Centric Maps and Realization of Diagrams in the Homotopy Category

W. G. Dwyer and D. M. Kan

University of Notre Dame Massachusetts Institute of Technology

§1. Introduction

Let D be a small category. Suppose that X̄ is a D-diagram in the homotopy category (in other words, a functor from D to the homotopy category of simplicial sets). The question of whether or not X̄ can be realized by a D-diagram of simplicial sets has been treated by [5]. The purpose of this note is to study a special situation in which the treatment can be simplified quite a bit. We look at two examples to which this simplified treatment is applicable; both examples involve homotopy decomposition diagrams for compact Lie groups. Our results show that in at least one of these examples ([13]) the decomposition diagram is completely determined by its underlying diagram in the homotopy category. It is possible that this “rigidity” result will eventually contribute to a general homotopy theoretic characterization theorem for classifying spaces of compact Lie groups (cf. [8]).

Before going any further we have to introduce some notation. The symbol S will denote the category of simplicial sets and hoS the associated homotopy category obtained by localizing with respect to (i.e. formally inverting) weak equivalences. If C is a category and D is a small category, then C^{D} will stand for the category of D-diagrams in C; the objects of C^{D} are functors D → C and the maps of C^{D} are natural transformations. There is a projection functor π : S → hoS; we will use the same symbol for induced functors S^{D} → (hoS)^{D}. Both S and S^{D} admit closed simplicial model category structures [19, II] [20, p. 233] [4, §2] , and we will sometimes require without loss of generality that chosen objects in these categories be fibrant.

Given a small category D and a diagram X̄ in (hoS)^{D}, a realization of X̄ [5, 3.1] is a pair (Y, f) such that Y is an object of S^{D} and f is an isomorphism f : πY ≅ X̄ in (hoS)^{D}. A weak equivalence between two such realizations (Y, f_{Y}) and (Z, f_{Z}) is a weak equivalence h : Y → Z ∈ S^{D} such that f_{Z} ∘ (πh) = f_{Y}.

This work was supported in part by the National Science Foundation.

If f : A → B is a map of simplicial sets, let Map(A, B)_{f} denote the component of the simplicial mapping space Map(A, B) containing f [17, p. 17]. Call a map f : A → B between fibrant simplicial sets centric if precomposition with f induces a homotopy equivalence Map(A, A)_{1} → Map(A, B)_{f}. Here “1” stands for the identity map of A and Map(A, A)_{1}, also denoted haut(A)_{1}, is the identity component of the simplicial monoid of homotopy self-equivalences of A. It is clear that whether or not a map is centric is a homotopy property. We will say that a homotopy class of maps ϕ : A → B is centric if any representative is, and that a diagram X in S^{D} (resp. X̄ in (hoS)^{D} ) is centric if X(g) (resp. X̄(g)) is centric for each morphism g of D. Remark: A map i : G → H of discrete groups induces a centric map BG → BH of classifying spaces iff i induces an isomorphism between the center of G and the centralizer in H of i(G). This explains the choice of the term “centric”.

Given a centric diagram X̄ in (hoS)^{D} construct functors α_{i}X̄ : D^{op} → Ab (where Ab is the category of abelian groups) by setting α_{i}X̄(D) = π_{i} haut(X̄(D))_{1} for D an object of D; there is no basepoint problem with these homotopy groups because haut(X̄(D))_{1} is a simple space. For each map g : D_{0} → D_{1} in D the induced map α_{i}X̄(g) : α_{i}X̄(D_{1}) → α_{i}X̄(D_{0}) is the composite

π_{i} haut(X̄(D_{1}))_{1} ⟶^{γ_{#}} π_{i} Map(X̄(D_{0}), X̄(D_{1}))_{γ} ⟶^{(γ^{#})^{−1}} π_{i} haut(X̄(D_{0}))_{1}

where γ is a representative of X̄(g), γ_{#} is induced by postcomposition with γ and γ^{#} by precomposition.

Our main technical result is the following one.

Theorem 1.1. Suppose that D is a small category and that X̄ ∈ (hoS)^{D} is a centric diagram.

(1) If the groups \underleftarrow{lim}^{i+2}α_{i}X̄ vanish for i ≥ 1 then at least one realization of X̄ exists.

(2) If in addition the groups \underleftarrow{lim}^{i+1}α_{i}X̄ vanish for i ≥ 1 then any two realizations of X̄ are in the same weak equivalence class.

Remark: See [3, XI, §6] for a discussion of the right derived functors \underleftarrow{lim}^{i} of the inverse limit functor \underleftarrow{lim} : Ab^{D} → Ab.

Theorem 1.1 is actually a consequence of a sharper and more geometric calculation which is described in §2.

We study two examples, both involving diagrams which are used to express the classifying space of a compact Lie group as a homotopy direct limit (or at least as a homotopy direct limit up to Z/p homology). We first show that a slight modification of the centralizer decomposition diagram of [12] is centric. We then show that the p-toral decomposition diagram of [13] is centric, and point out that the results of [13] imply that this diagram satisfies the hypotheses of parts (1) and (2) of Theorem 1.1. It follows from 1.1(2), then, that the diagram of [13] is determined up to weak equivalence by its underlying diagram in the homotopy category.

Organization of the Paper: Section 2 contains a discussion of realization complexes and some calculations in the centric case. Theorem 1.1 is proved in §3. Finally, §4 deals with the centrializer diagram of [12] and §5 with the p-toral diagram of [13].

Notation and Terminology: Some of the simplicial sets used in this note as in [5] are nerves of categories which are not necessarily small. Nevertheless these nerves are homotopically small and it is possible to make standard homotopy-theoretic constructions with them.

Since [3, XI] homotopy inverse limits only have homotopy meaning when applied to fibrant diagrams, we sometimes have to replace a given diagram Y ∈ S^{D} by a weakly equivalent fibrant one, such as, for instance, Ex^{∞} Y, where Ex^{∞} is the functor of [14]. To simplify notation, we will write Y^{f} instead of Ex^{∞} Y.

§2. Realization complexes

Let D be a small category, and suppose that X̄ is a diagram in (hoS)^{D}. Recall from [5, §3] that there is a natural realization complex rX̄ associated to X̄ such that the components of rX̄ are in 1-1 correspondence with weak equivalence classes {(Y, f)} of realizations of X̄ and such that the component corresponding to a particular (Y, f) is equivalent to the classifying space of an appropriate complex [5, 3.5] of homotopy automorphisms of Y. The complex rX̄ is defined as the nerve of the category whose objects are the realizations of X̄ and whose morphisms are weak equivalences between these realizations. Remark 2.4 below indicates that there is a simple characterization of centric maps in terms of realization complexes.

For any object D of D let D ↓ D denote the over category [3, XI, §2] [16, p. 46] of the identity functor. The objects of this category are pairs (D′, f) where D′ is an object of D and f : D′ → D is a map; a morphism (D′, f) → (D′′, g) is a map h : D′ → D′′ in D such that gh = f. There is a forgetful functor O_{D} : D ↓ D → D which maps a pair (D′, f) to D′. Given a diagram X̄ ∈ (hoS)^{D}, there is an induced diagram O_{D}^{∗}X̄ ∈ (hoS)^{D↓D} for each object D of D; there is also a functor r_{D^{op}} X̄ : D^{op} → S which maps an object D ∈ D^{op} to the realization complex r(O_{D}^{∗}X̄).

The following theorem is proven below.

Theorem 2.1. Suppose that D is a small category and that X̄ is a diagram in hoS^{D}. Then the natural map rX̄ = \underleftarrow{lim}^{D^{op}}r_{D^{op}} X̄ → ho\underleftarrow{lim}^{D^{op}}(r_{D^{op}} X̄)^{f} is a weak equivalence.

The main result of this section is a description of the diagram r_{D^{op}} X̄ in the special case in which X̄ is centric.

Theorem 2.2. Suppose that D is a small category and that X̄ ∈ (hoS)^{D} is a centric diagram. Then for each object D ∈ D the complex r_{D^{op}} X̄(D) has the weak homotopy type of the classifying complex W̄ haut(X̄(D))_{1}, and for each i ≥ 2 the functor π_{i}(r_{D^{op}} X̄) : D^{op} → Ab is naturally equivalent to α_{i−1}X̄.

The rest of this section is taken up with proofs. Proof of 2.1: For every integer n ≥ 0 let n denote the category with the integers 0, …, n as objects and with exactly one map i → j whenever i ≤ j. The division dD of D is the category which has as objects the functors n → D (n ≥ 0) and as maps (J_{1} : n_{1} → D) → (J_{2} : n_{2} → D) the functors K : n_{2} → n_{1} such that J_{1} ∘ K = J_{2}. There is a functor q : dD → D^{op} given by the formula (J : n → D) ↦ J(n). According to [5, 3.7] the natural map rX̄ = \underleftarrow{lim}^{dD}r_{dD}X̄ → ho\underleftarrow{lim}^{dD}(r_{dD}X̄)^{f} is a weak equivalence, where r_{dD}X̄ is the functor which assigns to J : n → D the classification complex r(J^{∗}X̄). Define two objects q_{#}r_{dD}X̄ and q_{∗}r_{dD}X̄ of S^{D^{op}} by the formulas

q_{#}r_{dD}X̄(D) = \underleftarrow{lim}^{D↓q}ϕ^{∗}r_{dD}X̄
q_{∗}r_{dD}X̄(D) = ho\underleftarrow{lim}^{D↓q}ϕ^{∗}(r_{dD}X̄)^{f}

where D ↓ q is an under category associated to the functor q [3, XI, §2] and ϕ : D ↓ q → dD is the forgetful functor. There are two things to notice here: first, that the diagram q_{#}r_{dD}X̄ is naturally isomorphic to r_{D^{op}} X̄ and second, that the natural map r_{D^{op}} X̄ ≅ q_{#}r_{dD}X̄ → q_{∗}r_{dD}X̄ is a weak equivalence. The first statement follows from inspection, and the second from [5, 3.7] and the fact that for any object D ∈ D^{op} the category D ↓ q is naturally isomorphic to the division d(D ↓ D) in such a way that the the functor ϕ^{∗}r_{dD}X̄ agrees with the classification functor r_{d(D↓D)}O_{D}^{∗}X̄. By [5, 3.7] and the pushdown theorem for homotopy inverse limits [6, 9.8] the composite map rX̄ → ho\underleftarrow{lim}^{dD}(r_{dD})^{f} → ho\underleftarrow{lim}^{D^{op}}q_{∗}r_{dD}X̄ is a weak equivalence. The desired result follows immediately.

Lemma 2.3. Suppose that D is a small category with a terminal object E, and that X̄ is a centric diagram in (hoS)^{D}. Let E be the trivial category with one object and one identity morphism, and ϵ : E → D the functor which takes the single object of E to E. Then the map rX̄ → r(ϵ^{∗}X̄) induced by ϵ is a weak equivalence.

Proof: We will use some of the notation from the proof of 2.1. First consider the case in which D is n for some integer n. Let X_{i} be the space X̄(i), i = 1, …, n, and assume without loss of generality that each X_{i} is fibrant. Let γ_{i} be a map X_{i−1} → X_{i} which represents the homotopy class X̄((i − 1) → (i)). The argument of [7, 9.3] then shows that the classification complex rX̄ is naturally weakly equivalent to the bar construction [7, 9.1]

B(haut(X_{0})_{1}, Map(X_{0}, X_{1})_{γ_{1}}, haut(X_{1})_{1}, …, haut(X_{n})_{1}) .

A straightforward induction using [7, 9.2(vii)], the definition of the bar construction, and the fact that X̄ is centric gives the desired result.

Now assume that D is an arbitrary small category with a terminal object. The result just proven for the categories n, n ≥ 0 shows that the classification diagram r_{dD}X̄ has the property that r_{dD}X̄(f) is a weak equivalence whenever f is a morphism of dD such that q(f) is an identity map of D^{op}. Let sdD be the subdivision of D [4, §5]; this is the category obtained from dD by turning all of the “degeneracy maps” (ie. maps (J_{1} : n_{1} → D) → (J_{2} : n_{2} → D) which correspond to an epimorphism K : n_{2} → n_{1}) into identity maps. The functor q can be factored as a composite q_{1} ∘ q_{0}, where q_{0} : dD → sdD is the natural projection and q_{1} : sdD → D^{op} is defined like q. According to [6, 6.11] the functor q_{0} is L-cofinal [6, 6.2], and according to [4, 5.5 and 5.6] or [6, 6.10(ii)] the functor q_{1} is R-cofinal [6, 6.15]. By [6, 6.5(iv)] and [6, 6.15], then, there is some fibrant diagram Y in S^{D^{op}} such that q^{∗}(Y) is in the same weak equivalence class as the classification diagram r_{dD}X̄. Moreover, since a functor which is either L-cofinal or R-cofinal is left cofinal ([6, 6.6] and opposite of [6, 6.7]) it follows from the cofinality theorem for homotopy inverse limits [6, 9.3] that ho\underleftarrow{lim}^{D^{op}}Y is weakly equivalent to ho\underleftarrow{lim}^{dD}r_{dD}X̄. The lemma now follows in a straightforward way from the fact that E is an initial object of D^{op}, so that the natural map ho\underleftarrow{lim}^{D^{op}}Y → Y (E) is a weak equivalence [3, XI, 4.1(iii)]. Proof of 2.2: The over category D ↓ D has a terminal object, so the first assertion is a consequence of Lemma 2.3 and [5, 3.5]. The second assertion is a routine calculation which uses the bar constructions which appear in the proof of Lemma 2.3; to make this calculation it is enough to restrict to the case in which D is the category n for n = 1. Remark 2.4: Let f : A → B be a map between fibrant simplicial sets and let 0 and 1 denote the categories n above for n = 0, 1. Consider the functor f̄: 1 → hoS which sends the map 0 → 1 in 1 to the homotopy class represented by f and the functor B̄ : 0 → hoS which sends the object 0 to B. It is an immediate consequence of the proof of Lemma 2.3 that f is centric iff the evident realization complex map rf̄ → rB̄ is an equivalence.

§3. Proof of 1.1

Given the definition of the realization complex rX̄ in §2, part (1) of Theorem 1.1 comes down to the statement that rX̄ is non-empty and part (2) to the statement that rX̄ is connected. According to 2.1 the complex rX̄ is weakly equivalent to the homotopy inverse limit ho\underleftarrow{lim}^{D^{op}}(r_{D^{op}} X̄)^{f}. Hence rX̄ is also weakly equivalent to Tot Y^{•}, where Y^{•} is the cosimplicial replacement of (r_{D^{op}} X̄)^{f} [3, XI, §5] and Tot is the “total space” or codiagonal functor [3, X, §3]. There is a spectral sequence

{E_{2}^{s,t} = π^{s}π_{t}Y^{•}} ⟹ π_{t−s} Tot Y^{•}

which has been studied by Bousfield [1] in great detail, especially in connection with its low-dimensional behavior. It is not necessary to specify basepoints in the above formula for the E_{2}-term because, in view of 2.2 and the way in which Y^{•} is constructed [3, XI, 5.1], each space Y^{s} is 1-connected. To work with the spectral sequence beyond E_{2} it is necessary to make successive choices which amount in the limit to choice of a basepoint for Tot Y^{•}, but this consideration has been suppressed from the above notation for the abutment of the spectral sequence. By 2.2, [3, XI, 6.2] and [1, 2.4] there are isomorphisms

E_{2}^{s.t} = \{ \begin{matrix}∗ & (s, t) = (0, 0) or (1, 1) \\ \{1\} & (s, t) = (0, 1) \\ \underleftarrow{lim}^{s}α_{t−1}X̄ & s ≥ 0 and t ≥ 2 \end{matrix}

and in fact these are the only cases in which E_{2}^{s,t} is in general defined. Statement (1) of Theorem 1.1 in now a consequence of the next-to-last sentence of [1, 6.1] in the special case r = 2, while (2) follows from [1, 6.5] in the special case q = 1.

§4. The centralizer diagram

Fix a prime number p. Let G be a compact Lie group, and let A_{G} be the category whose objects are the non-trivial elementary abelian p-subgroups of G; a morphism A → A′ in A_{G} is a homomorphism f : A → A′ of abelian groups with the property that there exists an element g ∈ G such that f(x) = gxg^{−1} for all x ∈ A. There is a functor α̂_{G} from A_{G}^{op} to the category of topological spaces which sends A to the Borel construction EG×_{G} (G/ζ(A)), where ζ(A) denotes the centralizer of A in G. (Notice that this Borel construction has the homotopy type of the classifying space Bζ(A).) The main result of [12] states that the natural map ho\underrightarrow{lim} α̂_{G} → EG ×_{G} ∗ = BG is an isomorphism on Z/p homology. In this section we will prove that a slight modification of this “centralizer diagram” α̂_{G} is centric. We do not know whether α̂_{G} or its modification satisfy the hypotheses of Theorem 1.1.

We will tacitly assume in this section as well as in §5 that the spaces which appear have been replaced by their singular complexes.

For any connected space X let CX denote the partial Z/p-completion of X in the sense of [3, 6.8]; there is a map X → CX which induces an isomorphism on Z/p homology, an isomorphism on fundamental groups, and the ordinary Z/p-completion map on universal covers. If G is a compact Lie group, then, there are maps BG → C(BG) and ho\underrightarrow{lim} C(α̂_{G}) → C(BG) which induce isomorphisms on Z/p homology.

Theorem 4.1. For any compact Lie group G the diagram C(α̂_{G}) is centric.

In view of the definition of α̂_{G}, Theorem 4.1 is an easy consequence of the following more general assertion.

Theorem 4.2. Let A be a finite abelian p-group which is a subgroup of the compact Lie group G, H the centralizer of A in G, and f : C(BH) → C(BG) the map derived from the inclusion H → G. Then f is centric.

The proof of 4.2 depends on a number of lemmas.

Lemma 4.3. If X is a 1-connected Z/p-complete space [3, I, §5] then the homotopy groups of X are uniquely q-divisible for every prime q ≠ p. If Y is a nilpotent space which has the Z/p homology of a point, then the homotopy groups of Y are uniquely p-divisible.

Proof: The first statment follows from [3, VI, 5.4(ii)]. The second can be proved by a standard induction on the refined Postnikov tower [3, III, 5.3] of Y, or by combining the algebraic calculation of the homotopy groups of the Z/p-completion of Y [3, VI, 5.1] with the fact that this Z/p-completion is contractible if Y has the Z/p homology of a point.

Lemma 4.4. Let X and Y be connected spaces, f : X → C(Y) a map, and W the universal cover of Map(X, C(Y))_{f}. Then W is Z/p-complete.

Proof: Let g : X → K(π_{1}Y, 1) be the composite of f with the map Y → K(π_{1}Y, 1) which classifies the universal covering fibration Ỹ → Y. For elementary reasons there is up to homotopy a fibration sequence

Γ → Map(X, C(Y))_{f} → Map(X, K(π_{1}Y, 1))_{g}

in which Γ is a union of components of the space of sections of a certain fibration E → X with homotopy fibre C(Ỹ). Since the space Map(X, K(π_{1}Y, 1)) has no homotopy above dimension 1, the space W is homotopy equivalent to the universal cover of one component of Γ. The space C(Y) is Z/p-complete and therefore H_{∗}(−, Z/p)-local in the sense of Bousfield [2, §4]. A slight variation on the construction of [3, X, 2.2(ii) and 3.3(i)] shows that the space of sections of E → X can be expressed as Tot of a fibrant cosimplicial space in which each constituent space has the homotopy type of a product of copies of C(Ỹ). Therefore [3, XI, 4.4] the space of sections of E → X is equivalent to the homotopy inverse limit of a diagram of H_{∗}(−, Z/p)-local spaces and is itself H_{∗}(−, Z/p)-local [2, 12.9]. It follows from [2, 5.5] that the universal cover of any component of Γ is H_{∗}(−, Z/p)-local and therefore [2, §4] Z/p-complete.

Lemma 4.5. Let A be a finite abelian p-group which is a subgroup of the compact Lie group G, H the centralizer of A in G, and v : BA ≅ C(BA) → C(BG) the map derived from the inclusion A → G. Then the natural map (see below)

C(BH) → Map(BA, C(BG))_{v}

is an equivalence

Remark: The multiplication map A × H → G induces a map BA × BH → BG and hence a map BA × C(BH) ≅ C(BA × BH) → C(BG). It is the adjoint of this last map which appears in 4.5. Proof of 4.5: Let G_{0} be the identity component of G and F the homotopy fibre of the completion map BG_{0} → C(BG_{0}). The space F is simple, because the action of π_{1}F on the homotopy groups of F extends to an action of the trivial group π_{1}BG_{0}. The Serre spectral sequence of the fibration F → BG_{0} → C(BG_{0}) immediately shows that F has the Z/p-homology of a point. It follows from Lemma 4.3 that the homotopy groups of F are uniquely p-divisible and that π_{1}F is q-divisible for any prime q ≠ p. It is clear that F is also the homotopy fibre of the partial completion map BG → C(BG).

Let u : BA → BG be derived from the inclusion A → G and let F′ be the homotopy fibre of the natural map U → V, where U is Map(BA, BG)_{u} and V is Map(BA, C(BG))_{v}. It follows from obstruction theory (cf. [10, proof of 2.3]) that F′ is connected and that π_{i}F′ is a direct summand of π_{i}F for i ≥ 1; in particular, π_{i}F′ is a uniquely p-divisible abelian group for i ≥ 1 and a divisible abelian group for i = 1. Consider the exact sequence π_{1}F′ → π_{1}U → π_{1}V → {1}. By [10, 1.1] the group π_{1}U is finite and so cannot accept a non-trivial map from the divisible group π_{1}F′. It follows that π_{1}U ≅ π_{1}V, that F′ is the homotopy fibre of the map Ũ → Ṽ of universal covers, and hence that Ũ → Ṽ induces an isomorphism on Z/p homology. By [10, 1.1], then, the natural map C(BH) → V induces an isomorphism on fundamental groups and a Z/p homology equivalence on universal covers. The lemma follows [3, I, 5.5] from the fact that the universal cover of C(BH) is Z/p-complete by construction, while the universal cover of V = Map(BA, C(BG))_{v} is Z/p-complete by 4.4.

Proof of 4.2: Let K be the quotient group H/A. By the fibre lemma [3, II] and a short low-dimensional calculation the principal fibration sequence BA → BH → BK gives rise to an induced principal fibration sequence BA → C(BH) → C(BK). Since the fibre BA in this principal fibration is connected, the monodromy action of π_{1}C(BK) on BA is trivial up to homotopy. It follows by elementary homotopy theory that there is an associated fibration e_{1} : E_{1} → C(BK) with homotopy fibre Map(BA, C(BG))_{v} (v is as in Lemma 4.5) such that the space of sections Γ(e_{1}) is naturally homotopy equivalent to the space of maps C(BH) → C(BG) which restrict to v on BA.

Let w : BA → C(BH) be induced by the inclusion A → H. In a similar way there is a fibration e_{2} : E_{2} → C(BK) with homotopy fibre Map(BA, C(BH))_{w} such that the space of sections Γ(e_{2}) is naturally homotopy equivalent to the space of maps C(BH) → C(BH) which restrict to w on BA. Composition with f gives a map E_{2} → E_{1} over C(BK) which, after passage to spaces of sections and appropriate component selection, induces the precomposition map haut(C(BH))_{1} → Map(C(BH), C(BG))_{f}. Now the natural map from the centralizer of A in H to the centralizer of A in G is an isomorphism (both groups are H); by Lemma 4.5, this implies that the map Map(BA, C(BH))_{w} → Map(BA, C(BG))_{v} is a homotopy equivalence and hence that the maps E_{2} → E_{1} and Γ(e_{2}) → Γ(e_{1}) are also homotopy equivalences. This proves the theorem.

§5. The p-toral diagram

Let p be a fixed prime as in §4. A compact Lie group H is said to be toral if its connected component H_{0} is a torus; H is p-toral if H is toral and H/H_{0} is a p-group. If G is a compact Lie group, P a ptoral subgroup of G, and N(P) the normalizer of P in G, then P is said to be p-stubborn if N(P)/P is finite and has no nontrivial normal p-subgroups. Let R_{p}(G) be the full subcategory of the category of Gspaces whose objects are the orbits G/P for p-stubborn P ⊂ G. There is a functor β̂_{G} from R_{p}(G) to spaces which assigns to G/P the Borel construction EG×_{G} G/P ∼ BP. One of the results in [13] states that if G is a compact Lie group the natural map ho\underrightarrow{lim} β̂_{G} → EG ×_{G} ∗ = BG induces an isomorphism on Z/p homology.

Theorem 5.1. For any connected compact Lie group G the “p-toral” diagram β̂_{G} is centric.

By [13, 2.7, 7.1], if G is connected the diagram π(β̂_{G}) in (hoS)^{R_{p}(G)} satisfies the hypotheses of parts (1) and (2) of Theorem 1.1. The conclusion of 1.1(1) is not interesting in this case, since π(β̂_{G}) is already known to have a realization, namely β̂_{G}. Statement 1.1(2), though, gives the following corollary.

Corollary 5.2. If G is a connected compact Lie group then any realization of the diagram π(β̂_{G}) is weakly equivalent, as an object of S^{R_{p}(G)}, to β̂_{G} itself.

This corollary implies that if G is a connected compact Lie group then BG is determined up to Z/p homology type by a certain relatively small and well-controlled diagram in the homotopy category. There is some hope that in the long run this observation will contribute to a generalization of the uniqueness result in [8].

The proof of Theorem 5.1 depends on the following proposition.

Proposition 5.3. Let j : K → L be a map of p-toral compact Lie groups and H the centralizer of j(K) in L. Then the natural map (cf. 4.5, [10, §1])

BH → Map(BK, BL)_{Bj}

is an equivalence.

Remark: It seems likely that the conclusion of this proposition remains valid under the weaker hypotheses that K is compact Lie and L is toral (see [15]). A slightly modified form of the proposition holds if K is assumed p-toral and L is assumed compact Lie [21] [18].

Proof of 5.3: The group L acts on itself by conjugation, and composition of this action with the homomorphism j gives a conjugation action of K on L. The fixed point set L^{K} of this action is the centralizer H of j(K) in L. As in [10, proof of 4.1], the proposition can be proved by showing that the natural map from L^{K} to the homotopy fixed point set L^{hK} of this conjugation action is an equivalence. (Here L^{hK} is the space of sections of the fibration over BK with fibre L associated to the conjugation action.)

Assume then that L and K are p-toral, and that K acts on L via group homomorphisms. We will show that the map L^{K} → L^{hK} is an equivalence. We can assume [10, proof of 2.2] that the action of K on the component group π_{0}L is trivial, since any component of L which is moved by the action of K contributes neither to L^{K} nor to L^{hK}. It is enough, then to show that for each component L_{i} of L the map L_{i}^{K} → L_{i}^{hK} is an equivalence. There is no difficulty if L_{i}^{hK} is empty, since then L_{i}^{K} is empty too. Assume then that L_{i}^{hK} is non-empty. It follows that for any finite p-subgroup τ of K the homotopy fixed point set L_{i}^{hτ} is non-empty and hence by [11] that the fixed point set L_{i}^{τ} is non-empty. Since K is p-toral it is possible to find a dense subgroup τ_{∞} of K which is an increasing union ⋃_{n} τ_{n} of finite p-subgroups; by a compactness argument, then, L_{i}^{τ_{∞}} is non-empty and hence L_{i}^{K} is nonempty. Pick x ∈ L_{i}^{K}. Multiplication by x then gives a K-equivariant homeomorphism between L_{i} and the identity component L_{0} of L, so we will be done if we can prove that L_{0}^{K} → L_{0}^{hK} is an equivalence. Assume for this purpose that L is connected. The group L is then a torus (S^{1})^{n} and the automorphism group of L is the discrete group GL(n, Z). The identity component of K thus acts trivially on L, so that the action of K on L factors through the quotient map K → π_{0}K = σ. It is obvious that the natural map L^{σ} → L^{K} is an isomorphism, and an elementary homotopy theory argument shows that the natural map L^{hσ} → L^{hK} is an equivalence (this comes down to the fact that the space of sections of the trivial fibration over B(K_{0}) with fibre L is homotopy equivalent to L itself). Let L̃ = R^{n} be the universal cover of L. The group σ acts on the short exact sequence {1} → π_{1}L → L̃ → L → {1} and the low dimensional end of the induced twisted cohomology exact sequence gives isomorphisms π_{0}(L^{σ}) ≅ H^{1}(σ, π_{1}L), π_{1}(L^{σ}) ≅ H^{0}(σ, π_{1}L), and π_{k}(L^{σ}) ≅ 0 for k ≥ 2. A straightforward calculation now shows that the homotopy groups of L^{hσ} are given by the same formulas and that the map L^{σ} → L^{hσ} in fact induces an isomorphism on homotopy.

Proof of 5.1: By [13, 5.1(i)] the centralizer in G of a p-stubborn P ⊂ G is the center of P. The desired result follows from 5.3.

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