A CLASSIFICATION OF THE STABLE TYPE

OF BG_{p}^{∧} FOR COMPACT LIE GROUPS

John Martino and Stewart Priddy

9/19/96

1. Introduction and Statement of Results.

In a previous paper [MP] we showed that after localization or completion at a prime p the stable type of the classifying space BG of a finite group G was abstractly determined by certain p-local group theoretic information, namely, conjugacy classes of homomorphisms of p-groups into G. The purpose of this paper is to generalize this result to compact Lie groups where the stable type is complicated by the appearance of tori and the consequent failure of the Segal conjecture to compute the stable homotopy classes of maps {BG_{p}^{∧}, BG_{p}^{∧}}. Nevertheless, analogous results continue to hold. To explain this we recall that in considering compact Lie groups G at a particular prime p, one is naturally lead to consider the p-toral subgroups of G, that is, extensions of tori (possibly trivial) by finite p-groups. These groups are the analogues of finite p-groups and as such they play a major role in this paper.

Let P be a p-toral group and let Inj(P, G) be the set of G-conjugacy classes of continuous injections α : P → G. Composition of maps gives the vector space F_{p}Inj(P, G) the structure of an Out(P) module.

Theorem 1.1. Let G and G′ be compact Lie groups such that for every p-toral group P there is an isomorphism

F_{p}Inj(P, G) ≈ F_{p}Inj(P, G′)

of Out(P) modules. Then BG_{p}^{∧} is stably homotopy equivalent to BG′_{p}^{∧}.

If N_{G}(P)/P is finite, there are finitely many conjugacy classes of subgroups isomorphic to P (Thm. 2.3(2)) in which case the modules of Theorem 1.1 are

Typeset by AMS-TEX

finitely generated. As we shall see in Section 5, p-toral subgroups of G satisfying this finiteness condition suffice to determine BG_{p}^{∧}. The converse of the theorem is true for finite groups [MP]; however it is not known to be true in general. One obstacle to proving the converse is there are homomorphisms that induce equivalances of classifying spaces, but not of F_{p}Out(P) modules. For example, if P = G = G′ = S^{1} and ρ : G → G′ is a p′ power homomorphism where p′ is prime to p then ρ induces an equivalence BG_{p}^{∧} → BG′_{p}^{∧} but does not induce a homomorphism F_{p}Inj(P, G) → F_{p}Inj(P, G′). However it does induce an isomorphism if P is replaced by a certain discrete subgroup approximating it at the prime p. Feshbach [F] has constructed such an approximation by means of a discrete subgroup P_{∞} = ⋃ P_{i} where P_{i} is an increasing sequence of finite p-subgroups such that H_{∗}(P_{∞}; F_{p}) = colimH_{∗}(P_{i}; F_{p}) ≈ H_{∗}(BP; F_{p}). We shall call P_{∞} a discrete approximation to Q. Let Inj_{P} (P_{∞}, G) be the set of G-conjugacy classes of injections β : P_{∞} → G such that β(P_{∞}) is contained in a subgroup isomorphic to P.

Theorem 1.2. Let p be an odd prime. For compact Lie groups G and G′ the following are equivalent:

(1) BG_{p}^{∧} and BG′_{p}^{∧} are stably homotopy equivalent.

(2) For every p-toral group P there is a discrete approximation P_{∞} and an isomorphism

F_{p}Inj_{P} (P_{∞}, G) ≈ F_{p}Inj_{P} (P_{∞}, G′)

of Out(P_{∞}) modules.

The stable type of BG_{p}^{∧} is also determined by finite groups. To describe this we let Inj_{P} (P_{i}, G) be the set of G-conjugacy classes of injections α : P_{i} → G such that α(P_{i}) is contained in a subgroup P′ isomorphic to P and α(P_{i})/(T′∩α(P_{i})) ≈ P′/T′. In what follows we use a generalized notion of Weyl group W_{G}(Q) = N_{G}(Q)/Q · C_{G}(Q) ≤ Out(Q) which is always finite for any closed subgroup Q ≤ G (Cor. 2.2). Finiteness implies that if P is a p-toral subgroup of G then for sufficiently large N the inclusion P_{N} ≤ P induces an exact sequence

0 → W_{P} (P_{N}) → W_{G}(P_{N}) → W_{G}(P) → 0

(Cor. 7.3). Furthermore for N large the number of conjugacy classes of p-toral subgroups Q, isomorphic to P, equals the number of conjugacy classes of subgroups Q_{K} (Lemma 7.4). If both conditions hold we say P_{N} is sufficient for W_{G}(P).

Theorem 1.3. Let p be an odd prime. For compact Lie groups G, G′ the following are equivalent

(1) BG_{p}^{∧} and BG′_{p}^{∧} are stably homotopy equivalent.

THE STABLE TYPE OF BG_{p}^{∧} FOR COMPACT LIE GROUPS 3

(2) For every p-toral group P there is a P_{N} sufficient for W_{G}(P) and W_{G′}(P)

such that

F_{p}Inj_{P} (P_{N}, G) ≈ F_{p}Inj_{P} (P_{N}, G′)

as Out(P_{N}) modules.

For each finite p-group Q there are finitely many G-conjugacy classes of subgroups isomorphic to Q (Thm. 2.3(1)). Therefore the modules of Theorem 1.3 are finitely generated.

It follows from Theorem 1.2 that G and G′ have isomorphic Sylow p-subgroups. This result, originally due to Lee [L1], generalizes the same result of Nishida [N] in the finite case. More generally

Theorem 1.4. Let p be an odd prime and G and G′ be compact Lie groups. If BG_{p}^{∧} and BG′_{p}^{∧} are stably equivalent then G and G′ have the same number of conjugacy classes of subgroups isomorphic to Q for every p-toral group Q.

This generalizes Theorem 1.3 of [MP].

Theorems 1.2, 1.3, and 1.4 are also true at the prime 2 if the generalized quaternion group of order 16, Q_{16}, is not a subquotient of the Weyl group of the Sylow 2-subgroup of either G or G′. That is, Theorems 1.2, 1.3, and 1.4 hold if the Sylow 2-subgroups are Roquette (see Section 2).

The paper is organized as follows: In Section 2 we collect most of the preliminaries needed in the proofs of our theorems. These include information on conjugacy classes of subgroups, discrete approximations, and consequences of the Segal conjecture for finite groups. We also show that a result of Lee [L] allows us to work with stable maps generated by transfers and homomorphisms. This is our main technique (Cor. 2.14) for circumventing problems arising from the failure of the Segal conjecture for compact Lie groups. In Section 3 we give a theory of original summands which generalizes that for finite groups. It follows from this that modular representation theory can be used to determine the multiplicity of summands in BG in a way analogous to that for finite groups [MP]. In the short Section 4, we show that some of the representation theory of [MP] carries over to non-artinian rings such as F_{p}Out(P_{∞}). The final Sections 5,6,7 are devoted to the proofs of Theorems 1.1, 1.2, 1.3 respectively. To show BG_{p}^{∧} and BG′_{p}^{∧} are stably equivalent, we show they have the same indecomposable summands with the same multiplicities.

The authors would like to thank Mark Feshbach, John Franks, and Bob Oliver for useful comments and examples.

Throughout the paper all homology is taken with simple F_{p} coefficients unless otherwise stated.

2. Preliminaries.

In this section we establish some results preliminary to proving our main results. First we examine finiteness properties of conjugacy classes of subgroups. These results are mostly folklore; some appeared in an early version of [JMO].

Theorem 2.1. [Lo; Thms. I.4.8 & I.4.9] Let G be a Lie group.

(1) If G/G_{0} is finitely generated then Aut(G) is a Lie group with corresponding Lie algebra the derivations of G.

(2) If G is compact or semi-simple then all the derivations of G are inner.

Corollary 2.2. Let Q be a closed subgroup of a compact Lie group G. Then Out(Q) is discrete and N_{G}(Q)/Q·C_{G}(Q) is finite. Further N_{G}(Q)/Q is finite if and only if C_{G}(Q)/Z(Q) is finite.

Proof. Since there is a 1-1 correspondence between connected subgroups of a Lie group and subalgebras of its Lie algebra, the identity component of Aut(Q) equals the identity component of Inn(Q). Therefore, Out(Q) is a quotient of Aut(Q)/Aut(Q)_{0}, hence discrete. NQ/Q·CQ ≤ Out(Q) is compact hence finite. The second assertion is immediate since C_{G}(Q)/Z(Q) is the kernel of the projection N_{G}(Q)/Q → N_{G}(Q)/Q·C_{G}(Q). □

We recall that a p-toral group is a compact Lie group P which is an extension of a torus T by a finite p-group π

1 → T → P → π → 1

Theorem 2.3. Let G be a compact Lie group.

(1) If Q ≤ G is a finite p-subgroup then there are finitely many conjugacy classes of subgroups isomorphic to Q.

(2) Let T_{p}(G) be the set of all positive dimensional p-toral subgroups Q ≤ G with NQ/Q finite. Then T_{p}(G) contains finitely many conjugacy classes of subgroups.

Proof. (1) Embed G in a unitary group U(n) for some n. This affords a representation of Q and every subgroup isomorphic to it. Each is U(n)-conjugate to one of the finitely many n-dimensional representations K which is a sum of irreducibles. The result now follows from the general fact that if H, K are subgroups of a compact Lie group G then the subgroups of H which are G-conjugate to K fall into finitely many H-conjugacy classes [Q; proof of Lemma 6.3].

(2) Let S(G) denote the space of all closed subgroups of G with the Hausdorff metric topology ([D;IV.3]). G acts on S(G) via conjugation. Then S(G) and S(G)/G are compact ([JMO; Prop. A6]). Suppose T_{p}(G)/G is infinite then there is a sequence (Q_{i}) in T_{p}(G)/G which converges to some (Q) ∈ S(G)/G different from (Q_{i}). By the proof of [JMO; Prop 1.6] we may assume

a) Q_{i} < Q for i sufficiently large,

b) Q is itself a p-toral subgroup of G,

c) N(Q_{i}) ≤ N(Q) for i sufficiently large. Suppose dim(Q_{i}) < dim(Q) for some sufficiently large i. By c) Q normalizes Q_{i}, thus NQ_{i}/Q_{i} is positive dimensional contradicting Q_{i} ∈ T_{p}(G). Therefore for sufficiently large i, Q_{i} and Q have the same maximal torus but possibly a different number of components. Since the component set of each is finite, Q_{i} = Q for i large enough, contradicting the definition of the Q_{i}. □

Feshbach [F] has constructed homological approximations of p-toral groups P by means of discrete subgroups P_{∞}. For a torus P = T, T_{∞} is the subgroup of T consisting of elements of order a power of p. In general we have the following basic result.

Proposition 2.4. [F] Let P be a p-toral group with maximal torus T and components π = P/T. Then there is an increasing sequence of finite p-groups P_{j} ≤ P_{j+1} ≤ P, with associated discrete group P_{∞} = ⋃ P_{j} such that

(1) inclusion induces an isomorphism

H_{∗}(BP_{∞}) = colimH_{∗}(BP_{j}) ⟶^{≈} H_{∗}(BP)

(2) if T_{j} = P_{j} ∩ T then P_{j}/T_{j} = π and T_{∞} = ⋃ T_{j}

(3) If Q is a subgroup of P satisfying Q ∩ T = T_{∞} and Q/(Q ∩ T) ≅ π, then Q is conjugate to P_{∞} in P.

(4) P_{j} = P_{0}·T_{j}

Proof. A nested sequence {P_{j}} ≤ P of finite p-subgroups satisfying conditions (1)

- (3) was constructed in [F; §1]. Lee observed [L; Prop.3.3] one could also require (4).

We shall call P_{∞} a discrete approximation to P.

Lemma 2.5. Let P_{∞} be a discrete approximation of the p-toral group P and α : P_{∞} → G a monomorphism. Then

(1) the closure \overline{α(P_{∞})} is a p-toral group with components π_{0}(\overline{α(P_{∞})}) ≈ π_{0}(P)

(2) \overline{α(P_{∞})} ≈ P if im(α) is contained in a subgroup isomorphic to P.

Proof. (1) Since \overline{α(T_{∞})} is a compact abelian group, \overline{α(T_{∞})} = T′ × A where T′ is a torus and A is a finite abelian group. The composite α(T_{∞}) → \overline{α(T_{∞})} → π_{0}(\overline{α(T_{∞})}) = A is surjective since α(T_{∞}) is dense in \overline{α(T_{∞})}. Hence A is a finite divisible abelian p-group, i.e., A = 0. Also α(T_{∞}) ◁ α(P_{∞}) implies \overline{α(T_{∞})} ◁ \overline{α(P_{∞})} resulting in the commutative diagram

0 ⟶ \overline{α(T_{∞})} ⟶ \overline{α(P_{∞})} ⟶ π_{0}(\overline{α(P_{∞})}) ⟶ 1
↑  ↑  ↑
0 ⟶ α(T_{∞}) ⟶ α(P_{∞}) ⟶  π  ⟶ 1

with exact rows. Since the left and middle vertical maps are monomorphisms so is the third. But this map is also a surjection by density.

(2) Since \overline{α(T_{∞})} is a torus, \overline{α(T_{∞})} ≤ T. However the elements of T_{∞} of order p form a vector space of rank at least that of T. Hence \overline{α(T_{∞})} = T. By (1), \overline{α(P_{∞})} is p−toral with the same components as P. The result follows. □

Remark. Of course, \overline{α(P_{∞})} is not necessarily isomorphic to P since the dimension of T′ = \overline{α(T_{∞})} can be greater than that of T, the maximal torus of P. If the dimensions are equal then \overline{α(P_{∞})} is isomorphic to P. This explains why in Inj_{P} (P_{∞}, G), we only allow monomorphisms α whose image lies in a subgroup isomorphic to P.

As an example of this phenomenon, consider the circles S_{n} of slope n in the torus S^{1} × S^{1} where n = (3^{k} + 1)/2, k = 1, 2, 3, . . .. Then S_{m} ∩ S_{m+1} ≈ Z/3^{m} for m ≥ 2 and ⋃ (S_{m} ∩ S_{m+1}) ≈ Z/3^{∞} = lim_{⟶} Z/3^{m}, but the closure \overline{⋃ (S_{m} ∩ S_{m+1})} is the entire torus.

Next we discuss some implications of the Segal conjecture in case the source group is a finite p-subgroup P. Let J(P, G) be the submodule of {BP, BG_{p}^{∧}} generated by maps BP → BP′ → BG_{p}^{∧} where P′ is any p-group satisfying |P′| < |P |. If only one group G is under consideration we shall simply write J(P) = J(P, G). By the generalization of the Segal conjecture due to May, Snaith, Zelewski [MSZ; Thm. 2.3], {BP, BG_{p}^{∧}} is a Z_{p} module generated by composites

BP →^{tr} BP′ →^{Bh} BG_{p}^{∧}

where tr denotes the transfer, P′ ≤ P, and h is a homomorphism. Hence there is an isomorphism

(2.6)  {BP, BG_{p}^{∧}}/J(P) ⟶^{≈} Z_{p}Inj(P, G)

where Inj(P, G) is the set of conjugacy classes of monomorphisms Q → G.

Remark 2.7. This isomorphism is natural in G; however, it is not natural in P because an inclusion i : Q → P fails, in general, to induce a map {BP, BG_{p}^{∧}}/J(P) → {BQ, BG_{p}^{∧}}/J(Q) since maps of J(P) may fail to restrict to J(Q). For example there may exist retractions P → Q which when composed with i are the identity. The double coset formula shows similar problems arise with transfer maps. For this reason it seems difficult to prove Theorem 1.1 by approximating p-toral groups with finite p-groups and then applying (2.6) together with a colimit argument.

Lemma 2.8. Taking classifying spaces induces an injection

i : Z_{p}Inj(Q_{∞}, G) → {BQ_{∞}_{p}^{∧}, BG_{p}^{∧}}.

Proof. The inclusions i_{k} : Q_{k} → Q_{∞} induce a commutative diagram

Z_{p}Inj(Q_{∞}, G) ⟶^{lim i_{k}^{∗}} lim_{k}Z_{p}Inj(Q_{k}, G)
i↓  ↓
{BQ_{∞}_{p}^{∧}, BG_{p}^{∧}} ⟶^{lim i_{k}^{#}} lim_{k}{BQ_{k}, BG_{p}^{∧}}

The top arrow is an injection since Inj(Q_{∞}, G) = lim_{k}Inj(Q_{k}, G). The right hand arrow is an injection by the Segal conjecture for finite p-groups and the fact that an inverse limit of injections is injective. □

Corollary 2.9. Taking classifying spaces induces an injection

i : Z_{p}Inj(Q, G) → {BQ_{p}^{∧}, BG_{p}^{∧}}

Proof. With the subspace topology Q_{∞} is dense in Q. In this case Inj(Q, G) ≈ Inj(Q_{∞}, G) and the result follows since BQ_{p}^{∧} ≈ BQ_{∞}_{p}^{∧} by Prop. 2.4(1). □

By Prop. 2.4(1), Q_{∞} ≤ Q induces an equivalence BQ_{∞}_{p}^{∧} → BQ_{p}^{∧}. In the following we shall identify BQ_{∞}_{p}^{∧} and BQ_{p}^{∧} by this equivalence.

Let {BQ_{p}^{∧}, BG_{p}^{∧}}_{th} be the Z_{p} submodule of {BQ_{p}^{∧}, BG_{p}^{∧}} generated by maps

(2.10)  BQ_{p}^{∧} →^{tr} BQ′_{p}^{∧} →^{Bψ} BG_{p}^{∧}

where Q′ ≤ Q are p-toral subgroups of G with N_{Q}(Q′)/Q′ finite and ψ : Q′ → G a homomorphism. Let J(Q, G)_{th} ≤ {BQ_{p}^{∧}, BG_{p}^{∧}}_{th} be the Z_{p} submodule generated by maps of the form 2.10 where Q′ ≨ Q is a proper subgroup or ker(ψ) contains an element of order p.

Proposition 2.11. J(Q, P)_{th} ∩ i(Z_{p}Inj(Q_{∞}, G)) = 0. Hence the induced homomorphism

j = proj^{∘}i : Z_{p}Inj(Q_{∞}, G) → {BQ_{p}^{∧}, BG_{p}^{∧}}/J(Q, P)_{th}

is injective.

Proof. Consider the composite

π_{k} : {BQ_{p}^{∧}, BG_{p}^{∧}} →^{i_{k}^{#}} {BQ_{k}, BG_{p}^{∧}} →^{proj} {BQ_{k}, BG_{p}^{∧}}/J(Q_{k}, G)

If α ∈ i(Z_{p}Inj(Q_{∞}, G)) then for k large π_{k}(α) ≠ 0. On the other hand consider an element of J(Q, G)_{th} which is a linear combination of elements x of the form 2.10 such that if Q′ = Q, then ker(ψ) contains an element of order p. In this case if k large enough then i_{k}^{#}^{∘}ψ also has a non-trivial kernel, i.e. π_{k}(x) = 0. Now suppose Q′ ≨ Q so that Q_{∞} ∩ Q′ ≨ Q_{∞}. Then for k large Q_{k} ∩ Q′ ≨ Q_{k} hence it follows from the double coset formula that π_{k}(x) = 0. □

Proposition 2.12. If f : BG_{p}^{∧} → BG′_{p}^{∧} is a map of the form

f : BG_{p}^{∧} →^{tr} BQ_{p}^{∧} →^{Bψ} BG′_{p}^{∧}

where Q ≤ G is p-toral and ψ is a homomorphism then (1) f_{∗}(J(Q, G)_{th}) ≤ J(Q, G′)_{th}.

(2) f_{∗} induces a commutative diagram

Z_{p}Inj_{Q}(Q_{∞}, G)  ⟶^{f′_{∗}}  Z_{p}Inj_{Q}(Q_{∞}, G′)
j↓  ^{j}↓
\{BQ_{p}^{∧}, BG_{p}^{∧}\}/J(Q, G)_{th} ⟶^{\overline{f}_{∗}} \{BQ_{p}^{∧}, BG′_{p}^{∧}\}/J(Q, G′)_{th}

Proof. (1) follows immediately from the double coset formula [F]. To prove (2) we consider an element α ∈ Inj_{Q}(Q_{∞}, G) and calculate the composite

BQ_{p}^{∧} →^{α} BG_{p}^{∧} →^{tr} BQ_{p}^{∧} →^{Bψ} BG′_{p}^{∧}

Applying the double coset formula and working mod J(Q, G′)_{th} we find the composite is a sum of maps each of which is in the image of j : Z_{p}Inj_{Q}(Q_{∞}, G′) → {BQ_{p}^{∧}, BG′_{p}^{∧}}/J(Q, G′)_{th}. This defines f′_{∗}(α). □

When G is a positive dimensional compact Lie group a potentially serious problem arises from the fact that the completed double Burnside ring no longer computes the stable self-maps of BG. That is, one can no longer describe maps in terms of transfers and homomorphisms. This is true even if G = P is a p-toral group; however from the point of view of homology Lee [L] has shown this difficulty does not arise for a large class of groups.

A p-toral group P will be called a Roquette group if every simple Q[W]-submodule of π_{1}(T) ⊗ Q remains simple after tensoring with Q_{p}. It is known [L] that this is true unless a subquotient of W is isomorphic to the quaternion group of order 16. In particular P is a Roquette group if p is an odd prime.

For spaces X, Y let

H_{A}(H_{∗}X, H_{∗}Y) ≤ Hom_{A}(H_{∗}X, H_{∗}Y)

denote the subgroup induced by maps X → Y.

Proposition 2.13. [L; Proposition 3.5] Suppose P is a Roquette group and G is a compact Lie group. Then any element in the image of

{BP_{p}^{∧}, BG_{p}^{∧}} → H_{A}(H_{∗}BP, H_{∗}BG)

is an F_{p} linear combination of elements of the form

H_{∗}BP ⟶^{tr_{∗}} H_{∗}BQ ⟶^{Bψ_{∗}} H_{∗}BG

where Q is a p-toral subgroup of G with N_{G}(Q)/Q finite and ψ : Q → G a homomorphism.

Thus the homology set H_{A}(H_{∗}BP, H_{∗}BG) is generated by the maps predicted by the Segal Conjecture for finite groups even though these maps do not generate {BP_{p}^{∧}, BG_{p}^{∧}}. A compact Lie group is said to be p-Roquette if has a Roquette Sylow p-subgroup.

Corollary 2.14. Suppose G, G′ are compact Lie groups with G p-Roquette. If BG_{p}^{∧} and BG′_{p}^{∧} are stably equivalent then there is a stable equivalence which is a Z_{p}^{∧} linear combination of maps of the form

BG →^{tr} BQ →^{Bψ} BG′

where Q and ψ are as in Proposition 2.13.

Proof. Let g : BG_{p}^{∧} → BG′_{p}^{∧} be an equivalence and let P be a Sylow p-subgroup of

G. Then BG_{p}^{∧} →^{tr} BP_{p}^{∧} →^{i} BG_{p}^{∧} is a stable equivalence. Using Prop. 2.13, we can express BP_{p}^{∧} →^{i} BG_{p}^{∧} →^{g} BG′_{p}^{∧}, at least in homology, as a linear combination of transfers followed by homomorphisms. Therefore the same is true of the composite equivalence

BG_{p}^{∧} →^{tr} BP_{p}^{∧} →^{i} BG_{p}^{∧} →^{g} BG′_{p}^{∧}

However a map which induces a homology isomorphism is itself an equivalence and so we can replace g^{∘}i^{∘}tr by an equivalence consisting of maps of the required form. □

3. Types of summands and their multiplicities.

In this section we describe the types of summands, together with their multiplicities, which appear in a stable decomposition of BG_{p}^{∧}. For finite groups this was given in [MP]. For compact Lie groups, Lee [L] has given a method for determining a stable decomposition and our approach builds on his work. For our purposes the principal additional idea is that of linkage of summands which we proceed to define.

We begin by recalling some background from [L]. Let G be a compact Lie group. By a finiteness result of Henn [H] there is a stable decomposition

BG_{p}^{∧} = X_{1} ∨ X_{2} ∨ ⋯ ∨ X_{N}

with a finite number of summands completed at p. This decomposition is unique up to order and equivalence of summands. Let T be the maximal torus of the identity component of G and N(T) the normalizer of T in G. Let W_{p} be a Sylow p-subgroup of the finite group N(T)/T. Then P, the inverse image of W_{p} in NT, is a p-toral subgroup of G, i.e., there is an extension

1 → T → P → W_{p} → 1.

It follows that P is unique up to conjugation in G and is called a Sylow p-subgroup of G. It shares many of the properties of the usual Sylow p-groups in the finite group case [JMO]. Similarly it is important for splittings since BG_{p}^{∧} is a summand of BP_{p}^{∧} by a standard transfer argument.

Definition 3.1 Let Q be a p-toral group. An indecomposable summand X of BQ_{p}^{∧} is said to originate in BQ_{p}^{∧} if X is not a summand of BR_{p}^{∧} for any proper p-toral subgroup R ≤ Q. In this case we also say X is an original summand of BQ_{p}^{∧}.

Lemma 3.2. If X is an indecomposable summand of BG_{p}^{∧} then X originates in BQ_{p}^{∧} for some p-toral subgroup of G which is unique up to isomorphism.

Proof. The first part of this was observed explicitly by Lee [L: §5]; the second part follows easily from his results.

Now we recall an indecomposable summand X of BQ_{p}^{∧} corresponds to a unique (up to conjugation) primitive idempotent e ∈ {BQ_{p}^{∧}, BQ_{p}^{∧}}, i.e. X = eBQ_{p}^{∧}. Furthermore from the point of view of stable summands nothing is lost by passing to homology.

Proposition 3.3. [Ml] If X is a connected p-complete spectrum then the ring homomorphism

h = H_{∗} : {X, X} → H_{A}(H_{∗}X)

satisfies the following idempotent lifting property: if e_{1} + ⋯ + e_{n} = 1 is a primitive orthogonal idempotent decomposition of 1 in H_{A}(H_{∗}X) then there is a primitive orthogonal decomposition f_{1} + ⋯ + f_{n} = 1 of 1 in {X, X} such that f_{i}_{∗} = e_{i}. Furthermore, if f′_{1} + ⋯ + f′_{n} = 1 is another such decomposition, then f_{i}X ≅ f′_{i}X, 1 ≤ i ≤ n.

For a p-toral group Q, we are interested in two ideals J(Q) and \overline{J}_{τ} (Q) of {BQ_{p}^{∧}, BQ_{p}^{∧}}. They are defined exactly as in the finite group case: J(Q) is the ideal generated by maps

BQ_{p}^{∧} → BR_{p}^{∧} → BQ_{p}^{∧}

where R is a proper p-toral subgroup of Q. \overline{J}_{τ} (Q) is the ideal generated by maps of the form

BQ_{p}^{∧} →^{tr} BR_{p}^{∧} →^{Bh} BQ_{p}^{∧}

where tr is the transfer, R a proper p-toral subgroup of Q, and h : R → Q is a homomorphism of Lie groups. Then \overline{J}_{τ} (Q) ≤ J(Q). We shall also consider the images of these ideals under the map

h = H_{∗} : {BQ_{p}^{∧}, BQ_{p}^{∧}} → H_{A}(H_{∗}BQ)

Proposition 3.4. [L; Prop. 2.1] Let \overline{e} be a primitive idempotent of H_{A}(H_{∗}BQ) such that \overline{e} ∉ h(J(Q)). If e is a primitive idempotent of \{BQ_{p}^{∧}, BQ_{p}^{∧}\} such that h(e) = \overline{e} then eBQ_{p}^{∧} is an original summand of BQ_{p}^{∧}. Conversely, if eBQ_{p}^{∧} is an original summand of BQ_{p}^{∧} for some primitive idempotent e of {BQ_{p}^{∧}, BQ_{p}^{∧}} then h(e) ∉ h(J(Q)).

As we shall now explain, the idempotents of F_{p}Out(Q_{∞}) determine the original summands of BQ_{p}^{∧}. A priori it would seem more natural to consider Out(Q), however, even in the simplest cases this group does not contain enough information. For example if Q = S^{1}, the circle, then Out(Q) = Z/2 while it is well known that BQ splits into p − 1 summands if p is an odd prime. Thus if p > 3, the idempotents of F_{p}Out(Q) can not give this splitting.

Lemma 3.5. The composite ring homomorphism

Z_{p}Out(Q_{∞}) →^{i} \{BQ_{∞}_{p}^{∧}, BQ_{∞}_{p}^{∧}\} →^{h} H_{A}(H_{∗}BQ) →^{π} H_{A}(H_{∗}BQ)/\sqrt{h(J(Q))}

is surjective.

Proof. According to Prop. 2.13, h(J(Q)) is generated over F_{p} by maps of the form

H_{∗}BQ ⟶^{tr_{∗}} H_{∗}BS ⟶^{Bψ_{∗}} H_{∗}(BQ)

where S is a proper p-toral subgroup of Q with N_{Q}(S)/S finite or S = Q and ψ : Q → Q a homomorphism which is not a surjection. Thus, if ψ : Q → Q is a surjection, we must show ψ_{∗} is in the image modulo h(J(Q)). By [L; Thm.4.1] if ker(ψ) contains an element of order p then for some N, ψ_{∗}^{N} ∈ h(J(Q)), i.e. ψ_{∗} ∈ \sqrt{h(J(Q))}. Thus it remains to show that if ker(ψ) has no elements of order p then ψ_{∗} ∈ im(πhi). However for such ψ, ψ(T) = T, hence ψ(T_{∞}) = T_{∞} and ψ(Q_{∞})/T_{∞} = W. Therefore by Proposition 2.8(3), ψ(Q_{∞}) is conjugate in Q to Q_{∞} . This defines an automorphism of Q_{∞} which maps to ψ_{∗}. □

Proposition 3.6. There are 1-1 correspondences (1) {homotopy classes of original summands of BP _{p}^{∧}}

(2) {conjugacy classes of primitive idempotents in F_{p}Out(P_{∞})}

(3) {isomorphism classes of simple F_{p}Out(P_{∞}) modules}

If X = eBQ_{p}^{∧} is an original summand of BQ_{p}^{∧} and f ∈ Z_{p}Out(Q_{∞}) is the corresponding idempotent then in {BQ_{p}^{∧}, BQ_{p}^{∧}}, f decomposes as a sum of primitive idempotents

f = e_{0} + e_{1} + ⋯ + e_{k}

where e_{0} = e and e_{i} ∈ J(Q). With this notation we define

Definition 3.7. If e_{i} ∉ \overline{J}_{τ} (Q) then Y_{i} = e_{i}BQ_{p}^{∧} is said to be linked to X in BQ_{p}^{∧}. In particular an original summand of BQ_{p}^{∧} is linked to itself in BQ_{p}^{∧}.

Example 3.8. Let Q = O(2) the orthogonal group. Then Q is 2-toral. ¿From [MiP] we have a 2-complete stable decomposition

BO(2)_{2}^{∧} ≅ BSO(3)_{2}^{∧} ∨ L(2) ∨ B(Z/2)

It is straightforward to see BSO(3)_{2}^{∧} originates in BO(2)_{2}^{∧}, L(2) originates in B(Z/2 × Z/2), and B(Z/2) is linked to L(2) in B(Z/2 × Z/2).

Let

R(Q) = F_{p} ⊗ \{BQ_{p}^{∧}, BQ_{p}^{∧}\}/\overline{J}_{τ} (Q)

and let X be an indecomposable summand which is linked to a summand in BQ_{p}^{∧}. Since X corresponds to a conjugacy class of primitive idempotents e ∈ {BQ_{p}^{∧}, BQ_{p}^{∧}} it also corresponds to a conjugacy class of primitive idempotents ê ∈ R(Q) via projection. In this way X corresponds to a simple R(Q) module. We will use this module below to determine the multiplicity of X in BG_{p}^{∧}.

The following lemma was proved for finite groups in [MP]. With the preceding notion of linkage for p-toral groups the same proof carries over for compact Lie groups.

Lemma 3.9 (Replacement Lemma). Let X be an indecomposable summand of BG_{p}^{∧}. Suppose there is a unit in {X, X} of the form

X → BG_{p}^{∧} →^{tr} BR_{p}^{∧} → X

where R is a p-toral subgroup of G and X is not linked in BR_{p}^{∧}. Then there is a unit of the form

X → BG_{p}^{∧} →^{tr} BS_{p}^{∧} → X

where S ≨ R is p-toral and X is linked in BS_{p}^{∧}.

Let X be an indecomposable summand of BG_{p}^{∧}. As in our paper [MP] we use a matrix A_{τ} (X) to calculate the multiplicity of X in BG_{p}^{∧}. As in [MP] we begin by defining the matrix a_{uv} to have entries

X ⟶^{i} BP_{u} ⟶^{Bi} BG ⟶^{tr} BP_{v} ⟶^{π} X

where i is an inclusion map, tr is the reduced transfer, π is a projection onto the summand X. P_{u} and P_{v} are representatives of conjugacy classes of p-toral subgroups of G, where N_{G}(P)/P is a finite group, in which X is a summand of the classifying space. The maps i and π need only be chosen so as to detect distinct copies of X. Hence the dimension of the matrix a_{kl} is the multiplicity of X in BP_{u} by the multiplicity of X in BP_{v}.

Since {X, X} is a local ring, k = {X, X}/Rad({X, X}) is a finite field [Mr]. For our purposes we consider the possibly larger ideal I_{uv} generated by Rad({X, X}) together with maps of the form

X → BP_{u} ⟶^{δ} BP_{u} → BP_{v} → X

where δ ∈ \overline{J}_{τ} (P_{u}). Then \{X, X\}/I_{uv} is either k or 0 ∈ k ; in either case it is a vector space over k. Define ā_{uv} to be the matrix a_{uv} with entries taken modulo I_{uv}.

Definition 3.10.

A_{τ} (X) = (ā_{uv})

Thus A_{τ} (X) is a block matrix over k composed of submatrices ā_{uv}.

Proposition 3.11.

a) ā_{uv} = 0 unless P_{u} ≤_{G} P_{v}.

b) If P_{u} ≤_{G} P_{v} then any row in ā_{uv} is a row in ā_{vv}.
c) ā_{uu} = 0 unless X is linked in BP_{u}.

Proof. (a) follows immediately from the double coset formula and the fact that the entries in ā_{uv} are modulo \overline{J}_{τ} (P_{u}). (b) follows from the fact that up to conjugation the inclusion of P_{u} in G factors through P_{v}. (c) follows since the map π : BP_{u} → X can be written π = π^{∘}e where e ∈ {BP_{u}, BP_{u}} is an idempotent for the summand X. If X is not linked in BP_{u} then by definition e ∈ \overline{J}_{τ} (P_{u}). □

Let m(X, BG) be the multiplicity of X in BG then:

Theorem 3.12.

m(X, BG) = rank_{k}A_{τ} (X)

Proof. Let r = rank_{k}A_{τ} (X) and s = m(X, BG). We will first show that r ≤ s. This follows immediately since the identity on X^{∨r} can be written

X^{∨r} → X^{∨n} ⟶^{A_{τ} (X)} X^{∨n} → X^{∨r}

where n is the size of A_{τ} (X). All the entries in A_{τ} (X) factor through BG so we have the identity factoring through BG.

We now want to show that s ≤ r. Since X^{∨s} is a summand of BG localized at p, we can write the s × s identity matrix as a matrix of maps of the form

(1)  X ⟶^{f} BG ⟶^{g} X.

If P is a Sylow p-subgroup then

α : BG ⟶^{tr} BP ⟶^{Bi} BG

is an equivalence. Thus id : BG → BG is homotopic to

BG ⟶^{tr} BP ⟶^{βBi} BG

where β is a homotopy inverse for α. It follows that (1) factors as

(2)  X ⟶^{f} BG ⟶^{tr} BP ⟶^{gβBi} X.

We now use the Replacement Lemma 3.6 to replace each entry 1 of form (2) in the identity matrix by a linear combination of entries from A_{τ} (X) of the form

X ⟶^{f} BG ⟶^{tr} BQ → X.

Consequently the space spanned by such maps is a subspace of the row space of A_{τ} (X) and so s ≤ r.

The matrix A_{τ} (X) is potentially infinite, but the rank must be finite or the above argument would show that there are infinitely many copies of X in BG_{p}^{∧}, which is false.

□

We have reduced the problem of determining m(X, BG) to determining the rank of A_{τ} (X). Recall from above Lemma 3.6 that if X is linked in BQ_{i}_{p}^{∧} then X corresponds to a simple R_{i}-module M_{i} where R_{i} = R(Q_{i}) = F_{p}⊗\{BQ_{i}_{p}^{∧}, BQ_{i}_{p}^{∧}\}/\overline{J}_{τ} (Q_{i}). Further End_{R_{i}}(M_{i}) = k = {X, X}/Rad. Let {Q_{i}} be a set of representatives of G-conjugacy classes of p-toral subgroups where X is linked in BQ_{i}_{p}^{∧}. Which leads us to the next theorem.

Theorem 3.13.

rank_{k}A_{τ} (X) = ∑ dim_{k} \overline{W}_{i}M_{i}

where the sum ranges over {Q_{i}} such that X is linked in BQ_{i}_{p}^{∧} and C_{G}(Q_{i})/Z(Q_{i}) is a finite p′-group. \overline{W}_{i} = ∑ c_{x} where x ∈ N_{G}(Q_{i})/Q_{i}.

Proof. By Proposition 3.9, A_{τ} (X) can be reduced to the diagonal matrix diag(ā_{ii}). Thus

(3.14)  rank_{k}A_{τ} (X) = ∑ rank_{k}ā_{ii}

where the sum can be taken over representatives of G-conjugacy classes of P_{i}’s where X is linked in BQ_{i}, also by Proposition 3.9.

Let m = m(X, BQ_{i}) then by the double coset formula for compact Lie groups [F1] ā_{ii} equals

X^{∨m} → BQ_{i} ⟶^{\overline{W}_{i}} BQ_{i} → X^{∨m}

modulo \overline{J}_{τ} (Q_{i}). Let π : \{X, X\} → End(M_{i}) be reduction by the radical then we get π(ā_{ii}) = \overline{W}_{i} ∈ Mat_{m}(M_{i}). It is now clear from representation theory that

rank_{k}ā_{ii} = dim_{k} \overline{W}_{i}M_{i}.

However if p divides |C_{G}(Q_{i})/Z(Q_{i})| then \overline{W}_{i}M_{i} = 0 modulo p. So the sum in (3.14) equals the sum in the statement of the theorem. □

4. Some results on the functor P Hom.

In order to use Theorems 3.12 & 3.13 to compute multiplicities of summands, we need an algebraic result, Proposition 4.3 below. We begin by recalling the functor P Hom of [MP; §4]. Let R be an algebra over F_{p} and let N be an R-module with projective cover P_{R}(N). We define a functor P Hom_{R}(M, N) for R-modules M to be the set of R-module homomorphisms ρ : M → N which factor through P_{R}(N)

P_{R}(N)
ψ↗  ↘ ^{π}
M ⟶_{ρ}  N

P Hom is a functor from R-modules to F_{p}-vector spaces. We establish some facts about P Hom used in the proofs of Theorems 1.1 and 1.2.

Lemma 4.1. Let ρ : R → S be a surjection of algebras over F_{p}, satisfying the unique idempotent lifting property, with S artinian. If N is a finitely generated S-module, then viewed as an R-module via ρ, N has a projective cover P_{R}(N).

Proof. Since S is artinian, N has a projective cover P_{S}(N) = ⨁ Sē_{i} for some idempotent in S. Choose an idempotent e_{i} so that ρ(e_{i}) = ē_{i}. Then P_{R}(N) = ⨁ Re_{i} is easily seen to be a projective cover of N viewed as an R-module. □

Remark 4.2. The homomorphism ρ induces homomorphisms from Re_{i} to Sē_{i} and hence an R-module homomorphism ρ_{∗} : P_{R}(N) ⟶ P_{S}(N). Note also that the projection π : P_{R}(N) ⟶ N factors through ρ_{∗}, i.e. π = π̄^{∘}ρ_{∗} where π̄ : P_{S}(N) ⟶ N is the projection for P_{S}(N).

Proposition 4.3. Let ρ : R → S be a surjection of algebras over F_{p}, satisfying the unique idempotent lifting property, with S artinian and quasi-frobenius. Let N be a finitely generated S-module. If x ∈ R there is an isomorphism of F_{p}-modules

F : P Hom_{R}(Rx, N) ⟶^{≈} xN = ϕ(x)N

given by F (ϕ) = ϕ(x) for ϕ ∈ P Hom_{R}(Rx, N).

First we show that R satisfies a sort of injectivity property.

Lemma 4.4. Assuming the hypotheses of Proposition 4.3, let f : Rx → P_{R}(N) be a homomorphism of R-modules. Then there exists a map g completing a diagram of R-module homomorphisms

P_{R}(N)
f↗  ↖ ^{g}
Rx ⟶_{i}  R

where i is inclusion and the diagram commutes modulo the ker ρ_{∗} (i.e., ρ_{∗}^{∘}f = ρ_{∗}^{∘}g^{∘}i. The homomorphism ρ_{∗} : P_{R}(N) ⟶ P_{S}(N) is induced from ρ as in Remark 4.2.

Proof. We adopt the convention x̄ = ρ(x). The projective module P_{R}(N) = ⨁ Re_{i}; define P_{S}(N) = ⨁ Sē_{i} then P_{S}(N) is a projective cover of N as an S-module. The surjection ρ induces an S-module homomorphism f̄ : Sx̄ ⟶ P_{S}(N), where ρ_{∗}^{∘}f = f̄^{∘}ρ|_{Rx}. Consider the inclusion ī: Sx̄ ⟶ S since P_{S}(N) is also an injective module there exists an S-module homomorphism ḡ : S ⟶ P_{S}(N) completing the diagram

P_{S}(N)
f̄↗  ↖ ḡ
Sx̄ ⟶_{ī}  S

Since R is a free (hence projective) R-module there exists a R-module homomorphism g : R ⟶ P_{R}(N) that completes the diagram below

R
g ↙ ↘ ^{ḡ∘ρ}
P_{R}(N) ⟶_{ρ_{∗}}  P_{S}(N)

The S-module P_{S}(N) is an R-module through ρ and the induced homomorphism ρ_{∗} is a surjection by construction.

To complete the proof we need only verify that ρ_{∗}^{∘}f = ρ_{∗}^{∘}g^{∘}i.

ρ_{∗}^{∘}g^{∘}i = ḡ^{∘}ρ^{∘}i
= ḡ^{∘}ī^{∘}ρ|_{Rx}
= f̄^{∘}ρ|_{Rx}
= ρ_{∗}^{∘}f.

Proof of Proposition 4.3. We first verify that F is well-defined. Let ϕ ∈ P Hom_{R}(Rx, N); we need to check that ϕ(x) is in xN. Since ϕ is in P Hom_{R}(Rx, N), ϕ factors ϕ = π^{∘}f where π : P_{R}(N) ⟶ N is the projection and f is a homomorphism from Rx to P_{R}(N). By Remark 4.2 π = π̄^{∘}ρ_{∗} which yields

ϕ = π^{∘}f = π̄^{∘}ρ_{∗}^{∘}f = π̄^{∘}ρ_{∗}^{∘}g^{∘}i = π^{∘}g^{∘}i.

Hence by Lemma 4.4 ϕ factors through R and ϕ(x) = π̄^{∘}ρ_{∗}^{∘}g(x1) = xπ̄^{∘}ρ_{∗}^{∘}g(1) ∈ xN. The rest of the proof is the same as that given in the finite group case [MP; Lemma 4.2].

5. Proof of Theorem 1.1.

We shall show BG_{p}^{∧} and BG′_{p}^{∧} have the same summands with the same multiplicities. This is accomplished by computing multiplicities with Theorems 3.12 & 3.13. We begin with some notation. Let Cen(P, G) ⊂ Inj(P, G) consist of those G-conjugacy classes of injections α : P → G such that C_{G}(α(P))/Z(α(P)) is a finite p′-group.

Lemma 5.1. Under the hypotheses of Theorem 1.1 there are isomorphisms

F_{p}Cen(P, G) ≈ F_{p}Cen(P, G′)

of Out(P) modules.

Proof. If S is a Sylow p-subgroup of G then F_{p}Inj(S, G) ≠ 0. Thus F_{p}Inj(S, G′) ≠ 0 and S ≤ G′. Reversing the argument shows G and G′ have isomorphic Sylow psubgroups. In this case the result holds since Inj(S, G) = Cen(S, G).

The proof now proceeds by downward induction on rank(P) followed by |π_{0}(P)| using the argument of Prop. 4.5 of [MP]; the requisite Lemma 4.6 holds for compact Lie groups. Induction works because of the following observation: if Q is p-toral and C_{G}(Q)/Z(Q) contains an element of order p then C_{G}(Q̃)/Z(Q̃) is a finite p′-group where Q̃ is a Sylow p-subgroup of Q · C_{G}(Q). □

Proof of Theorem 1.1. Let Q be a p-toral group with original summand X. Let L(X) be a set of representatives of isomorphism classes of p-toral subgroups in which the summand X is linked. Let M be the module associated to Q, X above Theorem 3.13. For each Q ∈ L(X) let {H_{i}} (resp. {H′_{j}}) be a set of representatives of G- (resp. G′-) conjugacy classes of subgroups isomorphic to Q satisfying the additional requirement that C_{G}(H_{i})/Z(H_{i}) (resp. C_{G′}(H′_{j})/Z(H′_{j})) is a finite p′ group. As an R = F_{p}Out(Q) module F_{p}Cen(Q, G) decomposes as a sum of permutation modules

F_{p}Cen(Q, G) ≈ ⨁_{i} R/RW_{G}(H_{j}) ≈ ⨁_{i} R · \overline{W}_{G}(H_{i}).

where \overline{W}_{G}(H_{i}) = ∑ c_{x}, x ∈ W_{G}(H_{i}) Combining this with Lemma 5.1 we have

⨁_{i} R · \overline{W}_{G}(H_{i}) ≈ ⨁_{j} R · \overline{W}_{G′}(H′_{j})

Thus

⨁_{i} P Hom_{R}(R · \overline{W}_{G}(H_{i}), M) ≈ ⨁_{j} P Hom_{R}(R · \overline{W}_{G′}(H′_{j}), M)

By Prop. 4.3 we obtain

∑_{i} dim_{k}\overline{W}_{G}(H_{i}) · M ≈ ∑_{j} dim_{k}\overline{W}_{G′}(H′_{j}) · M

Summing over Q ∈ L(X) we obtain the sum over a set of representatives {Q(l)} of G-conjugacy (equals G′-conjugacy) classes of all p-toral subgroups in which X is linked

∑_{l} dim_{k}\overline{W}_{G}(Q(l)) · M(l) ≈ ∑_{l} dim_{k}\overline{W}_{G′}(Q(i)) · M(l)

where M(l) is the module associated with Q(l),X. Since p ∤ |C_{G}(Q(l))/Z(Q(l))|,

\overline{W}_{G}(Q(l)) · M(l) = q ∑ c_{x} · M(l)

where q is prime to p and x ∈ N_{G}(Q(l))/Q(l). Thus by Theorems 3.12 & 3.13 we have m(X, G) = m(X, G′) which completes the proof.

6. Proof of Theorem 1.2.

(1) implies (2)

Proposition 6.1. Let G, G′ be p-Roquette compact Lie groups whose classifying spaces BG_{p}^{∧} and BG′_{p}^{∧} are stably equivalent. Then there is a stable equivalence f : BG_{p}^{∧} → BG′_{p}^{∧} which, for p-toral groups Q, induces an isomorphism of Out(Q_{∞}) modules

f′_{∗} : F_{p}Inj_{Q}(Q_{∞}, G) →^{≈} F_{p}Inj_{Q}(Q_{∞}, G′)

Proof. By Cor. 2.14, we may assume f has homotopy class in {BG_{p}^{∧}, BG′_{p}^{∧}}_{th}. Let A = Z_{p}Inj_{Q}(Q_{∞}, G), B = {BQ_{p}^{∧}, BG_{p}^{∧}}_{th}, and let A′, B′ be defined similarly with G replaced by G′. By Prop. 2.12 (1), f induces a homomorphism f_{∗} : J(Q, G)_{th} → J(Q, G′)_{th}. Since f is an equivalence f_{∗} is injective. It is also surjective; to see this recall that by Prop. 2.13 we can replace a homotopy inverse for f by a map g : BG′_{p}^{∧} → BG_{p}^{∧} where [g] ∈ {BG′_{p}^{∧}, BG_{p}^{∧}}_{th} such that g_{∗} = f^{−1}_{∗} in homology. Thus f^{∘}g = id + ρ where ρ ∈ R, where R = {BG′_{p}^{∧}, BG′_{p}^{∧}}_{th} and ρ_{∗} = 0 in homology. By Prop. 2.15, ρ ∈ rad(R) which implies f^{∘}g is a unit of R. Thus f_{∗}^{∘}g_{∗} : J(Q, G′)_{th} → J(Q, G′)_{th} equals multiplication by the unit f^{∘}g and so f_{∗} is surjective.

Thus the induced map

\overline{f}_{∗} : B/J(Q, G)_{th} → B′/J(Q, G′)_{th}

is an isomorphism. By Prop. 2.12, \overline{f}_{∗}^{∘}j factors

A  ⟶^{f′_{∗}}  A′
j↓  ^{j}↓
B/J(Q, G)_{th} ⟶^{\overline{f}_{∗}} B′/J(Q, G′)_{th}

Now suppose α ∈ Inj_{Q}(Q_{∞}, G). To check that f′_{∗}(α) ∈ F_{p}Inj_{Q}(Q_{∞}, G′) we observe that modulo J(Q, G′) the only maps of concern are homomorphisms Q → G′ with kernels not containing an element of order p. However, such homomorphisms are injective on Q_{∞} and they map Q to a p-toral subgroup of G′ isomorphic to Q.

Similarly there is an induced map g′_{∗} : A′ → A. Since j is injective so is f′_{∗}. Viewing A′ as an R submodule of B/J(Q, G′)_{th} via j, we conclude f′_{∗}^{∘}g′_{∗} : A′ → A′ equals multiplication by the unit f^{∘}g. Thus f′_{∗} is also surjective. □

Theorem 1.4 now follows from Proposition 6.1 and Lemma 2.5. The number of conjugacy classes of subgroups isomorphic to a p-toral group Q is number of transitive summands in F_{p}Inj_{Q}(Q_{∞}, G). The number of transitive summands is equal to the number of occurrences of the trivial module in a composition series, hence the number is an isomorphism invariant.

(2) implies (1)

We do this by showing BG_{p}^{∧} and BG′_{p}^{∧} have the same indecomposable summands with the same multiplicities. The proof is analogous to that of Theorem 1.1 in Section 5. We begin by proving an analog of Lemma 5.1. Let Cen_{Q}(Q_{∞}, G) ⊂ Inj_{Q}(Q_{∞}, G) be the set of classes represented by injections α : Q_{∞} → G such that if Q′ = \overline{im(α)}, Q′ ≈ Q then C_{G}(Q′)/Z(Q′) is a p′-group which is necessarily finite since N_{G}(Q′)/Q′ is finite by assumption.

Proposition 6.2. Under the hypotheses of Theorem 1.2 (2) there are isomorphisms

F_{p}Cen_{Q}(Q_{∞}, G) ≈ F_{p}Cen_{Q}(Q_{∞}, G′)

of Out(Q_{∞}, G′) modules.

Proof. If α ∈ Inj_{Q}(Q_{∞}, G) then \overline{im(α)} = Q′ where Q′ ≈ Q. Therefore the proof can proceed by downward induction starting with a Sylow p-subgroup exactly as the proof of Lemma 5.1 □

Lemma 6.3. There is a short exact sequence of groups

1 → W_{Q}(Q_{∞}) ⟶ W_{G}(Q_{∞}) ⟶^{ρ} W_{G}(Q) → 1

Proof. With the subspace topology Q_{∞} ≤ Q is dense. By continuity, N_{G}(Q_{∞}) ≤ N_{G}(Q), C_{G}(Q_{∞}) = C_{G}(Q). Let ρ : W_{G}(Q_{∞}) → W_{G}(Q) the induced homomorphism. Then W_{Q}(Q_{∞}) = ker(ρ) and it remains to show ρ is surjective. If g ∈ N_{G}(Q) then c_{g}(T) = T since T is a maximal torus. Thus c_{g}(T_{∞}) = T_{∞} which implies c_{g}(Q_{∞}) is Q-conjugate to Q_{∞} by Prop. 2.4(3). Thus modulo an element of Q, c_{g} maps Q_{∞} to itself; thus we have constructed an element of N_{G}(Q_{∞}) which maps to g. □

Let R = F_{p}Out(Q_{∞}) and let I be the ideal of R generated by the augmentation ideal, ker(ϵ) : F_{p}W_{Q}(Q_{∞}) → F_{p}. The quotient ring will be denoted by R̃ = R/I.

Lemma 6.4. Let N = F_{p}[Out(Q_{∞})/W_{G}(Q_{∞})]. As an R̃ module

N/I · N ≈ R̃ · \overline{W}_{G}(Q)

Proof. Let S = F_{p}W_{G}(Q_{∞}). Since S ∩ I = kerϵ, we have W_{G}(Q) ≤ R̃ by Cor. 5.4. Tensoring is right exact hence there is an exact sequence

I ⊗_{S} F_{p} → R ⊗_{S} F_{p} → R̃ ⊗_{S} F_{p} → 0

Now N = R ⊗_{S} F_{p} and R̃ ⊗_{S} F_{p} = R̃ ⊗_{W_{G}(Q)} F_{p} = R̃ · \overline{W}_{G}(Q). Therefore the result follows from the exact sequence. □

We can now prove the implication (2) ⟹ (1) of Theorem 1.2. We continue with the notation of Section 5. By Prop. 6.2.

F_{p}Cen_{Q}(Q_{∞}, G) ≈ F_{p}Cen_{Q}(Q_{∞}, G′)

as an R = F_{p}Out(Q_{∞}) module. F_{p}Cen_{Q}(Q_{∞}, G) decomposes as a sum of permutation modules

F_{p}Cen_{Q}(Q_{∞}, G) ≈ ⨁_{i} R/RW_{G}(H_{i}_{∞}).

Combining these isomorphisms and computing modulo I we have

⨁_{i} R̃ · \overline{W}_{G}(H_{i}) ≈ ⨁_{j} R̃ · \overline{W}_{G}(H_{j})

by Lemma 6.4. Thus

⨁_{i} P Hom_{R}(R̃ · \overline{W}_{G}(H_{j}), M) ≈ ⨁_{j} P Hom_{R}(R̃ · \overline{W}_{G′}(H′_{j}), M)

By Prop. 4.3, we obtain

∑_{i} dim_{k}\overline{W}_{G}(H_{j}) · M ≈ ∑_{j} dim_{k}\overline{W}_{G′}(H′_{j}) · M

Summing over Q ∈ L(X) we obtain the sum over the set {Q(l)} of representatives of G-conjugacy (equals G′-conjugacy) classes of p-toral subgroups in which X is linked

∑_{l} dim_{k}\overline{W}_{G}(Q(l)) · M(l) ≈ ∑_{l} dim_{k}\overline{W}_{G′}(Q(l)) · M(l)

It now follows from the argument of the proof of Section 5 that m(X, G) = m(X, G′) which completes the proof.

7. Proof of Theorem 1.3.

(1) implies (2)

Proposition 7.1. Let G, G′ be p-Roquette compact Lie groups whose classifying spaces BG_{p}^{∧} and BG′_{p}^{∧} are stably equivalent. Then the isomorphism f′_{∗} of Proposition 6.1 induces an isomorphism

f′′_{∗} : F_{p}Inj_{Q}(Q_{K}, G) →^{≈} F_{p}Inj_{Q}(Q_{K}, G′)

of Out(Q_{K}) modules for every p-toral group Q.

Proof. Let α ∈ Inj_{Q}(Q_{K}, G). By definition α(Q_{K}) ≤ Q′ where Q′ ≈ Q and α(Q_{K})/(α(Q_{K}) ∩ T′) ≈ Q′/T′ where T′ is the maximal torus of Q′. Inclusion Q_{K} ≤ Q induces a restriction homomorphism F_{p}Inj_{Q}(Q_{∞}, G) → F_{p}Inj_{Q}(Q_{K}, G) It follows from the proof of Prop. 6.1 that it suffices to show that the composite

Q_{K} →^{α} Q′ →^{h} G′

is an element of Inj_{Q}(Q_{K}, G′) where h : Q′ → G′ is a homomorphism whose kernel contains no element of order p. First we note h(Q′) ≈ Q′. Now let T be the maximal torus of h(Q′). Then h|_{T′}_{∞} : T′_{∞} →^{≈} T_{∞} Hence h induces isomorphisms of p-groups

α(Q_{K})/(α(Q_{K}) ∩ T′_{∞}) ≈ hα(Q_{K})/(hα(Q_{K}) ∩ T_{∞})
Q′/(Q′ ∩ T′) ≈ h(Q′)/(h(Q′) ∩ T)

The source groups are isomorphic so therefore are the targets. □

(2) implies (1)

We begin by recording an important observation with which we shall capture enough information about W_{G}(Q) from a single Q_{k}.

Corollary 7.2. For K large, depending on Q, the sequence of Lemma 6.1 restricts to an exact sequence

1 → W_{Q}(Q_{K}) → W_{G}(Q_{K}) → W_{G}(Q) → 1

Proof. Since W_{G}(Q) is finite we can choose a finite set F = {[g] ∈ W_{G}(Q_{∞})} which maps onto W_{G}(Q). Since c_{g}(Q_{0}) ≤ Q_{k} for some k we can choose K large enough so that c_{g}(Q_{0}) ≤ Q_{K} for all [g] ∈ F. By construction, g ∈ N(Q) ≤ N(T), hence c_{g}(T_{k}) = T_{k}. By Theorem 2.3(4), Q_{k} = Q_{0}·T_{k} implying g ∈ N(Q_{K}). Furthermore for K large enough N(Q_{K}) ≤ N(Q) (see (c) in the proof of Theorem 2.3) Thus [g] ∈ W_{G}(Q_{K}). The kernel is W_{G}(Q_{K}) ∩ W_{Q}(Q_{∞}) = W_{Q}(Q_{K}). □

Let R = F_{p}Out(Q_{K}) and let I be the ideal of R generated by the augmentation ideal, ker(ϵ), for the augmentation ϵ : F_{p}W_{Q}(Q_{K}) → F_{p}. The quotient ring will be denoted by R̃ = R/I.

Lemma 7.3. Let N = F_{p}[Out(Q_{K})/W_{G}(Q_{K})]. As a left R̃ module

N/I · N ≈ R̃ · \overline{W}_{G}(Q_{K})

Proof. The result follows by an argument analogous to that of Lemma 6.2.

Lemma 7.4. If Q is a p-toral subgroup of a compact Lie group G then for K large |{(P_{K}) : P ≤ G, P ≈ Q}| = |{(P) : P ≤ G, P ≈ Q}|.

Proof. Let A = {Q(1), …, Q(N)} be a complete set of representatives for the Gconjugacy classes of subgroups of G isomorphic to Q. We use ∼ to denote Gconjugacy. If H_{∞}, H′_{∞} are two discrete approximations of P ≈ Q then H_{∞} ∼ H′_{∞} by Prop. 2.4(3). Hence H_{0} ≤_{G} H′_{i} for some i and so H_{i} ≤_{G} H′_{i} by 2.4(4). It follows that H_{i} ∼ H′_{i} since both groups have the same order by 2.4(3). Thus for each K large enough, B = {Q(1)_{K}, …, Q(N)_{K}} is a set of not necessarily distinct representatives for {(P_{K}) : P ≈ Q}. However if Q(i)_{K} ∼ Q(j)_{K} for all sufficiently large K then Q(i)_{∞} ∼ Q(j)_{∞} and hence Q(i) = \overline{Q(i)_{∞}} ∼ \overline{Q(j)_{∞}} = Q(j) proving |A| = |B|. □

Let Cen_{Q}(Q_{K}, G) ⊂ Inj_{Q}(Q_{K}, G) be the subset of classes of injections α : Q_{K} → G such that (1) α(Q_{K}) is contained in a subgroup Q′, Q′ ≈ Q, (2) α(Q_{K})/(T′ ∩ α(Q_{K})) ≈ Q′/T′, and (3) C_{G′}(Q′)/Z(Q′) is a finite p′ group.

Proposition 7.5. Under the hypotheses of Theorem 1.3 (2) there are isomorphisms

F_{p}Cen_{Q}(Q_{K}, G) ≈ F_{p}Cen_{Q}(Q_{K}, G′)

of Out(Q_{K}) modules.

Proof. Requiring an element α ∈ Inj_{Q}(Q_{K}, G) to belong to Cen_{Q}(Q_{K}) only involves the isomorphism type of Q. Hence the argument proceeds in a manner similar to that of Lemma 5.1. □

We can now prove the implication (2) ⟹ (1) of Theorem 1.3. Let L(X) be a set of representatives of isomorphism classes of p-toral subgroups in which the summand X is linked. For each Q ∈ L(X) choose an integer K satisfying Cor. 7.2 and Lemma 7.4. By Prop.7.5 we have an isomorphism

F_{p}Cen_{Q}(Q_{K}, G) ≈ F_{p}Cen_{Q}(Q_{K}, G′)

of R modules. Let {H_{i}} (resp. {H′_{j}}) be a set of representatives of G- (resp. G′-) conjugacy classes of p-toral subgroups isomorphic to Q. Then F_{p}Cen_{Q}(Q_{K}, G) decomposes as a sum of permutation modules for R

F_{p}Cen_{Q}(Q_{K}, G) ≈ ⨁_{i} R/R · W_{G}(H_{i}).

Combining these isomorphisms and computing modulo I we have

⨁_{i} R̃ · \overline{W}_{G}(H_{i}) ≈ ⨁_{i} R̃ · \overline{W}_{G′}(H′_{i})

by Lemma 7.3. Thus

⨁_{j} P Hom_{R}(R̃ · \overline{W}_{G}(H_{j}), M) ≈ ⨁_{j} P Hom_{R}(R̃ · \overline{W}_{G′}(H′_{j}), M)

∑_{i} dim_{k}\overline{W}_{G}(H_{i}) · M ≈ ∑_{j} dim_{k}\overline{W}_{G′}(H′_{j}) · M

Summing over Q ∈ L_{X} we obtain the sum over a set of representatives {Q(l)}of G-conjugacy (equals G′-conjugacy) classes of all p-toral subgroups in which X is linked

∑_{l} dim_{k}\overline{W}_{G}(Q(l)) · M(l) ≈ ∑_{l} dim_{k}\overline{W}_{G′}(Q(l)) · M(l)

By Theorem 3.10 this implies m(X, G) = m(X, G′) which completes the proof.

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Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, MI 49008

   E-mail address: martino@math-stat.wmich.edu

Department of Mathematics, Northwestern University, Evanston, IL 60208

   E-mail address: s priddy@math.nwu.edu