Bredon homology and equivariant K-homology of SL(3, ℤ)

Rubén Sánchez-García^{∗}

April 2006

Abstract

We obtain the equivariant K-homology of the classifying space \underline{E}SL(3, ℤ) from the computation of its Bredon homology with respect to finite subgroups and coefficients in the representation ring. We also obtain the corresponding results for GL(3, ℤ). Our calculations give therefore the topological side of the Baum-Connes conjecture for these groups.

1 Introduction

Consider a discrete group G. The Baum-Connes conjecture [1] identifies the Ktheory of C_{r}^{∗}(G), the reduced C^{∗}-algebra of G, with the equivariant K-homology of a certain classifying space associated to G. This space is called the classifying space for proper actions, written \underline{E}G. The conjecture states that a particular map between these two abelian groups, the assembly map,

μ_{i} : K_{i}^{G}(\underline{E}G) ⟶ K_{i}(C_{r}^{∗}(G)) i ≥ 0 ,

is an isomorphism. The conjecture can be stated more generally, see [1, Conjecture 3.15].

The equivariant K-homology and the assembly map are usually defined in terms of Kasparov’s KK-theory. For a discrete group G, however, there is a more topological description due to Davis and Lück [3], and Joachim [9] in terms of spectra over the orbit category of G. We will keep in mind the topological picture of the Baum-Connes conjecture (see Mislin’s notes in [12]).

Part of the importance of this conjecture is due to the fact that it is related to many other relevant conjectures in different areas of mathematics [12]. Nevertheless, the conjecture itself allows the computation of the K-theory of C_{r}^{∗}(G) from the K^{G}-homology of \underline{E}G. In turn, this K-homology can be achieved by means of the Bredon homology of \underline{E}G, as we explain later.

The Baum-Connes conjecture has been proved for some large families of groups, yet remains unsolved in general. In particular, Higson and Kasparov [7] proved the conjecture for groups having the Haagerup property (or a-T-menable), that is, groups which admits a metrically proper isometric action on some affine Hilbert space. On the other side, a group has Kazhdan’s property T if every isometric action of G on an affine Hilbert space has a fixed point. Thus, infinite groups with the property T do not have the Haagerup property. There are not many infinite discrete groups with the property T for which the Baum-Connes conjecture has been proved (the first examples are due to Lafforgue [10]).

∗Funded by the EPSRC and the School of Mathematics, University of Southampton

Consequently, the group SL(3, ℤ) becomes relevant in this context since the Baum-Connes conjecture is unknown for SL(n, ℤ), n ≥ 3 and these groups have property T. On the other hand, the Baum-Connes assembly map is known to be injective for SL(n, ℤ) (in general, for all closed subgroups of a Lie group with a finite number of connected components, see [1, §7]). Finally, note that there are counterexamples to the Baum-Connes conjecture for groupoids that can be constructed from SL(3, ℤ), and more generally for a discrete group with property T and such that the Baum-Connes map is injective ([8, p. 338]).

In this paper we obtain the equivariant K-homology of \underline{E}G for G = SL(3, ℤ) and G = GL(3, ℤ) from the computation of its Bredon homology. The results amount to the topological side of the Baum-Connes conjecture, K_{i}^{G}(\underline{E}G). We state the results here.

Theorem 1. The Bredon homology of \underline{E}SL(3, ℤ) with coefficients in the representation ring is

H_{i}^{𝔉𝔦𝔫} (\underline{E}SL(3, ℤ); R) = \{ \begin{matrix}ℤ^{⊕8} & i = 0 \\ 0 & i ≠ 0 \end{matrix} .

Corollary 2. The equivariant K-homology of \underline{E}G for G = SL(3, ℤ) is

K_{0}^{G} (\underline{E}G) = ℤ^{⊕8} , K_{1}^{G} (\underline{E}G) = 0 .

The results for GL(3, ℤ) follow from a Künneth formula for Bredon homology since GL(3, ℤ) = SL(3, ℤ) × ℤ/2ℤ.

Theorem 3. The Bredon homology of \underline{E}GL(3, ℤ) with coefficients in the representation ring is

H_{i}^{𝔉𝔦𝔫} (\underline{E}GL(3, ℤ); R) = \{ \begin{matrix}ℤ^{⊕16} & i = 0 \\ 0 & i ≠ 0 \end{matrix} .

Corollary 4. The equivariant K-homology of \underline{E}G for G = GL(3, ℤ) is

K_{0}^{G} (\underline{E}G) = ℤ^{⊕16} , K_{1}^{G} (\underline{E}G) = 0 .

We start with a brief review on classifying spaces, Bredon homology and equivariant K-homology; then we describe a model of \underline{E}SL(3, ℤ); in the last section we compute its Bredon homology and prove the theorems above. These results are part of the author’s PhD thesis [14, Chapter 4].

I would like to thank my PhD supervisor Ian Leary for suggesting this problem and support throughout the work. Thanks also to Christine Vespa and Samuel Wüthrich for useful comments on a previous version of this paper.

2 Preliminaries

2.1 Classifying space for proper actions

Let G be a discrete group. A G-CW-complex is a CW-complex with a Gaction permuting the cells and such that if a cell is sent to itself, it is done by the identity map. We call the G-action proper if all cell stabilizers are finite subgroups of G.

Definition 1. A model for \underline{E}G is a proper G-CW-complex X such that for any proper G-CW-complex Y there is a unique G-map Y → X, up to G-homotopy equivalence.

One can prove that a proper G-CW-complex X is a model of \underline{E}G if and only if the subcomplex of fixed points X^{H} is contractible for each H ≤ G finite. It can be shown that classifying spaces proper actions always exists. They are clearly unique up to G-homotopy equivalence. We write \underline{B}G for the quotient \underline{E}G/G. Note that for free actions instead of proper, we recover the definition of EG, whose quotient BG is the classifying space for principal G-bundles. See [1, §2] or [11] for details and more information on classifying spaces.

2.2 Bredon (co)homology

Given a group G and a family 𝔉 of subgroups, we will write O_{𝔉}G for the orbit category. The objects are left cosets G/K, K ∈ 𝔉, and morphisms the Gmaps ϕ : G/K → G/L. Such a G-map is uniquely determined by its image ϕ(K) = gL, and we have g^{−1}Kg ⊂ L. Conversely, such g ∈ G defines a G-map.

A left (resp. right) Bredon module is a covariant (resp. contravariant) functor from O_{𝔉}G to the category of abelian groups. Bredon modules form a category, which is abelian, and we can use homological algebra to define Bredon homology (see [12, pp. 7-10]). Nevertheless, we give now a practical definition.

Consider a G-CW-complex X, a family 𝔉 of subgroups of G containing all cell stabilizers, and a left Bredon module M. The Bredon homology groups H_{i}^{𝔉} (X; M) are obtained as the homology of the following chain complex (C_{∗}, ∂_{∗}). Let {e_{α}} be orbit representatives of the d-cells (d ≥ 0) and write S_{α} for stab(e_{α}) ∈ 𝔉. Define

C_{d} = ⨁_{α} M (G/S_{α}) .

If ge′ is a typical (d−1)-cell in the boundary of e_{α} then g^{−1}stab(e_{α})g ⊂ stab(e′), giving a G-map (write S′ for stab(e′))

ϕ : G/S_{α} → G/S′ ,

which induces a homomorphism M(ϕ): M (G/S_{α}) → M (G/S′). This yields a differential ∂_{d}: C_{d} → C_{d−1}, and the Bredon homology groups H_{i}^{𝔉} (X; M) correspond to the homology of (C_{∗}, ∂_{∗}). Bredon cohomology is defined analogously, for M a right Bredon module.

2.3 Proper actions and the representation ring

We are interested in the case X = \underline{E}G, 𝔉 = 𝔉𝔦𝔫(G) —the family of all finite subgroups of G— and M = R the complex representation ring, considered as a Bredon module as follows. On objects we set

R(G/K) = R_{ℂ}(K), K ∈ 𝔉𝔦𝔫(G)

the ring of complex representations of the finite group K (viewed just as an abelian group). For a G-map ϕ : G/K → G/L we have g^{−1}Kg ⊂ L for some g ∈ G so define R(ϕ) : R_{ℂ}(K) → R_{ℂ}(L) as induction from g^{−1}Kg into L once we identify R_{ℂ}(g^{−1}Kg) with R_{ℂ}(K).

We state two useful results about the zero degree and the higher degree Bredon homology groups H_{i}^{𝔉𝔦𝔫} (\underline{E}G; R), and a Künneth formula.

Proposition 5. Let G be group and denote by FC(G) the set of conjugacy classes of elements of finite order in G. Then there is an isomorphism

H_{0}^{𝔉𝔦𝔫} (\underline{E}G; R) ⊗_{ℤ} ℂ ≅ ℂ[FC(G)] .

Proof. See Definition 3.18 and Theorem 3.19 in Mislin’s notes [12]. ♢

Consequently, the rank of the zeroth Bredon homology group above coincides with the number of conjugacy classes of elements of finite order in G.

Define the singular set X^{sing} of a G-set X as the subspace of the points with non-trivial stabilizers. The following is Lemma 3.21 in Mislin’s notes [12].

Proposition 6. Let G be an arbitrary group. Then there is a natural map

H_{i}^{𝔉𝔦𝔫} (\underline{E}G; R) ⟶ H_{i}(\underline{B}G; ℤ) ,

which is an isomorphism in dimensions i > dim (\underline{E}G^{sing}) + 1 and injective in dimension i = dim (\underline{E}G^{sing}) + 1.

There is a Künneth formula for the direct product of two groups. Given a group G, define H_{i}^{𝔉𝔦𝔫} (G; R) as H_{i}^{𝔉𝔦𝔫(G)}(X; R^{G}), where X is any model of \underline{E}G and R^{G} is the representation ring as a Bredon module over O_{𝔉𝔦𝔫(G)}G (cf. [12]).

Proposition 7. Let G and H be two groups. For every n ≥ 0 there is a split exact sequence

0 → ⨁_{i+j=n} (H_{i}^{𝔉𝔦𝔫}(G; R) ⊗ H_{j}^{𝔉𝔦𝔫}(H; R)) → H_{n}^{𝔉𝔦𝔫}(G×H; R) →

⨁_{i+j=n−1} Tor(H_{i}^{𝔉𝔦𝔫}(G; R), H_{j}^{𝔉𝔦𝔫}(H; R)) → 0 .

Proof. Details can be found in [13, §4] (see also [5]). ♢

2.4 Equivariant K-homology

There is an equivariant version of K-homology, denoted K_{i}^{G}(−) and defined in [3] (see also [9]) using spaces and spectra over the orbit category of G. It was originally defined in [1] using Kasparov’s KK-theory. We will only recall the properties we need.

Equivariant K-homology satisfies Bott mod-2 periodicity, so we only consider K_{0}^{G} and K_{1}^{G}. For any subgroup H of G we have

K_{i}^{G} (G/H) = K_{i} (C_{r}^{∗}(H)) ,

that is, its value at one-orbit spaces corresponds to the K-theory of the reduced C^{∗}-algebra of the typical stabilizer. If H is a finite subgroup then C_{r}^{∗}(H) = ℂH and

K_{i}^{G} (G/H) = K_{i}(ℂH) = \{ \begin{matrix}R_{ℂ}(H) & i = 0 , \\ 0 & i = 1 . \end{matrix}

This allows us to view K_{i}^{G}(−) as a Bredon module over O_{𝔉𝔦𝔫}G.

We can use an equivariant Atiyah-Hirzebruch spectral sequence to compute the K^{G}-homology of a proper G-CW-complex X from its Bredon homology (see [12, pp. 49-50] for details), as

E_{p,q}^{2} = H_{p}^{𝔉𝔦𝔫} (X; K_{q}^{G}(−)) ⇒ K_{p+q}^{G} (X) .

In the simple case when Bredon homology concentrates at low degree we deduce the following fact.

Proposition 8. Write H_{i} = H_{i}^{𝔉𝔦𝔫}(X; R) and K_{i} = K_{i}^{G}(X). If H_{i} = 0 for i ≥ 2 then K_{0} = H_{0} and K_{1} = H_{1}.

3 A model for \underline{E} SL(3, ℤ)

3.1 The symmetric space

We describe a first model of the classifying space \underline{E}SL(n, ℤ), for any n ≥ 2 (cf. [2, pp. 38-40]). Let Q(n) be the space of real, symmetric, positive definite n×n matrices (equivalently, positive definite quadratic forms on ℝ^{n}). Multiplication by positive scalars gives an action whose quotient X(n) = Q(n)/ℝ^{+} is called the symmetric space. Since the right action of GL(n, ℝ) on Q(n) given by A · g = g^{t}Ag is transitive, with typical stabilizer O(n), we can identify Q(n) with GL(n, ℝ)/O(n). On the other hand, the inclusion of SL(n, ℝ)/SO(n) into GL(n, ℝ)/O(n) is an SL(3, ℤ)-equivariant homotopy equivalence, also up to multiplication by positive scalars on the codomain. Therefore, the symmetric space X(n) is SL(3, ℤ)-equivariant homotopy equivalent to SL(n, ℝ)/SO(n).

The action of GL(n, ℝ) on Q(n) induces and action on X(n), which restricts to a SL(n, ℤ)-action. As a SL(n, ℤ)-space, X(n) has finite stabilizers, since SL(n, ℤ) is a discrete subgroup of GL(n, ℝ) and O(n) is compact. Moreover, we have the following.

Proposition 9. The symmetric space X(n) ≃ SL(n, ℝ)/SO(n) is a model of \underline{E}SL(n, ℤ), of dimension n(n + 1)/2 − 1.

Proof. The space Q(n) is clearly a (convex) cone, so contractible. Since the action is linear, the fixed point subspace Q(n)^{H}, H ≤ SL(n, ℤ), is also a cone. It is not empty whenever H is a finite subgroup; take for instance the ‘average point’ of the orbit of any A ∈ Q(n),

\frac{1}{|H|} ∑_{h∈H} A · h ∈ Q(n)^{H} .

Note that A ∈ Q(n)^{H} if and only if its class [A] ∈ X(n)^{H}, so that X(n)^{H} is neither empty if H if finite. Moreover, if we fix a representative of each class (for instance, choose a matrix norm ∥ · ∥ and define A_{u} = A/∥A∥), then there is a contracting homotopy H([B], t) = [tA_{u} + (1 − t)B_{u}] for any fixed [A] ∈ X^{H}. ♢

Remark 10. More generally, if Γ is a discrete subgroup of a Lie group G with finitely many connected components, take K a maximal compact subgroup of G. Then G/K is a model for \underline{E}G and, therefore, a model for \underline{E}Γ [1].

3.2 Deformation retractions

There is a better model of \underline{E}SL(n, ℤ), obtained as an SL(n, ℤ)-equivariant deformation retract of the symmetric space X(n), of dimension n(n−1)/2. It is obtained via reduction theory of quadratic forms (see [2, pp. 213-17]). However, finding an explicit cellular decomposition and describing the stabilizers is in general quite laborious. For SL(3, ℤ) this has been done by Soulé in [15] and also by Henn in [6]. We now describe (without proofs) the deformation retract and orbit space for n = 3, following Soulé.

Denote the elements of Q = Q(3), respectively of X = X(3), as

A = (a_{ij}) ∈ Q , [A] = {λA | λ ∈ ℝ^{+}} ∈ X .

Recall the right action of SL(3, ℤ) on Q, A · g = g^{t}Ag, which extends to an action on X. Each orbit in X has a total preorder as follows. Given A = B · g we say that [A] < [B] if the sequence of diagonal elements of A is smaller than the one of B with respect to the lexicographic order in ℝ^{3}. (This is well-defined : if λA = B ·g′ then λB = B ·(g′g^{−1}) so λ = 1 and g = g′.) An element [A] ∈ X is called reduced if it is minimal in its orbit. Define the subspace

Y = {[A] reduced and a_{11} = a_{22} = a_{33}} .

Proposition 11. The space Y is an SL(3, ℤ)-deformation retract of X and therefore a model of \underline{E}SL(3, ℤ), of dimension 3.

Proof. The result follows from Theorem 1 in [15]. ♢

Remark 12. The minimal dimension for a model of \underline{E}SL(3, ℤ) is actually three. In general, one can prove that the strict upper triangular group in SL(n, ℤ) has cohomological dimension n(n − 1)/2, so any model of \underline{E}SL(n, ℤ) has dimension at least that; see [2, Chp. VIII].

3.3 Description of the orbit space

In this section we give an equivariant cellular decomposition and describe the stabilizers for the orbit space SL(3, ℤ)\Y = \underline{B}SL(3, ℤ). We follow the approach and notation in Soulé’s paper [15], although an equivalent and detailed description of this model can be found in Henn’s [6].

Let C be the truncated cube of ℝ^{3} with centre (0, 0, 0) and side length 2, truncated at the vertices (1, 1, −1), (1, −1, 1), (−1, 1, 1,) and (−1, −1, −1), through the mid-points of the corresponding sides (Figure 1). An element in Y can be uniquely written as the class of a matrix

A = ( \begin{matrix}2 & z & y \\ z & 2 & x \\ y & x & 2 \end{matrix} ) = A(x, y, z) ,

so that we can identify [A] with the point (x, y, z) ∈ ℝ^{3}. It can be shown that A(x, y, z) is reduced if and only if (x, y, z) ∈ C. Let E be the subspace of C given by the points (x, y, z) satisfying

\begin{matrix}|z| ≤ y ≤ x ≤ 1 \\ z − x − y + 2 ≥ 0 . \end{matrix}

Figure 1: Truncated cube C

We can give an explicit triangulation of E as shown in Figure 2. The vertices are

O =  (0, 0, 0)  Q =  (1, 0, 0)
M =  (1, 1, 1)  N = (1, 1, 1/2)
M′ =  (1, 1, 0)  N′ = (1, 1/2, −1/2)
P = (2/3, 2/3, −2/3) .

Note that the elements of SL(3, ℤ)

q_{1} = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & −1 \end{matrix}) and q_{2} = (\begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & −1 \end{matrix})

send M, N, Q to M′, N′, Q and N, N′, M′, Q to N′, N, M′, Q respectively. Consequently, we must identify the following simplices in the quotient

\begin{matrix}M ≡ M′ , N ≡ N′ , QM ≡ QM′ , QN ≡ QN′ , MN ≡ M′N ≡ M′N′ \\ and QMN ≡ QM′N ≡ QM′N′ . \end{matrix}

This identifications correspond to folding over the triangles QMN, QNM′ and QM′N′ along the edges QN and QM′ respectively.

Figure 2: Triangulation of the fundamental domain E

Vertices  2-cells
v_{1} O  g_{2}, g_{3}  S_{4}  t_{1} OQM  g_{2}  C_{2}
v_{2} Q  g_{4}, g_{5}  D_{6}  t_{2} QM′N  g_{1}  {1}
v_{3} M  g_{6}, g_{7}  S_{4}  t_{3} M′N′P  g_{12}, g_{14} C_{2}×C_{2}
v_{4} N  g_{6}, g_{8}  D_{4}  t_{4} OQN′P g_{5}  C_{2}
v_{5} P  g_{5}, g_{9}  S_{4}  t_{5} OMM′P g_{6}  C_{2}
Edges  3-cell
e_{1} OQ g_{2}, g_{5} C_{2}×C_{2} T_{1} g_{1} {1}
e_{2} OM g_{6}, g_{10}  D_{3}
e_{3} OP g_{6}, g_{5}  D_{3}
e_{4} QM g_{2}  C_{2}
e_{5} QN′ g_{5}  C_{2}
e_{6} MN g_{6}, g_{11} C_{2}×C_{2}
e_{7} M′P g_{6}, g_{12}  D_{4}
e_{8} N′P g_{5}, g_{13}  D_{4}

Table 1: Orbits of cells

Theorem 13. The space E with the identifications above is a 3-dimensional model of \underline{B} SL(3, ℤ).

Proof. See Theorem 2 in [15] or Theorem 2.4 in [6]. ♢

We now describe the orbits of cells and corresponding stabilizers. This can be found in Theorem 2 of Soulé’s article (although we use a cellular structure instead of a simplicial one) or in Section 2.5 of Henn’s work. We have changed the chosen generators so that they agree with the presentations on page 10. We summarize the information on Table 1.

The first column is an enumeration of equivalence classes of cells; the second lists a representative of each class; the third column gives generating elements for the stabilizer of the given representative; and the last one is the isomorphism type of the stabilizer. We use the following notation: {1} denotes the trivial group, C_{n} the cyclic group of n elements, D_{n} the dihedral group with 2n elements and S_{n} the symetric group of permutations on n objects. The generating elements referred to above are

g_{1} = (\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) g_{2} = (\begin{matrix}−1 & 0 & 0 \\ 0 & 0 & −1 \\ 0 & −1 & 0 \end{matrix}) g_{3} = (\begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ −1 & 0 & 0 \end{matrix})

g_{4} = (\begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & −1 \end{matrix}) g_{5} = (\begin{matrix}−1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}) g_{6} = (\begin{matrix}0 & −1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & −1 \end{matrix})

g_{7} = (\begin{matrix}0 & 0 & −1 \\ −1 & 0 & 0 \\ 1 & 1 & 1 \end{matrix}) g_{8} = (\begin{matrix}−1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & −1 & −1 \end{matrix}) g_{9} = (\begin{matrix}0 & 0 & −1 \\ −1 & 0 & −1 \\ 0 & 1 & 1 \end{matrix})

\begin{matrix}g_{10} = (\begin{matrix}0 & 0 & −1 \\ 0 & −1 & 0 \\ −1 & 0 & 0 \end{matrix}) & g_{11} = (\begin{matrix}−1 & 0 & 0 \\ 0 & −1 & 0 \\ 1 & 1 & 1 \end{matrix}) & g_{12} = (\begin{matrix}0 & −1 & −1 \\ 0 & −1 & 0 \\ −1 & 1 & 0 \end{matrix}) \\ g_{13} = (\begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & −1 \end{matrix}) & g_{14} = (\begin{matrix}−1 & 0 & 0 \\ −1 & 0 & −1 \\ 1 & −1 & 0 \end{matrix}) . \end{matrix}

Finally, the cell boundaries can be easily read from Figure 2, once we fix an orientation; namely, the ordering of the vertices O < Q < M < M′ < N < N′ < P induces an orientation in E and also in \underline{B}SL(3, ℤ) = E/ ∼. Thus, the boundaries with respect to the orbit representatives are (recall the identifications given by q_{1} and q_{2})

∂e_{1} = v_{2} − v_{1}  ∂e_{2} = v_{3} − v_{1} ∂e_{3} = v_{5} − v_{1}  ∂e_{4} = v_{3} − v_{2}
∂e_{5} = v_{4} · q_{2} − v_{2} ∂e_{6} = v_{4} − v_{3} ∂e_{7} = v_{5} − v_{3} · q_{1} ∂e_{8} = v_{5} − v_{4} · q_{2}
∂t_{1} = e_{1} − e_{2} + e_{4}  ∂t_{2} = e_{4} · q_{1} − e_{5} · q_{2} + e_{6} · q_{1}q_{2}
∂t_{3} = e_{6} · q_{1} − e_{7} + e_{8}  ∂t_{4} = e_{1} − e_{3} + e_{5} + e_{8}
∂t_{5} = e_{2} − e_{3} + e_{6} − e_{6} · q_{1}q_{2} + e_{7} ∂T_{1} = −t_{1} + t_{2} − t_{3} + t_{4} − t_{5} .

We now have all the information needed to compute the corresponding Bredon homology, which is carried out in the next section.

4 Bredon homology of \underline{E}SL(3, ℤ)

The Bredon chain complex associated to Y = \underline{E}SL(3, ℤ) is

0 ⟶ R_{ℂ} (stab(T_{1})) ⟶^{Ψ_{3}} ⨁_{i=1}^{5} R_{ℂ} (stab(t_{i})) ⟶^{Ψ_{2}}

⨁_{j=1}^{8} R_{ℂ} (stab(e_{j})) ⟶^{Ψ_{1}} ⨁_{k=1}^{5} R_{ℂ} (stab(v_{k})) ⟶ 0 ,  (4.1)

where the Ψ_{i}’s are given by induction among representation rings: if the boundary of a d-cell e^{d} is, in terms of orbit representatives of (d − 1)-cells,

∂e^{d} = ∑_{j=1}^{n} e_{j}^{d−1} · g_{j}

and τ ∈ R_{ℂ}(stab(e^{d})), then Ψ_{d}(τ) = τ ↑ stab(e_{j}^{d−1}) , where ↑ represents induction with respect to g · stab(e^{d}) · g^{−1} ⊂ stab(e_{j}^{d−1}). Note that we write ρ ↑ H for induction into a supergroup H and ρ ↓ H for restriction into a subgroup (we omit the group when it is clear from the context).

To compute Ψ_{d}(τ), we use two basic facts:

• any representation (or character) can be uniquely written as a sum of irreducible ones τ = n_{1}ρ_{1} + … + n_{s}ρ_{s}, with n_{i} = (τ|ρ_{i}) and (·|·) the usual scalar product of characters;

• Frobenius reciprocity: (τ ↑ |ρ)_{H} = (τ|ρ ↓)_{H′} where H′ ≤ H.

Firstly, we note down the character tables of the groups appearing as cell stabilizers. We write ⟨ g ⟩ for the group generated by an element g, and ⟨ gens | rels ⟩ for a presentation of a group.

Trivial group {1} = ⟨ g_{1} ⟩

\begin{matrix}\{1\} & g_{1} \\ τ & 1 \end{matrix}

Cyclic group C_{2} = ⟨ g_{i} ⟩

\begin{matrix}C_{2} & 1 & g_{i} \\ ρ_{1} & 1 & 1 \\ ρ_{2} & 1 & −1 \end{matrix}

Dihedral group D_{n} = ⟨ g_{i}, g_{j} ⟩ = ⟨ g_{i}, g_{j} | (g_{i})^{2} = (g_{j})^{2} = (g_{i}g_{j})^{n} = 1 ⟩

\begin{matrix}D_{n} & (g_{i}g_{j})^{k} & g_{j}(g_{i}g_{j})^{k} \\ χ_{1} & 1 & 1 \\ χ_{2} & 1 & −1 \\ χ̂_{3} & (−1)^{k} & (−1)^{k} \\ χ̂_{4} & (−1)^{k} & (−1)^{k+1} \\ ϕ_{p} & 2 cos (2πpk/n) & 0 \end{matrix}

where 0 ≤ k ≤ n − 1, p varies from 1 to n/2 − 1 (n even) or (n − 1)/2 (n odd) and the hat denotes a character which only appears when n is even. Note that C_{2} × C_{2} ≅ D_{2}.

Symmetric group S_{4} = ⟨ g_{i}, g_{j} ⟩ where g_{i} is a transposition and g_{j} a cycle of length 4. The character table, in cycle type notation, is

\begin{matrix}S_{4} & 1 & (12) & (123) & (1234) & (12)(34) \\ π_{1} & 1 & 1 & 1 & 1 & 1 \\ π_{2} & 1 & −1 & 1 & −1 & 1 \\ π_{3} & 2 & 0 & −1 & 0 & 2 \\ π_{4} & 3 & 1 & 0 & −1 & −1 \\ π_{5} & 3 & −1 & 0 & 1 & −1 \end{matrix}

In the next sections we will need to agree on an ordering of the conjugacy classes for each cell stabilizer. The same applies for the irreducible characters. We fix both orderings as shown in the character tables above. Note that for a dihedral group the arranging of conjugacy classes (in terms of representatives) is

\begin{matrix}n odd: & 1, g_{i}g_{j}, (g_{i}g_{j})^{2}, …, (g_{i}g_{j})^{(n−1)/2}, g_{j} \\ n even: & 1, g_{i}g_{j}, (g_{i}g_{j})^{2}, …, (g_{i}g_{j})^{n/2}, g_{j} , g_{j}(g_{i}g_{j}). \end{matrix}

Considering the ranks of the corresponding representation rings, that is, the number of irreducible characters, we can view the Bredon chain complex (4.1) as

\xymatrix{0 \ar[r] & ℤ \ar[r]^{Ψ_{3}} & ℤ^{⊕11} \ar[r]^{Ψ_{2}} & ℤ^{⊕28} \ar[r]^{Ψ_{1}} & ℤ^{⊕26} \ar[r] & 0 .}  (4.2)

4.1 Computation of Ψ_{3}

Denote by ∂e the boundary of a d-cell e in terms of (d − 1)-cells. Let τ be the trivial representation of stab(T_{1}) = {1}. We have

∂T_{1} = ∑_{i=1}^{5}(−1)^{i}t_{i} ⇒ Ψ_{3}(τ) = ∑_{i=1}^{5}(−1)^{i} τ ↑ stab(t_{i}) .

Inducing the trivial representation gives the regular representation ρ_{1} + … + ρ_{s} so Ψ_{3} is, written as a matrix of an homomorphism of free abelian groups,

Ψ_{3}(1) = ( −1 −1 1 −1 −1 −1 −1 1 1 −1 −1 ) .

Here each group of ±1s indicates the elements corresponding to each representation ring R_{ℂ}(stab(t_{i})). This matrix reduces by elementary operations to

Ψ_{3} ≡ (1 0 ⋯ 0) .

In particular, H_{3} = ker Ψ_{3} = 0 and im Ψ_{3} ≅ ℤ.

4.2 Computation of Ψ_{2}

For each 2-cell, we work out the induction map for each inclusion (possibly after conjugation) of stabilizers. For t_{1}, we have ∂t_{1} = e_{1} − e_{2} + e_{4} so stab(t_{1}) ⊂ stab(e_{i}) for i = 1, 2, 4. The first inclusion is

C_{2} = stab(t_{1}) = ⟨g_{2}⟩ ⊂ ⟨g_{2}, g_{5}⟩ = stab(e_{1}) = C_{2} × C_{2} = D_{2} .

Consider the irreducible characters ρ_{1}, ρ_{2} in R_{ℂ}(stab(t_{1})) and χ_{1}, …, χ_{4} in R_{ℂ}(stab(e_{1})) as on the character tables on page 10. Then

1 g_{2} (ρ_{1} | χ_{j} ↓) (ρ_{2} | χ_{j} ↓)
χ_{1} ↓ 1 1  1  0
χ_{2} ↓ 1 −1  0  1
χ_{3} ↓ 1 −1  0  1
χ_{4} ↓ 1 1  1  0

Therefore,

\begin{aligned}ρ_{1} ↑ & = & χ_{1} + χ_{4} \\ ρ_{2} ↑ & = & χ_{2} + χ_{3} . \end{aligned}

The corresponding submatrix representing R_{ℂ}(stab(t_{1})) ⟶^{Ind} R_{ℂ}(stab(e_{1})) is

χ_{1} χ_{2} χ_{3} χ_{4}
ρ_{1} ↑ 1 0 0 1
ρ_{2} ↑ 0 1 1 0

For the inclusion of stab(t_{1}) = ⟨g_{2}⟩ into stab(e_{2}) = ⟨g_{6}, g_{10}⟩, we have g_{2} = g_{10}g_{6}g_{10}, and

1 g_{10}(g_{6}g_{10} (ρ_{1} | χ_{j} ↓) (ρ_{2} | χ_{j} ↓)
χ_{1} ↓ 1  1  1  0
χ_{2} ↓ 1  −1  0  1
ϕ_{1} ↓ 2  0  1  1

Consequently,

\begin{aligned}ρ_{1} ↑ & = & χ_{1} + ϕ_{1} \\ ρ_{2} ↑ & = & χ_{3} + ϕ_{1} . \end{aligned}

The corresponding submatrix is (note the minus sign, since the cell appears as −e_{2} in ∂t_{1})

e_{2}
t_{1} −1 0 −1
0 −1 −1

Here we have simplified the notation; a matrix with top label e and left label t will represent the coefficients of the induced characters of R_{ℂ} (stab(t)) into R_{ℂ} (stab(e)), possibly after conjugation.

Now, stab(t_{1}) = stab(e_{4}), so induction gives the identity map. In brief,

e_{1}  e_{2}  e_{4}
t_{1} 1 0 0 1 −1 0 −1 1 0
0 1 1 0 0 −1 −1 0 1

The process is analogous for the other 2-cells. All the inclusions among stabilizers are immediate except the ones listed below.

stab(t_{3}) ⊂ stab(e_{6} · q_{1}),  stab(e_{7}),  stab(e_{8})
g_{12} = q_{1}^{−1}g_{6}q_{1}  = g_{12}  = g_{13}(g_{5}g_{13})^{2}
g_{14} = q_{1}^{−1}g_{11}q_{1}  = g_{12}(g_{6}g_{12})^{2} = (g_{5}g_{13})^{2} .
stab(t_{5}) ⊂ stab (e_{6} · q_{1}q_{2})
g_{6} = (q_{1}q_{2})^{−1}g_{6}g_{11}(q_{1}q_{2}) .

We now show the results, which can be easily verified by a careful reader.

\begin{aligned}& & e_{4} & & & e_{5} & & & & e_{6} \\ t_{2} & 1 & & 1 & −1 & & −1 & 1 & 1 & & 1 & 1 \end{aligned}

e_{6}  e_{7}  e_{8}
t_{3} 1 0 0 0 −1 0 −1 0 0 1 0 1 0 0
0 1 0 0 0 −1 0 −1 0 0 0 0 0 1
0 0 1 0 0 0 0 0 −1 0 1 0 1 0
0 0 0 1 0 0 0 0 −1 0 0 0 0 1
e_{1}  e_{3}  e_{5}  e_{8}
t_{4} 1 0 1 0 −1 0 −1 1 0 1 0 0 1 1
0 1 0 1 0 −1 −1 0 1 0 1 1 0 1
e_{2}  e_{3}  e_{6}  e_{7}
t_{5} 1 0 1 −1 0 −1 0 −1 0 1 1 0 0 1 1
0 1 1 0 −1 −1 0 1 0 −1 0 1 1 0 1

The submatrices above amount to an 11×28 matrix representing Ψ_{2}. This matrix can be reduced to its normal form consisting of the identity of size 10 and zeroes elsewhere,

Ψ_{2} ≡ ( \begin{matrix}Id_{10} & 0 \\ 0 & 0 \end{matrix} ) .

4.3 Computation of Ψ_{1}

The computations for Ψ_{1} are similar and straightforward. The relevant inclusions among stabilizers are the following. We give a conjugacy representative as (∼ g_{i}) when necessary.

stab(e_{1}) ⊂ stab(v_{1}),  stab(v_{2})
g_{2} = g_{2}  = g_{5}(g_{4}g_{5})^{3}
g_{5} = g_{3}g_{2}g_{3}^{−1}g_{2}g_{3} (∼ g_{2})
stab(e_{2}) ⊂ stab(v_{1}),  stab(v_{3})
g_{6} = g_{3}g_{2}g_{3}^{−1}  =
g_{10} = g_{2}g_{3}g_{2}g_{3}^{−1}g_{2} (∼ g_{2}) = g_{7}^{−1}g_{6}g_{7}
stab(e_{3}) ⊂ stab(v_{1}),  stab(v_{5})
g_{6} = g_{3}g_{2}g_{3}^{−1}  = g_{9}^{−1}g_{5}g_{9}
g_{5} = g_{3}g_{2}g_{3}^{−1}g_{2}g_{3} (∼ g_{2})
stab(e_{4}) ⊂ stab(v_{2}),  stab(v_{3})
g_{6} = g_{5}(g_{4}g_{5})^{3}  = g_{6}g_{7}^{2}g_{6}g_{7}^{−1} (∼ g_{6})
stab(e_{5}) ⊂ stab(v_{4} · q_{2})
g_{5} = q_{2}^{−1}g_{8}q_{2}
stab(e_{6}) ⊂ stab(v_{3}),  stab(v_{4})
g_{11} = g_{7}g_{6}g_{7}^{−1}g_{6}g_{7} (∼ g_{6}) = (g_{6}g_{8})^{2}
stab(e_{7}) ⊂ stab(v_{3} · q_{1}),  stab(v_{5})
g_{6} = q_{1}^{−1}(g_{6}g_{7}^{2}g_{6})q_{1} (∼ g_{6}) = g_{9}^{−1}g_{5}g_{9}
g_{12} = q_{1}^{−1}g_{6}q_{1}  = g_{9}^{2} (∼ g_{5})
stab(e_{8}) ⊂ stab(v_{4} · q_{2}),  stab(v_{5})
g_{5} = q_{2}^{−1}g_{8}q_{2}
g_{13} = q_{2}^{−1}g_{6}q_{2}  = g_{5}g_{9}^{2}g_{5} (∼ g_{5}) .

The matrices representing induction among stabilizers are the following.

v_{1}  v_{2}
e_{1} −1 0 −1 −1 0 1 0 0 0 0 1
0 −1 −1 0 −1 0 1 0 0 0 1
0 0 0 −1 −1 0 0 1 0 1 0
0 0 0 −1 −1 0 0 0 1 1 0
v_{1}  v_{3}
e_{2} −1 0 0 −1 0 1 0 0 1 0
0 −1 0 0 −1 0 1 0 0 1
0 0 −1 −1 −1 0 0 1 1 1
v_{1}  v_{5}
e_{3} −1 0 0 −1 0 1 0 0 1 0
0 −1 0 0 −1 0 1 0 0 1
0 0 −1 −1 −1 0 0 1 1 1
v_{2}  v_{3}
e_{4} −1 0 0 −1 −1 −1 1 0 1 2 1
0 −1 −1 0 −1 −1 0 1 1 1 2

v_{2}  v_{4}
e_{5} −1 0 −1 0 −1 −1 1 0 1 0 1
0 −1 0 −1 −1 −1 0 1 0 1 1
v_{3}  v_{4}
e_{6} −1 0 −1 −1 0 1 0 0 1 0
0 −1 −1 0 −1 0 0 0 0 1
0 0 0 −1 −1 0 1 1 0 0
0 0 0 −1 −1 0 0 0 0 1
v_{3}  v_{5}
e_{7} −1 0 −1 0 0 1 0 1 0 0
0 0 0 0 −1 0 0 0 0 1
0 0 0 −1 0 0 1 1 0 0
0 −1 −1 0 0 0 0 0 1 0
0 0 0 −1 −1 0 0 0 1 1
v_{4}  v_{5}
e_{8} −1 0 0 0 0 1 0 1 0 0
0 −1 0 0 0 0 0 0 0 1
0 0 0 −1 0 0 1 1 0 0
0 0 −1 0 0 0 0 0 1 0
0 0 0 0 −1 0 0 0 1 1

Altogether they form a 28×26 matrix whose normal form is

( \begin{matrix}Id_{18} & 0 \\ 0 & 0 \end{matrix} ) .

Remark 14. The computations have been verified with the help of a computer. In fact, the author has implemented a program in GAP [4], which computes the Bredon homology with coefficients in the representation ring of any finite proper G-CW-complex, from the cell stabilizers and boundaries. Details and the code for the algorithm can be found in [14, Appendix A].

We can now determine the Bredon homology of SL(3, ℤ) from the chain maps above. Note that if we have a exact sequence of free abelian groups

⋯ ⟶ n · ℤ ⟶^{f} m · ℤ ⟶^{g} k · ℤ ⟶ ⋯

with f and g represented by matrices A and B for some fixed basis, then the homology at ℤ^{⊕m} is

ker (g) / im (f) ≅ ℤ/d_{1}ℤ ⊕ … ℤ/d_{s}ℤ ⊕ ℤ^{⊕(m−s−r)} ,

where r = rank (B) and d_{1}, …, d_{s} are the elementary divisors of A. In our case, we obtain that the Bredon homology of \underline{E}SL(3, ℤ) is

H_{0}^{𝔉𝔦𝔫} (\underline{E}Γ; R) ≅ ℤ^{⊕8}
H_{i}^{𝔉𝔦𝔫} (\underline{E}Γ; R) = 0 ∀ i ≠ 0 .

Remark 15. These results agree with the Bredon homology expected at degree 0 and 3. The dimension of the singular part of our model of \underline{E}Γ is 2, and \underline{B}Γ is contractible, so we have H_{i} = 0 for all i ≥ 3, by Proposition 6. On the other hand, rank (H_{0}) = 8 must be the number of conjugacy classes of elements of finite order of SL(3, ℤ), by Proposition 5; the latter can be deduced, for instance, from the list of all finite subgroups (up to conjugacy) of SL(3, ℤ) in [16]. Also, 8 is the alternating sum of the ranks in the Bredon chain complex (4.2).

4.4 Equivariant K-homology

Since the Bredon homology concentrates at degree 0, it coincides with the K^{G}- homology (Proposition 8), that is,

K_{0}^{G} (\underline{E}Γ) = ℤ^{⊕8}
K_{1}^{G} (\underline{E}Γ) = 0 .

This amounts to the topological side of the Baum-Connes conjecture and injects into the analytical side, that is, the K-theory of C_{r}^{∗}(SL(3, ℤ)).

4.5 Results for GL(3, ℤ)

We have the direct product decomposition GL(3, ℤ) = SL(3, ℤ) × C_{2}. We can therefore use the Künneth formula for Bredon homology from Proposition 7. Since C_{2} is finite, a one-point space is a model for \underline{E}C_{2} and its Bredon homology is R_{ℂ}(C_{2}) ≅ ℤ^{⊕2} at degree 0 and vanishes elsewhere. Consequently,

\begin{aligned}H_{0}^{𝔉𝔦𝔫} (\underline{E} GL(3, ℤ); R) & = & H_{0}^{𝔉𝔦𝔫} (\underline{E}SL(3, ℤ); R) ⊗ H_{0}^{𝔉𝔦𝔫} (\underline{E}C_{2}; R) ≅ ℤ^{⊕16} \\ H_{i}^{𝔉𝔦𝔫} (\underline{E} GL(3, ℤ); R) & = & 0 , if i ≠ 0 . \end{aligned}

As before, these groups coincide with the equivariant K-homology:

K_{0}^{G} (\underline{E} GL(3, ℤ)) ≅ ℤ^{⊕16}
K_{1}^{G} (\underline{E} GL(3, ℤ)) = 0 .

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