THE CENTRALIZER DECOMPOSITION OF BG. W.G. Dwyer University of Notre Dame

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1 THE CENTRALIZER DECOMPOSITION OF BG W.G. Dwyer University of Notre Dame 1. Introduction Let G be a compact Lie group and p a fixed prime number. Recall that an elementary abelian p-group is an abelian group isomorphic to (Z/p) r for some r. Jackowski and McClure showed in [10] how to decompose the classifying space BG at the prime p as a homotopy colimit of spaces of the form BC G (V ), where V is a nontrivial elementary abelian p-subgroup of G and C G (V ) is the centralizer of V in G (see 2). If the center of G is trivial then each of the centralizers C G (V ) is a proper subgroup of G, and so in this case the decomposition theorem gives an explicit way of gluing together BG, at least at p, from the classifying spaces of smaller groups. In this paper we will use this decomposition to give parallel inductive proofs of three theorems about BG; the first two theorems are already known but the third is probably new. The prime p will be fixed in everything that follows. If X is a space, let L Z/p X denote the HZ/p-localization of X constructed by Bousfield [2]. A space is said to be HZ/p-local if the natural map X L Z/p X is an equivalence, or alternatively if any map f : A B which induces an isomorphism on mod p homology also induces an equivalence f # : Map(B,X) Map(A, X). If W and X are spaces, say that X is W -null if every map from W to X is canonically homotopic to a constant, in the sense that the map (1.1) κ : X Map(W, X) given by inclusion of constant maps is an equivalence (see [3] and [7]). 1.2 Theorem. (cf. [9], [16]) LetG be a compact Lie group and X aspacewhich is HZ/p-local and BZ/p-null. Then X is BG-null. 1.3 Miller s theorem. Suppose that Y is a finite complex. Miller [13] shows that Y is BG-null for any finite group G (this is the Sullivan Conjecture ). Theorem 1.2 implies (see below) that the Bousfield-Kan p-completion Y ˆp is BG-null for any compact Lie group G, and in this sense gives a generalization of Miller s theorem The author was supported in part by the National Science Foundation. Typeset by AMS-TEX

2 2 W. DWYER to compact Lie groups. Some kind of completion is definitely necessary here; one way to see this is to use 1.2 and the arithmetic square [4] to compute that, if G is a connected compact Lie group, the 3-sphere S 3 is usually not BG-null. To derive the fact that Y ˆp is BG-null from 1.2, observe that Y ˆp is HZ/p-local by [2] and BZ/p-null by Miller s arguments. Note that Y ˆp is equivalent to L Z/p Y if Y is simply connected or more generally Z/p-good [1]. If Y is a finite complex which is not Z/p-good, it seems to be unknown whether or not L Z/p Y is BZ/p-null. If G is a (topological) group, Z is a space, and f :BG Z is a map, say that f is null on finite p-groups if f (Bρ) is null homotopic for every finite p-group P and homomorphism ρ : P G. 1.4 Theorem. (cf. [9, 3.3], [11, 3.11]) LetG be a compact Lie group and Z a pointed connected space such that Z is HZ/p-local and ΩZ is BZ/p-null. Then a map f :BG Z is null homotopic if and only if it is null on finite p-groups. Remark. If H be a compact Lie group such that π 0 H is a p-group, the hypotheses of 1.4 apply to the space Z = L Z/p (BH). To see this use 1.3 and note that BH is Z/p-good [1, VII, 5], so that by the fibre lemma [1, Ch. II] there is an equivalence ΩL Z/p (BH) = Ω((BH)ˆp) Hˆp. Theorem 1.4 thus gives a criterion for maps between classifying spaces to be null homotopic at p. 1.5 The functor P W. The statement of the final theorem requires some more terminology from Bousfield [3] and Farjoun [7]. Suppose that W is some fixed space. Amapf : A B is said to be a P W -equivalence if f induces an equivalence Map(B,X) f # Map(A, X) for every W -null space X. Bousfield and Farjoun show that for any space X there is an associated W -null space P W (X) together with a natural P W -equivalence X P W (X). It is easy to check from the definitions that if X is any other W -null space with a P W -equivalence X X, then up to homotopy there is a unique equivalence X P W (X) which makes the appropriate diagram involving X commute. This implies that a map f is a P W -equivalence if and only if P W (f) is an equivalence. 1.6 A natural map. Suppose now that W =BZ/p and that q is a prime different from p. Let L Z[1/p] X denote Bousfield s HZ[1/p]-localization of the space X [2]. Direct checking with the definition shows that the Eilenberg-Mac Lane spaces K(Z/q, n) andk(q,n)(n 0) are W -null. It follows that if f : X Y is a P W -equivalence then H (f,z/q) andh (f,q) are isomorphisms, hence that H (f,z[1/p]) is an isomorphism, and hence that L Z[1/p] f : L Z[1/p] X L Z[1/p] Y is an equivalence. In particular, L Z[1/p] X L Z[1/p] P W (X) is an equivalence, and so the natural map P W (X) L Z[1/p] P W (X) gives up to homotopy a natural map P W (X) L Z[1/p] X.

3 CENTRALIZER DECOMPOSITION Theorem. Let G be a compact Lie group such that π 0 G is a p-group, and let W =BZ/p. Then the natural map P W (BG) L Z[1/p] (BG) is an equivalence. Remark. Miller s theorem (1.3) implies that in the above situation the space BG is ΣW -null and hence that the map BG P ΣW (BG) is an equivalence. This shows that there is a large difference between the spaces P W (BG) andp ΣW (BG). The functors P W and P ΣW are related somewhat more closely if W itself is a suspension [8]. Remark. The three theorems above have corresponding forms that apply to p- compact groups. For a version of 1.2 see [5, 9] and for a version of 1.4 see [14, 5]. The analogue of 1.7 is proved by an argument very similar to the one in 6. Remark. It is an interesting exercise to derive 1.2 from 1.7 (and 6.3), at least in the case in which the group G involved has π 0 G a finite p-group. Notation and terminology. We assume that all spaces have been replaced if necessary by weakly equivalent CW-complexes (for instance, by the geometric realizations of their singular complexes). The word equivalence means homotopy equivalence. A space is said to be Z/p-acyclic if it has the mod p homology of a point; a map is a Z/p-equivalence if it induces an isomorphism on mod p homology. The author would like to thank E. Farjoun and the referee for their suggestions. 2. The Jackowski-McClure theorem In this section we will briefly describe the main theorem of [10] and indicate the general way in which it can be used in inductive arguments. Suppose that G is a compact Lie group. Let A G be the category in which the objects are the nontrivial elementary abelian p-subgroups V of G; a morphism V V in A G is a group homomorphism f : V V with the property that there exists g G such that f(x) =gxg 1 for all x V. Note that specifying a morphism in A G does not involve choosing a particular such g. There is a functor α G from the opposite category Aop G to G-spaces which assigns to V the coset space G/C G (V ); if f : V V is realized by conjugation with g G, thenα G (f) assigns to the coset xc G (V ) the coset xgc G (V ). Let EG be the total space of a universal principal G-bundle and α G : A G Top the functor (EG α G )/G. The following two properties of this functor are easy to check. (1) For each object V of A G, the space α G (V ) is homeomorphic to EG/C G (V ) and thus equivalent to BC G (V ). (2) The unique G-maps α G (V ) pass to compatible maps α G(V ) BG. These induce a map a G : hocolim α G BG. See [1, Ch. XII] for a discussion of homotopy colimits. Recall from [1, XII, 3.2] that if C is a category, the classifying space BC is defined to be hocolim( C ), where C : C Top assigns to each object of C the one-point space. The classifying spaces BC and BC op are homotopy equivalent [15, p. 86]. 2.1 Theorem. [10] Suppose that G is a compact Lie group. Then (1) the map a G : hocolim α G BG is a Z/p-equivalence, and (2) the classifying space BA G is Z/p-acyclic.

4 4 W. DWYER Remark. Jackowski and McClure actually show that a certain cohomology spectral sequence for H (hocolim α G, Z/p) collapses; this is much sharper than 2.1(1). In the course of this they show that the higher limits lim i H j (α G ) vanish for i>0. For j = 0 these are the higher limits of the constant functor on A G with value Z/p, which by [1, XI, 5] are just the mod p cohomology groups of BA G. This gives 2.1(2). Theorem 2.1 more or less states that at the prime p the homotopy type of BG can be constructed from the homotopy types of classifying spaces of smaller compact Lie groups, in such a way that the shape of the gluing diagram, in other words the classifying space of A G, is trivial at p. This suggests using the theorem to prove statements about BG by induction on the size (e.g., the dimension) of G. Thereis one minor problem with this: the values of the functor α G, which up to homotopy are the spaces BC G (V ) for elementary abelian p-subgroups of G, arenot necessarily the classifying spaces of Lie groups smaller than G. In fact,if G contains a central subgroup of order p then the space BG appears among the values of the functor α G and Theorem 2.1 amounts to a complicated but essentially circular construction of BG in terms of itself. This indicates that any inductive argument using 2.1 must treat centers in some special way. In practice this also involves treating disconnected groups in a special way, since in general it is only if G is connected that dividing out by the center of G gives a quotient group with trivial center. One inductive scheme that fits this situation is described in the following proposition. 2.2 Proposition. Suppose that C is class of (topological) groups which has the following three closure properties with respect to compact Lie groups G, H: (1) If G is connected, the center of G is trivial, and H Cfor all H of smaller dimension than G, theng C. (2) If G is connected with center C and G/C C,thenG C. (3) If G 0 is the identity component of G and G 0 C then G C. Then C contains every compact Lie group. Proof. We prove by induction on the dimension of the compact Lie group G that G C. Condition (1) implies that the trivial group is in C. Suppose that that G is of dimension d and assume inductively that H Cfor each compact Lie group H of dimension less than d (this is certainly true if d=0). Let G 0 be the identity component of G and C the center of G 0. Then G 0 /C Cby property (1) and induction, G 0 Cby (2), and hence G Cby (3). 2.3 Finite groups. Intheexamplesthatcomeupinthispaper,C will typically be the class of all topological groups G such that BG has some appropriate property. Statement 2.2(1) is then proved using 2.1. Statements 2.2(2)-(3) are proved by more direct arguments involving facts about finite groups; for instance, obtaining 2.2(3) usually involves knowing something about the finite group G/G 0. We will prove the necessary statements about finite groups by a subsidiary initial induction that depends on the following elementary proposition. 2.4 Proposition. Suppose that C is class of (topological) groups which has the following two closure properties with respect to finite groups G, H: (1) If G has no central subgroup of order p, andh Cfor all H of smaller

5 CENTRALIZER DECOMPOSITION 5 order than G, theng C. (2) If G has a central subgroup C of order p and and G/C C,thenG C. Then C contains every finite group. 3. The fibration principle In this section we will discuss some basic homotopy theoretic observations that turn out to be useful in proving Theorems 1.2, 1.4 and 1.7. We first discuss mapping spaces for which the domain is the total space of a fibration (3.1), and then mapping spaces for which the domain is a homotopy colimit (3.6). We end by explaining how to tie these two discussions together (3.11). 3.1 Maps from the total space of a fibration. 3.2 Proposition. (Fibration Principle) Suppose that f : E B is a fibration over a connected base B with fibre F, and that X is some space. Then there is another naturally associated fibration f X : E X B with fibre Map(F, X) such that the space of sections of f X is equivalent to Map(E,X). The way to understand this proposition is to picture E as a fibre bundle over B and notice that giving a map E X amounts to giving, for each point b B, a map from a copy of F to X. We will sketch a direct proof (see also 3.11). For a topological treatment of fibrewise function spaces, see [12, Ch. 9]. Proof of 3.2 (Sketch). Let G =Aut(F ) denote the monoid of self homotopy equivalences of F. The monoid G acts from the left on F, and associated to this action is a universal fibration u : E(F ) BG with fibre F. Since u is universal there is amapc : B BG, unique up to homotopy, such that the pullback of u over c is equivalent to the fibration f. We denote such a pullback B BG E(F ); the map c is understood in this notation. The monoid G also acts from the right (by composition) on M = Map(F, X). Associated to this action is a fibration v : E(M) BG with fibre M. The evaluation map F M X induces a map (3.3) e : E(F ) BG E(M) X. Consider now the category C of spaces over BG. An object in this category is aspacey together with a map g : Y BG; a morphism h : Y Y is a map of spaces such that g h = g. For any object Y of C let Ψ(Y ) denote the mapping space Map(Y BG E(F ),X) and Φ(Y ) the space of sections of the fibration Y BG E(M) Y. The map e above (3.3) induces a map t(y ):Φ(Y ) Ψ(Y ) which gives a natural transformation between the two indicated functors C op Top. To prove the proposition we will show that t(b) is an equivalence.

6 6 W. DWYER The map t(y ) is an equivalence if Y is a point or more generally if Y is a contractible space; in this case the domain and range of t(y ) are each equivalent to M = Map(F, X). Suppose that Y 1 Y 2 Y 3 Y 4 is a homotopy pushout diagram of spaces over BG. It is not hard to see that the induced diagram Y 1 BG E(F ) Y 2 BG E(F ) Y 3 BG E(F ) Y 4 BG E(F ) is also a homotopy pushout diagram. The natural transformation t then gives a map of squares Φ(Y 4 ) Φ(Y 2 ) Φ(Y 3 ) Φ(Y 1 ) t Ψ(Y 4 ) Ψ(Y 2 ) Ψ(Y 3 ) Ψ(Y 1 ) in which each square is a homotopy pullback square (because mapping constructions like Φ and Ψ convert homotopy pushouts to homotopy pullbacks [1, XII 4.1]). It follows that if t(y i ) is an equivalence for i 3thent(Y 4 ) is also an equivalence. Both Φ and Ψ convert disjoint unions to products, so if {Y α } is a collection of spaces over BG with disjoint union Y = α Y α,thent(y ) is an equivalence if each t(y α )is. LetC be the smallest homotopy invariant class of spaces over BG which contains all contractible spaces, is closed under homotopy pushouts, and is closed under disjoint unions. By the discussion above, t(y ) is an equivalence for each space Y in C. It is clear by induction on dimension that C contains every space Y over BG such that the underlying space of Y is a finite dimensional CW-complex. Suppose that Y is an infinite complex over BG, andlety n be the n-skeleton of Y. The fact that Y Cthen follows from the fact that there is a homotopy pushout diagram id + id ( n Y n) ( n Y n) n Y n id +s n Y n Y in which the map s is a shift map derived from the inclusions Y n Y n+1. One case is particularly interesting. The following proposition is closely related to work of Zabrodsky as reformulated by Miller [13, 9.5].

7 CENTRALIZER DECOMPOSITION Proposition. Suppose that f : E B is a fibration over a connected base B with fibre F, and that X is a space. If the map κ : X Map(F, X) is an equivalence (1.1), then the restriction map f # : Map(B,X) Map(E,X) is an equivalence. Proof. Let g : B B be the identity fibration. Under the stated hypotheses the map f itself induces an equivalence between the fibration f X of 3.2 and the fibration g X (which is the projection X B B). The induced map between spaces of sections, which is essentially f #, is then also an equivalence. Restricting attention to individual mapping space components gives a more specialized variant of 3.4. If F and X are spaces, let Map(F, X) [F ] denote the space of maps F X which are homotopic to constant maps. More generally, if f : E B is a fibration over a connected base B with fibre F,letMap(E,X) [F ] denote the space of those maps E X which are homotopic to constant maps when restricted to F. 3.5 Proposition. Suppose that f : E B is a fibration over a connected base B with fibre F, and that X is a space. If the map κ : X Map(F, X) [F ] is an equivalence, then the restriction map f # : Map(B,X) Map(E,X) [F ] is an equivalence. 3.6 Maps from a homotopy colimit. There are mapping space results roughly parallel to the above ones with the notion total space of a fibration replaced by the notion homotopy colimit. The analogue of 3.2 is the following proposition of Bousfield and Kan. 3.7 Proposition. [1, XII, 4] Suppose that C is a small category, X aspace,and γ : C Top a functor. Let Map(γ,X):C op Top be the functor which assigns to each object c the space Map(γ(c),X). Then there is an equivalence Map(hocolim γ,x) holim Map(γ,X). In a situation like that of 3.4 this gives the following. 3.8 Proposition. Let C be a small category, X aspace,andγ : C Top a functor. Assume that for each object c of C, themapκ : X Map(γ(c),X) is an equivalence (1.1). Then the unique natural transformation γ C induces an equivalence Map(BC,X) = Map(hocolim( C ),X) Map(hocolim γ,x). Proof. By assumption the functor Map(γ,X) (see 3.7) is equivalent to Map( C,X). The homotopy limit of this constant functor is Map(hocolim( C ),X) As above, restricting attention to individual mapping space components gives a specialized variant. If C is a small category and γ : C Top is a functor, then by the definition of homotopy colimit [1, XII, 2] thereisanaturalmap γ(c) hocolim γ for each object c of C. IfX is a space, let Map(hocolim γ,x) [γ] denote the space of maps f : hocolim γ X such that for each object c of C the restriction of f to γ(c) is homotopic to a constant map.

8 8 W. DWYER 3.10 Proposition. Let C be a small category, X aspace, andγ : C Top a functor such that for each object c of C, themapκ : X Map(γ(c),X) [γ(c)] is an equivalence. Then the unique natural transformation γ C induces an equivalence Map(BC,X) = Map(hocolim( C ),X) Map(hocolim γ,x) [γ] Total spaces vs. homotopy colimits. Propositions 3.2 and 3.7 are tied together by the fact that the total space of a fibration over B with fibre F is equivalent to the homotopy colimit of a functor whose values are all equivalent to F and whose domain category has classifying space equivalent to B. Suppose for simplicity that B is the geometric realization of a simplicial complex K. Let C B be the category in which an object is a (closed) simplex σ of B and there is a single morphism σ σ if σ is contained in σ (there are no other morphisms). The classifying space BC B is the geometric realization of the barycentric subdivision of K and so is homeomorphic to B. If f : E B is a fibration with fibre F,then there is a functor γ E : C B Top which sends σ to f 1 (σ),anditispossibleto check that hocolim γ E is equivalent to E. This is obvious if B itself is a simplex, and in general one can make an induction, based on homotopy pushouts, over the skeletal filtration of B (cf. proof of 3.2). By 3.7 there is an equivalence Map(E,X) holim Map(γ E,X). Now consider the following proposition Proposition. Suppose that C is a small category and that γ : C Top is a functor which sends each object of C to a space equivalent to Z and each morphism of C to an equivalence. Then (1) the natural map hocolim γ BC is up to homotopy a fibration with fibre Z, and (2) the space of sections of this fibration is equivalent to holim γ. Applying this proposition to the functor Map(γ E,X) shows that hocolim γ E is up to homotopy the total space of a fibration over B with fibre Map(F, X) and space of sections equivalent to Map(E,X). (Note [15, p. 91] that the classifying space BC op B is equivalent to BC B, and hence to B). This gives a proof of 3.2 which uses 3.7. Remark. The first statement of 3.12 is a form of Quillen s Theorem B [15, p. 97]. Statement 3.12(2) can be proved by using the interpretation of homotopy limit in [6, 2.12]. This identifies holim γ up to homotopy as the mapping space Map( C,γ), where C is a free resolution of the functor C (i.e, a CW-functor [6, 1.16] weakly equivalent to C ) and the maps are computed in the category of functors C Top. The space hocolim( C ) is equivalent to BC. One proves by skeletal induction ([6, 1.16], cf. proof of 3.2) that if A : C Top is any CW-functor then the space Map(A, γ) is equivalent in a natural way to the space of sections of the fibration E A hocolim A, wheree A is determined by the (homotopy) pullback diagram E A hocolim γ. hocolim A hocolim( C )=BC

9 CENTRALIZER DECOMPOSITION 9 4. Maps into spaces which are BZ/p-null In this section we will prove 1.2. Suppose that X is a space which is HZ/p-local and BZ/p-null. Let C be the class of all topological groups G with the property that κ : X Map(BG, X) is an equivalence; it is necessary to prove that C contains all compact Lie groups. Recall that, by the definition of what it means to be HZ/p-local, any Z/p-equivalence A B induces an equivalence Map(B,X) Map(A, X). 4.1 Lemma. If G is a discrete locally finite group (i.e. a union of finite groups) then G C. Remark. The following argument is a prototype of the argument below for compact Lie groups. Proofof4.1.We prove using 2.4 and induction on the order of G that any finite group G belongs to C. The group Z/p belongs to C by definition. If G has a central subgroup C of order p then applying the 3.4 to the fibration BC BG B(G/C) and using the induction hypothesis shows that G C.If G has no central subgroup C of order p, then for each object V of A G the space α G (V ) has the homotopy type of BH, whereh is of smaller order than G. By 2.1(1), the map hocolim α G BG induces an equivalence Map(BG, X) Map(hocolim α G,X). By induction and 3.8 the natural transformation α G AG induces an equivalence Map(BA G,X) Map(hocolim α G,X). However 2.1(2) guarantees that BA G is Z/p-acyclic, so that the map κ : X Map(BA G,X) is an equivalence. Tracing through the various identifications shows that κ : X Map(BG, X) is also an equivalence. The passage to general locally finite groups is by a standard homotopy colimit argument [13, proof of 9.8]. 4.2 Lemma. If G is an abelian compact Lie group, then G C. Proof. The group G is isomorphic to the product of a finite abelian group with atorus. LetD G be the group of elements of order a power of p, considered as a discrete group. It is not hard to see by explicit calculation that the map BD BG induces an isomorphism on mod p homology and therefore an equivalence Map(BG, X) Map(BD, X). The desired result follows from 4.1.

10 10 W. DWYER There are now three steps to carry out, which correspond to the three hypotheses of 2.2. Step I. Suppose that G is a connected compact Lie group of dimension d with a trivial center, and that H Cfor all compact Lie groups H of dimension less than d. It is necessary to prove that G C. By 2.1(1), the map hocolim α G BG induces an equivalence Map(BG, X) Map(hocolim α G,X). Since G is connected and has trivial center, for each object V of A G the space α G (V ) has the homotopy type of BH, whereh is of dimension less than d. By induction and 3.8 the natural transformation α G AG induces an equivalence Map(BA G,X) Map(hocolim α G,X). However 2.1(2) guarantees that BA G is Z/p-acyclic, so that the map κ : X Map(BA G,X) is an equivalence. Tracing through the various identifications shows that κ : X Map(BG, X) is also an equivalence. Step II. Suppose that G is connected with center C, andthatg/c C. necessary to show that G C. There is a fibration sequence It is BC BG B(G/C) in which by 4.2 the fibre BC belongs to C. By 3.4, then, the restriction map Map(B(G/C), X) Map(BG, X) is an equivalence. The result now follows from the fact that G/C C. Step III. Suppose that the identity component G 0 of G belongs to C; it is necessary to show that G C. There is a fibration sequence BG 0 BG Bπ 0 G. By 3.4 and the assumption on G 0 the restriction map Map(Bπ 0 G, X) Map(BG, X) is an equivalence. The result now follows from the fact (4.1) that π 0 G C.

11 CENTRALIZER DECOMPOSITION Maps null on finite p-groups In this section we will prove 1.4, or more accurately a slight generalization of it. If G is a topological group and X is a space, let Map(BG, X) [p] denote the space of all maps f :BG X which are null on finite p-groups. Let Z be a pointed connected space such that Z is HZ/p-local and ΩZ is BZ/p-null. Define C to be the class of all topological groups G with the property that the map κ : Z Map(BG, Z) [p] is an equivalence (1.1). What we will prove is the following. 5.1 Theorem. The class C contains all compact Lie groups G. In particular, if G is a compact Lie group then the space Map(BG, Z) [p] is connected. This implies that every map BG Z which is null on finite p-groups is homotopic to a constant map, which is Lemma. If G is a locally finite group (i.e., a union of finite p-groups) then G C. Remark. As in 4, this is a prototype of the proof below for compact Lie groups. Proof. We prove using 2.4 and induction on the order of G that any finite group G belongs to C. If G has a central subgroup C of order p then applying the 3.5 to the fibration BC BG B(G/C) and using the induction hypothesis shows that G C.If G has no central subgroup of order p, for each object V of A G the space α G (V ) has the homotopy type of BK for a group K of order smaller than G. By 2.1(1), the map hocolim α G BG induces an equivalence a # G :Map(BG, Z) Map(hocolim α G,Z). Moreover, a check with the definitions shows that the composite maps (cf. 3.9) α G (V ) hocolim α G a G BG are obtained up to homotopy by applying the classifying space construction to homomorphisms K G. It follows in the notation of 3.9 (with γ = α G )thata # G induces an equivalence from Map(BG, Z) [p] to a union of components of the space Map(hocolim α G,Z) [γ]. By induction, 3.10, and 2.1(2), the map κ : Z Map(hocolim α G,Z) [γ] is an equivalence. Tracing through the various identifications gives the desired equivalence Z Map(BG, Z) [p]. The result follows for arbitrary locally finite groups by a homotopy colimit calculation [13, proof of 9.8].

12 12 W. DWYER 5.3 Lemma. If G is an abelian compact Lie group, then G C. Proof. This follows from 5.2: as in 4.2, there is a locally finite p-group D and a homomorphism D G which induces an equivalence Map(BG, Z) Map(BD, Z). The proof of 5.1 now has three steps, which correspond to the three steps of 2.2. Step I. Suppose that G is a connected compact Lie group of dimension d with a trivial center, and that K Cforall compact Lie groups K of dimension less than d. It is necessary to prove that G C. By 2.1(1), the map hocolim α G BG induces an equivalence a # G :Map(BG, Z) Map(hocolim α G,Z). Since G is connected and has trivial center, for each object V of A G the space α G (V ) has the homotopy type of BK, wherek C. Moreover, a check with the definitions shows that the composite maps (cf. 3.9) α G (V ) hocolim α G a G BG are obtained up to homotopy by applying the classifying space construction to homomorphisms K G. It follows in the notation of 3.9 (with γ = α G )thata # G induces an equivalence from Map(BG, Z) [p] to a union of components of the space Map(hocolim α G,Z) [γ]. By induction, 3.10, and 2.1(2), the map κ : Z Map(hocolim α G,Z) [γ] is an equivalence. Tracing through the various identifications gives the desired equivalence Z Map(BG, Z) [p]. Step II. Suppose that G is connected with center C, andthatg/c C. Itis necessary to show that G C. This is the same as the second step in the proof of 1.2, but uses 5.3 and 3.5 instead of 4.2 and 3.4. Step III. Suppose that the identity component G 0 of G belongs to C; it is necessary to show that G C. This is the same as the third step in the proof of 1.2, but uses 5.2 and 3.5 instead of 4.1 and Calculating P W (BG) In this section W will denote the fixed space BZ/p. The goal is to compute P W (BG) wheng is a compact Lie group such that π 0 G is a p-group. 6.1 Remark. As described in [3], the space P W (BG) is built by constructing a nested collection of spaces X λ, one for each countable ordinal λ. The space X 0 is BG; ifλ is a limit ordinal then X λ = λ <λx λ ;ifλ = λ + 1 is a successor ordinal then X λ is obtained from X λ by adjoining cones on all maps of W and its suspensions into X λ. The space P W (BG) is then the union or colimit λ X λ (note that the number of spaces in this union is uncountable).

13 CENTRALIZER DECOMPOSITION Lemma. For a 1-connected space X the following three conditions are equivalent: (1) X is Z/p-acyclic. (2) For each i 2, π i X is a module over Z[1/p]. (3) The natural map X L Z[1/p] X is an equivalence. Moreover, these conditions imply (4) X is W -null. Proof. The equivalence of (1) and (2) follows from Serre mod-c theory, since X is Z/p-acyclic if and only if the reduced integral homology groups of X are uniquely p-divisible, which if π 1 X is trivial is the case if and only if the homotopy groups of X are uniquely p-divisible. The equivalence between (2) and (3) is from [1, V, 3] (note that since X is simply connected the Bousfield localization L Z[1/p] X is equivalent to the Bousfield-Kan space Z[1/p] (X) [2]). The fact that (2) implies (4) results for instance from a direct calculation with obstruction theory. Note that (4) does not imply the others, e.g., the n-sphere (n 2) is 1-connected and satisfies (4) (by 1.3), but does not satisfy (1). 6.3 Remark. If X is a connected space with the property that H 1 (X, Z[1/p]) = 0, e.g., X =BG with π 0 G a p-group, then L Z[1/p] X is 1-connected [1, VII, 3.2] [2] and satisfies the conditions given in 6.2 [1, V, 3]. In particular it follows from Theorem 1.7 that if G is a compact Lie group with π 0 G a p-group then all of the mod p homology of BG can be killed by the iterated cone adjunction process of Remark. Lemma 6.2 leads to the following recognition principle. Let X be a connected space such that H 1 (X, Z[1/p]) is trivial, and suppose that we can construct a P W -equivalence f : X Y such that Y is 1-connected and Z/pacyclic. Then the natural map P W (X) L Z[1/p] X is an equivalence. To see this, note first that H (f,z[1/p]) is an isomorphism (1.6). The desired result now follows from the commutative diagrams X L Z[1/p] X f Y L Z[1/p] Y X P W (X) f Y P W (Y ) in which by 6.2 the indicated vertical arrows are equivalences (for instance, 6.2(4) implies that Y P W (Y )). 6.5 Lemma. [3, 2.5] The class of P W -equivalences is closed under homotopy colimits, in the sense that if C is a small category, γ, γ : C Top are functors, and τ : γ γ is a natural transformation which gives a P W -equivalence τ c : γ(c) γ (c) for each object c of C, thenhocolim τ : hocolim γ hocolim γ is a P W -equivalence. 6.6 Remark. Note that 6.5 is proved with 3.7. This lemma implies in particular that P W -equivalences are stable under cobase change, i.e., if X X is a P W - equivalence and X Y is a map, then the natural inclusion of Y in the homotopy pushout of the diagram X X Y is also a P W -equivalence.

14 14 W. DWYER 6.7 Remark. The class of Z/p-equivalences is also closed under arbitrary homotopy colimits [1, XII, 5.7]. 6.8 Lemma. Let C be a small category such that BC is connected, γ, γ : C Top functors, and τ : γ γ a natural transformation. Suppose that for each object c of C the map τ c : γ(c) γ (c) is a map between connected spaces which induces a surjection of fundamental groups. Then the map hocolim τ : hocolim γ hocolim γ is also a map between connected spaces which induces a surjection of fundamental groups. Proof. This follows from the van Kampen theorem and the explicit construction of the homotopy colimit in [1, XII, 5]. 6.9 Lemma. [3, 2.9] Suppose that X is a connected space. Then P W (X) is connected, and the map X P W (X) induces a surjection of fundamental groups. Remark. This is clear from the description of P W (X) in6.1. Let C be the class of (topological) groups with the property that the map P W (BG) L Z[1/p] (BG) is an equivalence. As usual, there are three steps (cf. 2.2) involved in proving that C contains every compact Lie group G such that π 0 G is a p-group. We have to modify Step I slightly to handle a technical issue connected with the center (see Step II). The following theorem guarantees that the main inductive step does not leave the class of compact Lie groups G such that π 0 G is a p-group Theorem. [11, A.4] Suppose that G is a compact Lie group such that π 0 G is a p-group, and that K is a finite p-subgroup of G (for example, K might be an elementary abelian p-subgroup of G). Then π 0 C G (K) is also a p-group. Step I. Suppose that G is a connected compact Lie group of dimension d with no element of order p in its center. Assume that H Cfor all H of dimension less than d such that π 0 H is a p-group. We have to show that G C. Let Y be the space which fits into the homotopy pushout diagram hocolim α G a G BG hocolim P W (α G ) Y in which the left hand vertical arrow is induced by the natural map α G P W (α G ). By 6.9, 6.8, and the van Kampen theorem, Y is 1-connected. By 6.5 the left hand vertical map is a P W -equivalence, and so (6.6) the map BG Y also is. Part (1) of 2.1 gives that the upper horizontal arrow in the above diagram is a Z/p-equivalence. Since G has no central element of order p, eachspaceα G (V )(V an object of A G ) is of form BH for some H which by induction (6.10) belongs to C, and so (6.3) each space P W (α G (V )) is Z/p-acyclic. It follows from 6.7, applied to the unique natural transformation P W (α G ) AG, that the induced map hocolim P W (α G ) BA G is a Z/p-equivalence, and thus by 2.1(2) that hocolim P W (α G )isz/p-acyclic. A Meyer-Vietoris sequence calculation now gives that Y is Z/p-acyclic, and so the desired result is a consequence of 6.4.

15 CENTRALIZER DECOMPOSITION P W and fibrations. Recall from [3, 4] that given a fibration sequence F E B over a connected base B, it is possible to apply P W fibrewise to obtain another fibration sequence P W (F ) Ē B. ThereisaP W -equivalence E Ē of spaces over B which on fibres gives the natural map F P W (F ). If E B is a principal fibration, then so is Ē B (cf. [3, 3]); in particular, taking E = and F = G shows that if G is a topological group or more generally a loop space then P W (G) isalsoaloopspace Lemma. Let G E B be a principal fibration sequence over a connected base B. Assume that the classifying space BP W (G) is W -null. Then the sequence P W (G) P W (E) P W (B) is also up to homotopy a fibration sequence. Proof. This is a restatement of [3, 4.3] Lemma. If G is a locally finite p-group then P W (BG) is contractible. In particular, G C. Proof. Note that G Cif and only if P W (BG) is contractible, since H (BG, Z[1/p]) vanishes (G is a union of finite p-groups) and so L Z[1/p] (BG) is contractible. We first prove by induction on the order of G that if G is a finite p-group then P W (BG) is contractible. This is clear if G is trivial or if G = Z/p. Otherwise, there exists a cyclic group σ of order p in the center of G and a corresponding fibration sequence Bσ BG B(G/σ) =BK with P W (BK) contractible by induction. Applying P W fibrewise (6.11) thus gives a P W -equivalence BG BK, which shows that P W (BG) is contractible too. The statement for a general locally finite group G follows 6.5 and the fact that BG can be expressed as the filtered homotopy colimit of the classifying spaces of the finite subgroups of G (cf. [1, XII, 3.5] or [13, 9.8]) Lemma. If G is an abelian compact Lie group such that π 0 G is a p-group, then G C. Proof. As in the proof of 4.2, let D G be the group of elements of order a power of p, considered as a discrete group. Let f :BD BG be the map induced by the inclusion D G. ThenF is a Z/p-equivalence and (by the assumption on π 0 G) π 1 (f) is a surjection. Let Y be the space with fits into the homotopy pushout diagram BD BG P W (BD) in which P W (BD) is contractible by By the van Kampen theorem Y is 1-connected. Clearly Y is Z/p-acyclic, and by 6.6 the map BG Y is a P W - equivalence. The desired result follows from 6.4. Step II. Suppose that G is a connected compact Lie group. Let C be the center of G, andc C the inverse image in C of the p-torsion subgroup of π 0 C. Assume that G/C C. (Observe that the center of G/C, which is isomorphic to C /C, has Y

16 16 W. DWYER no nontrivial element of order p, so that in the modified inductive scheme we are following the group G/C will be handled in Step I.) It is necessary to prove that G C. Since C is abelian, the fibration sequence BC BG B(G/C) is principal. By 6.14 and 6.3, P W (BC) is a 1-connected Z/p-acyclic space. By a Serre spectral sequence argument, the classifying space BP W (BC) (see 6.11) is also Z/p-acyclic, and so (6.2) is W -null. Lemma 6.12 thus shows that there is a fibration sequence P W (BC) P W (BG) P W (B(G/C)) From 6.3 it is clear that the base and fibre here are Z/p-acyclic. This implies the space P W (BG) isz/p-acyclic. The space P W (BG) is 1-connected (6.9) and so the desired result follows from 6.4. Step III. Suppose that G is a compact Lie group such that π 0 G is a p-group, and assume that the identity component G 0 C. It is necessary to prove that G C. Consider the fibration (6.15) P W (BG 0 ) X Bπ 0 G obtained by applying P W fibrewise (6.11) to the fibration BG 0 BG Bπ 0 G. There is a P W -equivalence BG X. By induction and 6.3, the space P W (BG 0 ) satisfies the conditions given in 6.2, and in particular the higher homotopy groups of this space are uniquely p-divisible. This implies that the (twisted) cohomology groups H i (Bπ 0 G, π j P W (BG 0 )) vanish for i>0andj 2 and hence by obstruction theory that the fibration 6.15 has a section s :Bπ 0 G X. Since P W (BG 0 )is Z/p-acyclic (6.2), H (s, Z/p) is an isomorphism; since P W (BG 0 ) is 1-connected, π 1 (s) is an isomorphism. Let Y be the space which fits into the homotopy pushout diagram Bπ 0 G s X P W (Bπ 0 G) Y in which P W (Bπ 0 G) is contractible by Clearly Y is simply connected and Z/pacyclic; by 6.6 the map X Y is a P W -equivalence. The composite BG X Y is then also a P W -equivalence, and the desired result follows from 6.4. References [1] A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math. 304, Springer, Berlin, [2] A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), [3] A. K. Bousfield, Localization and periodicity in unstable homotopy theory, preprint (University of Illinois, Chicago) [4] W.G.Dwyer,E.Dror,andD.M.Kan,An arithmetic square for virtually nilpotent spaces, Illinois J. Math. 21 (1977),

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