SELF HOMOTOPY EQUIVALENCES OF CLASSIFYING SPACES

OF COMPACT CONNECTED LIE GROUPS

by Stefan Jackowski, James McClure, and Bob Oliver

In an earlier paper [JMO], we gave a complete description of all homotopy classes of self maps of the classifying space BG, when G is any compact connected simple Lie group. In this paper, we extend those results to the case where G is any compact connected Lie group, but only considering self maps of BG which are rational equivalences. Most of the paper deals with self maps of the p-adic completions BG_{p̂}; and the results are extended to global maps only at the end.

The first complete description of [BG, BG] for any nonabelian connected Lie group G was given by Mislin [Ms], for the group G = S^{3}. More recently, in [JMO] (and based on earlier work by Hubbuck [Hu] and Ishiguro [Is]), we extended Mislin’s result to a description of [BG, BG] for an arbitrary compact connected simple Lie group G. The assumption that G be simple was however crucial: examples were given in [JMO, §7] to show that a similar, simple description of all self maps is unlikely for arbitrary connected G.

When G is simple, any f : BG_{p̂} ⟶ BG_{p̂} is either a ℚ-equivalence or nulhomotopic. The most natural setting for obtaining similar strong results for semisimple or connected groups seems to be to restrict attention to the ℚ-equivalences. For example, we will see in Corollary 2.6 below that for connected G, two ℚ-equivalences f, f′ : BG_{p̂} ⟶ BG_{p̂} are homotopic if and only if H^{∗}(f; ℚ) = H^{∗}(f′; ℚ). This is not the case for arbitrary self maps, as was shown in [JMO, Example 7.1].

Throughout this paper, p will be a fixed prime, G a fixed compact connected Lie group, T ⊆G a maximal torus, and W = N(T)/T the Weyl group. The monoid of homotopy classes of ℚ-equivalences BG_{p̂} ⟶ BG_{p̂} will be denoted [BG_{p̂}, BG_{p̂}]_{ℚ}, and the subgroup of homotopy equivalences [BG_{p̂}, BG_{p̂}]_{h}. Let T_{∞}⊆T be the subgroup of elements of p-power order. A homomorphism ϕ : T_{∞} ⟶ T_{∞} will be called admissible [AM, p.5] if it is equivariant with respect to some endomorphism of W; i.e. if there is some \overline{ϕ} : W ⟶ W such that ϕ(wt) = \overline{ϕ}(w)ϕ(t) for all w ∈ W and t ∈ T. The monoid of admissible epimorphisms from T_{∞} to itself will be denoted AdmEpi(T_{∞}, T_{∞}).

Our first step (Propositions 1.2 and 1.4) is to construct a homomorphism of monoids

Θ : [BG_{p̂}, BG_{p̂}]_{ℚ} ⟶ AdmEpi(T_{∞}, T_{∞})/W,

where W acts on the admissible maps by composition. This is just the p-adic version of the construction of Adams & Mahmud [AM, Corollary 1.8] (in the special case of self maps). The map Θ is characterized by the property that Θ(f) = ϕ·W for any

Typeset by AMS-TEX

f ∈ [BG_{p̂}, BG_{p̂}]_{ℚ} and any ϕ : T_{∞} ↠ T_{∞} such that f|BT ≃Bϕ. And in this situation, f is a homotopy equivalence if and only if ϕ is an isomorphism.

The main result in Section 2 (Theorem 2.5) is that Θ is an injection, and its image is closed in the p-adic topology. The precise image of Θ is then described in Theorem 3.4. The formulation of of Theorem 3.4 is somewhat technical, but it leads to a very simple result about homotopy equivalences (Corollary 3.5): Θ restricts to a group monomorphism

Θ_{h} : [BG_{p̂}, BG_{p̂}]_{h} ⟶ N_{Aut(T_{∞})}(W)/W,

which is an isomorphism except when p = 2 and G contains a direct factor of the form SO(2n+1)×Sp(n). We also show that all components of map(BG_{p̂}, BG_{p̂})_{ℚ} have the homotopy type of BZ(G)_{p̂}.

At the end of the paper, Sullivan’s arithmetic square for completions and localizations is applied to obtain the analogous results for ℚ-equivalences from BG to itself.

Maps f : BG_{p̂} ⟶ BG_{p̂} (or, equivalently, BG ⟶ BG_{p̂}) are studied here using the p-local approximation of BG constructed in [JMO, §1]. More precisely, we showed there that BG is 𝔽_{p}-homology equivalent to the homotopy direct limit of classifying spaces BP for certain p-toral subgroups P ⊆G. The details of this approximation will be recalled in Section 2 below.

Our general reference for completion techniques and results is Bousfield & Kan [BK]. For any space X, we let X_{p̂} denote the 𝔽_{p}-completion of X. Then map(Y, X_{p̂})≃ map(Y′, X_{p̂}) for any 𝔽_{p}-homology equivalence Y ⟶ Y′ [BK, II.2.8]. Also, if X has finite type and π_{1}(X) is a finite p-group, then π_{1}(X_{p̂}) ≅ π_{1}(X), and π_{i}(X_{p̂}) ≅ π_{i}(X)_{p̂} ≅ π_{i}(X) ⊗ ℤ̂_{p} for each i ≥ 2. This last statement follows from [BK, VI.5.2] when X is 1-connected; and holds in the general case since the sequence (X̃)_{p̂} ⟶ X_{p̂} ⟶ K(π_{1}X, 1) is a homotopy fibration (see [BK, II.5.2(iv)]). Here, X̃ denotes the universal cover of X.

For any pair X, Y of spaces, and any map f : X ⟶ Y, map(X, Y)_{f} will denote the space of maps X ⟶ Y which are homotopic to f.

Partial results similar to those given here have also been obtained by Dietrich Notbohm [No2] and Jesper Møller [Mø].

We would like to thank Bill Dwyer for suggesting the formulation of some of the results in Section 3.

Section 1

We first recall the description by Dwyer-Zabrodsky [DZ] and Notbohm [No1] of the mapping spaces map(BP, BG), when P is p-toral and G is an arbitrary compact Lie group.

For any pair G and G′ of compact Lie groups, define

Rep(G, G′) := Hom(G, G′)/ Inn(G′);

the set of G′-conjugacy classes of homomorphisms from G to G′. For any ρ : G ⟶ G′, C_{G′}(ρ) will denote the centralizer in G′ of Im(ρ).

Theorem 1.1. Fix a p-toral group P and a compact connected Lie group G. Let T ⊆P denote the identity component of P, and let T_{∞}⊆T be the set of elements of p-power order. Then there exists a dense subgroup P_{∞}⊆P such that P_{∞}∩T = T_{∞}; and any two such subgroups are conjugate by some element of T. Furthermore, if we regard P_{∞} as a discrete group:

(i) The maps

Rep(P_{∞}, G) ⟶_{≅}^{B} [BP_{∞}, BG_{p̂}] ⟵_{≅}^{res} [BP, BG_{p̂}]

are bijections.

(ii) For any ρ : P_{∞} ⟶ G, the pairing BC_{G}(ρ)×BP_{∞} ⟶ BG induces a homotopy equivalence

ê_{ρ} : BC_{G}(ρ)_{p̂} ⟶^{≃} map(BP_{∞}, BG_{p̂})_{Bρ} (≃ map(BP, BG_{p̂})_{Bρ}).

Proof. This theorem is implicit in the work of Notbohm [No1]. In fact, it is much more elementary than his results, which involve the analogous description of maps from BP to BG (without completion).

A choice of P_{∞} is equivalent to a choice of splitting map for the extension 1 ⟶ T/T_{∞} ⟶ P/T_{∞} ⟶ P/T ⟶ 1. Hence the existence of P_{∞} will follow if H^{2}(P/T; T/T_{∞}) = 0, and its uniqueness will follow if H^{1}(P/T; T/T_{∞}) = 0. And these cohomology groups vanish because P/T is a finite p-group and T/T_{∞} is uniquely p-divisible.

By construction, P_{∞} is the union of an increasing sequence P_{1} ⊆ P_{2} ⊆ P_{3} ⊆ … of finite p-groups, and hence BP_{∞} ≃ \underrightarrow{hocolim}_{n}(BP_{n}). For each n ≥ 1, [BP_{n}, BG_{p̂}] ≅

Rep(P_{n}, G), and

map(BP_{n}, BG_{p̂})_{Bρ} ≃ BC_{G}(ρ)_{p̂}  (1)

for each ρ : P_{n} ⟶ G, by a theorem of Dwyer and Zabrodsky [DZ]. In particular, each component of map(BP_{n}, BG_{p̂}) has finite fundamental group, so the appropriate \underleftarrow{lim}^{1}π_{1}(−) all vanish, and

[BP_{∞}, BG_{p̂}] ≅ \underleftarrow{lim}[BP_{n}, BG_{p̂}] ≅ \underleftarrow{lim} Rep(P_{n}, G) ≅ Rep(P_{∞}, G).

Also, map(BP_{∞}, BG_{p̂}) is the homotopy inverse limit of the spaces map(BP_{n}, BG_{p̂}); and so by (1)

map(BP_{∞}, BG_{p̂})_{Bρ} ≃ (map(BP_{∞}, BG)_{Bρ}) _{ p̂} ≃ BC_{G}(ρ)_{p̂}

for any ρ : P_{∞} ⟶ G. And map(BP_{∞}, BG_{p̂})≃ map(BP, BG_{p̂}), since the inclusion BP_{∞}↪BP is a 𝔽_{p}-homology equivalence (cf Feshbach [Fe, Proposition 2.3]). □

By Theorem 1.1, for any p-toral P and any compact Lie group G, any two choices P_{∞}, P′_{∞}⊆P are conjugate by some element t∈T (the identity component of P). Also, if tP_{∞}t^{−1} = P_{∞}, then t has p-power order in T/(Z(P)∩T); and so conjugation by t is an inner automorphism of P_{∞}. It follows that there is a unique natural way to identify the sets Rep(P_{∞}, G) and Rep(P′_{∞}, G) (independently of the choice of t); or in other words that we can regard Rep(P_{∞}, G) as depending only on P and not on P_{∞}.

An element of Rep(P_{∞}, G) — i. e., a homomorphism from P_{∞} to G defined up to conjugacy in G — will be called a quasirepresentation from P to G. We let QRep(P, G) denote the set of all quasirepresentations φ : P_{∞} ⟶ G (note that the prime p is implicit in this definition). For example, if T is an n-dimensional torus, then QRep(T, T)≅M_{n}(ℤ̂_{p}).

In these terms, Theorem 1.1(i) says that [BP, BG_{p̂}]≅ QRep(P, G) for any p-toral P and any compact Lie group G. This is implicit in [No1], where Notbohm also proves the (much harder) corresponding global result that [BP, BG]≅ Rep(P, G).

The next proposition (except for part (iii)) is analogous to a theorem of Adams & Mahmud [AM, Corollary 1.11], but stated here for self maps of the p-completion of BG. The main point is that any self map of BG_{p̂} lifts to some essentially unique self map of BT_{ p̂}. This applies, in fact, to arbitrary maps between classifying spaces of distinct compact connected Lie groups (see [AM,Theorem 1.1]), but the following (simpler) case suffices for our purposes here.

Proposition 1.2. Fix a compact connected Lie group G, let T ⊆G be a maximal torus, and let W = N(T)/T be the Weyl group. Then for any ℚ-equivalence f : BG_{p̂} ⟶ BG_{p̂}, there exists an epimorphism ϕ : T_{∞}↠T_{∞}, inducing a ℚ-equivalence f_{T} = Bϕ : BT_{ p̂} ⟶ BT_{ p̂}, and such that the following square commutes up to homotopy:

BT_{ p̂} ⟶_{≃}^{f_{T}} BT_{ p̂}
↓^{incl}  ↓^{incl}
BG_{p̂} ⟶_{≃}^{f} BG_{p̂}.

(1)

Furthermore, the following hold.

(i) For any other f′_{T} = Bϕ′ : BT_{ p̂} ⟶ BT_{ p̂} for which (1) commutes up to homotopy, there exists w∈W such that ϕ′ = conj(w) ∘ ϕ.

(ii) ϕ is admissible: there is an automorphism β∈ Aut(W), which permutes the reflections in W, and such that ϕ ∘ conj(w) ≃ conj(β(w)) ∘ ϕ for all w∈W.

(iii) f_{T} is a homotopy equivalence if and only if f is.

Proof. Let T_{∞} be the subgroup in T of elements of p-power order. By Theorem 1.1 above, there is a homomorphism ϕ : T_{∞} ⟶ G such that f|BT ≃Bϕ : BT_{ p̂} ⟶ BG_{p̂}. After composing ϕ with an inner automorphism of G, if necessary, we may assume that Im(ϕ)⊆T. So from now on, we regard ϕ as a homomorphism T_{∞} ⟶ T_{∞}. Set

f_{T} = Bϕ : BT_{ p̂} ≃ (BT_{∞})_{p̂} ⟶ BT_{ p̂};

then square (1) commutes up to homotopy.

Let ι : T ⟶ G be the inclusion. For any w∈N(T), the commutative diagram

BT_{ p̂} ⟶^{w_{∗}} BT_{ p̂} ⟶_{=f_{T}}^{Bϕ} BT_{ p̂}
↓^{Bι}  ↓^{Bι}  ↓^{Bι}
BG_{p̂} ⟶_{≃Id}^{w_{∗}} BG_{p̂} ⟶^{f} BG_{p̂}.

shows that B(ι ∘ ϕ ∘ conj(w))≃B(ι ∘ ϕ). By Theorem 1.1 again, this means that

ϕ ∘ conj(w) = conj(β(w)) ∘ ϕ for some β(w) ∈ N(Im(ϕ)).  (2)

Set n = dim(T). Consider the group H_{2}(BT_{ p̂}; ℚ) ≅ (ℚ̂_{p})^{n}, regarded as a ℚ̂_{p}[W]- representation. By (2), Ker(H_{2}(f_{T}; ℚ)) is W -invariant. Also, if G_{i} is a simple factor in G with maximal torus T_{i} = T ∩G_{i}, then H_{2}(BT_{i}_{p̂}; ℚ) is W -irreducible (cf. [Bb1, p. 82, Proposition 5(v)]). Thus, if H_{2}(f_{T}; ℚ) is not an isomorphism, then either it vanishes on some such summand H_{2}(BT_{i}_{p̂}; ℚ), or it fails to be injective on H_{2}(BT_{ p̂}; ℚ)^{W} = H_{2}(BZ(G)_{p̂}; ℚ).And in either case, this contradicts the assumption that f is a ℚequivalence.

This shows that H_{2}(f_{T}; ℚ) is an isomorphism. In particular, since T_{∞} ≅ (ℤ[\frac{1}{p}]/ℤ)^{n}, ϕ sends T_{∞} onto itself with finite kernel, and hence f_{T} = Bϕ is a ℚ-equivalence. Also, since ϕ : T_{∞} ⟶ T_{∞} is onto, and since W acts effectively on T_{∞}, the element β(w)∈W which satisfies (2) is unique (for any w∈W); and β : W ⟶ W is an automorphism. Then ϕ is β-equivariant by construction, and this finishes the proof of (ii).

Assume now that f′_{T} : BG_{p̂} ⟶ BG_{p̂} is such that Bι∘f_{T} ≃Bι∘f′_{T}. Write f′_{T} = Bϕ′, where ϕ′(T_{∞}) = T_{∞}. Then by Theorem 1.1, ϕ and ϕ′ are conjugate via some element g∈G. In particular, gT_{∞}g^{−1} = T_{∞}, and so g∈N(T). In other words, if we set w = gT ∈W, then ϕ = conj(w) ∘ ϕ′, and hence f_{T} ≃f′_{T} ∘w_{∗}. This proves point (i).

It remains to prove point (iii): that f_{T} is a homotopy equivalence if and only if f is. If f is a homotopy equivalence, then let \overline{f}_{T} be a lifting of f^{−1} to BT_{ p̂}. The composites f_{T} ∘\overline{f}_{T} and \overline{f}_{T} ∘f_{T} are homotopy equivalences by (i) (the uniqueness of the lifting), and so f_{T} is also a homotopy equivalence.

Now assume that f_{T} is a homotopy equivalence. Each term in the fibration G/T ⟶^{i} BT ⟶ BG is simply connected, so the completed sequence G/T_{ p̂} ⟶ BT_{ p̂} ⟶ BG_{p̂} is again a fibration [BK, VI.6.5]. Since square (1) commutes up to homotopy, f_{T} restricts to a map f_{G/T} : G/T_{ p̂} ⟶ G/T_{ p̂}. Also, i_{∗} : H_{∗}(G/T; ℚ) ⟶ H_{∗}(BT; ℚ) is injective by [Br, Theorem 20.3(b)], and H_{∗}(G/T; ℤ) is torsion free by [Bt]. So for each n, H_{n}(f_{T}; ℤ) is an automorphism of the lattice H_{n}(BT; ℤ) which sends the sublattice H_{n}(G/T; ℤ) into itself, and hence it restricts to an automorphism of H_{n}(G/T; ℤ). Thus, H_{∗}(f_{G/T}; ℤ) is an automorphism, and f_{G/T} is a homotopy equivalence. Since f_{T} is a homotopy equivalence by assumption, f must also be a homotopy equivalence. □

Proposition 1.2 says that there is a well defined map from the monoid of ℚequivalences BG_{p̂} ⟶ BG_{p̂} to the monoid of admissible epimorphisms T_{∞}↠T_{∞} modulo the action of the Weyl group. This will be stated more precisely in Proposition 1.4 below, and the map will be shown to be injective in Section 2.

In [JMO, Theorem 3.4], we saw that when G is simple and p||W |, then any ℚequivalence BG_{p̂} ⟶ BG_{p̂} is a homotopy equivalence. This was in turn a generalization of Ishiguro’s theorem [Is], that unstable Adams operations on BG are defined only for degrees prime to |W |. The next proposition describes what happens for general connected G.

Proposition 1.3. Let G be a compact connected semisimple Lie group such that p divides the order of the Weyl group of each simple component of G. Then any ℚequivalence f : BG_{p̂} ⟶ BG_{p̂} is a homotopy equivalence.

Proof. Fix a ℚ-equivalence f : BG_{p̂} ⟶ BG_{p̂}. Let f_{T} = Bϕ : BT_{ p̂} ⟶ BT_{ p̂} be the map of Proposition 1.2, where ϕ : T_{∞} ⟶ T_{∞} is β-equivariant for some β∈ Aut(W). Since W is finite, there is some r > 0 such that ϕ^{r} is W -equivariant. Upon replacing f by f^{r}, we can assume that ϕ itself is W -equivariant. We will show that ϕ is an isomorphism: then f_{T} is a homotopy equivalence, and f is a homotopy equivalence by Proposition 1.2(iii).

Set n = rk(G) = dim(T). Then H_{2}(Bϕ; ℚ) = H_{2}(f_{T}; ℚ) is a W -equivariant automorphism of H_{2}(BT_{ p̂}; ℚ) ≅ (ℚ̂_{p})^{n}, and sends H_{2}(BT_{ p̂}; ℤ) ≅ (ℤ̂_{p})^{n} to itself. As a W -representation, H_{2}(BT_{ p̂}; ℚ) ≅ ℚ̂_{p} ⊗_{ℚ} H_{2}(BT; ℚ) splits as a sum of distinct irreducible summands, one for each simple factor in G (cf. [Bb1, p. 82, Proposition 5(v)]). Hence, for each simple factor G_{i}◁G with maximal torus T_{i} = T ∩G_{i}, H_{2}(Bϕ; ℚ) restricted to H_{2}(BT_{i}_{p̂}; ℚ) is multiplication by some constant k_{i} ∈ ℤ̂_{p}.

We must show that p∤k_{i} for all i. Assume otherwise: fix a simple factor G_{i}◁G such that p|k_{i}, set T_{i} = T ∩G_{i}, and let W_{i} = N(T_{i})/T_{i} be the Weyl group of G_{i}. Choose any w∈N_{G_{i}}(T_{i})∖T_{i} of p-power order (p||N(T_{i})/T_{i}| by assumption), and let t∈T_{i} be any element conjugate in G_{i} to w. Set P = ⟨T_{i}, w⟩, a p-toral subgroup. By Theorem 1.1, f|BP_{∞} ≃ B\overline{ϕ} for some \overline{ϕ} : P_{∞} = ⟨T_{i∞}, w⟩ ⟶ G. Then \overline{ϕ}(P_{∞}) = P_{∞}, since the restriction of ϕ sends T_{i∞} onto itself and is W_{i}-equivariant (and W_{i} acts effectively). Also, since w and t are conjugate in G, \overline{ϕ}^{i}(w) and \overline{ϕ}^{i}(t) are conjugate in G for all i (by Theorem 1.1 again). Since ϕ(t) = t^{k_{i}} and p|k_{i}, we have ϕ^{m}(t) = 1 for some m, and hence ϕ^{m}(w) = 1. On the other hand, ϕ : T_{i∞} ⟶ T_{i∞} is W_{i}-equivariant and onto, and W_{i} acts effectively on T_{i∞}; so \overline{ϕ}(wT_{i∞}) = wT_{i∞}. And this implies that \overline{ϕ}^{i}(w) ≠ 1 for all i; which is a contradiction. □

So far, we have associated, to each ℚ-equivalence f ∈ [BG_{p̂}, BG_{p̂}]_{ℚ}, an admissible map ϕ : T_{∞}↠T_{∞}. When formulating later results, it will be convenient to work with the integral lattice in T instead of with T itself.

Write T = L(T)/Λ, where L(T) ≅ ℝ^{n} is the Lie algebra (or universal covering group) for T, and where

Λ = Ker[exp : L(T) ↠ T]

is the integral lattice. Set Λ_{p̂} = ℤ̂_{p} ⊗ Λ = \underleftarrow{lim}(Λ/p^{n}Λ).

Consider the standard isomorphism Hom(T, T)≅ Hom(Λ, Λ), which sends ϕ : T ⟶ T to L(ϕ)|Λ. We want to define the corresponding isomorphism in the p-adic situation. One way to do this is to set

L(T_{∞}) = \underleftarrow{lim} (⋯ ⟶ T_{∞} ⟶^{p·} T_{∞} ⟶^{p·} T_{∞}) ≅ (ℚ̂_{p})^{n}

(where n = dim(T)); so that Λ_{p̂} = Ker[L(T_{∞}) ↠ T_{∞}]. Any ϕ∈ End(T_{∞}) lifts to the map L(ϕ) = \underleftarrow{lim}(ϕ)∈ End(L(T_{∞})). The isomorphism

Hom(T_{∞}, T_{∞}) ⟶^{≅} Hom(Λ_{p̂}, Λ_{p̂}) ≅ M_{n}(ℤ̂_{p})

is now defined by sending ϕ∈ End(T_{∞}) to L(ϕ)|Λ_{p̂}. Equivalently, this identification can be obtained by identifying H_{2}(BT_{ p̂}; ℤ) with Λ_{p̂}.

Under this identification, the monoid of epimorphisms from T_{∞} to itself is identified with Aut(L(T_{∞})) ∩ End(Λ_{p̂}). An admissible epimorphism is by definition one which is equivariant with respect to some endomorphism of the Weyl group W; and if we regard W as a subgroup of Aut(T_{∞}) this allows us to identify:

AdmEpi(T_{∞}, T_{∞}) ≅ N_{Aut(L(T_{∞}))}(W) ∩ End(Λ_{p̂}).

Also, if we let AdmIso(T_{∞}, T_{∞}) denote the group of admissible isomorphisms, then

AdmIso(T_{∞}, T_{∞}) ≅ N_{Aut(T_{∞})}(W).

As before, [BG_{p̂}, BG_{p̂}]_{ℚ} denotes the monoid of homotopy classes of ℚ-equivalences of BG_{p̂} to itself, and [BG_{p̂}, BG_{p̂}]_{h} denotes the subgroup of homotopy equivalences. The following reformulation of Propositions 1.2 and 1.3 was suggested to us by Bill Dwyer (see also [DW, Proposition 5.5]).

Proposition 1.4. There is a well defined homomorphism of monoids

Θ : [BG_{p̂}, BG_{p̂}]_{ℚ} ⟶ AdmEpi(T_{∞}, T_{∞})/W ≅ [N_{Aut(L(T_{∞}))}(W) ∩ End(Λ_{p̂})]/W,

where Θ(f) = H_{2}(f_{T}; ℚ) = L(ϕ) for any ℚ-equivalence f : BG_{p̂} ⟶ BG_{p̂} and any lifting of f to f_{T} = Bϕ : BT_{ p̂} ⟶ BT_{ p̂}. Furthermore, Θ restricts to a group homomorphism

Θ_{h} : [BG_{p̂}, BG_{p̂}]_{h} ⟶ AdmIso(T_{∞}, T_{∞})/W ≅ N_{Aut(T_{∞})}(W)/W

≅ N_{Aut(Λ_{p̂})}(W)/W;

and

[BG_{p̂}, BG_{p̂}]_{h} = Θ^{−1} (N_{Aut(T_{∞})}(W)/W ) ≅ Θ^{−1} (N_{Aut(Λ_{p̂})}(W)/W ) .

Proof. By definition, an epimorphism ϕ : T_{∞} ↠ T_{∞} is admissible if and only if H_{2}(Bϕ; ℚ) = L(ϕ)∈ Aut(L(T_{∞})) lies in the normalizer of W. Hence Θ and Θ_{h} are well defined by Proposition 1.2. And by Proposition 1.2(iii), a ℚ-equivalence f : BG_{p̂} ⟶ BG_{p̂} is a homotopy equivalence if and only if Θ(f) is an automorphism of

Λ_{p̂}. □

Section 2

Throughout this section, we concentrate on showing that Θ and Θ_{h} are injective. This requires the machinery developed in [JMO].

Recall that a compact Lie group P is called p-toral if the identity component P_{0} is a torus, and if P/P_{0} is a finite p-group. If W_{p} = N_{p}(T)/T is a Sylow p-subgroup of W, then N_{p}(T) is a maximal p-toral subgroup of G in the strong sense that any p-toral subgroup is conjugate to a subgroup of N_{p}(T) (cf. [JMO, Lemma A.1]). We now fix (for the rest of this section) such a maximal p-toral subgroup N_{p}(T).

Proving the injectivity of the map Θ above means showing that two ℚ-equivalences BG_{p̂} ⟶ BG_{p̂} are homotopic if they agree on BT. This will be done using an approximation of BG by a limit of classifying spaces of p-toral subgroups. So we must first show that maps f, f′ : BG_{p̂} ⟶ BG_{p̂} which are homotopic on BT are homotopic on BN_{p}(T); and hence (by maximality) on BP for all p-toral P ⊆G. This was shown in [JMO] in the special case where f, f′ are homotopy equivalences; and the next proposition extends this to the case of ℚ-equivalences.

Proposition 2.1. Let G be any compact connected Lie group. Assume that f, f′ : BG_{p̂} ⟶ BG_{p̂} are ℚ-equivalences, and that f|BT ≃f′|BT. Then for any p-toral subgroup P ⊆G, f|BP ≃f′|BP.

Proof. Since N_{p}(T) is a maximal p-toral subgroup, it suffices to show that f|BN_{p}(T)≃f′|BN_{p}(T).

By Theorem 1.1, there are quasihomomorphisms ρ, ρ′ : N_{p}(T) ⟶ G such that f|BN_{p}(T)≃Bρ and f′|BN_{p}(T)≃Bρ′. Since f|BT ≃f′|BT, we may assume that ρ|T = ρ′|T.

Let G′◁G be the product of those simple factors for which p divides the order of the Weyl group. Then it suffices to show that ρ and ρ′ are conjugate after restriction to N_{p}(T)∩G′. In other words, we are reduced to the case where G = G′. But then f and f′ are homotopy equivalences by Proposition 1.3, and so f|BN_{p}(T)≃f′|BN_{p}(T) by [JMO, Proposition 3.5]. □

We now recall some definitions from [JMO]. A p-toral subgroup P ⊆G is called pstubborn if N(P)/P is finite and contains no nontrivial normal p-subgroups. For example, if G = SO(3) and p = 2, then the subgroup P ≅ ℤ/2 × ℤ/2 is p-stubborn since the only normal 2-subgroup of N(P)/P ≅ Σ_{3} (the symmetric group of order 6) is the trivial group. Note that a maximal p-toral subgroup is always p-stubborn.

We let R_{p}(G) denote the category whose objects are orbits G/P for p-stubborn P ⊆G, and where Mor(G/P, G/P′) is the set of all G-maps between the orbits. By [JMO, Proposition 1.5], R_{p}(G) is a finite category, in the sense that it contains finitely many isomorphism classes of objects, and has finite morphism sets.

One of the main results in [JMO] is that the map

\underrightarrow{hocolim}_{G/P ∈R_{p}(G)}(EG/P ) ≅ EG ×_{G} ( \underrightarrow{hocolim}_{G/P ∈R_{p}(G)}(G/P )) ⟶ BG

(induced by projection) is an 𝔽_{p}-homology equivalence. In particular,

[BG_{p̂}, BG_{p̂}] ≅ [BG, BG_{p̂}] ≅ [ \underrightarrow{hocolim}_{G/P ∈R_{p}(G)}(EG/P ) , BG_{p̂}].

This makes it natural to study the restriction map

R : [BG_{p̂}, BG_{p̂}] ⟶ \underleftarrow{lim}_{G/P ∈R_{p}(G)}[EG/P, BG_{p̂}] ≅ \underleftarrow{lim}_{G/P ∈R_{p}(G)}[BP, BG_{p̂}]

≅ \underleftarrow{lim}_{G/P ∈R_{p}(G)} QRep(P, G)

(where the last step follows from Theorem 1.1).

An R_{p}-invariant quasirepresentation of G is defined here to be a homomorphism ρ : N_{p}(T)_{∞} ⟶ G′ into some compact connected Lie group G′ which extends (via restriction and conjugation) to an element in the inverse limit \underleftarrow{lim} QRep(P, G′). Thus, for any f : BG ⟶ BG′_{p̂}, f|BN_{p}(T)≃Bρ for some unique R_{p}-invariant quasirepresentation ρ.

For any R_{p}-invariant quasirepresentation ρ : N_{p}(T)_{∞} ⟶ G′ of G,

map(BG, BG′_{p̂})_{[ρ]}

will denote the space of “maps of type ρ”: i. e., the space of maps f : BG ⟶ BG′_{p̂} such that f|BN_{p}(T)≃Bρ. The brackets are added to emphasize that this is a (possibly empty) union of connected components in map(BG, BG′_{p̂}). We next need to describe the relationship between the homotopy groups of map(BG, BG′_{p̂})_{[ρ]}, and those of the spaces map(BP, BG′_{p̂})_{Bρ|P} for p-stubborn P ⊆G. When doing this, it is convenient to use the following functors.

Definition 2.2. For any R_{p}-invariant quasirepresentation ρ : N_{p}(T)_{∞} ⟶ G′ of G, where G′ is connected, define a contravariant functor Π_{∗}^{ρ} : R_{p}(G) ⟶ ℤ_{(p)}-mod by setting

Π_{∗}^{ρ}(G/P) := π_{∗} (map(EG/P, BG′_{p̂})_{Bρ|P} ) ≅ π_{∗}(BC_{G′}(ρ(P_{∞})))_{p̂}

for each p-stubborn P ⊆G.

When making Π_{∗}^{ρ} into a well defined functor to abelian groups, there are, of course problems with choosing base points, and with possibly nonabelian fundamental groups. This is discussed in detail in Wojtkowiak [Wo]. In all of the cases which occur in this paper, the centralizer C_{G′}(ρ(P_{∞})) is abelian, and so these difficulties do not arise.

It is convenient to think of the higher derived functors of inverse limits as “cohomology groups” of a category. For this reason, and to simplify notation, we write H^{i}(C; F) = \underleftarrow{lim}^{i}(F) for any contravariant functor F : C ⟶ Ab defined on a small category C. As one might expect, the obstructions to the restriction map R displayed above being a bijection are the higher limits of the functors Π_{∗}^{ρ}.

Theorem 2.3. Fix a compact connected Lie group G′, and an R_{p}-invariant quasirepresentation ρ : N_{p}(T) ⟶ G′. Then map(BG, BG′_{p̂})_{[ρ]} is nonempty if H^{i+1}(R_{p}(G); Π_{i}^{ρ}) = 0 for all i ≥ 1, and is connected if H^{i}(R_{p}(G); Π_{i}^{ρ}) = 0 for all i ≥ 1.

Proof. See Wojtkowiak [Wo], who deals more generally with spaces of the form map(\underrightarrow{hocolim}(X_{α}), Y). □

Theorem 2.3 is a special case of a second quadrant spectral sequence, which converges to the homotopy of map(\underrightarrow{hocolim}(X_{α}), Y)_{[f̂]} for f̂ ∈ \underleftarrow{lim}[X_{α}, Y]. See Bousfield & Kan [BK, Propositions XII.4.1 and XI.7.1] and Bousfield [Bf] for details.

We want to show, for a pair f, f′ : BG_{p̂} ⟶ BG_{p̂} of ℚ-equivalences, that f≃f′ if f|BT ≃f′|BT. We have already seen that f, f′ ∈ [BG, BG_{p̂}]_{[ρ]} for the same R_{p}-invariant quasirepresentation ρ; and so they are homotopic by Theorem 2.3 if the higher limits H^{i}(R_{p}(G); Π_{i}^{ρ}) all vanish.

In fact, it will suffice to consider the case where ρ is the inclusion. In this case, we write Π_{∗} for Π_{∗}^{ρ}. Thus, for any G/P in R_{p}(G),

Π_{i}(G/P) = π_{i} (map(BP, BG_{p̂})_{incl}) ≅ π_{i}(BC_{G}(P))_{p̂} ≅ π_{i−1}(C_{G}(P))_{p̂}

by Theorem 1.1. Also, for any p-stubborn P ⊆G, C_{G}(P) = Z(P) [JMO, Lemma 1.5(ii)], and is in particular abelian. The identity component of C_{G}(P) is thus a torus, and hence π_{i}(BC_{G}(P)) = 0 for i ≥ 3. In other words, Π_{i} = 0 for all i ≥ 3. And for i = 1 or 2,

Π_{i}(G/P) ≅ π_{i−1}(Z(P))

for each G/P in R_{p}(G).

We thus need the following computation of the higher limits of the functors Π_{i}.

Theorem 2.4. Fix a compact connected Lie group G. Then if π_{1}(G) is torsion free and i ≥ 1, or if G is arbitrary and i ≥ 2,

H^{∗}(R_{p}(G); Π_{i}) ≅ \{ \begin{matrix}π_{i−1}(Z(G))_{p̂} & if ∗ = 0 \\ 0 & if ∗ > 0. \end{matrix}  (1)

Proof. See [JMO, Theorem 4.1 and Sections 5–6]. When i = 1, this is shown in [JMO] only when G is simply connected. But if G′ ◁ G is the maximal semisimple component, then G′ is simply connected by assumption, R_{p}(G) ≅ R_{p}(G′) [JMO, Proposition 1.6(i,ii)]; and Π_{1} is the same on both categories. □

In fact, Theorem 2.4 holds for any compact connected Lie group G. For i ≥ 2, this is proven in [JMO]. For i = 1, a proof of this was given in the first version of [JMO], before Notbohm showed us how to get by with the result in the simply connected case.

We are now ready to apply these techniques. The following theorem really consists of two separate results. But we combine them here, since there is quite a bit of overlap in the proofs.

Theorem 2.5. The map

Θ : [BG_{p̂}, BG_{p̂}]_{ℚ} ⟶ N_{Aut(L(T_{∞}))}(W)/W

defined in Proposition 1.4 is an injection of monoids, and its image is closed in the p-adic topology.

Proof. The theorem will be shown in three steps. The injectivity of Θ is shown in Step 2, and the fact that Im(Θ) is closed in Step 3. The first step is somewhat more technical, and is needed (in part) to get around the problem that the higher limits H^{j}(R_{p}(G); Π_{i}) have been completely computed only when π_{1}(G) is torsion free.

Step 1 Let n be the exponent of π_{1}(G/Z(G)_{0}), where Z(G)_{0} is the identity component of the center. Let G̃ ⟶^{γ} G be the finite covering such that π_{1}(G̃) = n · π_{1}(G) (i.e., the subgroup of all n-th powers in π_{1}(G)). Then G̃ splits as a product G̃ = S×G′, where S is a torus and G′ is simply connected. For any f : BG_{p̂} ⟶ BG_{p̂}, π_{2}(f) sends π_{2}(BG̃_{p̂}) (⊆ π_{2}(BG_{p̂})) to itself, so that the composite

BG̃_{p̂} ⟶^{Bγ} BG_{p̂} ⟶^{f} BG_{p̂} ⟶ K([π_{1}(G)/π_{1}(G̃)]_{p̂}, 2)

is nullhomotopic. Since the fibration BG̃ ⟶ BG ⟶ K(π_{1}(G)/π_{1}(G̃), 2) is still a fibration after completion (cf. [BK, VI.6.5]), this shows that f lifts to a map f̃: BG̃_{p̂} ⟶ BG̃_{p̂}.

Set T̃ = γ^{−1}(T): a maximal torus in G̃. A homomorphism ϕ : T_{∞} ⟶ T_{∞} lifts to at most one homomorphism ϕ̃ : T̃_{∞} ⟶ T̃_{∞}, since two liftings of ϕ differ by an element of Hom(T_{∞}, Ker(γ)) = 0.

Write G′ = G_{1}×G_{2}, where G_{2} is the product of those simple factors for which p divides the order of the Weyl group. Then T̃ = S×T_{1}×T_{2}, where T_{i} is a maximal torus in G_{i}; and N_{p}(T̃) = S×T_{1}×N_{p}(T_{2}) for some maximal p-toral subgroup N_{p}(T_{2})⊆G_{2}. If P ⊆N_{p}(T̃) is a p-stubborn subgroup of G̃, then since N(P)/P is finite (by definition of p-stubborn), P must have the form P = S×T_{1}×P′ for some p-stubborn P′⊆G_{2}.

Now let ρ : N_{p}(T) ⟶ G be any R_{p}-invariant quasirepresentation, and set f_{P} = B(ρ|P) : BP ⟶ BG_{p̂} for each p-toral P ⊆N_{p}(T). In other words, the f_{P} define an element ρ̂ ∈ \underleftarrow{lim}[BP, BG_{p̂}]. By [BK, XII.4.1], map(BG_{p̂}, BG_{p̂})_{[ρ]} is the homotopy inverse limit (over R_{p}(G)) of the spaces map(BP, BG_{p̂})_{f_{P}}.

For each p-stubborn P ⊆G we write P̃ = γ^{−1}(P). By [JMO, Proposition 1.6(i)], the correspondence G/P ⟵ G̃/P̃ induces an isomorphism of categories R_{p}(G) ≅ R_{p}(G̃). Also, Theorem 1.1 applies to show that map(BP, BG_{p̂})_{f_{P}} ≃ map(BP̃, BG_{p̂})_{f_{P} ∘Bγ} for each P. Upon taking homotopy inverse limits [BK, XII.4.1], we now get a homotopy equivalence

map(BG_{p̂}, BG_{p̂})_{[ρ]} ⟶_{≃}^{−∘Bγ} map(BG̃_{p̂}, BG_{p̂})_{[ρ∘γ|P̃]}.  (1)

In particular, Bρ extends to a map BG_{p̂} ⟶ BG_{p̂} if and only if it extends to a map BG̃_{p̂} ⟶ BG_{p̂}; and the map defined on BG_{p̂} is unique if and only if it is unique when defined on BG̃_{p̂}.

Step 2 Fix a pair f, f′ : BG_{p̂} ⟶ BG_{p̂} of ℚ-equivalences such that f|BT ≃f′|BT. We want to show that f≃f′. Choose liftings of f and f′ to f̃, f̃′ : BG̃_{p̂} ⟶ BG̃_{p̂}. Then f̃|BT̃ ≃ f̃′|BT̃ (by the uniqueness of the lifting). By Proposition 2.1, f|BN_{p}(T)≃f′|BN_{p}(T)≃Bρ for some ρ∈ QRep(N_{p}(T), G). In other words,

f, f′ ∈ map(BG_{p̂}, BG_{p̂})_{[ρ]}.

If we can show that f̃ ≃ f̃′, then f∘Bγ≃Bγ ∘ f̃≃Bγ ∘ f̃′≃f′∘Bγ, and (1) applies to show that f≃f′. So to simplify the notation, we can just assume that G = G̃.

Let ρ∈ QRep(N_{p}(T), G) be an R_{p}-invariant quasirepresentation such that f|BN_{p}(T)≃Bρ (Theorem 1.1). We may assume (after conjugating, if necessary) that ρ(N_{p}(T)_{∞})⊆N_{p}(T)_{∞}.

Recall (Step 1) that G factors as a product G = S×G_{1}×G_{2}, where S is a torus, where G_{1} and G_{2} are simply connected, and where p divides the orders of the Weyl groups of all simple summands of G_{2} but not of any of the simple summands of G_{1}. Let f_{2} denote the composite

f_{2} : (BG_{2})_{p̂} ⟶ BG_{p̂} ⟶^{f} BG_{p̂} ⟶^{proj} (BG_{2})_{p̂}.

Then f_{2} is a ℚ-equivalence, and is a homotopy equivalence by Proposition 1.3. Upon composing with 1×f_{2}^{−1}, we can thus assume that f|BN_{p}(T_{2}) is homotopic to the inclusion; or equivalently that ρ|N_{p}(T_{2}) is the inclusion of N_{p}(T_{2}) into G.

Now fix a p-stubborn subgroup P ⊆G. We may assume that P ⊆N_{p}(T), and hence (by Step 1) that P = S×T_{1}×P′ for some P′⊆N_{p}(T_{2}). In particular, ρ(P) = P; and so upon referring to Definition 2.2, we see that Π_{∗}^{ρ} = Π_{∗}. Theorem 2.4 now applies to show that H^{i}(R_{p}(G); Π_{j}^{ρ}) = 0 for all i, j > 0. And so f≃f′ by Theorem 2.3.

Step 3 It remains to show that Im(Θ) is closed in N_{Aut(L(T_{∞}))}(W)/W in the padic topology. So consider an element ω = lim_{i→∞}(ω_{i}), where ω_{i}·W ∈ Im(Θ) for each i. Let ϕ, ϕ_{i} : T_{∞}↠T_{∞} be the corresponding admissible maps. We may assume that ϕ = lim(ϕ_{i}) in Hom(T_{∞}, T_{∞}) (i. e., not only mod W).

For each i, choose an extension f_{i} : BG_{p̂} ⟶ BG_{p̂} of Bϕ_{i} (i. e., ω_{i} = Θ(f_{i})); and let f̃_{i} : BG̃_{p̂} ⟶ BG̃_{p̂} be a lifting. Then f̃_{i}|BT̃≃Bϕ̃_{i}, where ϕ̃_{i}∈ End(T̃_{∞}) is a (unique) lifting of ϕ_{i}. Let p^{m} be the exponent of Ker(γ). If ϕ_{i} and ϕ_{j} agree on all p^{k+m}-torsion in T, for any k > 0, then ϕ̃_{i} and ϕ̃_{j} agree on all p^{k}-torsion in T̃. Hence, since the ϕ_{i} converge to ϕ, the ϕ̃_{i} converge to some ϕ̃∈ End(T̃_{∞}), and ϕ̃ is a lifting of ϕ.

Using Theorem 1.1, choose elements ρ_{i}∈ QRep(N_{p}(T̃), G̃) such that f̃_{i}|BN_{p}(T̃)≃Bρ_{i}. The ρ_{i} are R_{p}-invariant (by Theorem 1.1 again). We may assume (since the ρ_{i} are welldefined only up to conjugation) that ρ_{i} | T̃_{∞} = ϕ̃_{i} and Im(ρ_{i})⊆N_{p}(T̃)_{∞} for each i. And since ϕ̃_{i}(T̃_{∞}) = T̃_{∞}, a counting argument shows that Im(ρ_{i}) = N_{p}(T̃)_{∞}.

Now write N_{p}(T̃)_{∞} = ∪_{n=1}^{∞}P_{n}, where P_{1}⊆P_{2}⊆P_{3} ⊆ … are finite p-groups. For each n, the set

{ρ_{i}|P_{n} : i ≥ 1} ⊆ Rep(P_{n}, N_{p}(T̃))

is finite (since Rep(P_{n}, N_{p}(T̃)) is finite: this follows easily from [MZ, Theorems 1.10.5 & 5.3]). Hence, we can successively choose elements ρ_{ni}, where {ρ_{n1}, ρ_{n2}, … } is a subsequence of {ρ_{n−1,1}, ρ_{n−1,2}, … } for each i (and ρ_{0i} = ρ_{i}); and where ρ_{ni}|P_{n} and ρ_{nj}|P_{n} are conjugate for each n and each i, j.

Now set σ_{n} = ρ_{nn} for each n. Upon replacing the σ_{n} by conjugate homomorphisms, if necessary, we may assume that σ_{n}|P_{m} = σ_{n}|P_{n} for each m≥n. The σ_{n} thus converge to a homomorphism σ : N_{p}(T̃)_{∞} ⟶ G̃; and σ|T̃_{∞} is conjugate to ϕ̃ by construction. For any p-toral subgroup P ⊆N_{p}(T̃) and any g ∈ G̃ for which gP g^{−1}⊆N_{p}(T̃), (σ_{n}|gP_{∞}g^{−1})∘ conj(g) is (for each n) conjugate in G̃ to σ_{n}|P_{∞}; and so the same holds for σ. In other words, σ is an R_{p}-invariant quasirepresentation of G̃.

By Step 1 again, N_{p}(T̃) = S×T_{1}×N_{p}(T_{2}), and each p-stubborn subgroup P ⊆N_{p}(T̃) of G has the form P = S×T_{1}×P′ for some p-stubborn P′⊆G_{2}. Also, σ = ρ′ × ρ′′, where ρ′∈ End((S×T_{1})_{∞}) and ρ′′∈ Aut(N_{p}(T_{2})_{∞}). Set ρ = 1 × ρ′′: also an R_{p}-invariant quasirepresentation of G̃. For any p-stubborn P ⊆ G̃ contained in N_{p}(T̃), ρ(P) = σ(P) by construction, and hence Π_{∗}^{σ} ≅ Π_{∗}^{ρ} as functors defined on R_{p}(G̃).

We next want to show that Π_{∗}^{ρ} ≅ Π_{∗} as functors on R_{p}(G̃). Fix a p-stubborn subgroup P ⊆N_{p}(T̃). By [JMO, Lemma 1.5], C_{G}(P) = Z(P)⊆P. We claim that C_{G}(ρP) = Z(ρP). By [JMO, Proposition A.4], π_{0}(C_{G}(ρP)) is a p-group, and so C_{G}(ρP) is a union of p-toral subgroups. Hence, if C_{G}(ρP) ⊈ ρP, we can choose g∈C_{G}(ρP) ∖ ρP of ppower order, and thus such that ⟨g, ρP ⟩ is still p-toral. So after conjugating (in G), we may assume that ⟨g, ρP_{∞}⟩ ⊆ N_{p}(T̃)_{∞}. But then ρ^{−1}(g)∈C_{G}(P) = Z(P), and this is a contradiction. Thus, ρ induces isomorphisms

ρ_{P} : Π_{∗}(G/P) ≅ π_{∗}(BZ(P)) ⟶^{≅} π_{∗}(BZ(ρP)) ≅ Π_{∗}^{ρ}(G/P)

for each G/P in R_{p}(G); and this induces an isomorphism Π_{∗}^{ρ} ≅ Π_{∗} of functors on R_{p}(G̃).

Theorem 2.4 can now be applied to show that

H^{j}(R_{p}(G̃); Π_{i}^{σ}) ≅ H^{j}(R_{p}(G̃); Π_{i}^{ρ}) ≅ H^{j}(R_{p}(G̃); Π_{i}) = 0

for all i, j ≥ 1. Hence map(BG̃_{p̂}, BG̃_{p̂})_{[σ]} is nonempty (and connected) by Theorem 2.3. Choose some element f̃∈ map(BG̃_{p̂}, BG̃_{p̂})_{[σ]}.

Recall that we are working with the finite covering G̃ ⟶^{γ} G of Step 1. By construction, σ | T̃_{∞} = ϕ̃, and hence σ(Ker(γ)) = Ker(γ). So σ factors through a unique map τ∈ End(N_{p}(T)_{∞}), and τ is also an R_{p}-invariant quasirepresentation. Since Bγ ∘ f̃∈ map(BG̃_{p̂}, BG_{p̂})_{τ̂∘γ} ≠ ∅, formula (1) above applies to show that

map(BG_{p̂}, BG_{p̂})τ̂ ≠ ∅. And for any f∈ map(BG_{p̂}, BG_{p̂})τ̂, f|BT_{ p̂}≃Bτ|BT ≃Bϕ, so

Θ(f) = ω, and we are done. □

The following corollary to Theorem 2.5 is immediate.

Corollary 2.6. For any pair of ℚ-equivalences f, f′ : BG_{p̂} ⟶ BG_{p̂}, f≃f′ if and only if f|BT ≃f′|BT, if and only if H^{∗}(f; ℚ) = H^{∗}(f′; ℚ).

Proof. We have just seen that f≃f′ if and only if f|BT ≃f′|BT. And f|BT ≃f′|BT if and only if H^{∗}(f; ℚ) = H^{∗}(f′; ℚ) by a theorem of Notbohm [No1, Proposition 4.1]. □

Using the same techniques, we get the following description of the individual connected components of the space of ℚ-equivalences. This result has also been proven recently by Dwyer & Wilkerson [DW2, Theorem 1.3], using quite different methods.

Proposition 2.7. For any compact connected Lie group G, and any ℚ-equivalence f : BG_{p̂} ⟶ BG_{p̂}, the natural homomorphism Z(G) × G ⟶ G induces homotopy equivalences

BZ(G)_{p̂} ⟶^{≃} map(BG_{p̂}, BG_{p̂})_{Id} ⟶_{≃}^{f∘−} map(BG_{p̂}, BG_{p̂})_{f}.

Proof. We first show that the first map is a homotopy equivalence. By the obstruction theory of Wojtkowiak [Wo] (more precisely, by the version of his result given in [JMO, Theorem 3.9]), this will follow immediately if the formula for H^{j}(R_{p}(G); Π_{i}) in Theorem 2.4 (formula (1)) holds for all i > j ≥ 0, and i = j ≥ 2. So the only case which it remains to check is that

H^{0}(R_{p}(G); Π_{1}) := \underleftarrow{lim}_{R_{p}(G)}(Π_{1}) ≅ π_{1}(BZ(G)_{p̂}).  (1)

Whether or not this holds, we have

π_{1}(map(BG_{p̂}, BG_{p̂})_{Id}) ≅ \underleftarrow{lim}_{R_{p}(G)}(Π_{1}) ≅ \underleftarrow{lim}_{G/P ∈R_{p}(G)} π_{0}(Z(P))_{(p)}.

So by [JMO, Theorem 4.2], (1) does hold if G is simple. If G is a product of a torus and simple groups G_{i}, then R_{p}(G) is the product of the R_{p}(G_{i}) [JMO, Proposition 1.6(ii)]; and so (1) again holds. Finally, if G is arbitrary, then there are monomorphisms

π_{1}(BZ(G)) ≅ π_{0}(Z(G)) ↣ \underleftarrow{lim}_{G/P ∈R_{p}(G)} π_{0}(Z(P))

and

( \underleftarrow{lim}_{G/P ∈R_{p}(G)} π_{0}(Z(P)))/π_{0}(Z(G)) ↣ \underleftarrow{lim}_{G/P ∈R_{p}(G)} π_{0}(Z(P/Z(G))) ≅ π_{0}(Z(G/Z(G))) = 1

(since R_{p}(G/Z) ≅ R_{p}(G) by [JMO, Proposition 1.6(i)]). So (1) also holds in this case.

To see that the second map is a homotopy equivalence, let G_{s} ⊆ G be the maximal semisimple subgroup, and note that R_{p}(G_{s}) ≅ R_{p}(G) (any p-stubborn subgroup of G contains the connected component of Z(G), and its intersection with G_{s} is p-stubborn in G_{s}). Hence the obstructions to (f ∘ −) being a homotopy equivalence are the same for G and G_{s}, and we can assume that G = G_{s} is semisimple. In this case, we can write

BG_{p̂} ≃ (BG_{1})_{p̂} × (BG_{2})_{p̂} and f ≃ f_{1} × f_{2} :

where p divides the orders of all simple components of G_{1} but of none of the simple components of G_{2}. (The simple components for which p does not divide the order of the Weyl groups all have center and fundamental group of order prime to p.) Then f_{1} is a homotopy equivalence by Proposition 1.3, and we can assume f_{1} = Id. Write f|BN_{p}(T) ≃ Bρ. The only p-stubborn subgroups of G_{2} are the maximal tori, so Π_{∗}^{ρ} ≅ Π_{∗}, and (f ∘−) is a homotopy equivalence by the Bousfield-Kan spectral sequence again. □

Section 3

The main goal of this section is to describe the images of the maps

Θ : [BG_{p̂}, BG_{p̂}]_{ℚ} ↣ N_{Aut(L(T_{∞}))}(W)/W

and

Θ_{h} : [BG_{p̂}, BG_{p̂}]_{h} ↣ N_{Aut(Λ_{p̂})}(W)/W

of Proposition 1.4. The image of Θ is described explicitly in Theorem 3.4. In Corollary 3.5, we then show that Θ_{h} is an isomorphism if p ≠ 2, or if G contains no factor of the form Sp(n)×SO(2n+1). The following example shows why this restriction is necessary.

Example 3.1. Assume that p = 2, and that G = Sp(n)×SO(2n + 1) for some n ≥ 1. Let T = T_{1}×T_{2} be the product of the standard maximal tori; where

T_{1} = Im[α : ℝ^{n} ⟶ Sp(n)]  and  T_{2} = Im[β : ℝ^{n} ⟶ SO(2n + 1)],

and

α(θ_{1}, …, θ_{n}) = diag(e^{2πiθ_{1}}, …, e^{2πiθ_{n}}) ∈ Sp(n),

β(θ_{1}, …, θ_{n}) = ( \begin{matrix}cos(2πθ_{1}) & −sin(2πθ_{1}) \\ sin(2πθ_{1}) & cos(2πθ_{1}) \end{matrix} ) ⊕ ⋯ ⊕ ( \begin{matrix}cos(2πθ_{n}) & −sin(2πθ_{n}) \\ sin(2πθ_{n}) & cos(2πθ_{n}) \end{matrix} ) ⊕ (1).

Let Λ = Λ_{1} × Λ_{2} ⊆ T̃ be the integral lattice for T, and let ϕ∈ Aut(T)≅ Aut(Λ) be the involution which switches the factors (sending α(θ_{1}, …, θ_{n}) to β(θ_{1}, …, θ_{n}) and vice versa). Then ϕ is an admissible map for G, but Bϕ does not extend to any self map of BG_{ˆ2}. In other words,

Θ_{h} : Aut(BG_{p̂}) ⟶ N_{Aut(Λ_{p̂})}(W)/W

is not onto in this case.

Proof. Note first that ϕ is admissible, since Sp(n) and SO(2n + 1) have the same Weyl groups under this identification of their maximal tori. If Bϕ could be extended to a self map of BG_{ˆ2}, then it would be a homotopy equivalence by Proposition 1.2(iii), and hence would restrict to a homotopy equivalence between BSp(n)_{ˆ2} and BSO(2n + 1)_{ˆ2}. But BSp(n)_{ˆ2} is 3-connected, while π_{2}(BSO(2n + 1)_{ˆ2}) ≅ ℤ/2. □

The roots in G play an important role in both the statements and the proofs of the results later in this section. In the next proposition, we collect some of the facts about roots which will be needed later. But we first set up some notation.

Let R denote the set of roots of G: regarded as a subset of L(T)^{∗} = Hom(L(T), ℝ). Then the Lie algebra L(G), under the adjoint (conjugation) action of T, splits as a sum

L(G) = L(T) ⊕ ∑_{±θ∈R} V_{θ},

where each V_{θ} is a 2-dimensional irreducible T -representation with character χ_{θ} : T ⟶ S^{1}, and where χ_{θ}∘ exp(v) = exp(2πi · θ(v)) for each θ∈R and each v∈L(T). (Note that V_{θ} defines θ only up to sign.) In other words, the roots are defined to be the liftings to Hom(L(T), ℝ) of the irreducible characters for the adjoint action of T on L(G). In particular, since Λ = Ker[exp : L(T) ⟶ T], θ(Λ) ⊆ ℤ for each θ∈R; and θ(v) ∈ ℤ (for v∈L(T)) if and only if exp(v) commutes with all elements of exp(V_{θ}).

Proposition 3.2. Let R be the set of roots of G, regarded as a subset of Hom(T, S^{1})≅ Hom(Λ, ℤ) ⊆ L(T)^{∗}. Let W = N(T)/T be the Weyl group of G, with the induced action on L(T). Let R_{s} ⊆ R be the set of simple roots with respect to some fixed Weyl chamber. Assume that L(T) has been given a fixed W -invariant inner product. Then the following hold.

(i) W is the group generated by the reflections in the hyperplanes Ker(θ) for θ∈R_{s}. Conversely, for any w∈W, if w is a reflection (if T̃^{w} has codimension one), then T̃^{w} is the kernel of some root θ∈R.

(ii) R = W ·R_{s}: each root is in the W -orbit of some simple root.

(iii) The inclusion T ⊆G induces a surjection Λ ≅ π_{1}(T) ↠ π_{1}(G).

(iv) If Z(G) = 1, then Λ = {x∈L(T) : R(x) ⊆ ℤ}.

(v) For each θ ∈ ℝ, there exists a unique element v_{θ} ∈ Λ such that v_{θ}⊥ Ker(θ) and θ(v_{θ}) = 2 (v_{θ} is the nodal vector of θ in the notation of Bourbaki). And π_{1}(G) ≅ Λ/⟨v_{θ} : θ∈R⟩.

(vi) For any θ ∈ ℝ, Λ∩ Ker(θ)^{⊥} = ℤ·v_{θ} or ℤ · 1 ¯2v_{θ}, where the second possibility occurs if and only if G contains a direct factor SO(2n + 1) for some n ≥ 1, and θ is a short root of such a factor.

Proof. The first statement in (i) — that W is generated by reflections by simple roots — is shown in [Ad, 5.13(iv) & 5.34]. To see the second statement, note that if wT ∈N(T)/T is a reflection in T̃, then w∈C_{G}(S) (the centralizer) for some codimension one subtorus S⊆T. By [Bb2, p.31, Lemma 2], C_{G}(S) ⫌ ⟨T, w⟩ is connected (and nonabelian). So it must have at least one root θ∈R, and Ker(θ) = S.

Point (ii) is shown in [Bb1, p.154, Prop. 15]. Point (iii) is shown in [Ad, 5.47] or [Bb2, p.34, Prop. 11]. Point (iv) is shown in [Ad, Proposition 5.3], and is equivalent to [Bb2, Proposition 8(b)].

Fix some θ∈R, let v_{θ}∈L(T) be the unique element such that v_{θ}⊥ Ker(θ) and θ(v_{θ}) = 2, and let w_{θ}∈N(T) be such that w_{θ}T ∈W is the reflection in Ker(θ). Then w_{θ}∈H_{θ} := C_{G}(exp(Ker(θ))), and H_{θ} is a connected subgroup by [Bb2, p.31] again. Also, L(H_{θ}) = L(T)⊕V_{θ} in the notation above; and so Z(H_{θ}) = Ker(χ_{θ}) = exp(θ^{−1}(ℤ)). In particular, if v∈L(T) and θ(v) = 1; then exp(v)∈Z(H_{θ}), v_{θ} = v − w_{θ}(v), and so exp(v_{θ}) = [exp(v), w_{θ}] = 1.

This shows that v_{θ} ∈ Λ = Ker(exp) for each θ∈R. The formula π_{1}(G) ≅ Λ/⟨v_{θ} : θ∈R⟩ is shown in [Ad, 5.47 & 5.48] and [Bb2, p.34, Prop. 11].

It remains to check point (vi). Fix a root θ∈R. Since θ(Λ) ⊆ ℤ and θ(v_{θ}) = 2, we must have

Λ ∩ Ker(θ)^{⊥} = ℤ · v_{θ} or ℤ · 1 ¯2v_{θ}.

Assume that the second possibility occurs; i. e., that 1 ¯2v_{θ} ∈ Λ. Fix any other root η∈R∖\{±θ\}. Since the projection of η to the line ℝ·θ is \frac{<η,θ>}{<θ,θ>}·θ, and since (ℝ·θ)^{⊥}(v_{θ}) = 0, we have

\frac{< η, θ >}{< θ, θ >} = (\frac{< η, θ >}{< θ, θ >} · θ) (1 ¯2v_{θ}) = η(1 ¯2v_{θ}) ∈ ℤ.

In other words, the projection of η to ℝ·θ is an integral multiple of θ. A quick check of the different possibilities (cf. [Ad, Proposition 5.25]) shows that either η ⊥ θ, or the angle between η and θ is 45^{∘} or 135^{∘} and θ is the shorter root.

By point (ii), we can assume that θ∈R_{s}. The Dynkin diagram of G thus contains a node which is not connected to any other node by single or triple lines; and which, if connected to another node by a double line, represents the shorter of those two roots. The classification of simple Lie groups now shows that G contains a normal simple subgroup of type B_{n} (i. e., H≅SO(2n + 1) or Spin(2n + 1)) for some n ≥ 1.

Assume H◁G, where H≅SO(2n + 1) or Spin(2n + 1). Identify L(T ∩H) with ℝ^{n} via the map β of Example 3.1. Let x_{i}∈ Hom(ℝ^{n}, ℝ) denote projection to the i-th coordinate, and let ϵ_{i} ∈ ℝ^{n} denote the i-th element of the standard basis. Then the integral lattice of H is ℤ^{n} if H≅SO(2n + 1), and < ϵ_{i} ± ϵ_{j} > if H≅Spin(2n + 1). The roots of H are the elements ±x_{i} and ±x_{i}±x_{j} (for i≠j). Then v_{θ} = ±2ϵ_{i} if θ = ±x_{i}, and v_{θ} = ±ϵ_{i} ± ϵ_{j} if θ = ±x_{i}±x_{j}. Hence 1 ¯2v_{θ} ∈ Λ = ℤ^{n} if and only if H≅SO(2n + 1) and θ = ±x_{i}.

Thus, 1 ¯2v_{θ} ∈ Λ for some θ∈R if and only if θ is a short root of some simple component H◁G, where H≅SO(2n + 1) for some n ≥ 1. Since Z(SO(2n + 1)) = 1, this subgroup is in fact a direct factor of G. □

We now regard a root θ∈R⊆ Hom(Λ, ℤ) also as an element of Hom(Λ_{p̂}, ℤ̂_{p}) or Hom(L(T_{∞}), ℚ̂_{p}) (recall that L(T_{∞}) ≅ ℚ ⊗_{ℤ} Λ_{p̂}). We can assume that the W -invariant inner product on L(T) takes integer values on Λ (take any integer valued inner product on Λ and sum over its W -orbit). Hence the inner product extends to a ℚ̂_{p}-valued inner product on L(T_{∞}). Note that the cosine of angles between elements is no longer defined in this setting (since there’s no unique choice of square roots); but the square of the cosine is defined. It will be convenient to say that two angles α_{1}, α_{2} are “the same up to sign” if cos^{2}(α_{1}) = cos^{2}(α_{2}); i. e., if α_{1} = α_{2} or α_{1} = π − α_{2}.

Lemma 3.3. Let ω∈ Aut(L(T_{∞})) be any admissible map; i. e., ω lies in the normalizer of the Weyl group W ⊆ Aut(Λ)⊆ Aut(L(T_{∞})). Then ω permutes the components of L(T_{∞}) corresponding to simple factors of G. The dual map ω^{∗} sends each root in G to some ℚ̂_{p}-multiple of another root; and preserves the angles between roots up to sign. Furthermore, after replacing ω by w ∘ ω for some w ∈ W if necessary, we may assume that ω permutes any given set of simple roots of G (again up to scalar multiples); and hence induces an automorphism of the Dynkin diagram (though possibly reversing arrows).

Proof. By Proposition 3.2(i), the kernels of the roots of G are precisely those hyperplanes in L(T) (hence in L(T_{∞})) for which the corresponding reflection lies in W. Since ω is equivariant with respect to some automorphism of W, it permutes the reflections, hence the kernels of roots, and hence the roots themselves up to scalar multiple.

Write ℚ ⊗ Λ = V_{0}×V_{1} × … ×V_{m}, where V_{0} comes from the torus factor in G, and the V_{1}, …, V_{m} from the simple factors. Then V_{0} = (ℚ ⊗ Λ)^{W}, and the spaces ℂ ⊗_{ℚ} V_{1}, …, ℂ⊗_{ℚ}V_{m} are distinct irreducible representations of W (cf. [Bb1, p.82, Proposition 5(v)]). Hence the ℚ̂_{p}[W]-representations V_{i}_{p̂} = ℚ̂_{p} ⊗_{ℚ} V_{i} (1≤i≤m) are also irreducible. Since ω is equivariant with respect to some automorphism of W, it must permute the V_{i}_{p̂}. Furthermore, since the W -invariant inner product on each irreducible summand is unique up to scalar multiple, ω preserves the inner product on each of the V_{1}_{p̂}, …, V_{m}_{p̂} up to scalar multiple (and V_{i}_{p̂}⊥V_{j}_{p̂} for i≠j). Since each root θ∈R lies in one of the (V_{i}_{p̂})^{∗} for 1≤i≤m, this shows that ω preserves angles between roots up to sign.

Now choose a permutation σ ∈ Σ(R), and elements 0≠a_{θ} ∈ ℚ̂_{p}, such that ω^{∗}(θ) = a_{θ} · σθ for each root θ∈R. Fix a Weyl chamber C, and let R_{s} be the corresponding set of simple roots. Since the Dynkin diagram is a union of trees, we can arrange (by switching pairs of roots ±θ) that for any pair θ_{1} ≠ θ_{2} in R_{s}, the angle between σθ_{1} and σθ_{2} is between π/2 and π. Then for each θ_{1}, θ_{2}∈R_{s}, θ_{1} and θ_{2} form the same angle as σθ_{1} and σθ_{2}.

Let T_{s} be the intersection of T with the maximal semisimple subgroup of G. Then L(T_{s}) = ∑_{i=1}^{m} ℝ ⊗_{ℚ} V_{i}; and R_{s} and σ(R_{s}) are bases of L(T_{s}) (cf. [Ad, Prop. 5.33]). There is thus a unique μ : L(T_{s}) ⟶^{≅} L(T_{s}) such that μ^{∗}(θ/∥θ∥) = σθ/∥σθ∥ for each θ∈R_{s}; and μ is orthogonal since σ preserves angles between the simple roots. Also,

μ^{−1}(C) = {x ∈ L(T_{s}) : σθ(x) ≥ 0 for all θ ∈ R_{s}};  (1)

and hence μ^{−1}(C) is a union of Weyl chambers. Since the Weyl chambers are permuted transitively by the orthogonal action of W (cf. [Ad, Theorem 5.13]), they and μ^{−1}(C) all have the same volume after intersection with the unit ball; and hence μ^{−1}(C) is itself a Weyl chamber. We can thus assume (after composing ω and μ by some element of W) that μ(C) = C. But then (1) implies that

{Ker(θ) : θ ∈ R_{s}} = {Ker(σθ) : θ ∈ R_{s}}

(the walls of C = μ(C)); and hence that σ(R_{s}) = R_{s}.

In particular, since the nodes in the Dynkin diagram correspond to simple roots of G, and the connectors correspond to the angles between roots, this shows that σ induces an automorphism (possibly reversing arrows) of the Dynkin diagram of G. □

We are now ready to show which admissible maps extend to ℚ-equivalences of BG_{p̂} to itself. In other words, we will describe the image of the homomorphism

Θ : [BG_{p̂}, BG_{p̂}]_{ℚ} ↣ AdmEpi(T_{∞}, T_{∞})/W ≅ [N_{Aut(L(T_{∞}))}(W) ∩ End(Λ_{p̂})]/W

of Proposition 1.4. Roughly, the following theorem says that the only necessary conditions for an admissible epimorphism ϕ to lift to a self map of BG_{p̂} are the ones imposed by Proposition 1.3 and Example 3.1.

Theorem 3.4. Let G_{⟨p⟩}◁G be the product of all simple summands whose Weyl group has order a multiple of p. Let R⊆ Hom(Λ, ℤ) be the set of roots of G, and let R_{⟨p⟩}⊆R be the subset of roots in G_{⟨p⟩}. For each n ≥ 1, let G_{n}◁G be the product of all normal subgroups isomorphic to SO(2n + 1). Then, for any admissible epimorphism ϕ : T_{∞} ↠ T_{∞}, ϕW ∈ Im(Θ) if and only if the following two conditions are satisfied

(a) Ker(ϕ) ∩ G_{⟨p⟩} = 1 and

(b) (if p = 2) ϕ(G_{n} ∩ T_{∞}) ⊆ G_{n} ∩ T_{∞} ∀n ≥ 1;

if and only if ω := L(ϕ)|Λ_{p̂}∈ End(Λ_{p̂}) satisfies the two conditions

(i) ω^{∗}(R) ⊆ ℤ̂_{p} · R and

(ii) ω^{∗}(R_{⟨p⟩}) ⊆ (ℤ̂_{p})^{∗} · R_{⟨p⟩}.

Proof. We will prove the implications

(ϕW ∈ Im(Θ)) ⟹^{Step 1} (a,b) ⟹^{Step 2} (i,ii) ⟹^{Step 3} (ωW ∈ Im(Θ)).

Step 1 Assume that ϕW = Θ(f); i. e., f : BG_{p̂} ⟶ BG_{p̂} is such that f|BT_{ p̂}≃Bϕ. We will show that ϕ satisfies conditions (a) and (b) above.

Write G′ = G_{⟨p⟩} for short, set G′′ = G/G′, and let α : G′ ⟶ G and β : G ⟶ G′′ denote the induced maps. Let T′ = T ∩G′ and T′′ = T/T′ be their maximal tori, and set ϕ′ = ϕ|T′_{∞} : T′_{∞} ⟶ T′_{∞}. The composite

BG′_{p̂} ⟶^{Bα} BG_{p̂} ⟶^{f} BG_{p̂} ⟶^{Bβ} BG′′_{p̂}

is null homotopic, since it is trivial in rational cohomology (cf. [JMO, Theorem 3.11]). Hence f∘Bα pulls back along the fibration BG′_{p̂} ⟶ BG_{p̂} ⟶ BG′′_{p̂} (cf. [BK, VI.6.5]) to a map f′ : BG′_{p̂} ⟶ BG′_{p̂}. By Proposition 1.3, f′ is a homotopy equivalence, and so Ker(ϕ′) = Ker(ϕ)∩G_{⟨p⟩} = 1 by Proposition 1.2(iii). This proves point (a).

Now assume p = 2; we must prove condition (b). We have just seen that f restricts to a self map (and homotopy equivalence) of G_{⟨2⟩}. So we can assume that G = G_{⟨2⟩}; i. e., that G is semisimple and ϕ∈ Aut(T_{∞}).

By Lemma 3.3, ϕ permutes the simple summands of G. Fix some SO(2n + 1)≅H◁G, and let H′◁G be the simple summand such that ϕ(T_{∞}∩H) = T_{∞}∩H′. Since Z(H) = 1, H must be a direct factor of G (i. e., not just up to a finite covering). Hence T_{∞}∩H′ is a direct factor of T_{∞}, and so BH′_{ˆ2} is a direct factor of BG_{ˆ2} (i. e., H′ is a direct factor of G up to odd degree covering). The composite

BH_{ˆ2} ⟶^{incl} BG_{ˆ2} ⟶^{f} BG_{ˆ2} ⟶^{proj} BH′_{ˆ2}

is a homotopy equivalence; so π_{2}(BH′_{ˆ2}) ≅ π_{2}(BSO(2n + 1)_{ˆ2}) ≅ ℤ/2. Also, by Lemma 3.3, H′ has the same Dynkin diagram as H, except possibly for the direction of the arrows; and hence (since π_{1}(H′) ≠ 1) must be isomorphic to one of the groups SO(2n+1) or P Sp(n). These two are isomorphic if n ≤ 2; while if n ≥ 3 then

π_{5}(BSO(2n + 1)) ≅ π_{5}(BO) ≅ 0 and π_{5}(BP Sp(n)) ≅ π_{5}(BSp) ≅ ℤ/2

(cf. [Ml, p.142]). Thus, H′≅H≅SO(2n+1). Alternatively, this last point can be proven by showing that any admissible homomorphism between the (2-adic) integral lattices of SO(2n + 1) and P Sp(n) (for n ≥ 3) must (after composing with a Weyl group element) be a scalar multiple of the map used in Example 3.1 — and hence is not an isomorphism.

We have now shown that (for any n ≥ 1), ϕ permutes those simple factors isomorphic to SO(2n + 1) among themselves. This proves condition (b).

Step 2 Let R_{s}⊆R be the simple roots with respect to some Weyl chamber. Then R_{⟨p⟩}∩R_{s} is a ℚ̂_{p}-basis for L(Λ_{p̂}_{⟨p⟩}) (cf. [Ad, Proposition 5.33]).

By Lemma 3.3, there is a permutation σ ∈ Σ(R) such that for each θ∈R, ω^{∗}(θ) = a_{θ} · σθ for some a_{θ} ∈ (ℚ̂_{p})^{∗}. We may assume (by Lemma 3.3 again) that σ(R_{s}) = R_{s}. For each θ∈R, let v_{θ} ∈ Λ be the element defined in Proposition 3.2(v,vi): v_{θ}⊥ Ker(θ) and θ(v_{θ}) = 2. Since ω preserves angles (and in particular orthogonality), we see that ω(v_{σθ}) = a_{θ}·v_{θ} for each θ.

Assume now that conditions (a) and (b) hold. If p is odd, then by Proposition 3.2(vi), (ℚ̂_{p}·v_{θ}) ∩ Λ_{p̂} = ℤ̂_{p}·v_{θ} for each θ∈R. Hence, since ω(v_{σθ}) = a_{θ}·v_{θ} and ω(Λ) ⊆ Λ, we get a_{θ} ∈ ℤ̂_{p} for each θ, and so ω^{∗}(R) ⊆ ℤ̂_{p}·R.

If p = 2, then let R_{0}⊆R be the set of those roots θ such that 1 ¯2v_{θ} ∈ Λ. By 3.2(vi) again, the elements θ∈R_{0} are precisely the short roots of summands SO(2n + 1)◁G; i. e., the short roots in the G_{n} (for n ≥ 1). Also, for each i, ϕ(G_{n}∩T_{∞}) = G_{n}∩T_{∞} by condition (b), and ϕ|G_{n}∩T_{∞} is injective by (a). Thus, ω restricts to an admissible automorphism of the 2-adic integral lattice of G_{n}, which permutes the simple factors by Lemma 3.3. Also, the only admissible automorphisms of the 2-adic integral lattice of SO(2n + 1) are given by multiplication by scalars a ∈ (ℤ̂_{2})^{∗}, and so we can conclude that σ(R_{0}) = R_{0}. The same argument as for odd p now shows that a_{θ} ∈ ℤ̂_{2} for all θ; and so condition (i) also holds in this case.

Finally, as noted above, the elements of R_{s}∩R_{⟨p⟩} form a ℚ̂_{p}-basis for L(Λ_{p̂}_{⟨p⟩}). Hence by (a),

p ∤ det(ω | L(G_{⟨p⟩} ∩ T_{∞})) = ± ∏_{θ∈R_{s}∩R_{⟨p⟩}} a_{θ};  (2)

and so p∤a_{θ} for θ∈R_{⟨p⟩} = W · (R_{s}∩R_{⟨p⟩}). And this proves condition (ii).

Step 3a Assume that G = S×H, where S is a torus and H is a semisimple Lie group with trivial center. We show here that for such G, conditions (i) and (ii) suffice to imply that ω·W ∈ Im(Θ).

Write G = S×H_{1} × … ×H_{m}, where the H_{m} are simple. Let Λ = Λ_{0} × Λ_{1} × ⋯ × Λ_{m} and W = W_{1} × … ×W_{m} be the corresponding decompositions of Λ and W. By Lemma 3.3, there is some τ ∈ Σ_{m} such that ω(Λ_{i}_{p̂}) = (Λ_{τi})_{p̂} for all 1≤i≤m (and ω(Λ_{0}) = Λ_{0}); and H_{i} and H_{τi} have the same Dynkin diagram (up to arrow reversal) for each i. Also, since ω(R_{⟨p⟩}) ⊆ (ℤ̂_{p})^{∗}·R_{⟨p⟩} (by condition (ii)), the arrow on a double connector can be reversed only if p ≠ 2, and the arrow on a triple connector can be reversed only if p ≠ 3.

Recall that Z(H_{i}) = 1 for all i ≥ 1. Thus, for each i, either H_{i} and H_{τi} are isomorphic, or one of them is isomorphic to SO(2n + 1) and the other to P Sp(n) for some n ≥ 3. And by the remark on reversing arrows in the Dynkin diagram, this last case can occur only if p ≠ 2. By a result of Friedlander [Fr], BSO(2n+1)_{p̂}≃BSp(n)_{p̂} for any n and any odd p. So we can compose ω by Θ_{h}(Bα) for some appropriate α∈ Aut(G), to arrange that ω sends each simple factor to itself.

We can now write ω = ∏_{i=0}^{m} ω_{i}, where ω_{i} : Λ_{i}_{p̂} ↣ Λ_{i}_{p̂} for each i; and where ϕ_{0} ∈ Θ([BS_{p̂}, BS_{p̂}]_{ℚ}) by Theorem 1.1. We will be done upon showing that ω_{i} ∈ Θ([BH_{i}_{p̂}, BH_{i}_{p̂}]_{ℚ}) for each i. In particular, we can simplify the notation, and assume that G = H_{i} is simple.

We have seen that ω^{∗} permutes the roots and simple roots of G up to scalar multiple, and hence induces an automorphism of the Dynkin diagram of G; possibly reversing arrows. The only simple groups whose Dynkin diagrams have arrow reversing automorphisms are B_{2} (= SO(5)≅P Sp(2)), G_{2}, and F_{4}. Also, as noted above, such arrow reversing can occur only if p ≠ 2 and G≅B_{2} or F_{4}; or if p ≠ 3 and H_{i}≅G_{2}. In all of these cases, self maps BG_{p̂} ⟶ BG_{p̂} have been constructed by Friedlander (in [Fr] again), to realize the arrow reversing automorphisms. So if necessary we can compose with one of these maps, to arrange that ω acts on the Dynkin diagram preserving arrows.

Since Z(G) = 1, any arrow preserving automorphism of the Dynkin diagram can be realized by some automorphism α∈ Aut(G) (cf. [Bb2, p.42, Corollaire]). So upon replacing ω by L(α|T) ∘ ω for some α, we are reduced to the case where ω^{∗} acts on the Dynkin diagrams via the identity, and sends each root to some scalar multiple of itself. In particular, since the Weyl group W is generated by reflections in the kernels of the roots (3.2(i)), ω∈ End(Λ) is W -equivariant; and is multiplication by some k ∈ ℤ̂_{p} since L(T_{∞}) = ℚ̂_{p} ⊗ Λ is irreducible as a W -representation (cf. [Bb1, p.82]). Also, by (ii), k ∈ (ℤ̂_{p})^{∗} if p||W |.

For such k, unstable Adams operations ψ^{k} : BG_{p̂} ⟶ BG_{p̂}, have been constructed by Sullivan [Su] (when G = SU(n)) or Wilkerson [Wi] (in general). And since the restriction of ψ^{k} to BT_{ p̂} is induced by the k-th power map on T, we see that Θ(ψ^{k}) = ω.

Step 3b Now let G be arbitrary, and fix some admissible map ω∈ End(Λ_{p̂}) such that ω^{∗}(R) ⊆ (ℤ̂_{p}·R) and ω^{∗}(R_{⟨p⟩}) ⊆ ((ℤ̂_{p})^{∗}·R_{⟨p⟩}). We will show that ω extends to a ℚ-equivalence BG_{p̂} ⟶ BG_{p̂}.

Let n be the exponent of the center of the semisimple part of G, and let π = {z∈Z(G) : z^{n} = 1\}. Set \overline{G} = G/π: a quotient group which satisfies the condition in Step 2. Let \overline{Λ} ⊇ Λ be the integral lattice in \overline{G}; then

\overline{Λ} = \{x ∈ ℚ ⊗ Λ : R(x) ⊆ ℤ, nx ∈ Λ\}.

by Proposition 3.2(iv). For any x ∈ \overline{Λ}_{p̂}, n · ωx = ω(nx) ∈ Λ_{p̂} and

R(ωx) = (ω^{∗}R)(x) ⊆ (ℤ̂_{p} · R)(x) ⊆ ℤ̂_{p}

(using (i)). So ω(\overline{Λ}_{p̂}) ⊆ \overline{Λ}_{p̂}; and ω extends to a map \overline{f} : B\overline{G}_{p̂} ⟶ B\overline{G}_{p̂} by Step 2.

Identify \overline{Λ}_{p̂}/Λ_{p̂} ≅ π_{p} (the Sylow p-subgroup of π), and let ω′∈ Aut(π_{p}) be the map induced by ω. The composite

BG_{p̂} ⟶ B\overline{G}_{p̂} ⟶^{\overline{f}} B\overline{G}_{p̂} ⟶ K(π_{p}, 2)

is nullhomotopic, since

H^{2}(BG; π_{p})≅ Hom(π_{1}(G), π_{p})⊆ Hom(π_{1}(T), π_{p})  (Prop. 3.2(iii))
≅ Hom(Λ, π_{p}) ≅ Hom(Λ_{p̂}, π_{p}).

So \overline{f} pulls back along the fibration BG_{p̂} ⟶ B\overline{G}_{p̂} ⟶ K(π_{p}, 2) to a map f : BG_{p̂} ⟶ BG_{p̂}; and f extends the original admissible map Bϕ. □

An inspection of the proof of Theorem 3.4 shows, at least when G is semisimple with trivial center, that [BG_{p̂}, BG_{p̂}]_{ℚ} is generated by products of unstable Adams operations on the separate simple factors of G, by automorphisms of G, and by the “exceptional isogenies” of Friedlander. This is the generalization to connected groups of the theorem of Hubbuck [Hu], which says that for simple G, [BG,BG] is generated by automorphisms and unstable Adams operations.

The following description of the self homotopy equivalences of BG_{p̂} is now easy.

Corollary 3.5. If p is odd, then for any compact connected Lie group G, any admissible map ω∈ Aut(Λ_{p̂}) extends to a homotopy equivalence f : BG_{p̂} ⟶ BG_{p̂}. In other words,

Θ_{h} : [BG_{p̂}, BG_{p̂}]_{h} ⟶ N_{Aut(T_{∞})}(W)/W ≅ N_{Aut(Λ_{p̂})}(W)/W

is an isomorphism of groups in this case. If p = 2, then Θ_{h} is onto if and only if G contains no direct factor of the form Sp(n)×SO(2n+1) (for some n ≥ 1). And if G does contain such a factor, then Im(Θ_{h}) is the subgroup of all elements which send factors SO(2n + 1) to factors of the same type.

Proof. Recall that [BG_{p̂}, BG_{p̂}]_{h} = Θ^{−1} (N_{Aut(T_{∞})}(W)/W ) (Proposition 1.4). Using this, Theorem 3.4 implies that Θ_{h} is onto (and hence an isomorphism) if p is odd, or if p = 2 and G has no direct factor SO(2n + 1).

If p = 2 and G does contain a factor SO(2n + 1), then it can only be sent to another direct factor which is either isomorphic to SO(2n + 1), or which has the same integral lattice (2-adically) and root system \overline{R} (restricted to this summand). And a check of the root systems shows that the only other possibility is for it to be sent to a direct factor Sp(n). Thus, if Θ_{h} is not onto, then G must contain a direct factor SO(2n + 1)×Sp(n); and Im(Θ_{h}) is the group of all admissible maps which send factors SO(2n+1) to factors of the same type.

Finally, Example 3.1 shows that Θ_{h} is never onto when G has a direct factor SO(2n+ 1)×Sp(n). □

Using Sullivan’s arithmetic pullback square for completions and localizations of simply connected spaces, these results can now be converted to results about global self maps of BG.

Theorem 3.6. There is a monomorphism

Θ : [BG, BG]_{ℚ} ↣ AdmEpi(T, T) ≅ [N_{Aut(ℚ⊗Λ)}(W) ∩ End(Λ)]/W

such that for any ℚ-equivalence f : BG ⟶ BG, Θ(f) = ϕW for some ϕ : T ↠T with f|BT ≃Bϕ.

For each prime p, let G_{⟨p⟩}◁G be the product of all simple summands whose Weyl group has order a multiple of p, and set Λ_{⟨p⟩} = Λ∩L(G_{⟨p⟩}∩T). For each n ≥ 1, let H_{n}◁G be the product of all normal subgroups isomorphic to SO(2n + 1). Then, for any ϕ∈ AdmEpi(T, T), ϕW ∈ Im(Θ) if and only if

(a) Ker(ϕ) ∩ G_{⟨p⟩} = 1 for all p||W |, and

(b) ϕ(G_{n} ∩ T) = G_{n} ∩ T for all n ≥ 1.

Proof. For any f : BG ⟶ BG, f|BT ≃ Bϕ for some ϕ : T ⟶ T by Notbohm’s theorem [No1]; and ω = L(ϕ)|Λ satisfies conditions (a,b) since it satisfies them after p-completion for each p (Theorem 3.4). Thus, there is a well defined homomorphism Θ as above, and ϕW ∈ Im(Θ) only if ϕ satisfies the three given conditions.

If Θ(f) = Θ(f′), then Θ(f) = Θ(f′) for each p, and so f_{ p̂}≃f′_{p̂} for each p by the injectivity of Θ (Theorem 2.5). And by [JMO, Theorem 3.1], this implies that f≃f′.

Now fix some admissible map ϕ : T ↠T which satisfies conditions (a) and (b). For each prime p, Theorem 3.4 applies to show that Bϕ extends to a ℚ-equivalence f_{p} : BG_{p̂} ⟶ BG_{p̂}. And then by [JMO, Theorem 3.1] (applied with f_{T} = B(incl ∘ϕ)), there exists f : BG ⟶ BG such that f_{ p̂}≃f_{p} for each p, and hence such that f|BT ≃Bϕ.

Theorem 3.1 in [JMO] was used here to show both uniqueness and existence of maps f : BG ⟶ BG. Its proof is based on the homotopy pullback square of mapping spaces

map(BG, BG) ⟶ ∏_{p} map(BG, BG_{p̂})
↓  ↓
map(BG, BG_{ℚ}) ⟶ map(BG, (∏_{p} BG_{p̂})_{ℚ}),

which is induced by Sullivan’s arithmetic pullback square for BG [BK, VI.8.1]. It also uses the fact that BG_{ℚ} and (∏_{p} BG_{p̂})_{ℚ} are both products of Eilenberg-Maclane spaces. □

As a final application of these results, we get the following (disappointing) result about the global self homotopy equivalences of BG.

Corollary 3.7. For any compact connected Lie group G, any homotopy equivalence f : BG ⟶ BG is homotopic to Bα for some α∈ Aut(G).

Proof. Assume f|BT ≃Bϕ, where ϕ∈ AdmEpi(T, T). Then ϕ∈ Aut(T), since f is a homotopy equivalence. Set ω = L(ϕ)|Λ∈ Aut(Λ). By Theorem 3.4, ω^{∗} sends each root of G to an integral multiple of some other root; and since the simple roots are linearly independent those integers must be ±1. In other words, ω^{∗} permutes the roots; and so ω is an automorphism of the root system with integral lattice. It follows that ϕ = α|T for some α∈ Aut(G) (cf. [Bb2, p.41, Prop. 17]); and f≃Bα since Θ is injective (Theorem 3.6). □

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