The Type of the Classifying Space of a Topological Group for the Family of Compact Subgroups by Wolfgang L"uck and David Meintrup* Abstract Let G be a locally compact topological group. We investigate the type* * of the classifying space of G for the family of compact subgroups. We give crite* *ria for this space to have a d-dimensional G-CW -model, a finite G-CW -model or a G-CW* * -model 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. Key words: classifying space of a group for a family, topological group 1991 mathematics subject classification: 55R35 Introduction Throughout this paper we denote by G a locally compact topological group (w* *here locally compact_always_includes Hausdorff). We denote by G0 the component of th* *e identity element and by_G_ = G=G0 its component group. Notice that G0 is locally compac* *t and connected and G is locally compact and totally disconnected, i.e. each componen* *t consists of exactly one point. Subgroup will always mean closed subgroup. Sometimes we m* *ake the additional assumption For any closed subgroup H G the projection p : G ____-G=H has a (S) local cross section, i.e. there is a neighborhood U of eH togeth* *er with a map s : U ! G satisfying p O s = idU. 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 Ha* *usdorff 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 t* *hat for any H; K 2 F and g 2 G, the subgroups g-1Hg and H \ K belong to F. Notice that we d* *o not require that F is closed under taking subgroups. A classifying space of G for t* *he family F ___________________________________ *David Meintrup is supported by the Graduiertenkolleg "Algebraische Geometri* *e und Zahlentheorie" in M"unster. Parts of this paper grew out of his Dissertation. 1 is a G-CW -complex E(G; F) such that the fixed point set E(G; F)H is weakly con* *tractible for H 2 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 the induced map f* : ssn(X; x) ! * *ssn(Y; f(x)) is an isomorphism for all base points x 2 X and n 0 and that a space X is weak* *ly contractible if and only if the projection X ! {*} to the space consisting of o* *ne 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-hom* *otopy precisely one G-map X ____-E(G; F) and in particular it is unique up to G-homot* *opy. The notion of a G-CW -complex can be found for instance in [10, Definition 1.2], th* *e 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 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 this notion and put it into context with the s* *imilar notion due to tom Dieck [4, section I.6]. We mention that these spaces E(G; F) and in part* *icular E_G play an important role in the formulation of the Baum-Connes Conjecture [3, Con* *jecture 3.15 on p. 254], the Isomorphism Conjecture in algebraic K- and L-theory of Far* *rell 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 E_G can be * *found for instance in [3]. We want to get information about the possible 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* *. A G- CW -complex X is finite if it is built by finitely many equivariant cells or, e* *quivalently, G\X is compact. It is called of finite type if each skeleton Xn is finite. For disc* *rete groups the type of E_G has been investigated in [9], [11] and [16]. We will give in Section 2 a* * necessary and sufficient algebraic criterion which not only applies to FIN but to any famil* *y 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 const* *ant Z Or(G; F)-module 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 Z_has a p* *rojective 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 Z_has a finite * *free resolution over Or(G; F); (d) There is a G-CW -model with finite 2-skeleton for E_G = E(G; FIN ) if an* *d only if there are only finitely many conjugacy classes of finite subgroups H G a* *nd for any finite subgroup H G its Weyl group WH := NH=H is finitely presented. 2 In Section 3 we will reduce the case of a totally disconnected group to the* * one of a discrete group. Throughout the paper we will denote the discretization of a to* *pological group G by Gd, i.e. the same group but now with the discrete topology. Given a * *family F of (closed) subgroups of G, denote by Fd the same set of subgroups, but now in * *connection with Gd. Notice that Fd 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 d-dimensiona* *l (resp. finite, resp. of finite type) if and only if there is a Gd-CW -model for E(Gd;* * Fd) that is d-dimensional (resp. finite, resp. of finite type). __ The case of an almost connected group G, i.e. G is compact, has already be* *en treated by Abels [2, Corollary 4.14]. Namely, for an almost connected (locally compact)* * group G there is a model for 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 pro* *perty 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 G := G=G* *0. Then there is a G-CW_-model_for_E_G that is d-dimensional (resp. finite, resp. of fi* *nite type) if and only if E_G has a G -CW -model that is d-dimensional (resp. finite, resp. o* *f 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 COM * * the family of compact subgroups of its component group G and let d 3. Then (a) There__is _a__d-dimensional_ G-CW -model for E_G if and only if th* *e constant Z Or(G d; COM d)-module Z_has a d-dimensional projective resolution; __ _____* *_ __ (b) There is a G-CW -model for E_G of finite type if and only_if_E(G_d;_COM * *d) has a G d- CW -model with finite 2-skeleton and the constant Z Or(G d; COM d)-modul* *e Z_has a projective resolution of finite type; __ ______ * * __ (c) There is a finite G-CW -model for E_G if and only if_E(G_d;_COM_ d) has a* * G d-CW - model with finite 2-skeleton and the constant Z Or(G d; COM d)-module Z_* *has a finite free resolution. In particular we see from Theorem 0.3 that, for a Lie group G, type questio* *ns_about E_G are equivalent to the corresponding type questions_of_E_ss0(G),_since ss0(G) = * *G is discrete (cf. [11, Problem 7.1]). In this case the family COM d appearing in Theorem 0.* *4 is just the family FIN of finite subgroups of ss0(G). 3 1 Review of Classifying Spaces of a Group for a Family Recall from the introduction the G-CW -complex E(G; F). In particular, notice * *that we do not work with the stronger condition that E(G; F)H is contractible but only* * weakly contractible. If G is discrete, then each fixed point_set E(G; F)H has the hom* *otopy type of a CW -complex and is contractible for H 2 F. If G is discrete and F = COM * * , then E(G; F)H is contractible for H 2 COM by Proposition 4.3. In general E(G; F)H * *need not to be contractible as the following example shows. Let G be totally disconnected and let F be the trivial family TR consistin* *g of one element, namely the trivial group. We claim that then E(G; TR ) is contractibl* *e if and only if G is discrete. If G is discrete, we already know that E(G; TR ) is con* *tractible. 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 fac* *t that the quotient G\E(G; TR ) is a CW -complex and hence paracompact. This implies t* *hat 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 (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, (G\E(G* *; TR )) has the homotopy type of a CW -complex [13]. Let f : G ! X be a homotopy equiva* *lence from G to a CW -complex X. Then the map induced between the set of path compone* *nts ss0(G) ! ss0(X) is bijective. Hence any preimage of a path component of X is a* * point since G is totally disconnected. Since X is locally path-connected each path co* *mponent of X is open in X. We conclude that G is the disjoint union of the preimages of t* *he path components of X and each of these preimages is open in G and consists of one po* *int. Hence G is discrete. There is another variant of the classifying 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 constr* *ucted by a variant of Milnor's infinite join construction. Namely, a model for J(G; F) is * **1n=1Z, where Z is a disjoint union of homogeneous spaces G=Hi such that each G=Hi is G-isomo* *rphic to G=H for one H 2 F and each H 2 F occurs this way. This is an F-numerable G- space with the universal property that for any F-numerable G-space X there is u* *p to homotopy precisely one G-map X ! J(G; F). Again J(G; F) is unique up to G-homot* *opy. In contrast to E(G; F) the H-fixed point set J(G; F)H is always contractible fo* *r H 2 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. fH : E(G; F)H ! J(G; F)H is a weak homotopy equivalence f* *or each H G. In other words, E(G; F) is a G-CW -approximation of J(G; F). We know th* *at f cannot be a G-homotopy equivalence in general since E(G; F)H is not contract* *ible 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-homot* *opy over the G-map f : E(G; F) ! J(G; F). Moreover, if G is discrete or if G is a L* *ie group and F contained in COM , then f : E(G; F) ! J(G; F) is a G-homotopy equivalenc* *e and these concepts agree. This can be seen as follows. Under the assumptions on G and F, *kn=0Z has the G-homotopy type of a G-CW - 4 complex and hence *1n=1Zweakhas the G-homotopy type of a G-CW -complex, where ** *1n=1Zweak is equipped with the weak topology with respect to the filtration by the subspa* *ces *kn=0Z 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 su* *bordinate partition of unity the G-map X ! J(G; F) constructed in [4, Lemma 6.13 on p. 4* *9 and Lemma 6.9 on p. 48] actually factorizes through *1n=1Zweak, since locally this * *maps takes val- ues in one of the subspaces *kn=0Z. In particular we obtain a G-map J(G; F) ! ** *1n=1Zweak. Since *1n=1Zweakis a G-CW -complex we obtain a G-map h : J(G; F) ! E(G; F). By * *the universal properties both compositions h O f and f O h are G-homotopic to the i* *dentity. In the case F = COM there is another model for the universal classifying* * space of G, well known from harmonic analysis, described in [2, x2]. Denote by C0(G) th* *e space of complex-valued continuous functions on G vanishing at infinity, endowed with* * the sup- norm-topology. By g . f(x) := f(g-1x), g 2 G, f 2 C0(G), G acts isometrically o* *n C0(G). Denote by P C0(G) the subspace of real-valued functions f 6= 0 that only take n* *on-negative values. Then P C0(G) is a final object in the homotopy category of numerably pr* *oper G- spaces. As [2] and [4] work over the same category ([2, Prop. 3.9]), the models* * P C0(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 questions about E(G; F) for G and a* * family of subgroups F to homological algebra of modules over the associated orbit categor* *y Or(G; F). We begin by recalling some basic definitions. The orbit category Or (G) of G is the small category whose objects are homo* *geneous G-spaces G=H and whose morphisms are G-maps. Let Or(G; F) be the full subcatego* *ry 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 Z-modules. A morp* *hism of such modules is a natural transformation. The category of Z Or(G; F)-modules i* *nherits 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 mapG (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 ZOr(G;F)(Z mapG (G=?; G=K); N) ____-N(G=K); OE 7! OE(G=K)(idG=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 i2IZ map(G=?; G=Ki) for appropriate choice of objects G=Kiand i* *ndex 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 i2IZ map(G=?; G=Ki) with a finite index set I. Notice that a lot o* *f standard facts for Z-modules carry over to Z Or(G; F)-modules. For instance, a Z Or(G; F* *)-module is 5 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 resp* *ectively. The notion of a projective resolution P* of a Z Or(G; F)-module is obvious. We * *call P* of finite type if each Pn 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 w* *hose isotropy groups belong to the family F. The contravariant functor Cc*(X; A) : Or(G; F) ____- Z - Chain complexes G=H 7- ! Cc*(XH ; AH ) is called the cellular Z Or(G; F)-chain complex of (X; A). Functoriality comes from the fact that XH = map G(G=H; X). Notice that (XH ; * *AH ) is canonically a CW -complex, hence we can speak of its cellular chain complex Cc** *(XH ; AH ). As in the nonequivariant situation the chain modules are free with basis given * *by the (equiv- ariant) cells. Namely, we have Lemma 2.2 For any n 2 Z the n-th chain module Ccn(X) : Or(G; F) ____-Z - Mod* *ules is a free Z Or(G; F)-module. Proof: Let the n-skeleton of X be given by a pushout a G=Hix Sn-1 ____- Xn-1 In | \ || idxi || | | || a ?| ?| G=Hix Dn ______-Xn: In Since idxi is a cofibration, we get by excision in G=H natural isomorphisms ` Ccn(X; A)(G=H) ~=Hn(XHn; XHn-1) ~=Hn( In(G=Hi)H x (Dn; Sn-1)) ~= L I H ((G=H )H x (Dn; Sn-1)) ~= L H ((G=H )H ) ~= L Z[(G=H )H ] n n i L In 0 i In i ~= I Z[map (G=H; G=H )]: | (* *2.3) n G i From the explicit description (2.3) of the n-th chain module Ccn(X) we imme* *diately get the following corollary, linking the type of the G-CW -complex to the type of i* *ts cellular chain complex. Corollary 2.4 Let X be a G-CW -complex whose isotropy groups belong to the fam* *ily F. Then X is d-dimensional (resp. finite, resp. of finite type) if and only if i* *ts cellular Z Or(G; F)-chain complex Cc*(X) is d-dimensional (resp. finite, resp. of finite* * type). 6 Proposition 2.5 Let h : Z ____-Y be a G-map between G-CW -complexes such tha* *t both ZH and Y H are simply-connected for H 2 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 g* *iven. Finally, suppose that there is a chain homotopy equivalence f* : D* ____-Cc*(Y;* * B) such that D*|r = Cc*(Z)|r and f*|r = Cc*(h)|r. Then there is a G-CW -complex X with Xr = Z and a cellular G-homotopy equiv* *alence g : X ____-Y such that: i) g|Z = h; ii)D* = Cc*(X); iii)Cc*(g) = f*. Proof: The proof is exactly the same as in [10, Theorem 13.19 on p. 268]. Th* *ere only proper actions are considered but the same methods go through because here we a* *re dealing with the easy case where all fixed point sets are simply connected, the isotrop* *y 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 * *Cc*(E) is a free resolution over the orbit category Or(G; F) of the constant Or(G; F)-mod* *ule Z_with value Z. Proof: The modules are free by Lemma 2.2. It remains to show that Cc*(EH ), H 2* * F has the homology of a point. But this follows from the fact that for discrete H the* * space EH has a canonical CW -structure whose n-skeleton in exactly EHn and EH is weakly con* *tractible and hence contractible. | To shorten the next proof we start with the following lemma. Its proof is p* *urely 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 com* *plex and f* : C* _____-D* a chain map with Hi(cone(f*)*) = 0; i 2: Then there is a free* * chain complex C0*and a chain homotopy equivalence g* : C0*____-D* with C0*|2 = C* and* * g*|2 = f*. If C* is finite and D* is homotopic to a finite free chain complex, resp. a fre* *e complex of finite type, then C0*can be chosen to be finite, resp. of finite type. If D* is* * homotopic to a finite-dimensional free complex, then C0*can be chosen to be finite-dimensional. Proof of Theorem 0.1: The "only if"-case is clear for the first three assertion* *s 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 Z_and let E = E(G; F) be* * a G- CW -model with finite 2-skeleton in the second and third case. By adding elemen* *tary chain complexes, i.e. complexes concentrated in two consecutive dimensions with the i* *dentity as 7 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 Cc*(E) also gives a free resolution of Z_by Lemma 2.6, we have * *a homotopy equivalence g* : P* _____-Cc*(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* __* *__-Cc*(E) which induces the identity in dimensions 0,1 and 2. Therefore we can apply Prop* *osition 2.5 to the inclusion i : E2 ____-E and f*. The result is a G-CW -complex X with X2* * = E2 together with a homotopy equivalence k : X ____-E and Cc*(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 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 con* *sists of exactly one point. In this section we want to show that there is a close relation betwe* *en classifying spaces of a totally disconnected group G and its discretization Gd. The reason * *for this is, that homotopy does not see the difference between a totally disconnected group * *G and Gd, i.e. the canonical map Gd=Hd ____- resGdGG=H is a weak Gd-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 neighborhoo* *ds. Lemma 3.1 For a locally compact group the conditions (T), (D) and (FS) are e* *quivalent. Proof: The implications (T ) ) (D) ) (F S) are shown in [7, Theorem 7.7 on p. 6* *2]. The implication (F S) ) (T ) is done as follows: Let U be a set containing two dist* *inct points x and y. We show that U is disconnected. Let V be an open and compact neighborhoo* *d of x, not containing y. Then (V \ U) q (V c\ U) = U is a disjoint union of two nonemp* *ty 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 has Y . In particula* *r, 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. 8 Lemma 3.3 Let f : X ____-Y be a map. If f-1(y) is weakly contractible for al* *l y 2 Y and Y is totally disconnected, then f is a weak homotopy equivalence. If f-1(y) is * *contractible for all y 2 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. Th* *en the map iX : X ____- resGdGG xGd X; x 7! [e; x] is a weak Gd-homotopy equivalence. Proof: We begin with the case, where X is a homogeneous space Gd=Hd for a clos* *ed subgroup H G. Then G xGd Gd=Hd ____-G=H; [g; g0Hd] 7! gg0H is a G-homeomorphism. The obvious map Gd=Hd ! resGdGG=H is a weak Gd-homotopy equivalence by Lemma 3.2 and Lemma 3.3. Hence iGd=Hd : Gd=Hd ____- resGdGG xGd * *Gd=Hd is a weak Gd-homotopy equivalence. Next we prove the claim for all skeleta Xn by induction over n. The inducti* *on begin- ning n 0 follows from the case of a homogeneous space since X0 is a disjoint u* *nion of homogeneous spaces. In the induction step from n - 1 to n one chooses a Gd-push* *out a Gd=Hix Sn-1 ____-Xn-1 In | \ | | | | | | || a ?| ?| Gd=Hix Dn ______-Xn In and checks using the fact that G is locally compact that the induced diagram is* * a Gd-pushout a resGdGG xGd Gd=Hix Sn-1 ___-resGdGG xGd Xn-1 In || \ | | | | | | | a ?| ?| resGdGG xGd Gd=Hix Dn ____-resGdGG xGd Xn: In Notice that in both`diagrams the left`vertical arrows are Gd-cofibrations and t* *he various maps iY for Y = InGd=Hix Sn-1, Y = InGd=Hix Dn, Y = Xn-1 and Y = Xn map the two diagrams to one another. By the induction hypothesis the first three a* *re weak Gd-homotopy equivalences. Hence iXn is a weak Gd-homotopy equivalence [10, Lemm* *a 2.13 on p. 38]. Let K Gd be a subgroup. Since XK has the weak topology withirespect to t* *hej K filtration given by the subspaces XKn and G is locally compact, resGdGG xGd X * * has the 9 i * * jK weak topology with respect to the filtration given by the subspaces resGdGG xG* *d Xn . Since (iXn)K is a weak homotopy equivalence for n 0, the same follows for (iX * *)K . | Corollary 3.5 Let G be a totally disconnected locally compact group and F a fa* *mily of subgroups of G. If E(Gd; Fd) is a Gd-CW -model for the classifying space of Gd* * for the family Fd, then G xGd E(Gd; Fd) is a G-CW -model for E(G; F). Proof: We have for any K 2 F by Lemma 3.4: (G xGd E(Gd; Fd))K = (resGdGG xGd E(Gd; Fd))K 'w E(Gd; Fd)Kd 'w {*}: | Proposition 3.6 Let G be totally disconnected and let X be a G-CW -complex th* *at is d- dimensional (resp. finite, resp. of finite type). Then resGdGX has a Gd-CW -app* *roximation Y that is d-dimensional (resp. finite, resp. of finite type) and whose isotropy* * groups are the same as the one of X. If X is a G-CW -model for E(G; F), then Y is a Gd-CW -mod* *el for E(Gd; Fd). Proof: For n -1 we construct by induction a Gd-CW -complex Yn together with a * *Gd- approximation fn : Yn ____- resGdGXn such that Yn-1 is a subcomplex of Yn and f* *n|Yn-1= fn-1. The induction begin n = -1 is given by the empty set. For the induction step fr* *om n - 1 to n we proceed as follows. Choose a pushout a r G=Hix Sn-1 ____- Xn-1 In | \ | | | | | | || a ?| ?| G=Hix Dn ______-Xn In ` ` G for the n-skeleton of Xn. Let i : InGd=(Hi)d ! InresdGG=Hi be the obvious w* *eak Gd- homotopy equivalence (see Lemma 3.4). Since by the induction hypothesis fn-1 is* * a weak Gd-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 Gd-map g : InGd=(Hi)dxSn-1 ____-Yn-1 and a Gd-homotopy h : InGd=(Hi)d x Sn-* *1 x I ______- resGdGXn-1 between fn-1 O g and r O (i x idSn-1). Let f0n-1: cyl(g) _* *___- resGdGXn-1 be`the obvious map given by h and fn-1. Its restriction to Yn-1 cyl(g) is fn-* *1 and to n-1 InGd=(Hi)d x S is r O (i x idSn-1). Thus we obtain a commutative diagram a a Gd=(Hi)d x Dn oe__ Gd=(Hi)d x Sn-1 _____-cyl(g) In In || | i x id|| i x id|| f0n-1|| | | | a ?| a ?| r ?| resGdGG=Hix Dn oe_ resGdGG=Hix Sn-1 ___-resGdGXn-1: In In 10 Taking the pushout of the upper row yields a n-dimensional Gd-CW -complex Yn wh* *ich contains Yn-1 as Gd-CW -subcomplex and is finite if Xn is finite. Moreover, we * *get by the pushout property a Gd-map fn : Yn ! resGdGXn which extends fn-1 : Yn-1 ! resGdG* *Xn-1 and is a weak Gd-homotopy equivalence, since all vertical maps are weak Gd-homotopy* * equiva- lences ([10, Lemma 2.13 on p. 38]). Now put Y := colimYn. Then f := colimfn : Y* * ! resGdGX n!1 n!1 is the Gd-CW -approximation we look for. | Proof of Theorem 0.2: Follows from Corollary 3.5 and Proposition 3.6. | 4 Locally Compact Groups The strategy of our study of locally compact groups is the following. Any local* *ly compact p __ group G gives rise to a short sequence of the_form 1 ____-G0 ____-G ____-G ___* *_-1 with G0 locally compact and connected and with G locally compact and totally discon* *nected. This reduces the study of G to the study of locally compact connected groups, t* *hat are very similar to Lie groups by the solution of Hilbert's fifth problem (cp. [6])* *, and to locally compact totally disconnected groups, that are similar to their discrete underly* *ing 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 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 * *E_G and 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 necessaril* *y almost connected. From now on any locally compact group G is supposed to satisfy condi* *tion (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 pr* *ojection prH : (G=K)H ____-(G=L)H is a weak homotopy equivalence. If G=L is discrete, th* *en for any compact H G the projection prH : (G=K)H ____-(G=L)H is a homotopy equivale* *nce. Proof: If H is not subconjugated to L, all spaces are empty. So, let H be subco* *njugated 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 2 G. Then in the ass* *ociated pr fiber bundle Y ____-G xL Y ____-G=L, the typical fiber maps homeomorphically o* *nto the preimage p-1(wL) of wL 2 G=L by sending y to the class of (w; y). If wL is in (* *G=L)H then -1 H -1 this implies wHw-1 L and we get an induced homeomorphism Y wHw ____-(pr ) * *(wL) for prH : (G xL Y )H ____-G=LH . Now let Y be L=K which is a model for J(L; COM* * ) by -1 H Theorem 4.1. Therefore Y wHw is contractible. Hence by Lemma 3.3 the map pr * *is a weak 11 homotopy equivalence if G=L and hence (G=L)H is totally disconnected, and a hom* *otopy equivalence, if G=L and hence (G=L)H is discrete. | __ __ Proposition 4.3 _Given_a G -CW -model E_G, there is a G-CW -model_E_G_and a G-* *map f : E_G ____-p*E_G with the following_properties (where p*E_G is E_G viewed as * *a G-space by the projection p : G ____-G ): __ (a) If E_G is d-dimensional (resp. finite, resp. of finite type), then E_G is* * d-dimensional (resp. finite, resp. of finite type); __ H (b) If G is discrete, then E_G is contractible for all compact H G; __ __ (c) G0\f : G0\E_G ____-E_G is a G -homotopy equivalence. Proof: We will construct_for each n -1 an n-dimensional_G-CW -complex Xn and a* * G- map fn : Xn ____-p*E_Gn to the n-skeleton of p*E_G such that Xn-1 is the (n - 1* *)-skeleton of Xn and fn|Xn-1 = fn-1 with the following properties: __ (a) fHn_:_XHn _____-(p*E_Gn)H is a weak homotopy_equivalence for all compact* * H G. If G is discrete, fHn : XHn ____-(p*E_Gn)H is a homotopy equivalence for * *all compact H G; (b) The isotropy groups of Xn are all compact; (c) Xn-1 is the n-skeleton of Xn and fn|Xn-1 = fn-1. There is a_bijective_cor* *respondence between the equivariant n-dimensional cells of Xn and of E_Gn; __ __ (d) G0\fn : G0\Xn ____-E_G nis a G -homotopy equivalence. Notice that we then can define E_G := colimXn and f := colimfn, and check that * *E_G and n!1 n!1 f have the desired properties. We proceed by induction over n. The induction begin n = -1 is given by X-1 * *:= ;. For the induction step from n - 1 to n we choose a G-pushout ` a __ I qi __ G =Hix Sn-1 ______n-E_Gn-1 In || \ | | | | | | | a __ ?| __?| G =Hix Dn _______-E_Gn: In Put Li:= p-1(Hi) G for i 2 In. Obviously Li is almost connected. Let Ki be a m* *aximal__ compact subgroup of Li for i 2 In (see Theorem 4.1). Since the projection p : G* * ____-G ~= __ induces a homeomorphism G=Li ____-G =Hi, G=Li is totally disconnected. Hence Le* *mma 4.2 implies that prHi : (G=Ki)H _____-(G=Li)H is a weak homotopy equivalence f* *or all 12 compact subgroups_H_of G and is a homotopy equivalence for all compact subgroup* *s H of G, provided that G is discrete. The same is true for fn-1 by induction hypothes* *is. Therefore we have a bijection induced by fn-1 ([10, Prop. 2.3 on p.35]) (fn-1)* n-1 * __ [G=Kix Sn-1; Xn-1]G ______- [G=Kix S ; p E_Gn-1]G: Using the Equivariant Cellular Approximation Theorem [10, Theorem 2.1 on p. 32]* * we get a cellular_G-map ri: G=KixSn-1 ! Xn-1 together with a G-homotopy hi: G=Kix Sn-1 x* * [0; 1] ! p*E_Gn-1 from fn-1 O ri to qiO (prix idSn-1). Consider the following commuta* *tive G- diagram ` a __ a __ I qi __ p*(G =Hi) x Dnoe____ p*(G =Hi) x Sn-1_____________________n-p*E_G* *n-1 In In 6|| ` 6| ` 6| | Inprix idDn | Inprix idSn-1| id|| | | ` | a | a | I qiO (prix idSn-1) |__ G=Kix Dn oe________ G=Kix Sn-1 _______________________n-p*E_G* *n-1 In In || | i1|| i1|| id || | | ` | a ?| a ?| I hi ?_* *_| G=Kix Dn x [0; 1] oe_ G=Kix Sn-1 x [0; 1]____________________n-p*E_G* *n-1 In In 6|| | i06|| i06|| fn-1 | | | ` || a | a | I ri | G=Kix Dn oe________ G=Kix Sn-1 _________________________n-Xn-* *1: In In __ Notice that the pushout of the first row is p*E_Gn. Denote the pushout of t* *he second, third and fourth row respectively by X0n, X00nand_Xn._The diagram above togethe* *r with the pushout property induces G-maps f0n: X0n____-p*E_Gn, f00n: X0n____-X00nand f000* *n: Xn ! X00n. The map f00nis a G-homotopy equivalence_and the maps (f0n)H and(f000n)H are wea* *k homotopy equivalences (homotopy equivalence if G is discrete) for each compact subgroup* * H G ([10, Lemma 2.13_on_p. 38]). We can choose a G-map (f00n)-1 : X00n! X0nwhich in* *duces the_ identity on p*E_Gn-1 and is a G-homotopy inverse of f00n. Now define fn : Xn __* *__-p*E_Gn by the composition f0nO (f00n)-1_O f000n. By construction fHn is a weak homotop* *y equivalence (homotopy equivalence if G is discrete) for all compact subgroups H G and Xn * *is a G-CW -complex with Xn-1 as (n - 1)-skeleton and has only compact isotropy group* *s. __ __ It remains to show that G0\fn : G0\Xn ! E_Gn is a G -homotopy equivalence. * *Since Li inherits the property (S) from G, we get a locally trivial fiber bundle Ki ____* *-Li ____-Li=Ki which is automatically a Serre fibration and hence induces a long exact homotop* *y sequence [14, Theorem 2.11 on p. 60, Theorem 3.6 on p. 65 and Corollary 3.11 on p. 67]. * *Thus we 13 get the following diagram Ki | | ??| ~ = 1 = ss1(Li=Ki)___-ss0(Ki)____-ss0(Li)____-ss0(Li=Ki) = 1: | | ??| G0\Li= Hi __ We_conclude_that p(Ki) = Hi holds for all_i 2 In. Hence G0\pri: G0\G=Ki ____-G_* *=Hi is a G -homeomorphism. Therefore G0\f0nis a G -homeomorphism. G0\(f00n)-1 is a G -ho* *motopy_ equivalence, since (f00n)-1_is_a G-homotopy equivalence, and G0\(f000n) is a G-* *homotopy equiv- alence, since_G0\fn-1 is a G -homotopy equivalence by the induction hypothesis.* * Hence G0\fn is a G -homotopy equivalence. | Proof of Theorem 0.3: Is implied by Proposition 4.3. | References [1] Abels, H.: "Parallelizability of proper actions, global K-slices and maxi* *mal compact subgroups", Math. Ann 212, 1 - 19 (1974). [2] Abels, H.: "A universal proper G-space", Math. Z. 159, 143 - 158 (1978). [3] Baum, P., Connes, A., and Higson, N.: "Classifying space for proper acti* *ons and K-theory of group C*-algebras", in: Doran, R.S. (ed.), C*-algebras, Co* *ntemporary Mathematics 167, 241 - 291 (1994). [4] tom Dieck, T.: "Transformation groups", Studies in Math. 8, de Gruyter (19* *87). [5] Farrell, F.T. and Jones, L.E.: "Isomorphism conjectures in algebraic K-the* *ory", J. of the AMS 6, 249 - 298 (1993). [6] Gluskow, V.M.: "The structure of locally compact groups and Hilbert's fift* *h problem", AMS Transl. (2) 15, 55 - 93 (1960). [7] Hewitt, E., Ross, K.A.: "Abstract Harmonic Analysis I", Springer (1979). [8] Illman, S.: "Smooth proper G-manifolds have unique equivariant triangulati* *ons and unique G-CW complex structures, Applications to equivariant Whitehead tor* *sion", preprint, Helsinki (1997). [9] Kropholler, P.H. and Mislin, G.: "Groups acting on finite dimensional spac* *es with finite stabilizers", Comment. Math. Helv. 73, 122 - 136 (1998). 14 [10] L"uck, W.: "Transformation groups and algebraic K-theory", Lecture Notes i* *n Math- ematics 1408 (1989). [11] L"uck, W.: "The type of the classifying space for a family of subgroups", * *Preprintreihe SFB 478 _ Geometrische Strukture in der Mathematik, M"unster, Heft 12, to * *appear in J. of Pure and Applied Algebra (1998). [12] L"uck, W. and Oliver, R.: "The completion theorem in K-theory for proper * *ac- tions of a discrete group", Preprintreihe SFB 478 _ Geometrische Strukture* *n in der Mathematik, M"unster, Heft 1 (1998). [13] Milnor, J.: "On spaces having the homotopy type of a CW -complex", Trans. * *of the AMS 90, 272 - 280 (1959). [14] Mimura, M. and Toda, H.: "Topology of Lie Groups, I and II", Translations* * of mathematical monographs 91, AMS, Providence, Rhode Island (1991). [15] Mostert, P.S: "Local cross sections in locally compact groups", Proc. Amer* *. Math. Soc. 4, 645 - 649 (1953). [16] Nucinkis, B.A.: "Is there an algebraic characterization for proper actions* *?", preprint (1998). [17] Whitehead, G.W.: "Elements of homotopy theory", Graduate Texts in Mathemat* *ics 61, Springer (1978). Addresses: Wolfgang L"uck and David Meintrup Institut f"ur Mathematik und Informatik Westf"alische Wilhelms-Universtit"at Einsteinstr. 62, 48149 M"unster, Germany lueck@math.uni-muenster.de, meintrd@math.uni-muenster.de http://wwwmath.uni-muenster.de/math/u/lueck 15