Contemporary Mathematics

On the Universal Space for Group Actions with Compact Isotropy

Wolfgang Lück and David Meintrup

Abstract. Let G be a locally compact topological group and \underline{E}G its universal space for the family of compact subgroups. We give criteria for this space to be G-homotopy equivalent to a d-dimensional G-CW -complex, a finite G-CW -complex or a G-CW -complex of finite type. Essentially we reduce these questions to discrete groups, and to the homological algebra of the orbit category of discrete groups with respect to certain families of subgroups.

Introduction

Throughout this paper we denote by G a locally compact topological group (where locally compact always includes Hausdorff). We denote by G_{0} the component of the identity element and by \overline{G} = G/G_{0} its component group. Notice that G_{0} is locally compact and connected and \overline{G} is locally compact and totally disconnected, i.e. each component consists of exactly one point. Subgroup will always mean closed subgroup. Sometimes we make the additional assumption

(S)  For any closed subgroup H ⊂ G the projection p : G  G/H has a local cross section, i.e. there is a neighborhood U of eH together with a map s : U → G satisfying p ∘ s = id_{U}.

Condition (S) is automatically satisfied if G is discrete, if G is a Lie group, or more generally, if G is locally compact and second countable and has finite covering dimension [15]. The metric needed in [15] follows under our assumptions, since a locally compact Hausdorff space is regular and regularity in a second countable space implies metrizability.

A family F consists of a set of (closed) subgroups of G with the property that for any H, K ∈ F and g ∈ G, the subgroups g^{−1}Hg and H ∩K belong to F. Notice that we do not require that F is closed under taking subgroups. A universal space of G for the family F is a G-CW -complex E(G, F) such that the fixed point set E(G, F)^{H} is weakly contractible for H ∈ F and all its isotropy groups belong to F. Recall that a map f : X → Y of spaces is a weak homotopy equivalence if and only if

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

2000 Mathematics Subject Classification. 55R35.

David Meintrup is supported by the Graduiertenkolleg “Algebraische Geometrie und Zahlentheorie” in Münster. Parts of this paper grew out of his Dissertation.

the induced map f_{∗} : π_{n}(X, x) → π_{n}(Y, f(x)) is an isomorphism for all base points x ∈ X and for all n ≥ 0 and that a space X is weakly contractible if and only if the projection X → {∗} onto the space consisting of one point is a weak homotopy equivalence. The G-CW -complex E(G, F) has the universal property that for any G-CW -complex X whose isotropy groups belong to F, there is up to G-homotopy precisely one G-map X E(G, F). In particular, E(G, F) is unique up to G-homotopy. The notion of a G-CW -complex can be found for instance in [10, Definition 1.2], the existence of E(G, F) follows for instance from [10, 2.2], and the universal property of E(G, F) is a consequence of the Equivariant Whitehead Theorem [10, Theorem 2.4]. If F is the family of compact subgroups COM, we will often abbreviate E(G, COM) by \underline{E}G. Notice that for a discrete group COM is the same as the family FIN of finite subgroups.

In Section 1 we will explain our notion of E(G, F) and compare it with the similar notion due to tom Dieck [4, section I.6]. We mention that these spaces E(G, F) and in particular \underline{E}G play an important role in the formulation of the Baum-Connes Conjecture [3, Conjecture 3.15 on p. 254], the Isomorphism Conjecture in algebraic K- and L-theory of Farrell and Jones [5], the generalization of the completion theorem of Atiyah and Segal for finite groups to infinite discrete groups [12] and in the construction of classifying spaces for equivariant bundles [4, Section I.8 and I.9]. More information about models for \underline{E}G can be found for instance in [3].

We call any G-CW -complex in the G-homotopy class of E(G, F) a G-CW - model for E(G, F). In this paper, we investigate the type of E(G, F), i.e. whether there is an m-dimensional G-CW -model, a finite G-CW -model or a G-CW -model of finite type for E(G, F). A G-CW -complex X is finite if it is built by finitely many equivariant cells or, equivalently, if G\X is compact. It is called of finite type if each skeleton X_{n} is finite. For discrete groups the type of \underline{E}G has been investigated in [9], [11] and [16]. In Section 2, we will give a necessary and sufficient algebraic criterion which not only applies to FIN but to any family F. Namely, in Section 2 we will explain and prove

Theorem 0.1. Let G be a discrete group and let d ≥ 3. Then we have:

(a) There is a d-dimensional G-CW -model for E(G, F) if and only if the constant ℤ Or(G, F)-module \underline{ℤ} has a d-dimensional projective resolution;

(b) There is a G-CW -model for E(G, F) of finite type if and only if E(G, F) has a G-CW -model with finite 2-skeleton and the constant ℤ Or(G, F)-module \underline{ℤ} has a projective resolution of finite type;

(c) There is a finite G-CW -model for E(G, F) if and only if E(G, F) has a G-CW -model with finite 2-skeleton and the constant ℤ Or(G, F)-module \underline{ℤ} has a finite free resolution over Or(G, F);

(d) There is a G-CW -model with finite 2-skeleton for \underline{E}G = E(G, FIN) if and only if there are only finitely many conjugacy classes of finite subgroups H ⊂ G and for any finite subgroup H ⊂ G its Weyl group WH := NH/H is finitely presented.

In Section 3 we will reduce the case of a totally disconnected group to the one of a discrete group, as summarized in Theorem 0.2. Throughout the paper we will denote the discretization of a topological group G by G_{d}, i.e. the same group but now with the discrete topology. Given a family F of (closed) subgroups of G, denote by F_{d} the same set of subgroups, but now in connection with G_{d}. Notice that F_{d} is again a family.

Theorem 0.2. Let G be a locally compact totally disconnected group and let F be a family of subgroups of G. Then there is a G-CW -model for E(G, F) that is ddimensional (resp. finite, resp. of finite type) if and only if there is a G_{d}-CW -model for E(G_{d}, F_{d}) that is d-dimensional (resp. finite, resp. of finite type).

The case of an almost connected group G, i.e. \overline{G} is compact, has already been treated by Abels [2, Corollary 4.14]. Namely, for an almost connected (locally compact) group G there is a model for \underline{E}G consisting of one equivariant cell G/K. Notice that K is then necessarily a maximal compact subgroup of G and uniquely determined by this property up to conjugation. In Section 4 we use this result to reduce the case of a locally compact group G to a totally disconnected group. We show

Theorem 0.3. Let G be a locally compact group satisfying (S) and let \overline{G} := G/G_{0}. Then there is a G-CW -model for \underline{E}G that is d-dimensional (resp. finite, resp. of finite type) if and only if \underline{E}\overline{G} has a \overline{G}-CW -model that is d-dimensional (resp. finite, resp. of finite type).

If we combine Theorem 0.1, Theorem 0.2 and Theorem 0.3 we get

Theorem 0.4. Let G be a locally compact group satisfying (S). Denote by \overline{COM} the family of compact subgroups of its component group \overline{G} and let d ≥ 3. Then

(a) There is a d-dimensional G-CW -model for \underline{E}G if and only if the constant ℤ Or(\overline{G}_{d}, \overline{COM}_{d})-module \underline{ℤ} has a d-dimensional projective resolution;

(b) There is a G-CW -model for \underline{E}G of finite type if and only if E(\overline{G}_{d}, \overline{COM}_{d}) has a \overline{G}_{d}-CW -model with finite 2-skeleton and the constant ℤ Or(\overline{G}_{d}, \overline{COM}_{d})- module \underline{ℤ} has a projective resolution of finite type;

(c) There is a finite G-CW -model for \underline{E}G if and only if E(\overline{G}_{d}, \overline{COM}_{d}) has a \overline{G}_{d}- CW -model with finite 2-skeleton and the constant ℤ Or(\overline{G}_{d}, \overline{COM}_{d})-module \underline{ℤ} has a finite free resolution.

In particular we see from Theorem 0.3 that, for a Lie group G, type questions about \underline{E}G are equivalent to the corresponding type questions of \underline{E}π_{0}(G), since π_{0}(G) = \overline{G} is discrete (cf. [11, Problem 7.1]). In this case the family \overline{COM}_{d} appearing in Theorem 0.4 is just the family FIN of finite subgroups of π_{0}(G).

1. Review of the Universal Space for a Family of Subgroups

Recall from the introduction the G-CW -complex E(G, F). In particular, notice that we only assume that the fixed point sets E(G, F)^{H} for H ∈ F are weakly contractible, and not necessarily contractible. If G is discrete, then each fixed point set E(G, F)^{H} has the homotopy type of a CW -complex and is contractible for H ∈ F. If \overline{G} is discrete and F = COM, then E(G, F)^{H} is contractible for H ∈ COM by Proposition 4.3. In general E(G, F)^{H} need not be contractible as the following example shows.

Let G be totally disconnected and let F be the trivial family TR consisting of one element, namely the trivial group. We claim that then E(G, TR) is contractible if and only if G is discrete. If G is discrete, we already know that E(G, TR) is contractible. Suppose now that E(G, TR) is contractible. We obtain a numerable G-principal bundle G → E(G, TR) → G\E(G, TR) by the Slice Theorem [10, Theorem 1.37] and the fact that the quotient G\E(G, TR) is a CW -complex and hence paracompact. This implies that it is a fibration by a result of Hurewicz [17, Theorem on p. 33]. Since E(G, TR) is contractible, G and the loop space Ω(G\E(G, TR)) are homotopy equivalent [17, 6.9^{∗} on p. 137, 6.10^{∗} on p. 138, Corollary 7.27 on p. 40]. Since G\E(G, TR) is a CW -complex, Ω(G\E(G, TR)) has the homotopy type of a CW -complex [13]. Let f : G → X be a homotopy equivalence from G to a CW -complex X. Then the induced map π_{0}(G) → π_{0}(X) between the set of path components is bijective. Hence any preimage of a path component of X is a point since G is totally disconnected. Since X is locally pathconnected each path component of X is open in X. We conclude that G is the disjoint union of the preimages of the path components of X and each of these preimages is open in G and consists of one point. Hence G is discrete.

There is another variant of the universal space of a group G for a family F which we review next ([4, Theorem 6.6. on p. 47]. The assumption that G is a compact Lie group is not needed). We denote the space considered there by J(G, F) since it is constructed by a variant of Milnor’s infinite join construction. Namely, a model for J(G, F) is ∗_{n=1}^{∞}Z, where Z is a disjoint union of homogeneous spaces G/H_{i} such that each G/H_{i} is G-isomorphic to G/H for one H ∈ F and each H ∈ F occurs this way. This is an F-numerable G-space with the universal property that for any Fnumerable G-space X there is up to homotopy precisely one G-map X → J(G, F). Again J(G, F) is unique up to G-homotopy. In contrast to E(G, F) the H-fixed point set J(G, F)^{H} is always contractible for H ∈ F. Since E(G, F) is a G-CW - complex and hence an F-numerable G-space, there is a G-map f : E(G, F) → J(G, F) unique up to G-homotopy. Obviously f is a weak G-homotopy equivalence, i.e. f^{H} : E(G, F)^{H} → J(G, F)^{H} is a weak homotopy equivalence for each H ⊂ G. In other words, E(G, F) is a G-CW -approximation of J(G, F). We know that f cannot be a G-homotopy equivalence in general since E(G, F)^{H} is not contractible in general. Hence these concepts are different. However, for any G-CW -complex X whose isotropy groups belong to F, any G-map X → J(G, F) lifts uniquely up to G-homotopy over the G-map f : E(G, F) → J(G, F). Moreover, if G is discrete or if G is a Lie group and F is contained in COM, then f : E(G, F) → J(G, F) is a G-homotopy equivalence and these concepts agree. This can be seen as follows.

Under the assumptions on G and F, ∗_{n=0}^{k}Z has the G-homotopy type of a G-CW -complex and hence ∗_{n=1}^{∞}Z_{weak} has the G-homotopy type of a G-CW -complex, where ∗_{n=1}^{∞}Z_{weak} is equipped with the weak topology with respect to the filtration by the subspaces ∗_{n=0}^{k}Z for k = 1, 2, . . .. This follows for instance from [10, section 7]. (See also [8].) One checks that for a G-space X with a G-invariant G-covering and locally finite G-invariant subordinate partition of unity, the G-map X → J(G, F) constructed in [4, Lemma 6.13 on p. 49 and Lemma 6.9 on p. 48] actually factorizes through ∗_{n=1}^{∞}Z_{weak}, since locally this map takes values in one of the subspaces ∗_{n=0}^{k}Z. In particular we obtain a G-map J(G, F) → ∗_{n=1}^{∞}Z_{weak}. Since ∗_{n=1}^{∞}Z_{weak} is a G-CW -complex we obtain a G-map h : J(G, F) → E(G, F). By the universal properties both compositions h ∘ f and f ∘ h are G-homotopic to the identity.

In the case F = COM there is another model for the universal space of G, well known from harmonic analysis, described in [2, §2]. Denote by C_{0}(G) the space of complex-valued continuous functions on G vanishing at infinity, endowed with the sup-norm-topology. By g · f(x) := f(g^{−1}x), g ∈ G, f ∈ C_{0}(G), G acts isometrically on C_{0}(G). Denote by P C_{0}(G) the subspace of real-valued functions f ≠ 0 that only take non-negative values. Then P C_{0}(G) is a final object in the homotopy category of numerably proper G-spaces. As [2] and [4] work over the same category ([2, Prop. 3.9]), the models P C_{0}(G) and J(G, COM) are G-homotopy equivalent.

2. Discrete Groups

Throughout this section G denotes a discrete group. Finiteness conditions for E(G, F) focussing on the family F = FIN of finite subgroups have already been studied in [9], [11] and [16]. In this section we translate type questions about E(G, F) for G and a family of subgroups F to homological algebra of modules over the associated orbit category Or(G, F). We begin by recalling some basic definitions.

The orbit category Or(G) of G is the small category whose objects are homogeneous G-spaces G/H and whose morphisms are G-maps. Let Or(G, F) be the full subcategory of Or(G) consisting of those objects G/H for which H belongs to F. A ℤ Or(G, F)-module is a contravariant functor from Or(G, F) to the category of ℤmodules. A morphism of such modules is a natural transformation. The category of ℤ Or(G, F)-modules inherits the structure of an abelian category from the standard structure of an abelian category on the category of ℤ-modules. In particular the notion of a projective ℤ Or(G, F)-module is defined. The free ℤ Or(G, F)-module ℤ map(G/?, G/K) based at the object G/K is the ℤ Or(G, F)-module that assigns to an object G/H the free ℤ-module ℤ map_{G}(G/H, G/K) generated by the set map_{G}(G/H, G/K). The key property of it is that for any ℤ Or(G, F)-module N there is a natural bijection of ℤ-modules

hom_{ℤ Or(G,F)}(ℤ map_{G}(G/?, G/K), N) ^{≅} N(G/K), ϕ ↦ ϕ(G/K)(id_{G/K})

which is an application of the Yoneda Lemma. A ℤ Or(G, F)-module is free if it is isomorphic to a direct sum ⊕_{i∈I}ℤ map(G/?, G/K_{i}) for appropriate choice of objects G/K_{i} and index set I. A ℤ Or(G, F)-module is called finitely generated if it is a quotient of a ℤ Or(G, F)-module of the shape ⊕_{i∈I}ℤ map(G/?, G/K_{i}) with a finite index set I. Notice that a lot of standard facts for ℤ-modules carry over to ℤ Or(G, F)-modules. For instance, a ℤ Or(G, F)-module is projective or finitely generated projective respectively if and only if it is a direct summand in a free ℤ Or(G, F)-module or a finitely generated free ℤ Or(G, F)-module respectively. The notion of a projective resolution P_{∗} of a ℤ Or(G, F)-module is obvious. We call P_{∗} of finite type if each P_{n} is finitely generated projective. We call P_{∗} finite if P_{∗} is both of finite type and finite-dimensional. Each ℤ Or(G, F)-module has a projective resolution.

Definition 2.1. Let G be a discrete group and (X, A) a relative G-CW -complex whose isotropy groups belong to the family F. The contravariant functor

C_{∗}^{c}(X, A) : Or(G, F)  \overline{}ℤ − Chain complexes
G/H ⟼ C_{∗}^{c}(X^{H}, A^{H})

is called the cellular ℤ Or(G, F)-chain complex of (X, A).

Functoriality comes from the fact that X^{H} = map_{G}(G/H, X). Notice that (X^{H}, A^{H}) is canonically a CW -complex, hence we can speak of its cellular chain complex C_{∗}^{c}(X^{H}, A^{H}). As in the nonequivariant situation the chain modules are free with basis given by the (equivariant) cells. Namely, we have

Lemma 2.2. For any n ∈ ℤ the n-th chain module C_{n}^{c}(X) : Or(G, F) ℤ− Modules is a free ℤ Or(G, F)-module.

Proof. Let the n-skeleton of X be given by a pushout

[figure: ∐_{I_{n}} G/H_{i} × S^{n−1}; X_{n−1}; ∐_{I_{n}} G/H_{i} × D^{n}; id ×i; ∩; X_{n}.]

Since id ×i is a cofibration, we get by excision in G/H natural isomorphisms

C_{n}^{c}(X, A)(G/H) ≅ H_{n}(X_{n}^{H}, X_{n−1}^{H}) ≅ H_{n}(∐_{I_{n}}(G/H_{i})^{H} × (D^{n}, S^{n−1}))

≅ ⨁_{I_{n}} H_{n}((G/H_{i})^{H} × (D^{n}, S^{n−1})) ≅ ⨁_{I_{n}} H_{0}((G/H_{i})^{H}) ≅ ⨁_{I_{n}} ℤ[(G/H_{i})^{H}] ≅ ⨁_{I_{n}} ℤ[map_{G}(G/H, G/H_{i})]. (2.3)

From the explicit description (2.3) of the n-th chain module C_{n}^{c}(X) we immediately get the following corollary, linking the type of the G-CW -complex to the type of its cellular chain complex.

Corollary 2.4. Let X be a G-CW -complex whose isotropy groups belong to the family F. Then X is d-dimensional (resp. finite, resp. of finite type) if and only if its cellular ℤ Or(G, F)-chain complex C_{∗}^{c}(X) is d-dimensional (resp. finite, resp. of finite type).

Proposition 2.5. Let h : Z Y be a G-map between G-CW -complexes such that both Z^{H} and Y^{H} are simply-connected for H ∈ F and all their isotropy groups belong to the family F. Let r ≥ 2, r ≥ dim Z and a free ℤ Or(G, F)-chain complex (D_{∗}, d_{∗}) be given. Finally, suppose that there is a chain homotopy equivalence f_{∗} : D_{∗} C_{∗}^{c}(Y, B) such that D_{∗}|_{r} = C_{∗}^{c}(Z)|_{r} and f_{∗}|_{r} = C_{∗}^{c}(h)|_{r}.

Then there is a G-CW -complex X with X_{r} = Z and a cellular G-homotopy equivalence g : X Y such that:

i) g|_{Z} = h; ii) D_{∗} = C_{∗}^{c}(X); iii) C_{∗}^{c}(g) = f_{∗}.

Proof. The proof is exactly the same as in [10, Theorem 13.19 on p. 268]. There only proper actions are considered but the same methods go through because here we are dealing with the easy case where all fixed point sets are simply connected, the isotropy groups belong to F, and G is discrete.

Lemma 2.6. If G is a discrete group and E is a G-CW -model for E(G, F), then C_{∗}^{c}(E) is a free resolution over the orbit category Or(G, F) of the constant Or(G, F)-module \underline{ℤ} with value ℤ.

Proof. The modules are free by Lemma 2.2. It remains to show that C_{∗}^{c}(E^{H}), H ∈ F has the homology of a point. But this follows from the fact that for discrete H the space E^{H} has a canonical CW -structure whose n-skeleton in exactly E_{n}^{H} and E^{H} is weakly contractible and hence contractible.

To shorten the next proof we start with the following lemma. Its proof is purely technical and hence left out. Details of the proof can be found in [10, p. 279-280]. All modules are supposed to be over the orbit category.

Lemma 2.7. Let C_{∗} be a free, 2-dimensional chain complex, D_{∗} a free chain complex and f_{∗} : C_{∗} D_{∗} a chain map with H_{i}(cone(f_{∗})_{∗}) = 0, i ≤ 2. Then there is a free chain complex C′_{∗} and a chain homotopy equivalence g_{∗} : C′_{∗} D_{∗} with C′_{∗}|_{2} = C_{∗} and g_{∗}|_{2} = f_{∗}. If C_{∗} is finite and D_{∗} is homotopic to a finite free chain complex, resp. a free complex of finite type, then C′_{∗} can be chosen to be finite, resp. of finite type. If D_{∗} is homotopic to a finite-dimensional free complex, then C′_{∗} can be chosen to be finite-dimensional.

Proof of Theorem 0.1: The “only if”-case is clear for the first three assertions by Lemma 2.6 and Corollary 2.4. In the “if”-case for the first three assertions, let P_{∗} be the given projective resolution of the constant ℤOr(G, F)-module \underline{ℤ} and let E = E(G, F) be a G-CW -model with finite 2-skeleton in the second and third case. By adding elementary chain complexes, i.e. complexes concentrated in two consecutive dimensions with the identity as only non-trivial differential, we can get P_{∗} to be a free resolution. (In the d-dimensional case we use the Eilenberg trick for the last module. Notice that in the finite case P_{∗} is assumed to be free.) Since C_{∗}^{c}(E) also gives a free resolution of \underline{ℤ} by Lemma 2.6, we have a homotopy equivalence g_{∗} : P_{∗} C_{∗}^{c}(E). Using Lemma 2.7, we get a new free complex Q_{∗} with inherited finiteness property of P_{∗} and a chain homotopy equivalence f_{∗} : Q_{∗} C_{∗}^{c}(E) which induces the identity in dimensions 0,1 and 2. Therefore we can apply Proposition 2.5 to the inclusion i : E_{2} ^{⊂} E and f_{∗}. The result is a G-CW -complex X with X_{2} = E_{2} together with a G-homotopy equivalence k : X E and with C_{∗}^{c}(X) = Q_{∗}. So X is a G-CW -model for E(G, F) and has the desired properties by Corollary 2.4.

The same proof as that of [11, Theorem 4.2], replacing the words “of finite type” by “with finite 2-skeleton”, yields the last assertion of Theorem 0.1.

3. Totally Disconnected Groups

Recall that a topological space X is totally disconnected if any component consists of exactly one point. In this section we want to show that there is a close relation between universal spaces of a totally disconnected group G and those of its discretization G_{d}. The reason for this is that homotopy does not see the difference between a totally disconnected group G and G_{d}, i.e. the canonical map G_{d}/H_{d} res_{G}^{G_{d}} G/H is a weak G_{d}-homotopy equivalence. The different topologies will only appear in the family of subgroups that has to be considered. We start by collecting some elementary facts about totally disconnected spaces.

Let X be a topological space. Consider the following 3 conditions.

(T) X is totally disconnected; (D) The covering dimension of X is 0; (FS) Any element of X has a fundamental system of open and compact neigh-

borhoods.

Lemma 3.1. For a locally compact group the conditions (T), (D) and (FS) are equivalent.

Proof. The implications (T) ⇒ (D) ⇒ (F S) are shown in [7, Theorem 7.7 on p. 62]. The implication (F S) ⇒ (T) is done as follows: Let U be a set containing two distinct points x and y. We show that U is disconnected. Let V be an open and compact neighborhood of x, not containing y. Then (V ∩ U) ⨿ (V^{c} ∩ U) = U is a disjoint union of two nonempty open subsets of U.

The elementary proofs of the next two lemmas are left to the reader.

Lemma 3.2. Let f : X Y be a surjective and open map . If X is locally compact, then Y is locally compact. If X has property (FS), then so does Y. In particular, if G is a totally disconnected locally compact group, then (G/H)^{K} is totally disconnected and locally compact for all (closed) subgroups H, K ⊂ G.

Lemma 3.3. Let f : X Y be a map. If f^{−1}(y) is weakly contractible for all y ∈ Y and Y is totally disconnected, then f is a weak homotopy equivalence. If f^{−1}(y) is contractible for all y ∈ Y and Y is discrete, then f is a homotopy equivalence.

Lemma 3.4. Let G be a totally disconnected locally compact group and X be a G_{d}-CW -complex whose isotropy groups are all closed when viewed as subgroups of G. Then the map

i_{X} : X res_{G}^{G_{d}} G ×_{G_{d}} X, x ↦ [e, x]

is a weak G_{d}-homotopy equivalence.

Proof. We begin with the case where X is a homogeneous space G_{d}/H_{d} for a closed subgroup H ⊂ G. Then the map

G ×_{G_{d}} G_{d}/H_{d}  G/H,  [g, g′H_{d}] ↦ gg′H

is a G-homeomorphism. The obvious map G_{d}/H_{d} → res_{G}^{G_{d}} G/H is a weak G_{d}- homotopy equivalence by Lemma 3.2 and Lemma 3.3. Hence the map i_{G_{d}/H_{d}} : G_{d}/H_{d} res_{G}^{G_{d}} G ×_{G_{d}} G_{d}/H_{d} is a weak G_{d}-homotopy equivalence.

Next we prove the claim for all skeleta X_{n} by induction over n. The base case n ≤ 0 of the induction follows from the case of a homogeneous space since X_{0} is a disjoint union of homogeneous spaces. In the induction step from n − 1 to n one chooses a G_{d}-pushout

[figure: ∐_{I_{n}} G_{d}/H_{i} × S^{n−1}; X_{n−1}; ∐_{I_{n}} G_{d}/H_{i} × D^{n}; ∩; X_{n}]

and checks, using the fact that G is locally compact, that the induced diagram is a G_{d}-pushout

∐_{I_{n}} res_{G}^{G_{d}} G ×_{G_{d}} G_{d}/H_{i} × S^{n−1} \overline{}res_{G}^{G_{d}} G ×_{G_{d}} X_{n−1}

[figure: ∐_{I_{n}} res_{G}^{G_{d}} G ×_{G_{d}} G_{d}/H_{i} × D^{n}; ∩; res_{G}^{G_{d}} G ×_{G_{d}} X_{n}.]

Notice that in both diagrams the left vertical arrows are G_{d}-cofibrations and the various maps i_{Y} for Y = ∐_{I_{n}} G_{d}/H_{i} ×S^{n−1}, Y = ∐_{I_{n}} G_{d}/H_{i} ×D^{n}, Y = X_{n−1} and Y = X_{n} map the two diagrams to one another. By the induction hypothesis the first three are weak G_{d}-homotopy equivalences. Hence i_{X_{n}} is a weak G_{d}-homotopy equivalence [10, Lemma 2.13 on p. 38].

Let K ⊂ G_{d} be a subgroup. Since X^{K} has the weak topology with respect to the

filtration given by the subspaces X_{n}^{K} and G is locally compact, (res_{G}^{G_{d}} G ×_{G_{d}} X)^{K}

has the weak topology with respect to the filtration given by (res_{G}^{G_{d}} G ×_{G_{d}} X_{n})^{K}.

Since (i_{X_{n}})^{K} is a weak homotopy equivalence for n ≥ 0, the same follows for (i_{X})^{K}.

Corollary 3.5. Let G be a totally disconnected locally compact group and F a family of subgroups of G. If E(G_{d}, F_{d}) is a G_{d}-CW -model for the universal space of G_{d} for the family F_{d}, then G ×_{G_{d}} E(G_{d}, F_{d}) is a G-CW -model for E(G, F).

Proof. We have for any K ∈ F by Lemma 3.4:

(G ×_{G_{d}} E(G_{d}, F_{d}))^{K} = (res_{G}^{G_{d}} G ×_{G_{d}} E(G_{d}, F_{d}))^{K} ≃_{w} E(G_{d}, F_{d})^{K_{d}} ≃_{w} {∗}.

Proposition 3.6. Let G be totally disconnected and let X be a G-CW -complex that is d-dimensional (resp. finite, resp. of finite type). Then res_{G}^{G_{d}} X has a G_{d}- CW -approximation Y that is d-dimensional (resp. finite, resp. of finite type) and whose isotropy groups are the same as those of X. If X is a G-CW -model for E(G, F), then Y is a G_{d}-CW -model for E(G_{d}, F_{d}).

Proof. For n ≥ −1 we construct by induction a G_{d}-CW -complex Y_{n} together with a G_{d}-approximation f_{n} : Y_{n} res_{G}^{G_{d}} X_{n} such that Y_{n−1} is a subcomplex of Y_{n} and f_{n}|_{Y_{n−1}} = f_{n−1}. The base case n = −1 of the induction is given by the empty set. For the induction step from n − 1 to n we proceed as follows. Choose a pushout

[figure: ∐_{I_{n}} G/H_{i} × S^{n−1}; r; X_{n−1}; ∐_{I_{n}} G/H_{i} × D^{n}; ∩; X_{n}]

for the n-skeleton of X_{n}. Let i : ∐_{I_{n}} G_{d}/(H_{i})_{d} → ∐_{I_{n}} res_{G}^{G_{d}} G/H_{i} be the obvious weak G_{d}-homotopy equivalence (see Lemma 3.4). Since by the induction hypothesis f_{n−1} is a weak G_{d}-homotopy equivalence, we can find using [10, Proposition 2.3 on p. 35] and a version of the Cellular Approximation Theorem (see for instance [10, Theorem 2.1 on p. 32]) a cellular G_{d}-map g : ∐_{I_{n}} G_{d}/(H_{i})_{d} × S^{n−1} Y_{n−1} and a G_{d}-homotopy h : ∐_{I_{n}} G_{d}/(H_{i})_{d} × S^{n−1} × I res_{G}^{G_{d}} X_{n−1} between f_{n−1} ∘ g and r∘(i×id_{S^{n−1}}). Let f′_{n−1} : cyl(g) res_{G}^{G_{d}} X_{n−1} be the obvious map given by h and f_{n−1}. Its restriction to Y_{n−1} ⊂ cyl(g) is f_{n−1} and to ∐_{I_{n}} G_{d}/(H_{i})_{d} × S^{n−1} is r ∘ (i × id_{S^{n−1}}). Thus we obtain a commutative diagram

[figure: ∐_{I_{n}} G_{d}/(H_{i})_{d} × D^{n}; ⊃; ∐_{I_{n}} G_{d}/(H_{i})_{d} × S^{n−1}; ⊂; cyl(g)]

[figure: ∐_{I_{n}} res_{G}^{G_{d}} G/H_{i} × D^{n}; i × id; ⊃; ∐_{I_{n}} res_{G}^{G_{d}} G/H_{i} × S^{n−1}; i × id; r; res_{G}^{G_{d}} X_{n−1}.; f′_{n−1}]

Taking the pushout of the upper row yields a n-dimensional G_{d}-CW -complex Y_{n} which contains Y_{n−1} as G_{d}-CW -subcomplex and which is finite if X_{n} is finite. Moreover, we get by the pushout property a G_{d}-map f_{n} : Y_{n} → res_{G}^{G_{d}} X_{n} which extends f_{n−1} : Y_{n−1} → res_{G}^{G_{d}} X_{n−1} and is a weak G_{d}-homotopy equivalence, since all vertical maps are weak G_{d}-homotopy equivalences ([10, Lemma 2.13 on p. 38]). Now put Y := colim_{n→∞} Y_{n}. Then f := colim_{n→∞} f_{n} : Y → res_{G}^{G_{d}} X is the G_{d}-CW -approximation we look for.

Proof of Theorem 0.2: Follows from Corollary 3.5 and Proposition 3.6.

4. Locally Compact Groups

Our strategy for locally compact groups is the following. Any locally compact group G gives rise to a short sequence of the form 1 G_{0} G ^{p} \overline{G} 1 with G_{0} locally compact and connected and with \overline{G} locally compact and totally disconnected. This reduces the study of G to the study of locally compact connected groups, which are very similar to Lie groups by the solution of Hilbert’s fifth problem (cp. [6]), and to locally compact totally disconnected groups, which are similar to their discrete underlying group, as we saw in the preceding section. We start with some remarks on locally compact groups G which are almost connected, i.e. whose component group \overline{G} is compact.

Theorem 4.1. Let G be a almost connected locally compact group. Then G has a maximal compact subgroup K which is unique up to conjugacy and G/K is a model for both \underline{E}G and \underline{J}G := J(G, COM).

Proof. [1, Appendix, Theorem A.5], [2, Corollary 4.14].

We now turn our attention to locally compact groups that are not necessarily almost connected. From now on any locally compact group G is assumed to satisfy the condition (S) defined in the introduction.

Lemma 4.2. Let L be an almost connected subgroup of G and let K be a maximal compact subgroup of L. If G/L is totally disconnected, then for any compact H ⊂ G the projection pr^{H} : (G/K)^{H} (G/L)^{H} is a weak homotopy equivalence. If G/L is discrete, then for any compact H ⊂ G the projection pr^{H} : (G/K)^{H} (G/L)^{H} is a homotopy equivalence.

ON THE UNIVERSAL SPACE FOR GROUP ACTIONS WITH COMPACT ISOTROPY 11

Proof. If H is not subconjugated to L, all spaces are empty. So, let H be subconjugated to L. Since we assumed the existence of local cross sections, we know that G G/L is a principal L-bundle. Let Y be any L-space. Fix an element w ∈ G. Then in the associated fiber bundle Y ^{⊂} G×_{L} Y ^{pr} G/L, the typical fiber maps homeomorphically onto the preimage p^{−1}(wL) of wL ∈ G/L by sending y to the class of (w, y). If wL is in (G/L)^{H} then this implies wHw^{−1} ⊂ L and we get an induced homeomorphism Y^{wHw^{−1}} (pr^{H})^{−1}(wL) for pr^{H} : (G ×_{L} Y)^{H} G/L^{H}. Now let Y be L/K, which is a model for J(L, COM) by Theorem 4.1. Therefore Y^{wHw^{−1}} is contractible. Hence by Lemma 3.3 the map pr^{H} is a weak homotopy equivalence if G/L and hence (G/L)^{H} is totally disconnected, and a homotopy equivalence, if G/L and hence (G/L)^{H} is discrete.

Proposition 4.3. Given a \overline{G}-CW -model \underline{E}\overline{G}, there is a G-CW -model \underline{E}G and a G-map f : \underline{E}G p^{∗}\underline{E}\overline{G} with the following properties (where p^{∗}\underline{E}\overline{G} is \underline{E}\overline{G} viewed as a G-space by the projection p : G \overline{G}):

(a) If \underline{E}\overline{G} is d-dimensional (resp. finite, resp. of finite type), then \underline{E}G is ddimensional (resp. finite, resp. of finite type);

(b) If \overline{G} is discrete, then \underline{E}G^{H} is contractible for all compact H ⊂ G;

(c) G_{0}\f : G_{0}\\underline{E}G \underline{E}\overline{G} is a \overline{G}-homotopy equivalence.

Proof. We will construct for each n ≥ −1 an n-dimensional G-CW -complex X_{n} and a G-map f_{n} : X_{n} p^{∗}\underline{E}\overline{G}_{n} to the n-skeleton of p^{∗}\underline{E}\overline{G} with the following properties:

(a) f_{n}^{H} : X_{n}^{H} (p^{∗}\underline{E}\overline{G}_{n})^{H} is a weak homotopy equivalence for all compact H ⊂ G. If \overline{G} is discrete, f_{n}^{H} : X_{n}^{H} (p^{∗}\underline{E}\overline{G}_{n})^{H} is a homotopy equivalence for all compact H ⊂ G;

(b) The isotropy groups of X_{n} are all compact;

(c) X_{n−1} is the (n − 1)-skeleton of X_{n} and f_{n}|_{X_{n−1}} = f_{n−1}. There is a bijective correspondence between the equivariant n-dimensional cells of X_{n} and of \underline{E}\overline{G}_{n};

(d) G_{0}\f_{n} : G_{0}\X_{n} \underline{E}\overline{G}_{n} is a \overline{G}-homotopy equivalence. Notice that we then can define \underline{E}G := colim_{n→∞} X_{n} and f := colim_{n→∞} f_{n}, and check that \underline{E}G and f have the desired properties.

We proceed by induction over n. The base case n = −1 of the induction is given by X_{−1} := ∅. For the induction step from n − 1 to n we choose a G-pushout

[figure: ∐_{I_{n}} \overline{G}/H_{i} × S^{n−1}; ∐_{I_{n}} q_{i}; \underline{E}\overline{G}_{n−1}; ∐_{I_{n}} \overline{G}/H_{i} × D^{n}; ∩; \underline{E}\overline{G}_{n}.]

Put L_{i} := p^{−1}(H_{i}) ⊂ G for i ∈ I_{n}. Obviously L_{i} is almost connected. Let K_{i} be a maximal compact subgroup of L_{i} for i ∈ I_{n} (see Theorem 4.1). Since the projection p : G \overline{G} induces a homeomorphism G/L_{i} ^{≅} \overline{G}/H_{i}, G/L_{i} is totally disconnected. Hence Lemma 4.2 implies that pr_{i}^{H} : (G/K_{i})^{H} (G/L_{i})^{H} is a weak homotopy equivalence for all compact subgroups H of G and is a homotopy equivalence for all compact subgroups H of G, provided that \overline{G} is discrete. The same is true for f_{n−1} by the induction hypothesis. Therefore we have a bijection induced by f_{n−1} ([10, Prop. 2.3 on p.35])

[G/K_{i} × S^{n−1}, X_{n−1}]_{G} ^{(f_{n−1})_{∗}} [G/K_{i} × S^{n−1}, p^{∗}\underline{E}\overline{G}_{n−1}]_{G}.

Using the Equivariant Cellular Approximation Theorem [10, Theorem 2.1 on p. 32] we get a cellular G-map r_{i} : G/K_{i} ×S^{n−1} → X_{n−1} together with a G-homotopy h_{i} : G/K_{i} × S^{n−1} × [0, 1] → p^{∗}\underline{E}\overline{G}_{n−1} from f_{n−1} ∘r_{i} to q_{i} ∘(pr_{i} ×id_{S^{n−1}}). Consider the following commutative G-diagram

[figure: ∐_{I_{n}} p^{∗}(\overline{G}/H_{i}) × D^{n}; ⊃; ∐_{I_{n}} p^{∗}(\overline{G}/H_{i}) × S^{n−1}; ∐_{I_{n}} q_{i}; p^{∗}\underline{E}\overline{G}_{n−1}]

[figure: ∐_{I_{n}} G/K_{i} × D^{n}; ∐_{I_{n}} pr_{i} × id_{D^{n}}; ⊃; ∐_{I_{n}} G/K_{i} × S^{n−1}; ∐_{I_{n}} pr_{i} × id_{S^{n−1}}; ∐_{I_{n}} q_{i} ∘ (pr_{i} × id_{S^{n−1}}); p^{∗}\underline{E}\overline{G}_{n−1}; id]

[figure: ∐_{I_{n}} G/K_{i} × D^{n} × [0, 1]; i_{1}; ⊃; ∐_{I_{n}} G/K_{i} × S^{n−1} × [0, 1]; i_{1}; ∐_{I_{n}} h_{i}; p^{∗}\underline{E}\overline{G}_{n−1}; id]

[figure: ∐_{I_{n}} G/K_{i} × D^{n}; i_{0}; ⊃; ∐_{I_{n}} G/K_{i} × S^{n−1}; i_{0}; ∐_{I_{n}} r_{i}; X_{n−1}.; f_{n−1}]

Notice that the pushout of the first row is p^{∗}\underline{E}\overline{G}_{n}. Denote the pushout of the second, third and fourth row respectively by X′_{n}, X′′_{n} and X_{n}. The diagram above together with the pushout property induces G-maps f′_{n} : X′_{n} p^{∗}\underline{E}\overline{G}_{n}, f′′_{n} : X′_{n} X′′_{n} and f′′′_{n} : X_{n} → X′′_{n}. The map f′′_{n} is a G-homotopy equivalence and the maps (f′_{n})^{H} and(f′′′_{n})^{H} are weak homotopy equivalences (homotopy equivalence if \overline{G} is discrete) for each compact subgroup H ⊂ G ([10, Lemma 2.13 on p. 38]). We can choose a G-map (f′′_{n})^{−1} : X′′_{n} → X′_{n} which induces the identity on p^{∗}\underline{E}\overline{G}_{n−1} and is a G-homotopy inverse of f′′_{n}. Now define f_{n} : X_{n} p^{∗}\underline{E}\overline{G}_{n} by the composition f′_{n} ∘ (f′′_{n})^{−1} ∘ f′′′_{n}. By construction f_{n}^{H} is a weak homotopy equivalence (homotopy equivalence if \overline{G} is discrete) for all compact subgroups H ⊂ G and X_{n} is a G-CW -complex with X_{n−1} as its (n − 1)-skeleton and has only compact isotropy groups.

It remains to show that G_{0}\f_{n} : G_{0}\X_{n} → \underline{E}\overline{G}_{n} is a \overline{G}-homotopy equivalence. Since L_{i} inherits the property (S) from G, we get a locally trivial fiber bundle K_{i} L_{i} L_{i}/K_{i} which is automatically a Serre fibration and hence induces a long exact homotopy sequence [14, Theorem 2.11 on p. 60, Theorem 3.6 on p. 65 and Corollary 3.11 on p. 67]. Thus we get the following diagram

K_{i}

[figure: 1 = π_{1}(L_{i}/K_{i}); π_{0}(K_{i}); ≅; π_{0}(L_{i}); π_{0}(L_{i}/K_{i}) = 1.]

[figure: G_{0}\L_{i} = H_{i}]

We conclude that p(K_{i}) = H_{i} holds for all i ∈ I_{n}. Hence G_{0}\pr_{i} : G_{0}\G/K_{i} \overline{G}/H_{i} is a \overline{G}-homeomorphism. Therefore G_{0}\f′_{n} is a \overline{G}-homeomorphism. G_{0}\(f′′_{n})^{−1} is a \overline{G}-homotopy equivalence, since (f′′_{n})^{−1} is a G-homotopy equivalence, and G_{0}\(f′′′_{n}) is a \overline{G}-homotopy equivalence, since G_{0}\f_{n−1} is a \overline{G}-homotopy equivalence by the induction hypothesis. Hence G_{0}\f_{n} is a \overline{G}-homotopy equivalence.

Proof of Theorem 0.3: Is implied by Proposition 4.3.

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Institut für Mathematik und Informatik, Westf. Wilhelms-Universität, Einsteinstr. 62, 48149 Münster, Germany, http://wwwmath.uni-muenster.de/u/lueck

E-mail address: lueck@math.uni-muenster.de, meintrd@math.uni-muenster.de