TORAL GROUPS AND CLASSIFYING

SPACES OF p–COMPACT GROUPS

by Kenshi Ishiguro

A compact Lie group G is called a p–toral group if G is an extension of a torus T by a finite p–group π, so that there is a short exact sequence T \overline{}G \overline{}π. Analogously, a p–compact toral group X [5, §6] is defined in such a way that there is a fibration (BT)_{p}^{∧} \overline{}BX \overline{}Bπ, where (BT)_{p}^{∧} is the p–completion of the classifying space of a torus T.

For instance, suppose T is a maximal torus of a connected compact Lie group K, and N_{p}T denotes the inverse image in the normalizer NT of a p–Sylow subgroup W_{p} of the Weyl group W (K). Then N_{p}T is a p–toral group, and the loop space of (BN_{p}T)_{p}^{∧} is a p–compact toral group.

T  \overline{}NT  \overline{}W (K) [figure: ‖]
T  \overline{}N_{p}T  \overline{}W_{p}

It is knwon [23] that if p is odd, N_{p}T is isomorphic to the semidirect product T ⋊ W_{p}, and the fibration (BT)_{p}^{∧} \overline{}(BN_{p}T)_{p}^{∧} \overline{}BW_{p} has a section. The classifying spaces of N_{p}T and other p–toral subgroups of K have been extensively used for the study of BK at the prime p, [13] etc. We note, however, that for the normalizer NT the loop space of (BNT)_{p}^{∧} need not be a p–compact toral group. In fact, if p does not divide the order of the Weyl group, then (BNT)_{p}^{∧} ≃ (BK)_{p}^{∧}. We will discuss more about (BNT)_{p}^{∧} later.

In this paper, we study some properties of the classifying spaces of p–toral groups BG and the classifying spaces of p–compact toral group BX. Since BT and Bπ are Eilenberg–MacLane spaces, the n–th homotopy groups of (BG)_{p}^{∧} and BX are zero for n ≥ 3. We consider a converse to this fact.

Typeset by AMS-TEX

Theorem 1. Suppose G is a compact Lie group, and X is a p–compact group. Then we have the following:

(i) If there is a positive integer k such that π_{n}((BG)_{p}^{∧}) = 0 for any n ≥ k, then (BG)_{p}^{∧} is the classifying space of a p–compact toral group.

(ii) If there is a positive integer k such that π_{n}(BX) = 0 for any n ≥ k, then X is a p–compact toral group.

The loop space of the p–completed classifying space (BG)_{p}^{∧} need not be a p–compact group unless π_{0}G is a p–group. For instance, if Σ_{3} denotes the symmetric group of order 6, then the loop space Ω(BΣ_{3})_{3}^{∧} is not a 3– compact group. On the other hand, at p = 2, we have (BΣ_{3})_{2}^{∧} ≃ Bℤ/2 so that Ω(BΣ_{3})_{2}^{∧} is a 2–compact group.

We will obtain the necessary and sufficient conditions that (BG)_{p}^{∧} be the classifying space of a p–compact toral group. Recall that a finite group γ is p–nilpotent if and only if γ is expressed as the semidirect product ν ⋊ γ_{p}, where ν is the subgroup generated by all elements of order prime to p, and where γ_{p} is the p–Sylow subgroup. For such a group γ, we see (Bγ)_{p}^{∧} ≃ Bγ_{p}. The converse is a result of Casacuberta [3]. Recall also that a fibration of connected spaces F \overline{}E \overline{}B is said to be preserved by the p–completion if F_{p}^{∧} \overline{}E_{p}^{∧} \overline{}B_{p}^{∧} is again a fibration. According to a result of [2], if π_{1}B is a finite p–group, then π_{1}B acts nilpotently on H_{∗}(F; 𝔽_{p}) and hence the fibration F \overline{}E \overline{}B is preserved by the p–completion. We note, however, that the fibration Bℤ/3 \overline{}BΣ_{3} \overline{}Bℤ/2 is not preserved by the p–completion when p = 3.

Theorem 2. Suppose G is a compact Lie group, and G_{0} is the connected component with the identity. The loop space of (BG)_{p}^{∧} is a p–compact toral group if and only if the following conditions (a) and (b) hold:

(a) G_{0} is a torus T and G/G_{0} = π_{0}G is p–nilpotent.

(b) The fibration BT \overline{}BG \overline{}Bπ_{0}G is preserved by

the p–completion. Moreover, the p–completed fibration (BT)_{p}^{∧} \overline{}(BG)_{p}^{∧} \overline{}(Bπ_{0}G)_{p}^{∧} splits if and only if T is a central subgroup of G.

Next we turn to another property of the classifying spaces of p–toral groups. It is well–known that if G is a p–toral group, for any compact Lie group K the canonical map Rep(G, K) \overline{}[BG, BK] obtained from delooping the classes of homomorphisms is bijective . This is due to work of Dwyer–Zabrodsky [7] and Notbohm [21]. The converse stated as follows will be proved in §2.

Theorem 3. Suppose G is a compact Lie group. If the canonical map Rep(G, K) \overline{}[BG, BK] is bijective for any compact connected Lie group K, then G is a p–toral group.

The theorem of Dwyer–Zabrodsky and Notbohm also implies that for such compact Lie groups G and K and for a map BG \overline{}BK induced from ρ ∈ Rep(G, K), the mapping space map(BG, BK)_{Bρ} is p–equivalent to BC(ρ), [22, Theorem 2.1]. Here C(ρ) denotes the centralizer of the homomorphism ρ : G \overline{}K. Suppose the loop space of (BK)_{p}^{∧} is a p– compact group. This holds if π_{0}(K) is a p–group, as mentioned before. A result of Dwyer–Wilkerson [5] then shows that the loop space of the mapping space is a p–compact group. Consequently Ω(BC(ρ))_{p}^{∧} should also be a p– compact group. Note that the centralizer C(ρ) need not be connected. For this reason, in §3 we will investigate the conditions of a compact Lie group that its loop space of the p–completed classifying space be a p–compact group. As an application, for a connected compact Lie group K with a maximal torus T, we will see that (BNT)_{p}^{∧} ≃ (BK)_{p}^{∧} if and only if p does not divide the order of the Weyl group.

1. Classifying spaces of p–compact toral groups.

We will prove Theorem 1 and Theorem 2 in this section. First we recall some results about K–theory. A result of Anderson–Hodgkin [1] on the Ktheory of Eilenberg–MacLane spaces states that K̃(K(π, n)) = 0 if either n ≥ 2 and π is a countable torsion group, or n ≥ 3 and π is finitely generated. Another result we need is due to Zabrodsky [24], and states that a map f : BG \overline{}BG′ is null homotopic if and only if K(f) = 0. This implies that if the K–theory of BG is null, the classifying space BG is contractible.

Lemma 1.1. Suppose G is a compact Lie group, and X is a p–compact group. Let Y = (BG)_{p}^{∧} or BX. Assume the following:

(a) Y is 1–connected.

(b) π_{2}(Y) is a torsion group.

(c) There is an integer k ≥ 3 such that π_{n}(Y) = 0 for any n ≥ k. Then Y is contractible.

Proof. Consider the Postnikov system {Y_{n}} of the space Y so that

π_{i}(Y_{n}) = \{ \begin{matrix}0 & if i > n \\ π_{i}(Y) & if i ≤ n \end{matrix}

and we have the fibrations K(π_{n}(Y), n) \overline{}Y_{n} \overline{}Y_{n−1} for all n. A result of Anderson–Hodgkin [1] on the K-theory of Eilenberg–MacLane spaces implies K̃(K(π_{n}(Y), n)) = 0 for all n under our assumption. Applying the Atiyah–Hirzebruch spectral sequence repeatedly, we see that

K̃(Y_{k}) = ⋯ = K̃(Y_{2}) = 0

Since Y_{k} = Y, the K–theory of Y is null. One can show that the space Y is contractible, [24] and [14]. □

Proof of Theorem 1. (i) Let π = π_{1}((BG)_{p}^{∧}) so that π is a finite p–group. Consider the Postnikov fibration F_{1} \overline{}(BG)_{p}^{∧} \overline{}K(π, 1). Since the composite map BG ^{\overline{}^{j}} (BG)_{p}^{∧} ^{\overline{}^{q}} K(π, 1) = Bπ factors through Bπ_{0}G, there is a homomorphism ρ : G \overline{}π such that q · j ≃ Bρ. Note that ρ is surjective. Consequently, if H = Ker ρ, then we obtain the following commutative diagram of fibrations:

BH \overline{}F_{1} [figure]

BG ^{^{j} \overline{}}(BG)_{p}^{∧} [figure: Bρ; q]

Bπ = Bπ

We notice that π_{2}(F_{1}) is a finitely generated module over the ring ℤ_{p}^{∧} of p–adic integers. Let ζ denote the free part of this module so that K(ζ, 2) = K(⊕^{n}ℤ_{p}^{∧}, 2) = (BT^{n})_{p}^{∧} for a torus T^{n}. Consider a map f : F_{1} \overline{}K(ζ, 2) such that f_{∗} : π_{2}(F_{1}) \overline{}π_{2}(K(ζ, 2)) is onto. Note that [F_{1}, (BT^{n})_{p}^{∧}] = ⊕^{n}H^{2}(F_{1}, ℤ_{p}^{∧}) and that the universal coefficient theorem says H^{2}(F_{1}, ℤ) ⊗ ℤ_{p}^{∧} ≅ H^{2}(F_{1}, ℤ_{p}^{∧}). Since T^{n} is abelian, we can find such a map f and a homomorphism ξ : H \overline{}T^{n} with f ≃ (Bξ)_{p}^{∧}. If K = Ker ξ and F_{2} is the homotopy fiber of f, the composite map φ : BK \overline{}(BH)_{p}^{∧} ≃ F_{1} factors through the space F_{2}.

BK F_{2} [figure]

(BH)_{p}^{∧} \overline{}F_{1} [figure: (Bξ)_{p}^{∧}; f]

(BT^{n})_{p}^{∧} = K(ζ, 2)

From our assumption, we see that there is an integer k ≥ 3 such that π_{n}(F_{2}) = 0 for any n ≥ k. We see also that F_{2} is 1–connected and π_{2}(F_{2}) is a finite abelian group. A result of Anderson–Hodgkin [1] together with the Atiyah–Hirzebruch spectral sequence implies that the K–theory of F_{2} is null. The argument is similar to the proof of Lemma 1.1. So the induced homomorphism K(φ) must be trivial, and hence φ is null homotopic, [24]. Consequently (BK)_{p}^{∧} is contractible so that (BH)_{p}^{∧} is a p–compact torus.

(ii) We begin with the fibration BX_{0} \overline{}BX \overline{}Bπ_{0}X where X_{0} denotes the identity component of the p–compact group X. Since π_{0}X is a finite p–group, it suffices to show that BX_{0} is a p–compact torus.

A result of Møller–Notbohm [20, Theorem 1.4] shows that we can find a simply–connected p–compact group X_{s}, a p–compact torus T_{p}^{∧} and a finite abelian p–group κ that give us a fibration Bκ \overline{}BX_{s} × BT_{p}^{∧} \overline{} BX_{0}. Consider a map f : BX_{0} \overline{}BT_{p}^{∧} which induces an isomorphism H^{2}(BX_{0}, ℤ_{p}^{∧}) ≅ H^{2}(BT_{p}^{∧}, ℤ_{p}^{∧}). We claim that the homotopy fiber F of the map f is a p–compact group. For suitable finite abelian p–groups α and β, in fact, we obtain the following commutative diagram of fibrations:

Bα  \overline{}Bκ  \overline{}Bβ [figure]
BX_{s}  \overline{}BX_{s} × BT_{p}^{∧}  \overline{}BT_{p}^{∧} [figure]
F  \overline{}BX_{0}  ^{^{f} \overline{}}BT_{p}^{∧}

We see that the map Bα \overline{}BX_{s} is central and that the space F is homotopy equivalent to B(X_{s}/α). Thus ΩF is a p–compact group, [5, Proposition 8.3].

The rest of our argument is similar to the one used in the previous case. From our assumption, we see that the K–theory of F is null. Thus Lemma 1.1 shows that the p–compact group F must be contractible. Consequently we obtain the desired result. □

This work was motivated by some comments from Chuck McGibbon on H–structures of the classifying spaces of p–compact groups. If BX is an Hspace, Hubbuck’s theorem implies π_{n}(BX) = 0 for n ≥ 3. Technically, if Y is a 1–connected p–local space such that the loop space of the 2–connected cover is a mod p finite CW–complex which has a C^{p}–space structure [16], then π_{n}(Y) = 0 for n ≥ 3. Coming back to p–compact groups, it is shown [11] that BX is an H–space if and only if X is abelian. This means that X is equivalent to the product of a p–compact torus and a finite abelian p–group. Another proof of the result can be given by our work.

Proof of Theorem 2. Assume first that the conditions (a) and (b) hold. So we have the p–completed fibration (BT)_{p}^{∧} \overline{}(BG)_{p}^{∧} \overline{}(Bπ_{0}G)_{p}^{∧}. Since π_{0}G is p–nilpotent, if π denote the p–Sylow subgroup of the finte group π_{0}G, then (Bπ_{0}G)_{p}^{∧} is homotopy equivalent to Bπ. Consequently the loop space Ω(BG)_{p}^{∧} is a p–compact toral group.

Conversely, assume that (BG)_{p}^{∧} is the classifying space of a p–compact toral group. So we have a fibration (BT_{G})_{p}^{∧} \overline{}(BG)_{p}^{∧} \overline{}Bπ where π = π_{1}((BG)_{p}^{∧}) is a finite p–group. Since G_{0} is connected, we see [BG_{0}, Bπ] = 0. Hence we obtain the following commutative diagram of fibrations:

(BT_{G})_{p}^{∧}  \overline{}(BG)_{p}^{∧}  \overline{}Bπ [figure]
BG_{0} \overline{}BG \overline{}Bπ_{0}G

For the compact connected Lie group G_{0}, there is a finite covering α \overline{} G_{s}×T_{0} \overline{}G_{0}, where G_{s} is a simply–connected Lie group, T_{0} is a torus, and α is a central subgroup. Notice that the composite map BG_{0} \overline{}(BG)_{p}^{∧} is induced by a monomorphism, and the map factors through (BT_{G})_{p}^{∧}. Consequently G_{s} must be trivial, since [BG_{s}, (BT_{G})_{p}^{∧}] = 0. This implies that G_{0} is a torus T.

Next we will prove that the fibration BT \overline{}BG \overline{}Bπ_{0}G is preserved by the p–completion. From the above diagram of fibrations, we see that the map BT = BG_{0} \overline{}(BT_{G})_{p}^{∧} is p–equivalent, and that π_{1}(Bπ_{0}G) acts nilpotently on H_{∗}(BT; 𝔽_{p}). The Mod–R fiber lemma [2] implies the desired result.

Since BT \overline{}BG \overline{}Bπ_{0}G is preserved by the p–completion and the loop space of (BG)_{p}^{∧} is a p–compact toral group, we see (Bπ_{0}G)_{p}^{∧} ≃ Bγ for some finite p–group γ. The homotopy equivalence Bγ \overline{}(Bπ_{0}G)_{p}^{∧} is induced by a homomorphism ρ : γ \overline{}π_{0}G. Notice that the image of ρ is contained in a p–Sylow subgroup π_{p} of π_{0}G. We can show that π_{p} is isomorphic to γ. Therefore π_{0}G is p–nilpotent.

Next we will discuss the splitting problem of the p–completed fibration (BT)_{p}^{∧} ^{\overline{}^{Bi}} (BG)_{p}^{∧} \overline{}(Bπ_{0}G)_{p}^{∧}. If this fibration splits, then we see that map((BT)_{p}^{∧}, (BG)_{p}^{∧})_{Bi} ≃ (BG)_{p}^{∧}. A result of Mislin–Thévenaz [17] shows T is central in G.

It remains to show the converse. Recall [5, §9] that, for a p–compact group X with a maximal torus T_{X}, the Weyl group W_{T_{X}} (X) is defined as the connected components of the Weyl space W_{T_{X}} (X) = \{w : BT_{X} \overline{} BT_{X} | id·ι ≃ ι·w\} for the maps id : BX \overline{}BX and ι : BT_{X} \overline{}BX. We see that, if T is central in G, the map [(BT)_{p}^{∧}, (BT)_{p}^{∧}] \overline{}[(BT)_{p}^{∧}, (BG)_{p}^{∧}] is injective. This implies that the action of the Weyl group W_{T_{p}^{∧}}(Ω(BG)_{p}^{∧}) on (BT)_{p}^{∧} is trivial. Thus (BG)_{p}^{∧} = (BT)_{p}^{∧} × (Bπ_{0}G)_{p}^{∧} and this completes the proof. □

2. Converse to the theorem of Dwyer–Zabrodsky and Notbohm.

We will prove Theorem 3 in this section. To begin with, we consider the converse when G is a finite group. Our argument goes as follows. First we see that if the canonical map Rep(G, K) \overline{}[BG, BK] is surjective, the finite group must be nilpotent. Next it is shown that any finte nilpotent group satisfying the assumption of this theorem must be a p–group. And then we discuss the general case using unstable Adams operations ψ^{k}, [22, p1064].

Lemma 2.1. For a finite group π, if there is an epimorphism from π to its p–Sylow subgroup π_{p} for each p, then the group π is nilpotent.

Proof. Suppose |π| = p_{1}^{e_{1}} ⋯ p_{r}^{e_{r}} where each p_{i} is a prime and e_{i} ≥ 1, (1 ≤ i ≤ r). From the assumption, we can find epimorphisms φ_{i} : π \overline{}π_{p_{i}}. Define a homomorphism φ : π \overline{}π_{p_{1}} ×⋯×π_{p_{r}} by φ(x) = (φ_{1}(x), ⋯, φ_{r}(x)). Notice that Ker φ = ⋂_{i} Ker φ_{i} is trivial. Since |π| = |π_{p_{1}} × ⋯ × π_{p_{r}}|, we conclude that π ≅ π_{p_{1}} × ⋯ × π_{p_{r}} and hence π is nilpotent. □

The following result will imply that Theorem 3 holds when G is a finite group.

Theorem 2.2. Suppose π is a finite group and K is a compact connected Lie Group.

(a) If there is a monomorphism π ↪ K and the map Rep(π, K) \overline{} [Bπ, BK] is onto, then π is nilpotent.

(b) If π is nilpotent, the Lie group K contains the unitary group U(2) as a subgroup, and the map Rep(π, K) \overline{}[Bπ, BK] is one–to–one, then π is a p–group.

Proof. (a) We see that [Bπ, BK] = ⊕_{p | |π|}[(Bπ)_{p}^{∧}, (BK)_{p}^{∧}] since K is connected. If ρ : π \overline{}K is a monomorphism, for a prime q which divides |π|, we define f_{q} : Bπ \overline{}BK as follows:

f_{q} = \{ \begin{matrix}Bρ & if p = q \\ ∗ & otherwise \end{matrix}

Since Rep(π, K) \overline{}[Bπ, BK] is onto, there is a homomorphism θ : π \overline{}K such that f_{q} = Bθ. We see that θ(π_{p}) = ∗ if p ≠ q since Bθ|_{Bπ_{p}} is trivial. Consequently any element in π whose order is prime to q is contained in the kernel of θ. On the other hand, since f_{q} = Bρ at q, it follows that θ|_{π_{q}} = ρ|_{π_{q}} up to isomorphism. Because ρ is injective, so is θ on π_{q}. This implies that θ factors through an epimorphism π \overline{}π_{q}. Lemma 2.1 shows that π is nilpotent.

(b) Since π is nilpotent, the finite group is the product of its p–Sylow subgroups; π = π_{p_{1}} × ⋯ × π_{p_{r}}. We will show that if r ≥ 2, then the map Rep(π, K) \overline{}[Bπ, BK] is not injective. Note that there is an epimorphism π \overline{}ℤ/p_{1}×ℤ/p_{2}. Here, let p = p_{1} and q = p_{2} and let a and b be generators of ℤ/p and ℤ/q respectively. Define two homomorphisms

ρ′_{1} , ρ′_{2} : ℤ/p × ℤ/q \overline{}U(2) ↪ K

by

ρ′_{1}(a) = ( \begin{matrix}ζ & 0 \\ 0 & 1 \end{matrix} ), ρ′_{2}(a) = ( \begin{matrix}1 & 0 \\ 0 & ζ \end{matrix} )

ρ′_{1}(b) = ρ′_{2}(b) = ( \begin{matrix}η & 0 \\ 0 & 1 \end{matrix} )

where ζ and η are primitive p–th and q–th roots of unity respectively. Let ρ_{i} be the composite π \overline{}ℤ/p × ℤ/q \overline{}K for i = 1, 2. Then we see that Bρ_{1} = Bρ_{2}. On the other hand, it’s easy to see that the trace of ρ_{1}(ab) is not equal to the trace of ρ_{2}(ab), and hence ρ_{1} ≠ ρ_{2} as representations. Therefore the map Rep(π, K) \overline{}[Bπ, BK] is not injective. This completes the proof. □

Proof of Theorem 3. First we claim that π_{0}G is a finite p–group. Consider the following commutative diagram:

Rep(π_{0}G, K) ^{\overline{}^{Φ}} [Bπ_{0}G, BK] [figure: α; β]

Rep(G, K) \overline{}[BG, BK] Note that the vertical map α is injective, and so is β by [9, Lemma 1]. Thus the delooping map Φ is also injective by the assumption. We next show that Φ is surjective. Since the canonical map Rep(G, K) \overline{}[BG, BK] is bijective, for any f ∈ [Bπ_{0}G, BK], there is ρ̂ ∈ Rep(G, K) such that β(f) = Bρ̂. Notice β(f)|_{BG_{0}} ≃ ∗ so that ρ̂|_{G_{0}} = ∗. So there is ρ ∈ Rep(π_{0}G, K) with α(ρ) = ρ̂. Since β(Φ(ρ)) = β(f), the injectivity of β implies Φ(ρ) = f. Hence Φ is onto. Theorem 2.2 shows that π_{0}G must be a finite p–group.

Next suppose the connected component G_{0} is not a torus so that either S^{3} or SO(3) is contained in G_{0}. Recall that, for some n, there is a monomorphism i : G ↪ U(n). Let ψ^{k} be the unstable Adams operation on BU(n) for k with (k, n!) = 1. Since the map Rep(G, U(n)) \overline{}[BG, BU(n)] is bijective, there is a homomorphism ρ : G \overline{}U(n) such that ψ^{k} · Bi = Bρ. Thus ψ^{k} · Bi|_{BS^{3}} or ψ^{k} · Bi|_{BSO(3)} is induced by a homomorphism. For large k, however, there is no non-trivial homomorphism from S^{3} or SO(3) into U(n) whose kernel contains ℤ/k. This contradiction implies that G_{0} must be a torus. Consequently G is a p–toral group. □

3. Compact Lie groups and p–compact groups.

For the reasons mentioned at the end of the introduction, we consider the problem that, for what G, the loop space of (BG)_{p}^{∧} can be a p–compact group. In particular, we will determine whether Ω(BNT)_{p}^{∧} is a p–compact group for the normalizer NT of a connected compact Lie group K. The following result implies that (BNT)_{p}^{∧} ≃ (BK)_{p}^{∧} if and only if p does not divide the order of the Weyl group W (K).

Proposition 3.1. Suppose G is a compact Lie group. If the loop space of (BG)_{p}^{∧} is homotopy equivalent to a p–compact group, then the following hold:

(a) π_{0}G is p–nilpotent.

(b) π_{1}((BG)_{p}^{∧}) is isomorphic to a p–Sylow subgroup of π_{0}G.

Proof. Suppose (BG)_{p}^{∧} ≃ BX for a p–compact group X. Let f : (BG)_{p}^{∧} \overline{} BX be the homotopy equivalence and f^{−1} : BX \overline{}(BG)_{p}^{∧} its homotopy inverse. The composite map

BG \overline{}(BG)_{p}^{∧} ^{\overline{}^{f}} BX \overline{}Bπ_{0}X

factors through Bπ_{0}G, since π_{0}X is a finite p–group. Say this map α : Bπ_{0}G \overline{}Bπ_{0}X. If F denotes the homotopy fiber of the map (BG)_{p}^{∧} \overline{} (Bπ_{0}G)_{p}^{∧}, we obtain the following commutative diagram

F \overline{}BX_{0} [figure]

(BG)_{p}^{∧} ^{^{f} \overline{}}BX [figure]

(Bπ_{0}G)_{p}^{∧} ^{\overline{}^{(α)_{p}^{∧}}} Bπ_{0}X

Since the restriction on the classifying space of the maximal torus of the composite map BX_{0} \overline{}BX ^{\overline{}^{f^{−1}}} (BG)_{p}^{∧} \overline{}(Bπ_{0}G)_{p}^{∧} is null homotopic, so is this composition. Consequently we obtain the following commutative diagram

BX_{0} \overline{}F [figure]

BX ^{^{f^{−1}} \overline{}}(BG)_{p}^{∧} [figure]

Bπ_{0}X \overline{}(Bπ_{0}G)_{p}^{∧}

Notice that π_{0}X is a finite p–group. So the map Bπ_{0}X \overline{}(Bπ_{0}G)_{p}^{∧}, say β, is induced by a homomorphism ρ : π_{0}X \overline{}π_{0}G, and ρ factors through a p–Sylow subgroup π_{p} of π_{0}G. Since f · f^{−1} ≃ id, the homomorphism ρ is injective. Next consider the diagram:

BG \overline{}(BG)_{p}^{∧} ^{^{f} \overline{}}BX ^{^{f^{−1}} \overline{}}(BG)_{p}^{∧} [figure]

Bπ_{0}G  ^{^{α}  \overline{}}Bπ_{0}X ^{^{β}\overline{}}(Bπ_{0}G)_{p}^{∧}

The map of homotopy sets [Bπ_{0}G, (Bπ_{0}G)_{p}^{∧}] \overline{}[BG, (Bπ_{0}G)_{p}^{∧}] is injective, [9, Lemma 1]. So the composition β · α is homotopy equivalent to the completion map Bπ_{0}G \overline{}(Bπ_{0}G)_{p}^{∧}. Thus the homomorphism ρ with β = Bρ is surjective, and hence ρ is an isomorphism. Therefore Bπ_{p} ≃ Bπ_{0}X ≃ (Bπ_{0}G)_{p}^{∧}. Thus π_{0}G is p–nilpotent, and we see that π_{1}((BG)_{p}^{∧}) is isomorphic to the p–Sylow subgroup π_{p} of π_{0}G, since F ≃ BX_{0}. This completes the proof. □

Theorem 3.2. Suppose NT is the normalizer of a maximal torus T of a connected compact Lie group. The loop space of (BNT)_{p}^{∧} is homotopy equivalent to a p–compact group if and only if the following hold:

(a) The Weyl group W is p–nilpotent so that W = Γ ⋊ W_{p}, where W_{p} is the p–Sylow subgroup of W.

(b) The action of Γ on BT is represented as a pseudo-reflection group.

Proof. If the Weyl group W is p–nilpotent, we obtain the following commutative diagram of fibrations:

BT = BT \overline{}∗ [figure]

BH \overline{}BNT \overline{}BW_{p} [figure: ‖]

BΓ \overline{}BW ^{^{f} \overline{}}BW_{p}

where H is the kernel of the epimorphism NT \overline{}W_{p}.

First assume that (BNT)_{p}^{∧} is homotopy equivalent to BX for a p– compact group X. Proposition 3.1 says that W is p–nilpotent and that π_{1}((BNT)_{p}^{∧}) = W_{p}. Since W_{p} is a finite p–group, the fibration BH \overline{} BNT \overline{}BW_{p} is preserved by the p–completion, [2, Chapter II §5]. We have a diagram of fibrations:

(BH)_{p}^{∧}  \overline{}(BNT)_{p}^{∧}  \overline{}BW_{p} [figure: ≃; ≃]
BX_{0}  \overline{}BX  \overline{}Bπ_{0}X

and hence the p–compact group Ω(BH)_{p}^{∧} ≃ X_{0} must be connected. Applying a result of [5, §9] to BT \overline{}BH \overline{}BΓ, we see that the condition (b) holds.

Conversely, assume that the conditions (a) and (b) hold. Since p does not divide the order of Γ, the mod p cohomology H^{∗}((BH)_{p}^{∧}, 𝔽_{p}) is a finitely generated polynomial algebra, and Ω(BH)_{p}^{∧} is 𝔽_{p}–finite. Consequently, we see that both (BH)_{p}^{∧} and BW_{p} are the classifying spaces of p–compact groups. Thus Ω(BNT)_{p}^{∧} must be a p–compact group. □

It is easy to show the following.

Lemma 3.3. Every subgroup of a p–nilpotent group is p–nilpotent.

Proposition 3.4. Suppose a connected compact Lie group G is simple. Then the Weyl group W (G) is p–nilpotent if and only if one of the following holds:

(a) The prime p does not divide the order of the Weyl group.

(b) p = 2 and W (G) is isomorphic to one of the following:

(i) W (A_{n}) with n = 1, 2  (ii) W (B_{n}) = W (C_{n}) with n = 2, 3
(iii) W (D_{n}) with n = 2, 3  (iv) W (G_{2})

Proof. We need to use the classification of simple Lie group consisting of infinite families A_{n}, B_{n}, C_{n}, D_{n} with n ≥ 1 and five exceptional cases G_{2}, F_{4}, E_{6}, E_{7} and E_{8}. Our proof is homotopy theoritic and sketchy. We recall that a finite group γ with p–Sylow subgroup γ_{p} is p–nilpotent if and only if (Bγ)_{p}^{∧} ≃ Bγ_{p}.

First notice that W (A_{n}) is isomorphic to the symmetric group Σ_{n+1}. Suppose a prime p divides (n + 1)!. According to [8, Theorem 1 and Proposition 2.2], we see that, for n ≥ 3 any map f from (BΣ_{n+1})_{p}^{∧} to the classifying space of the p–Sylow subgroup of Σ_{n+1} is null homotopic, since so is the restriction of the map f on the classifying space of the alternationg group. Thus W (A_{n}) cannot be p–nilpotent for these cases. If n = 1, then W (A_{1}) = ℤ/2. For W (A_{n}) it remains to consider the cases n = 2 and p = 2, 3. It is known that (BΣ_{3})_{2}^{∧} ≃ Bℤ/2 so that Σ_{3} is 2–nilpotent. On the other hand, from [8, Theorem 2.3], we see [(BΣ_{3})_{3}^{∧}, Bℤ/3] = 0. Consequently Σ_{3} cannot be 3–nilpotent.

Next consider W (B_{n}) = W (C_{n}) = Σ_{n} ⋊ (ℤ/2)^{n}. From Lemma 3.3, the cases we need to check are (n, p) = (2, 2), (3, 2). For n = 2, the group is a 2–group. For n = 3, we can show that the group is 2–nilpotent since Σ_{3} = ℤ/3 ⋊ ℤ/2. The rest of the argument goes similarly by using the Weyl groups and their subgroups given as follows:

(i) W (D_{n}) = Σ_{n} ⋊ (ℤ/2)^{n−1}  (ii) W (A_{2}) < W (G_{2}) = D_{12}
(iii) W (B_{4}) < W (F_{4})  (iv) W (D_{5}) < W (E_{6})
(v) W (A_{7}) < W (E_{7})  (vi) W (A_{8}) < W (E_{8})

This completes the proof. □

Proposition 3.5. Suppose a connected compact Lie group G is simple, and NT denotes the normalizer of a maximal torus T of G. The loop space of (BNT)_{p}^{∧} is homotopy equivalent to a p–compact group if and only if one of the following holds:

(a) The prime p does not divide the order of the Weyl group.

(b) p = 2 and W (G) is isomorphic to one of the following:

(i) W (A_{1})  (ii) W (B_{2}) = W (C_{2})  (iii) W (D_{2})

Proof. It suffices to show that, from Theorem 3.2 and Proposition 3.4, that the loop space of (BNT)_{p}^{∧} cannot be a p–compact group if p = 2 and G is one of the following types of simple Lie groups: A_{2}, B_{3}, C_{3}, D_{3}, G_{2}. In each of the cases, the Weyl group W is is isomorphic to the semidirect product ℤ/3 ⋊ W_{2}. Theorem 3.2 says that if Ω(BNT)_{p}^{∧} was a p–compact group, the group ℤ/3 must act on the torus as a pseudo-reflection group. This is impossible since the representation of the Weyl group is integral. This contradiction completes the proof. □

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Fukuoka University, Fukuoka 814-0180, Japan e-mail: kenshi@ ssat.fukuoka-u.ac.jp