A Survey on classifying

spaces for families

Wolfgang Lück^{∗}

Fachbereich Mathematik und Informatik

Westfälische Wilhelms-Universität

Münster Einsteinstr. 62 48149 Münster

Germany

lueck@math.uni-muenster.de http://www.math.unimuenster.de/u/lueck

June 2004

1. The G-CW -version

Group means always locally compact Hausdorff topological group with a countable base for its topology. Definition 1. (G-CW -complex) A G-CW - complex X is a G-space together with a G-invariant filtration

∅ = X_{−1} ⊆ X_{0} ⊆ … ⊆ X_{n} ⊆ … ⊆ _{n≥0}^{⋃} X_{n} = X

such that X carries the colimit topology with respect to this filtration (i.e. a set C ⊆ X is closed if and only if C ∩ X_{n} is closed in X_{n} for all n ≥ 0) and X_{n} is obtained from X_{n−1} for each n ≥ 0 by attaching equivariant n-dimensional cells, i.e. there exists a G-pushout

∐_{i∈I_{n}} G/H_{i} × S^{n−1} \underrightarrow{∐_{i∈In} q_{i}^{n}} X_{n−1}
↓  ↓
∐_{i∈I_{n}} G/H_{i} × D^{n} \overrightarrow{∐_{i∈In} Q_{i}^{n}} X_{n}

Remark 2. (Proper G-CW -complexes) A G-space X is called proper if for each pair of points x and y in X there are open neighborhoods V_{x} of x and W_{y} of y in X such that the closure of the subset {g ∈ G | gV_{x} ∩ W_{y} ≠ ∅} of G is compact. A G-CW -complex X is proper if and only if all its isotropy groups are compact. Definition 3. (Family of subgroups) A family F of subgroups of G is a set of (closed) subgroups of G which is closed under conjugation and finite intersections.

Examples for F are

TR  = {trivial subgroup};
FIN  = {finite subgroups};
VCYC  = {virtually cyclic subgroups};
COM  = {compact subgroups};
COMOP = {compact open subgroups};
ALL  = {all subgroups}.

Definition 4. (Classifying G-CW -complex for a family of subgroups) Let F be a family of subgroups of G. A model E_{F}(G) for the classifying G-CW -complex for the family F of subgroups is a G-CW -complex E_{F}(G) which has the following properties:

1. All isotropy groups of E_{F}(G) belong to F;

2. For any G-CW -complex Y, whose isotropy groups belong to F, there is up to Ghomotopy precisely one G-map Y → X.

We abbreviate EG := E_{COM}(G) and call it the universal G-CW -complex for proper G-actions. We also write EG = E_{TR}(G).

The notion of a classifying space for a family is due to tom Dieck.

Theorem 5. (Homotopy characterization of E_{F}(G)) Let F be a family of subgroups.

1. There exists a model for E_{F}(G) for any family F;

2. Two model for E_{F}(G) are G-homotopy equivalent;

3. A G-CW -complex X is a model for E_{F}(G) if and only if all its isotropy groups belong to F and for each H ∈ F the H-fixed point set X^{H} is weakly contractible.

2. The numerable version

Definition 6. (F-numerable G-space) A F-numerable G-space is a G-space, for which there exists an open covering {U_{i} | i ∈ I} by G-subspaces such that there is for each i ∈ I a G-map U_{i} → G/G_{i} for some G_{i} ∈ F and there is a locally finite partition of unity {e_{i} | i ∈ I} subordinate to {U_{i} | i ∈ I} by G-invariant functions e_{i}: X → [0, 1].

Notice that we do not demand that the isotropy groups of a F-numerable G-space belong to F. If f: X → Y is a G-map and Y is F-numerable, then X is also Fnumerable. Lemma 7. Let F be a family. Then a G-CW -complex is F-numerable if each isotropy group is a subgroup of some element in F.

Proof. This follows from the Slice Theorem and the fact that G\X is a CW - complex and hence paracompact.

Definition 8. (Classifying numerable Gspace for a family of subgroups) Let F be a family of subgroups of G. A model J_{F}(G) for the classifying numerable Gspace for the family F of subgroups is a G-space which has the following properties:

1. J_{F}(G) is F-numerable;

2. For any F-numerable G-space X there is up to G-homotopy precisely one Gmap X → J_{F}(G).

We abbreviate JG := J_{COM}(G) and call it the universal numerable G-space for proper G-actions, or briefly the universal space for proper G-actions. We also write JG = J_{TR}(G)

Remark 9. (Proper G-spaces) A COMnumerable G-space X is proper. Not every proper G-space is COM-numerable. But a G-CW -complex X is proper if and only if it is COM-numerable. Theorem 10. (Homotopy characterization of J_{F}(G)) Let F be a family of subgroups.

1. For any family F there exists a model for J_{F}(G) whose isotropy groups belong to F;

2. Two models for J_{F}(G) are G-homotopy equivalent;

3. For H ∈ F the H-fixed point set J_{F}(G)^{H} is contractible.

3. Comparision of the two versions

There is always a G-map

ϕ_{F}: E_{F}(G) → J_{F}(G)

which is unique up to G-homotopy. Example 11. Let G be totally disconnected. Then

ϕ_{TR}: EG → JG

is a G-homotopy equivalence if and only if G is discrete. Theorem 12. (EG = JG)) The G-map ϕ_{F} is a G-homotopy equivalence if F = COM, i.e. we get a G-homotopy equivalence

ϕ_{COM}: EG ⟶^{≃} JG.

Lemma 13. Let G be a totally disconnected group Then the following square commutes up to G-homotopy and consists of G-homotopy equivalences

E_{COMOP}(G) ⟶ J_{COMOP}(G)
↓  ↓
EG  ⟶  JG

4. Special models

There are interesting special models • Operator theoretic models;

• G/K for an almost connected group G with K ⊆ G maximal compact subgroup;

• Actions on CAT(0)-spaces;

• Actions on affine buildings;

• The Rips complex for word-hyperbolic groups;

• The Borel-Serre compactification and arithmetic groups; • Mapping class groups and Teichmüller

space;

• Out (F_{n}) and outer space.

4.1. Operator Theoretic Model

Let C_{0}(G) be the Banach space of complex valued functions of G vanishing at infinity with the supremum-norm. The group G acts isometrically on C_{0}(G) by (g ·f)(x) := f(g^{−1}x) for f ∈ C_{0}(G) and g, x ∈ G. Let P C_{0}(G) be the subspace of C_{0}(G) consisting of functions f such that f is not identically zero and has non-negative real numbers as values. Theorem 14. (Operator theoretic model) The G-space P C_{0}(G) is a model for JG. Example 15. Let G be discrete. Another model for JG is the space

X_{G} = {f: G → [0, 1] | f has finite support, _{g∈G}^{∑} f(g) = 1}

with the topology coming from the supremum norm.

Remark 16. (Simplicial Model) Let G be discrete. Let P_{∞}(G) be the geometric realization of the simplicial set whose ksimplices consist of (k+1)-tupels (g_{0}, g_{1}, …, g_{k}) of elements g_{i} in G. This also a model for EG.

The spaces X_{G} and P_{∞}(G) have the same underlying sets but in general they have different topologies. The identity map induces a (continuous) G-map P_{∞}(G) → X_{G} which is a G-homotopy equivalence, but in general not a G-homeomorphism

4.2. Almost Connected Groups

The following result is due to Abels. Theorem 17. Almost connected groups) Let G be a (locally compact Hausdorff) topological group. Suppose that G is almost connected, i.e. the group G/G^{0} is compact for G^{0} the component of the identity element. Then G contains a maximal compact subgroup K which is unique up to conjugation. The G-space G/K is a model for JG. Theorem 18. (Discrete subgroups of almost connected Lie groups) Let L be a Lie group with finitely many path components. Then L contains a maximal compact subgroup K which is unique up to conjugation. The L-space L/K is a model for EL.

If G ⊆ L is a discrete subgroup of L, then L/K with the obvious left G-action is a finite dimensional G-CW -model for EG.

4.3. Actions on CAT(0)-spaces

Theorem 19. (Actions on CAT(0)-spaces) Let G be a (locally compact Hausdorff) topological group. Let X be a proper G-CW -complex. Suppose that X has the structure of a complete CAT(0)-space for which G acts by isometries. Then X is a model for EG. Remark 20. This result contains as special case isometric G actions on simplyconnected complete Riemannian manifolds with non-positive sectional curvature and G-actions on trees.

4.4. Affine Buildings

Theorem 21. (Affine buildings) Let G be a totally disconnected group. Suppose that G acts on the affine building by simplicial automorphisms such that each isotropy group is compact. Then Σ is a model for both J_{COMOP}(G) and JG and the barycentric subdivision Σ′ is a model for both E_{COMOP}(G) and EG. Example 22 (Bruhat-Tits building). An important example is the case of a reductive p-adic algebraic group G and its associated affine Bruhat-Tits building β(G). Then β(G) is a model for JG and β(G)′ is a model for EG by Theorem 21.

4.5. The Rips Complex of a Word-Hyperbolic Group

The Rips complex P_{d}(G, S) of a group G with a symmetric finite set S of generators for a natural number d is the geometric realization of the simplicial set whose set of k-simplices consists of (k+1)-tuples (g_{0}, g_{1}, … g_{k}) of pairwise distinct elements g_{i} ∈ G satisfying d_{S}(g_{i}, g_{j}) ≤ d for all i, j ∈ {0, 1, …, k}. The obvious G-action by simplicial automorphisms on P_{d}(G, S) induces a G-action by simplicial automorphisms on the barycentric subdivision P_{d}(G, S)′ Theorem 23. (Rips complex) Let G be a (discrete) group with a finite symmetric set of generators. Suppose that (G, S) is δ-hyperbolic for the real number δ ≥ 0. Let d be a natural number with d ≥ 16δ + 8. Then the barycentric subdivision of the Rips complex P_{d}(G, S)′ is a finite G-CW - model for EG.

4.6. Arithmetic Groups

Arithmetic groups in a semisimple connected linear ℚ-algebraic group possess finite models for EG. Namely, let G(ℝ) be the ℝpoints of a semisimple ℚ-group G(ℚ) and let K ⊆ G(ℝ) a maximal compact subgroup. If A ⊆ G(ℚ) is an arithmetic group, then G(ℝ)/K with the left A-action is a model for E_{FIN} (A) as already explained in Theorem 18. The A-space G(ℝ)/K is not necessarily cocompact. Theorem 24. Borel-Serre compactification) The Borel-Serre completion of G(ℝ)/K is a finite A-CW -model for E_{FIN} (A).

4.7. Mapping Class groups

Let Γ_{g,r}^{s} be the mapping class group of an orientable compact surface F of genus g with s punctures and r boundary components. We will always assume that 2g + s + r > 2, or, equivalently, that the Euler characteristic of the punctured surface F is negative. It is well-known that the associated Teichmüller space T_{g,r}^{s} is a contractible space on which Γ_{g,r}^{s} acts properly. Actually Theorem 25. (Teichmüller space) The Γ_{g,r}^{s}-space T_{g,r}^{s} is a model for E_{FIN} (Γ_{g,r}^{s}).

4.8. Outer Automorphism Groups of Free groups

Let F_{n} be the free group of rank n. Denote by Out (F_{n}) the group of outer automorphisms of F_{n}, i.e. the quotient of the group of all automorphisms of F_{n} by the normal subgroup of inner automorphisms. Culler and Vogtmann have constructed a space X_{n} called outer space on which Out (F_{n}) acts with finite isotropy groups. It is analogous to the Teichmüller space of a surface with the action of the mapping class group of the surface.

The space X_{n} contains a spine K_{n} which is an Out (F_{n})-equivariant deformation retraction. This space K_{n} is a simplicial complex of dimension (2n − 3) on which the Out (F_{n})-action is by simplicial automorphisms and cocompact. Actually the group of simplicial automorphisms of K_{n} is Out (F_{n}) by results due to Bridson and Vogtman. Theorem 26. The barycentric subdivision K′_{n} is a finite (2n − 3)-dimensional model of E Out (F_{n}).

5. Relevance and Applications

of Classifying Spaces for Families

5.1. Baum-Connes Conjecture

The goal of the Baum-Connes Conjecture is the computation of the topological Ktheory K_{n}(C_{r}^{∗}(G)) of the reduced group C^{∗}-algebra of G. Conjecture 27 (Baum-Connes Conjecture). The assembly map defined by taking the equivariant index

asmb : K_{n}^{G}(JG) ⟶^{≅} K_{n}(C_{r}^{∗}(G))

is bijective for all n ∈ ℤ.

5.2. Farrell-Jones Conjecture

Let G be a discrete group. Let R be a associative ring with unit. The goal of the Farrell-Jones Conjecture is to compute the algebraic K-groups K_{n}(RH) and the algebraic L-groups L_{n}^{−∞}(RG). Conjecture 28 (Farrell-Jones Conjecture). The assembly maps induced by the projection E_{VCYC}(G) → G/G

asmb : H_{n}^{G}(E_{VCYC}(G), K) → K_{n}(RG);
asmb : H_{n}^{G}(E_{VCYC}(G), L^{−∞}) → L_{n}^{−∞}(RG),

are bijective for all n ∈ ℤ.

5.3. Completion Theorem

Let G be a discrete group. For a proper finite G-CW -complex let K_{G}^{∗}(X) be its equivariant K-theory defined in terms of equivariant finite dimensional complex vector bundles over X. Let I ⊆ K_{G}^{0}(EG) be the augmentation ideal, i.e. the kernel of the map K^{0}(EG) → ℤ sending the class of an equivariant complex vector bundle to its complex dimension. Let K_{G}^{∗}(EG)Î be the I-adic completion of K_{G}^{∗}(EG) and let K^{∗}(BG) be the topological K-theory of BG. Theorem 29 (Completion Theorem for discrete groups). Let G be a discrete group such that there exists a finite model for EG. Then there is a canonical isomorphism

K^{∗}(BG) ⟶^{≅} K_{G}^{∗}(EG)Î.

5.4. Classifying Spaces for Equivariant Bundles

The equivariant K-theory for finite proper G-CW -complexes appearing above can be extended to arbitrary proper G-CW -complexes (including the multiplicative structure) using Γ-spaces in the sense of Segal and involving classifying spaces for equivariant vector bundles. These classifying spaces for equivariant vector bundles are again classifying spaces of certain Lie groups and certain families

5.5. Equivariant Homology and Cohomology

Classifying spaces for families play a role in computations of equivariant homology and cohomology for compact Lie groups such as equivariant bordism. Rational computations of equivariant (co-)-homology groups are possible in general using Chern characters for discrete groups and proper G-CW - complexes

6. Finiteness Conditions

The questions whether there exists finite models, models of finite type or models or finite-dimensional models for EG or wthat is the minimal value of dim (EG) is quite interesting and an obvious extention of the same question for BG. Remark 30 (Algebraic criterion). Let G be discrete. In the classical case one can read off the possible dimension of BG from the homological algebra of ℤG, in particular in terms of the cohomological dimension of the trivial ℤG-modul ℤ. There are analogous results for E_{F}(G) if one considers modules over the orbit category Or(G), in particular the constant contravariant ℤOr(G)- module ℤ_{F} whose value is ℤ on G/H for H ∈ F and {0} on G/H for H ∉ F. This gives in principle a complete answer in algebraic terms but is often hard to apply in concrete situations.

6.1. Some conditions for finite-dimensional models

As an illustration we give a small selection of results on this topic to due to Connolly, Dunwoody, Kropholler, Kozniewsky, L., Leary, Meintrup, Mislin and Nucinkis and others. Theorem 31 (Discrete subgroups of Lie groups). Let L be a Lie group with finitely many path components. Let K ⊆ L be a maximal compact subgroup K. Let G ⊆ L be a discrete subgroup of L.

Then L/K with the left G-action is a model for EG.

Suppose additionally that G contains a torsionfree subgroup Δ ⊆ G of finite index. Then we have

vcd (G) ≤ dim (L/K)

and equality holds if and only if G\L is compact.

Theorem 32 (A criterion for 1-dimensional models). Let G be a discrete group. Then there exists a 1-dimensional model for EG if and only the cohomological dimension of G over the rationals ℚ is less or equal to one. Theorem 33. Virtual cohomological dimension and dim (EG) Let G be a discrete group which contains a torsionfree subgroup of finite index and has virtual cohomological dimensionvcd (G) ≤ d. Let l ≥ 0 be an integer such that the length l(H) of any finite subgroup H ⊂ G is bounded by l.

Then we have vcd (G) ≤ dim (EG) for any model for EG and there exists a model for EG of dimension max {3, d} + l.

Example 34 (Virtually poly-cyclic groups). Let the group Δ be virtually poly-cyclic, i.e. Δ contains a subgroup Δ′ of finite index for which there is a finite sequence {1} = Δ′_{0} ⊆ Δ′_{1} ⊆ … ⊆ Δ′_{n} = Δ′ of subgroups such that Δ′_{i−1} is normal in Δ′_{i} with cyclic quotient Δ′_{i}/Δ′_{i−1} for i = 1, 2, …, n. Denote by r the number of elements i ∈ {1, 2, …, n} with Δ′_{i}/Δ′_{i−1} ≅ ℤ. The number r is called the Hirsch rank. The group Δ contains a torsionfree subgroup of finite index. We call Δ′ poly-ℤ if r = n, i.e. all quotients Δ′_{i}/Δ′_{i−1} are infinite cyclic. Then

1. r = vcd (Δ);

2. r = max {i | H_{i}(Δ′; ℤ/2) ≠ 0} for one (and hence all) poly-ℤ subgroup Δ′ ⊂ Δ of finite index;

3. There exists a finite r-dimensional model for EΔ and for any model EΔ we have dim (EΔ) ≥ r.

6.2. Reduction to discrete groups

The discretization G_{d} of a topological group G is the same group but now with the discrete topology. Theorem 35 (Passage from topological groups to totally disconnected groups). Let G be a locally compact second countable Hausdorff group. Put \overline{G} := G/G^{0}. Then there is a G-CW -model for EG that is d-dimensional or finite or of finite type respectively if and only if E\overline{G} has a \overline{G}-CW - model that is d-dimensional or finite or of finite type respectively. Theorem 36 (Passage from totally disconnected groups to discrete groups). Let G be a locally compact totally disconnected Hausdorff group and let F be a family of subgroups of G. Then there is a G-CW -model for E_{F}(G) that is d-dimensional or finite or of finite type respectively if and only if there is a G_{d}-CW -model for E_{F}(G_{d}) that is d-dimensional or finite or of finite type respectively.

7. Counterexamples

The following problem is stated by Brown It created a lot of activities and many of the results stated above were motivated by it. Problem 37. For which discrete groups G, which contain a torsionfree subgroup of finite index and has virtual cohomological dimension ≤ d, does there exist a ddimensional G-CW -model for EG?

Leary and Nucinkis have constructed many very interesting examples of discrete groups some of which are listed below. Their main technical input is an equivariant version of the constructions due to Bestvina and Brady. These examples show that the answer to the Problems 37 and to other problems appearing in the literature is not positive in general. A group G is of type VF if it contains a subgroup H ⊆ G of finite index for which there is a finite model for BH.

1. For any positive integer d there exist a group G of type VF which has virtually cohomological dimension ≤ 3d, but for which any model for EG has dimension ≥ 4d;

2. There exists a group G with a finite cyclic subgroup H ⊆ G such that G is of type VF but the centralizer C_{G}H of H in G is not of type FP_{∞};

3. There exists a group G of type VF which contains infinitely many conjugacy classes of finite subgroups;

4. There exists an extension 1 → Δ → G → π → 1 such that EΔ and EG have finite G-CW -models but there is no G-CW -model for Eπ of finite type.

8. The Orbit Space of EG

We will see that in many computations of the group (co-)homology, of the algebraic K- and L-theory of the group ring or the topological K-theory of the reduced C^{∗}- algebra of a discrete group G a key problem is to determine the homotopy type of the quotient space G\EG of EG. The following result shows that this is a difficult problem in general and can only be solved in special cases where som extra geometric input is available. It was proved by Leary and Nucinkis based on ideas due to Baumslag-Dyer-Heller and Kan and Thurston. Theorem 38 (The homotopy type of G\EG). Let X be a connected CW -complex. Then there exists a group G such that G\EG is homotopy equivalent to X.