ℤ → E_{VCYC}(D_{∞} × ℤ); L^{∞}(ℤ?)) and H_{n}^{D_{∞}⋊_{a}ℤ} (\underline{E}D_{∞} ⋊_{a} ℤ → E_{VCYC}(D_{∞} ⋊ ℤ); L^{∞}(ℤ?)). There is an obvious model for \underline{E}D_{∞}, namely ℝ with the trivial ℤ-action and the action of D_{∞} = ℤ ⋊_{a} ℤ/2, which comes from the ℤ-action by translation and the ℤ/2-action given by − id_{ℝ}. From this we obtain an exact sequence

0 → L_{n}^{⟨−∞⟩}(R) ⟶^{i} L_{2}^{⟨−∞⟩}(R[ℤ/2]) ⨁ L_{2}^{⟨−∞⟩}(R[ℤ/2]) ⟶^{f} H_{n}^{D_{∞}}(\underline{E}D_{∞}; L(R?)) → 0

such that the composition of f with the obvious map

H_{n}^{D_{∞}}(\underline{E}D_{∞}; L(R?)) → H_{n}^{D_{∞}}(\{∗\}; L(R?) = L_{n}^{⟨−∞⟩}(R[D_{∞}])

is given by the two obvious inclusions ℤ/2 → D_{∞} = ⟨s_{1}, s_{2} | s_{1} = s_{2}^{2} = 1⟩. Thus we obtain an isomorphism

H_{n}^{D_{∞}} (\underline{E}D_{∞} → E_{VCYC}(D_{∞}); L^{⟨−∞⟩}(R?)) = UNil_{n}(ℤ/2 ∗ ℤ/2; R), (4.8)

where UNil_{n}(ℤ/2 ∗ ℤ/2; R) is the UNil-term appearing in the short split exact sequence

0 → L_{n}^{⟨−∞⟩}(R) → L_{n}^{⟨−∞⟩}(R[ℤ/2]) ⊕ L_{n}^{⟨−∞⟩}(R[ℤ/2]) ⊕ UNil_{n}(ℤ/2 ∗ ℤ/2; R)

→ L_{n}^{⟨−∞⟩}(R[ℤ/2 ∗ ℤ/2]) → 0

due to Cappell [7, Theorem 10]. For the computation of these terms UNil_{n}(ℤ/2∗ ℤ/2; R) we refer to [2], [8] and [9]. They have exponent four and they are either trivial or are infinitely generated as abelian groups.

We can take as model for \underline{E}(D_{∞} × ℤ) the product \underline{E}D_{∞} × ℝ, where ℤ acts on ℝ by translation. We get from (4.8) and Lemma 4.2 using a Mayer-Vietoris argument an isomorphism

H_{n}^{ℤ×D_{∞}} (\underline{E}(D_{∞} × ℤ) → E_{VCYC}(D_{∞} × ℤ); L^{⟨−∞⟩}(R?))

≅ H_{n}^{ℤ×D_{∞}} (\underline{E}D_{∞} × ℝ → E_{VCYC}(D_{∞}) × ℝ; L^{⟨−∞⟩}(R?))
≅ H_{n}^{D_{∞}} (\underline{E}D_{∞} × S^{1} → E_{VCYC}(D_{∞}) × S^{1}; L^{⟨−∞⟩}(R?))

≅ UNil_{n}(ℤ/2 ∗ ℤ/2; R) ⨁ UNil_{n−1}(ℤ/2 ∗ ℤ/2; R). (4.9)

Next we investigate D_{∞}⋊_{a}ℤ. Let VCYC(D_{∞}) be the family of virtually cyclic subgroups of D_{∞} ⋊_{a} ℤ which lie in D_{∞} and let VCYC_{f} be the family of virtually cyclic subgroups of VCYC(D_{∞}) whose intersection with D_{∞} is finite. Then the family VCYC of virtually cyclic subgroups of D_{∞} ⋊_{a} ℤ is the union of VCYC(D_{∞}) and VCYC_{f} and the family FIN of finite subgroups of D_{∞} ⋊_{a} ℤ is the intersection of VCYC(D_{∞}) and VCYC_{f}. Hence we get a pushout of D_{∞} ⋊_{a} ℤ-spaces

\underline{E}(D_{∞} ⋊_{a} ℤ) ⟶ E_{VCYC(D_{∞})}(D_{∞} ⋊_{a} ℤ)
↓  ↓
E_{VCYC_{f}} (D_{∞} ⋊_{a} ℤ) ⟶ E_{VCYC}(D_{∞} ⋊_{a} ℤ)

Any finite subgroup of D_{∞} is trivial or isomorphic to ℤ/2. Any group V which can be written as extension 1 → ℤ/2 → V → ℤ → 1 is isomorphic to ℤ/2 × ℤ. Hence any infinite group V occurring in VCYC_{f} is isomorphic to ℤ or ℤ/2 × ℤ/2. We conclude from Lemma 4.2 and the D_{∞} ⋊_{a} ℤ-pushout above

H_{n}^{D_{∞}⋊_{a}ℤ)} (E_{VCYC(D_{∞})}(D_{∞} ⋊_{a} ℤ) → E_{VCYC}(D_{∞} ⋊_{a} ℤ); L^{⟨−∞⟩}(R?))

≅ H_{n}^{D_{∞}⋊_{a}ℤ)} (\underline{E}(D_{∞} ⋊_{a} ℤ) → E_{VCYC_{f}} (D_{∞} ⋊_{a} ℤ); L^{⟨−∞⟩}(R?)) ≅ 0.

Hence we get an isomorphism

H_{n}^{D_{∞}⋊_{a}ℤ)} (\underline{E}(D_{∞} ⋊_{a} ℤ) → E_{VCYC}(D_{∞} ⋊_{a} ℤ); L^{⟨−∞⟩}(R?))

≅ H_{n}^{D_{∞}⋊_{a}ℤ)} (\underline{E}(D_{∞} ⋊_{a} ℤ) → E_{VCYC(D_{∞})}(D_{∞} ⋊_{a} ℤ); L^{⟨−∞⟩}(R?)) .

We can take as model for \underline{E}(D_{∞} ⋊_{a} ℤ) the to both sides infinite mapping telescope of the (a: D_{∞} → D_{∞})-equivariant map \underline{E}a: \underline{E}D_{∞} → \underline{E}D_{∞} with the D_{∞} ⋊_{a} ℤ-action for which ℤ acts by shifting the telescope to the right. A model for E_{VCYC(D_{∞})}(D_{∞} ⋊_{a} ℤ) is the to both sides infinite mapping telescope of the (a: D_{∞} → D_{∞})-equivariant map {∗} → {∗}. Of course this is the same as ℝ with the D_{∞} ⋊_{a} ℤ-action, for which D_{∞} acts trivially and ℤ by translation. The long Mayer-Vietoris sequence together with (4.8) yields a long exact sequence

… → UNil_{n}(ℤ/2 ∗ ℤ/2; R) ⟶^{id − UNil_{n}(a)} UNil_{n}(ℤ/2 ∗ ℤ/2; R)

→ H_{n}^{D_{∞}⋊_{a}ℤ} (\underline{E}(D_{∞} ⋊_{a} ℤ) → E_{VCYC}(D_{∞} ⋊_{a} ℤ); L^{⟨−∞⟩}(R?))

→ UNil_{n−1}(ℤ/2 ∗ ℤ/2; R) ⟶^{id − UNil_{n−1}(a)} UNil_{n−1}(ℤ/2 ∗ ℤ/2; R) → …. (4.10)

Theorem 4.11. Let G be the group Hei ⋊ℤ/4 introduced in Section 2. Then

(i) There is a short exact sequence which splits after inverting 2

0 → L_{n}^{⟨−∞⟩}(ℤ) ⨁ L̃_{n}^{⟨−∞⟩}(ℤ[ℤ/2]) ⨁ L̃_{n−1}^{⟨−∞⟩}(ℤ[ℤ/2])

⨁ L̃_{n}^{⟨−∞⟩}(ℤ[ℤ/4]) ⨁ L̃_{n−1}^{⟨−∞⟩}(ℤ[ℤ/4])

⟶^{j} H^{G}(\underline{E}G; L^{⟨−∞⟩}(ℤ)) → L_{n−3}^{⟨−∞⟩}(ℤ) → 0;

(ii) There is for n ∈ ℤ an isomorphism

H_{n}^{G}(\underline{E}G; L^{⟨−∞⟩}(ℤ)) ⨁ H_{n}^{G} (\underline{E}G → E_{VCYC}(G); L^{⟨−∞⟩}(ℤ?)) ⟶^{≅} L^{⟨−∞⟩}(ℤG);

(iii) Let IC^{+} be the set {(n_{1}, n_{2}) ∈ ℤ^{2} | n_{1} > 0, n_{2} ≥ 0, (n_{1}, n_{2}) = 1}. Then there is an isomorphism

( ⨁_{\begin{matrix}(n_{1},n_{2})∈IC^{+}, \\ n_{1} or n_{2} even \end{matrix}} ⨁_{i=1}^{4} (UNil_{n}(ℤ/2 ∗ ℤ/2; ℤ) ⨁ UNil_{n−1}(ℤ/2 ∗ ℤ/2; ℤ))) ⨁

( ⨁_{\begin{matrix}(n_{1},n_{2})∈IC^{+}, \\ n_{1} and n_{2} odd \end{matrix}} ⨁_{i=1}^{2} H_{n}^{D_{∞}⋊_{a}ℤ} (\underline{E}(D_{∞} ⋊_{a} ℤ) → E_{VCYC}(D_{∞} ⋊_{a} ℤ); L^{⟨−∞⟩}(ℤ?)))

⟶^{≅} H^{G} (\underline{E}G → E_{VCYC}(G); L^{⟨−∞⟩}(ℤ?)) ,

where the term H_{n}^{D_{∞}⋊_{a}ℤ} (\underline{E}(D_{∞} ⋊_{a} ℤ) → E_{VCYC}(D_{∞} ⋊_{a} ℤ); L^{⟨−∞⟩}(ℤ?)) is analyzed in (4.10);

(iv) The canonical map L^{⟨−∞⟩}(ℤG) ⟶^{≅} L^{ϵ}(ℤG) is bijective for all decorations ϵ = p, h, s.

Proof. (i) follows from Theorem 4.1 (i) since the groups L̃_{n}^{p}(ℤ/2) = L̃_{n}^{⟨−∞⟩}(ℤ/2) and L̃_{n}^{p}(ℤ/4) = L̃_{n}^{⟨−∞⟩}(ℤ/4) are torsionfree [1, Theorem 1]. (ii) The Farrell-Jones Conjecture for algebraic L-theory with coefficients in R = ℤ is true for G = Hei ⋊ℤ/4 and p^{−1}(V) for V ⊆ Q virtually cyclic since G is a discrete cocompact subgroup of the virtually connected Lie group Hei(ℝ)⋊_{a}ℤ/4 (see [12]). Since for a virtually cyclic group V we have K_{n}(ℤV) = 0 for n ≤ −2 [13], we can apply Theorem 4.1 (ii). (iii) This follows from Lemma 4.5, the lists (4.6) and (4.7) and the isomorphism (4.9). (iv) Because of the Rothenberg sequences it suffices to show that the Tate cohomology groups Ĥ^{n}(ℤ/2, Wh_{q}(G)) vanish for q ≤ 1 and n ∈ ℤ. If q ≤ −1, then Wh_{q}(G) = 0, and, if q = 0, 1, then Wh_{q}(G) = NK_{q}(ℤ[ℤ/4]) ⨁ NK_{q}(ℤ[ℤ/4]) by Corollary 3.9. One easily checks that the involution on Wh_{q}(G) corresponds under this identification to the involution on NK_{q}(ℤ[ℤ/4]) ⨁ NK_{q}(ℤ[ℤ/4]) which sends (x_{1}, x_{2}) to (x_{2}, τ(x_{1})) for τ: NK_{q}(ℤ[ℤ/4]) → NK_{q}(ℤ[ℤ/4]) the involution on the Nil-Term. Hence the ℤ[ℤ/2]-module Wh_{q}(G) is isomorphic to the ℤ[ℤ/2]-module ℤ[ℤ/2] ⊗_{ℤ} NK_{q}(ℤ[ℤ/4]), which is obtained from the ℤ-module NK_{q}(ℤ[ℤ/4]) by induction with the inclusion of the trivial group into ℤ/2. This implies Ĥ^{n}(ℤ/2, Wh_{q}(G)) = 0 for q ≤ 1 and n ∈ ℤ.

Remark 4.12. If one inverts 2, then the computation for L_{n}(ℤ[Hei ⋊ℤ]) simplifies drastically as explained in the introduction because of Lemma 4.2. In general this example shows how complicated it is deal to deal with the infinite virtually cyclic subgroups which admit an epimorphism to D_{∞} and the resulting UNil-terms.

5 Group homology

Finally we explain what the methods above give for the group homology

Theorem 5.1. Let G be the group appearing in (0.1) and assume that conditions (M), (NM), and (T) are satisfied. Then we obtain from the pushout (1.3) a long exact Mayer-Vietoris sequence

… → H_{n+1}(G\\underline{E}G) ⟶^{∂_{n+1}} ⨁_{i∈I} H_{n}(p^{−1}(M_{i})) \underrightarrow{(⨁_{i∈I} H_{n}(l_{i})) ⨁(⨁_{i∈I} H_{n}(p^{−1}(M_{i})\s_{i}))}

H_{n}(G) ⨁ (⨁_{i∈I} H_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i})))

\underrightarrow{H_{n}(G\s) ⨁(⨁_{i∈I} H_{n}(d_{i}))} H_{n}(G\\underline{E}G) ⟶^{∂_{n}}

⨁_{i∈I} H_{n−1}(p^{−1}(M_{i})) \underrightarrow{(⨁_{i∈I} H_{n−1}(l_{i})) ⨁(⨁_{i∈I} H_{n−1}(p^{−1}(M_{i})\s_{i}))} …,

where l_{i}: p^{−1}(M_{i}) → G is the inclusion, s_{i}: Ep^{−1}(M_{i}) → \underline{E}p^{−1}(M_{i}), s: EG → \underline{E}G are the obvious equivariant maps and d_{i}: p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}) → G\\underline{E}G is the map induced by the l_{i}-equivariant map \underline{E}p^{−1}(M_{i}) → \underline{E}G.

Remark 5.2. Often there are finite-dimensional models for \underline{E}G as discussed in [16], [17]. For instance, if there is a k-dimensional model for BK and a mdimensional model for \underline{E}Q and d is a positive integer such that the order of any finite subgroup of Q divides d, then there is a (dk + n)-dimensional model for \underline{E}G [17, Theorem 3.1]. If Q is an extension 0 → ℤ^{n} → Q → F → 1 for a finite group F and there is a k-dimensional model for BK, then there is a (|F | · k + n)-dimensional model for \underline{E}G.

Suppose that there is a N-dimensional model for \underline{E}G. Then there is also a N-dimensional model for \underline{E}p^{−1}(M_{i}) for each i ∈ I and under the assumptions of Theorem 5.6 we obtain for n ≥ N + 1 an isomorphism

⨁_{i∈I} H_{n}(l_{i}): ⨁_{i∈I} H_{n}(p^{−1}(M_{i})) ⟶^{≅} H_{n}(G).

Next we compute the group homology H_{∗}(Hei ⋊ℤ/4). We start with the computation of H_{n}(Hei). The Atiyah-Hirzebruch spectral sequence associated

to the central extension 1 → ℤ ⟶^{i′} Hei ⟶^{p′} ℤ^{2} → 0 yields the isomorphism

H_{2}(ℤ^{2}) ⟶^{≅} H_{3}(Hei)

and the long exact sequence

0 → H_{1}(ℤ^{2}) → H_{2}(Hei) ⟶^{H_{2}(p′)} H_{2}(ℤ^{2}) → H_{0}(ℤ^{2}) = H_{1}(S^{1}) = H_{1}(ℤ)

⟶^{H_{1}(i′)} H_{1}(Hei) ⟶^{H_{1}(p′)} H_{1}(ℤ^{2}) → 0.

Since z ∈ Hei is a commutator, namely [u, v], the map H_{1}(i′): H_{1}(ℤ) → H_{1}(Hei) is trivial. This implies Lemma 5.3. There are natural isomorphisms

H_{1}(p′): H_{1}(Hei) ⟶^{≅} H_{1}(ℤ^{2});
H_{1}(ℤ^{2}) ⟶^{≅} H_{2}(Hei);
H_{2}(ℤ^{2}) ⟶^{≅} H_{3}(Hei);
H_{n}(Hei) = 0  for n ≥ 4.

Next we analyze the Atiyah-Hirzebruch spectral sequence associated to the split extension 1 → Hei ⟶^{k} G := Hei ⋊ℤ/4 ⟶^{pr} ℤ/4 → 1. The isomorphisms above appearing in the computation of the homology of Hei are compatible with the ℤ/4-actions. Thus we get

H_{p}(ℤ/4; H_{q}(Hei)) = H_{p}(ℤ/4; H_{q}(ℤ^{2})) = ℤ/2  for q = 1, 2, p ≥ 0, p even;
H_{p}(ℤ/4; H_{q}(Hei)) = H_{p}(ℤ/4; H_{q}(ℤ^{2})) = 0  for q = 1, 2, p ≥ 0, p odd;
H_{p}(ℤ/4; H_{q}(Hei)) = H_{p}(ℤ/4)  for q = 0, 3;
H_{p}(ℤ/4; H_{q}(Hei)) = = 0  for q ≥ 4.

Hence the E^{2}-term looks like

\begin{matrix}ℤ & ℤ/4 & 0 & ℤ/4 & 0 & ℤ/4 & 0 \\ ℤ/2 & 0 & ℤ/2 & 0 & ℤ/2 & 0 & ℤ/2 \\ ℤ/2 & 0 & ℤ/2 & 0 & ℤ/2 & 0 & ℤ/2 \\ ℤ & ℤ/4 & 0 & ℤ/4 & 0 & ℤ/4 & 0 \end{matrix}

Using the model for \underline{E}G of Lemma 2.4 we see that the map BHei → G\\underline{E}G can be identified with the quotient map BHei → ℤ/4\BHei of an orientation preserving smooth ℤ/4-action on the closed orientable 3-manifold BHei, where the quotient is again a closed orientable 3-manifold and the action has at least one free orbit. Since we can compute the degree of a map by counting preimages of a regular value, the degree must be ±4. Recall that G → ℤ/4 is split surjective. These remarks imply together with the spectral sequence above

Lemma 5.4. The composition H_{3}(BHei) ⟶^{H_{3}(Bk)} H_{3}(BG) ⟶^{H_{3}(G\s)} H_{3}(G\\underline{E}G) is an injective map of infinite cyclic subgroups whose cokernel has order four. The map H_{3}(Hei) → H_{3}(G) is injective and the order of the cokernel of the induced map H_{3}(G)/ tors(H_{3}(G)) → H_{3}(G\\underline{E}G) divides four;

Moreover, there are the following possibilities

(i) The differential d_{2,1}^{2} : E_{2,1}^{2} ≅ ℤ/2 → E_{0,2}^{2} ≅ ℤ/2 is trivial. Then H_{2}(G) is ℤ/2. Moreover, either the group H_{3}(G) is ℤ × ℤ/4 and the induced map H_{3}(Hei) → H_{3}(G)/ tors(H_{3}(G)) is an injective homomorphism of infinite cyclic groups whose cokernel has order two, or the group H_{3}(G) is ℤ × ℤ/2 × ℤ/4 and the induced map H_{3}(Hei) → H_{3}(G)/ tors(H_{3}(G)) is an isomorphism of infinite cyclic groups.

(ii) The differential d_{2,1}^{2} : E_{0,2}^{2} ≅ ℤ/2 → E_{2,1}^{2} ≅ ℤ/2 is non-trivial. Then H_{2}(G) is 0.

It is not obvious how to compute the homology groups H_{n}(G) for G = Hei ⋊ℤ/4) from the Atiyah-Hirzebruch spectral sequence. Let us try Theorem 5.1. It yields the long exact Mayer Vietoris sequence

… → H_{n+1}(G\\underline{E}G) ⟶^{∂_{n+1}} ⨁_{i=0}^{2} H_{n}(p^{−1}(M_{i}))

\underrightarrow{(⨁_{i=0}^{2} H_{n}(l_{i})) ⨁(⨁_{i=0}^{2} H_{n}(p^{−1}(M_{i})\s_{i}))} H_{n}(G) ⨁ (⨁_{i=0}^{2} H_{n}(p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i})))

\underrightarrow{H_{n}(G\s) ⨁(⨁_{i=0}^{2} H_{n}(d_{i}))} H_{n}(G\\underline{E}G) ⟶^{∂_{n}}

⨁_{i=0}^{2} H_{n−1}(p^{−1}(M_{i})) \underrightarrow{(⨁_{i=0}^{2} H_{n−1}(l_{i})) ⨁(⨁_{i=0}^{2} H_{n−1}(p^{−1}(M_{i})\s_{i}))} …,

where the maximal finite subgroups M_{0}, M_{1} and M_{2} of Q have been introduced in Lemma 2.2. The map s_{i}: p^{−1}(M_{i})\Ep^{−1}(M_{i}) → p^{−1}(M_{i})\\underline{E}p^{−1}(M_{i}) can be identified with

s_{0} : B⟨t, z⟩ = B⟨t⟩ × B⟨z⟩ ⟶^{pr} B⟨z⟩;
s_{1} : B⟨ut⟩ ⟶^{id} B⟨ut⟩;
s_{2} : B⟨ut^{2}, z⟩ = B⟨ut^{2}⟩ × B⟨z⟩ ⟶^{pr} B⟨z⟩.

Hence we obtain the exact sequence

… → H_{n+1}(G) ⨁ H_{n+1}(⟨z⟩) ⨁ H_{n+1}(⟨ut⟩) ⨁ H_{n+1}(⟨z⟩)

\underrightarrow{H_{n+1}(G\s) ⨁_{i=0}^{2} H_{n+1}(d_{i})} H_{n+1}(G\\underline{E}G)

⟶^{∂′_{n+1}} H_{n}(⟨t⟩ × ⟨z⟩) ⨁ H_{n}(⟨ut⟩) ⨁ H_{n}(⟨ut^{2}⟩ × ⟨z⟩)

(H_{n}(incl′_{0}) ⨁ H_{n}(incl′_{1}) ⨁ H_{n}(incl′_{2})) ⨁ H_{n}(pr) ⨁ id ⨁ H_{n}(pr)

⟶

H_{n}(G) ⨁ H_{n}(⟨z⟩) ⨁ H_{n}(⟨ut⟩) ⨁ H_{n}(⟨z⟩) \underrightarrow{H_{n}(G\s) ⨁_{i=0}^{2} H_{n}(d_{i})} H_{n}(G\\underline{E}G) ⟶^{∂′_{n}} …

where

incl′_{0} : ⟨t, z⟩ = ⟨t⟩ × ⟨z⟩ → G;
incl′_{1} : ⟨ut⟩ → G;
incl′_{2} : ⟨ut, z⟩ = ⟨ut⟩ × ⟨z⟩ → G

are the inclusions. This yields the exact sequence

… → H_{n+1}(G) ⟶^{H_{n+1}(G\s)} H_{n+1}(G\\underline{E}G)

⟶^{∂′′_{n}} H̃_{n}(⟨t⟩) ⨁ H̃_{n−1}(⟨t⟩) ⨁ H̃_{n}(⟨ut^{2}⟩) ⨁ H̃_{n−1}(⟨ut^{2}⟩)

→ H_{n}(G) ⟶^{H_{n}(G\s)} H_{n}(G\\underline{E}G) ⟶^{∂′′_{n}} …. (5.5)

Recall that G\\underline{E}G is S^{3}. We conclude from Lemma 5.4 that the order of the cokernel of the map H_{3}(G\s): H_{3}(BG) → H_{3}(G\\underline{E}G) divides four. Since the order of H̃_{2}(⟨t⟩) ⨁ H̃_{1}(⟨t⟩) ⨁ H̃_{2}(⟨e_{1}t^{2}⟩) ⨁ H̃_{2}(⟨e_{1}t^{2}⟩) is eight, the long exact sequence above implies that the group H_{2}(G) is different from zero and that the group H_{3}(G) is isomorphic to ℤ × ℤ/2 × ℤ/4. Now Lemma 5.4 and the long exact sequence (5.5) above imply

Theorem 5.6. For G = Hei ⋊ℤ/4 we have isomorphisms

H_{n}(G) = ℤ/2 × ℤ/4 for n ≥ 1, n ≠ 2, 3;
H_{2}(G) = ℤ/2;
H_{3}(G) = ℤ × ℤ/2 × ℤ/4.

The map H_{3}(Hei) → H_{3}(G)/ tors(H_{3}(G)) is an isomorphism.

One can compute the group cohomology analogously or derive it from the homology by the universal coefficient theorem.

6 Survey over other extensions

There are other prominent extensions of Hei which can be treated analogously to the case Hei ⋊ℤ/4. We give a brief summary of the topological K-theory and the algebraic K-theory below. In all cases G\\underline{E}G is S^{3}.

6.1 Order six symmetry

Consider the following automorphism ω: Hei → Hei of order 6 which sends u to v, v to u^{−1}v, and z to z.

Theorem 6.1. For the group

G = Hei ⋊ℤ/6 = ⟨u, v, z, t | [u, v] = z, t^{6} = 1, [u, z] = [v, z] = [t, z] = 1,

tut^{−1} = v, tvt^{−1} = u^{−1}v⟩

there is a short exact sequence

0 → R̃_{ℂ}(⟨t⟩) → K_{1}(C_{r}^{∗}(G)) → K̃_{1}(S^{3}) → 0

and an isomorphism

R_{ℂ}(⟨t⟩) ⟶^{≅} K_{0}(C_{r}^{∗}(G)).

There are isomorphisms

Wh_{n}(G) ≅ \{ \begin{matrix}NK_{1}(ℤ[ℤ/6]) ⨁ NK_{1}(ℤ[ℤ/6]) & for n = 1; \\ Wh_{−1}(ℤ/6) ≅ ℤ & for n = −1, 0; \\ 0 & for n ≤ −2. \end{matrix}

6.2 Order three symmetry

Next we deal with the ℤ/3-action on Hei given by ω^{2}, where ω is the automorphism of order six investigated in Subsection 6.1.

Theorem 6.2. For the group

G = Hei ⋊ℤ/3 = ⟨u, v, z, t | [u, v] = z, t^{3} = 1, [u, z] = [v, z] = [t, z] = 1,

tut^{−1} = u^{−1}v, tvt^{−1} = u^{−1}z^{−1}⟩

there is a short exact sequence

0 → R̃_{ℂ}(⟨t⟩) → K_{1}(C_{r}^{∗}(G)) → K̃_{1}(S^{3}) → 0

and an isomorphism

R_{ℂ}(⟨t⟩) ⟶^{≅} K_{0}(C_{r}^{∗}(G)).

We have Wh_{n}(G) = 0 for n ≤ 2.

The L-groups L_{n}ϵ(ℤG) are independent of the choice of decoration ϵ = −∞, p, h, s and the reduced ones fit into a short split exact sequence

0 → L̃_{n}^{⟨−∞⟩}(ℤ⟨t⟩) ⨁ L̃_{n−1}^{⟨−∞⟩}(ℤ⟨t⟩) → L̃_{n}^{⟨−∞⟩}(ℤG) → L_{n−3}^{⟨−∞⟩}(ℤ) → 0.

6.3 Order two symmetry

Next we deal with the ℤ/2-action on Hei given by u ↦ u^{−1}, v ↦ v^{−1} and z ↦ z. This is the square of the automorphism of order four used in the ℤ/4-case.

Theorem 6.3. For the group

G = Hei ⋊ℤ/2 = ⟨u, v, z, t | [u, v] = z, t^{2} = 1, [u, z] = [v, z] = [t, z] = 1,

tut^{−1} = u^{−1}, tvt^{−1} = v^{−1}⟩

there is a short exact sequence

0 → ⊕_{i=0}^{2}R̃_{ℂ}(M_{i}) → K_{1}(C_{r}^{∗}(G)) → K̃_{1}(S^{3}) → 0

and an isomorphism

0 → K_{0}({∗}) ⨁ ⨁_{i=0}^{2} R̃_{ℂ}(M_{i}) ⟶^{≅} K_{0}(C_{r}^{∗}(G)),

where

M_{0} = ⟨t⟩;
M_{1} = ⟨ut⟩;
M_{2} = ⟨vt⟩.

We have Wh_{n}(G) = 0 for n ≤ 2.

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