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]) Let G be a compact Lie group and X a space which 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 _{ p̂} is BG-null for any compact Lie group G, and in this sense gives a generalization of Miller’s theorem

Typeset by AMS-TEX

The author was supported in part by the National Science Foundation.

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 _{ p̂} is BG-null from 1.2, observe that Y _{ p̂} is HZ/p-local by [2] and BZ/p-null by Miller’s arguments. Note that Y _{ p̂} 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]) Let G 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)_{p̂}) ≃ H_{p̂} .

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. A map f : 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) and K(Q, n) (n ≥ 0) are W -null. It follows that if f : X ⟶ Y is a P_{W} -equivalence then H^{∗}(f, Z/q) and H^{∗}(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.

1.7 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) and P_{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 pcompact 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 A_{G}^{op} 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, then α′_{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.

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. There is 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, are not 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 ∈ C for all H of smaller dimension than G, then G ∈ C.

(2) If G is connected with center C and G/C ∈ C, then G ∈ 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 ∈ C for 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 ∈ C by property (1) and induction, G_{0} ∈ C by (2), and hence G ∈ C by (3). □

2.3 Finite groups. In the examples that come up in this paper, 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, and H ∈ C for all H of smaller

order than G, then G ∈ C.

(2) If G has a central subgroup C of order p and and G/C ∈ C, then G ∈ 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 a map c : 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 a space Y 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.

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

\begin{matrix}Φ(Y_{4}) & ⟶ & Φ(Y_{2}) \\ ↓ & & ↓ \\ Φ(Y_{3}) & ⟶ & Φ(Y_{1}) \end{matrix} ⟶^{t} \begin{matrix}Ψ(Y_{4}) & ⟶ & Ψ(Y_{2}) \\ ↓ & & ↓ \\ Ψ(Y_{3}) & ⟶ & Ψ(Y_{1}) \end{matrix}

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 ≤ 3 then t(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}, then t(Y) is an equivalence if each t(Y_{α}) is. Let C 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, and let Y_{n} be the n-skeleton of Y. The fact that Y ∈ C then follows from the fact that there is a homotopy pushout diagram

(∐_{n} Y_{n}) ∐ (∐_{n} Y_{n}) ⟶^{id + id} ∐_{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].

3.4 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, let Map(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 a space, 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 a space, and γ : C ⟶ Top a functor. Assume that for each object c of C, the map κ : 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). □

3.9. 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] there is a natural map γ(c) ⟶ hocolim γ for each object c of C. If X 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.

3.10 Proposition. Let C be a small category, X a space, and γ : C ⟶ Top a functor such that for each object c of C, the map κ : 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]} .

3.11 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), and it is possible to 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.

3.12 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_{B}^{op} 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, where E_{A} is determined by the (homotopy) pullback diagram

E_{A}  ⟶  hocolim γ ↓  ↓ hocolim A ⟶ hocolim(∗_{C}) = BC .

§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.

Proof of 4.1. 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, where H 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} ⟶ ∗_{A_{G}} 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 a torus. Let D ⊂ 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. □

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 ∈ C for 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, where H is of dimension less than d. By induction and 3.8 the natural transformation α_{G} ⟶ ∗_{A_{G}} 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, and that G/C ∈ C. It is necessary to show that G ∈ C. There is a fibration sequence

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. □

§5. 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 1.4.

5.2 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}) that a_{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]. □

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 ∈ C for all 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, where K ∈ 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}) that a_{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, and that G/C ∈ C. It is 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 3.4. □

§6. Calculating P_{W} (BG)

In this section W will denote the fixed space BZ/p. The goal is to compute P_{W} (BG) when G 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; if λ is a limit ordinal then X_{λ} = ∪_{λ′<λ}X_{λ′}; if λ = 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).

6.2 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 6.1.

6.4 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

\begin{aligned}X & ⟶^{f} & Y \\ ↓ & & ^{∼}↓ \\ L_{Z[1/p]}X & ⟶^{∼} & L_{Z[1/p]}Y \end{aligned}  \begin{aligned}X & ⟶^{f} & Y \\ ↓ & & ^{∼}↓ \\ P_{W} (X) & ⟶^{∼} & P_{W} (Y) \end{aligned}

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, then hocolim 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.

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) in 6.1.

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.

6.10 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 ∈ C for 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, each space α_{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}) ⟶ ∗_{A_{G}}, 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}) is Z/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. □

6.11 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. There is a P_{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) is also a loop space.

6.12 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]. □

6.13 Lemma. If G is a locally finite p-group then P_{W} (BG) is contractible. In particular, G ∈ C.

Proof. Note that G ∈ C if and only if P_{W} (BG) is contractible, since H̃_{∗}(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]). □

6.14 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. Then F 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) ∼ ∗ ⟶ Y

in which P_{W} (BD) is contractible by 6.13. 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, and C ⊂ 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 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) is Z/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}) ⟶ X̄ ⟶ 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 ⟶ X̄. 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 > 0 and j ≥ 2 and hence by obstruction theory that the fibration 6.15 has a section s : Bπ_{0}G ⟶ X̄. 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} X̄
↓  ↓
P_{W} (Bπ_{0}G) ∼ ∗ ⟶ Y

in which P_{W} (Bπ_{0}G) is contractible by 6.13. Clearly Y is simply connected and Z/pacyclic; by 6.6 the map X̄ ⟶ Y is a P_{W} -equivalence. The composite BG ⟶ X̄ ⟶ 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, 1972.

[2] A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150.

[3] A. K. Bousfield, Localization and periodicity in unstable homotopy theory, preprint (University of Illinois, Chicago) 1992.

[4] W. G. Dwyer, E. Dror, and D. M. Kan, An arithmetic square for virtually nilpotent spaces, Illinois J. Math. 21 (1977), 242–254.

[5] W. G. Dwyer and C. W. Wilkerson, The center of a p-compact group, preprint (Notre Dame) 1993.

[6] E. Dror Farjoun, Homotopy and homology of diagrams of spaces, Algebraic Topology, Proceedings, Seattle (1985), H. R. Miller and D. C. Ravenel, eds.,, Lect. Notes in Math. 1286, Springer, Berlin, 1987, pp. 93–134.

[7] E. Dror Farjoun, Localization with respect to a map and v_{1}-periodicity, Proceedings 1990 Barcelona Conference on Algebraic Topology, Lect. Notes in Math. 1509, Springer, Berlin, 1991.

[8] E. Dror Farjoun and J. H. Smith, Homotopy localization nearly preserves fibrations, Topology (to appear).

[9] E. Friedlander and G. Mislin, Locally finite approximations of Lie groups, II, Math. Proc. Camb. Phil. Soc. 106 (1986), 505–517.

[10] S. Jackowski and J.E. McClure, Homotopy decomposition of classifying spaces via elementary abelian p-groups, Topology 31 (1992), 113–132.

[11] S. Jackowski, J.E. McClure and R. Oliver, Homotopy classification of self-maps of BG via G-actions, Annals of Math. 135 (1992), 227–270.

[12] I. M. James, Fibrewise Topology, Cambridge University Press, Cambridge, 1989.

[13] H. R. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984), 39–87.

[14] J. M. Møller, Rational isomorphisms of p-compact groups, preprint (Mathematisk Institut, Kobenhavn) 1994.

[15] D. G. Quillen, Higher algebraic K-theory: I, Algebraic K-theory I, Lect. Notes in Math. 341, Springer, Berlin, 1973, pp. 85–147.

[16] A. Zabrodsky, Maps between classifying spaces, Algebraic topology and algebraic K-theory, Ann. Math. Studies 113, Princeton Univ. Press, Princeton, 1987, pp. 228–246.

University of Notre Dame, Notre Dame, Indiana 46556

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