KERNELS OF MAPS BETWEEN CLASSIFYING SPACES

by Dietrich Notbohm

Abstract. For homomorphisms between groups, one can divide out the kernel to get an injection. Here, we develop a notion of kernels for maps between classifying spaces of compact Lie groups. We show that the kernel is a normal subgroup in a modified sense and prove a generalization of a theorem of Quillen, namely, a map f: BG⟶BH_{p}^{∧} is injective, iff the induced map in mod-p cohomology is finite. Moreover, for compact connected Lie groups, every map f: BG ⟶ BH_{p}^{∧} factors over a quotient of G in a modified sense and this factorisation is an injection.

1. Introduction.

For a homomorphism ρ: G ⟶ H between groups, we know that the kernel ker(ρ) of ρ is a normal subgroup of G, which gives rise to an exact sequence

ker(ρ) ⟶ G ⟶ G/ker(ρ) .

The induced homomorphism

\overline{ρ}: G/ker(ρ) ⟶ H

is an injection. In this paper we will develop a analoguos concept for maps between classifying spaces.

To investigate the topological situation we pass to the p–adic completion. We also allow a more general target. Let G be a compact connected Lie group. A space X is called BG–local if the evaluation induces an equivalence map(BG, X) ≃ X between the mapping space and X. The space X is called almost BG–local if the evaluation induces an equivalence map(BG, X)_{const} ≃ X between the component of the constant map const: BG⟶X and X. This is equivalent to the condition that the loop space ΩX is BG–local.

The situation, we are interested in, is the following

(S) f: BG ⟶ X_{p}^{∧} is a map, where G is a compact Lie group, where

X_{p}^{∧} is p–complete and almost Bℤ/p-local and where

H^{∗}(X; 𝔽_{p}) is of finite type.

When talking about the kernel of a map f: BG⟶X_{p}^{∧} as in (S), one has to look for elements g ∈ G of order a power of p, such that f|_{B⟨g⟩} is homotopic to the constant map. Here, ⟨g⟩ denotes the subgroup generated by g. This leads to

Typeset by AMS-TEX

a definition of the prekernel, due to Ishiguro [7, 8], and of the kernel, which we explain now.

For every compact Lie group G there exists a maximal p–toral subgroup S_{p}G, unique up to conjugation, and every p–toral subgroup is subconjugated to S_{p}G [9]. The group S_{p}G is called the p–toral Sylow group of G. If G is finite, S_{p}G is the usual p–Sylow group.

For a compact Lie group G, we denote by T_{G} the maximal torus, by NT_{G} the normalizer of T_{G}, and by W_{G} the Weyl group. Then, S_{p}G is the counterimage of S_{p}W_{G} in NT_{G}. We also denote S_{p}G by N_{p}T_{G} to indicate that S_{p}G is the p–toral Sylow subgroup of NT_{G}, too. T_{G} is the component of the unit of N_{p}T_{G} = S_{p}G.

We define a subgroup S_{p^{∞}}G ⊂ S_{p}G by the commutative diagram

T_{p^{∞}} ⟶ S_{p^{∞}}G ⟶ π_{0}(S_{p}G)
↓  ↓  ↓
T_{G} ⟶ S_{p}G ⟶ π_{0}(S_{p}G) ,

where T_{p^{∞}} ⊂ T_{G} denotes the subgroup generated by all elements of order a power of p. For a map f: BG⟶X_{p}^{∧}, we define the prekernel of f as

preker(f) := {g ∈ S_{p^{∞}}G: f|_{B⟨g⟩} ≃ const} ,

and the kernel of f as

ker(f) := cl(preker(f)) ,

where cl( ) denotes the closure in S_{p}G or G.

1.1 Theorem. Let f: BG ⟶ X_{p}^{∧} be a map as in (S).

(1) preker(f) is a subgroup of S_{p^{∞}}G.

(2) ker(f) is a p–toral subgroup of S_{p}G.

(3) f|_{Bker(f)} is nullhomotopic.

As the proof in the next section shows, the theorem is true without the aasuption that H^{∗}(X_{p}^{∧}; 𝔽_{p}) is of finite type.

These results lead to the following definitions: A map f: BG ⟶ X_{p}^{∧} as in (S) is called injective if ker(f) is the trivial group, and, following [4] or [13], monic or finite if H^{∗}(BG; 𝔽_{p}) is finitely generated over H^{∗}(X_{p}^{∧}; 𝔽_{p}). In [13] is proved that the kernel of a homomorphism ρ: G⟶H between two compact Lie groups is finite and of order coprime to p if and only if the induced map Bρ: BG⟶BH_{p}^{∧} is monic. The following statement also generalizes a result of Dwyer and Wilkerson [4, Proposition 4.4]

1.2 Theorem. Let f: BG⟶X_{p}^{∧} be a map as in (S). Then, f is injective, if and only if f is monic.

Let O(G) be the orbit category of G, and let O_{p}(G) be the full subcategory, whose objects are homogenous spaces G/P, where P is a p–toral subgroup of S_{p}G. Usually, O_{p}(G) is defined to be the full subcategory of the spaces G/P, where P is any p–toral subgroups of G. Our definition is more convenient for our purpose and gives a homotopy equivalent category. This follows because, up to conjugation, every p–toral subgroup is contained in S_{p}G [9]. For a subgroup K ⊂ S_{p}G and a p–toral subgroup P ⊂ S_{p}G, we define K_{P} := K ∩ P.

1.3 Lemma. For a p–toral subgroup Γ ⊂ S_{p}G the following conditions are equivalent:

(1) For every x ∈ Γ of order a power of p and for every g ∈ G, we have gxg^{−1} ∈ S_{p}G if and only if gxg^{−1} ∈ Γ.

(2) For every pair P, P′ ⊂ S_{p}G of p–toral subgroups and for every g ∈ G, such that gP g^{−1} ⊂ P′, we have gΓ_{P} g^{−1} ⊂ Γ_{P′}.

Proof. S_{p^{∞}}Γ ⊂ Γ is a dense subset and contains only elements of order a power of p. If Γ satisfies the first condition and if gP g^{−1} ⊂ P′, then g(S_{p^{∞}}Γ ∩ P)g^{−1} ⊂ S_{p^{∞}}Γ ∩ P′. Because P′ is closed, this is also true for Γ, which is condition (2). Every element of order a power of p generates a finite p–group. Hence, (2) implies (1) obviously. □

A p–toral subgroup Γ ⊂ S_{p}G is called O_{p}(G)–normal, if Γ satisfies one of the conditions of Lemma 1.3.

1.4 Proposition. Let f: BG⟶X_{p}^{∧} be a map as in (S). Then, ker(f) ⊂ S_{p}G is O_{p}(G)–normal.

Proof. In [7] is shown that for two subgroups Γ, Γ′ ⊂ S_{p}G, which are conjugated in G, the restriction f|_{BΓ} is nullhomotopic if and only if f|_{BΓ′} is nullhomotopic. Hence, for every x ∈ ker(f), conjugated elements are also contained in ker(f).

For a map f: BG ⟶ X_{p}^{∧} as in (S), the kernel ker(f) ⊂ S_{p}G is not a normal subgroup of G in general. But, for G connected, ker(f) is the right invariant, which tells us, on which part the map f: BG⟶X_{p}^{∧} is trivial. To avoid discussions and arguments about homotopy colimits, we study O_{p}(G)–normal subgroups of N_{p}T_{G}. This investigation allows to prove the following theorem:

1.5 Theorem. Let G be a compact connected Lie group and let f: BG⟶X_{p}^{∧} be a map as in (S). Then, there exists a compact Lie group H and a commutative diagram

BG ⟶^{f} X_{p}^{∧}
q↓  ∥
BH_{p}^{∧} ⟶^{\overline{f}} X_{p}^{∧}

such that \overline{f}: BH_{p}^{∧}⟶X_{p}^{∧} is injective.

Using a weak form of Theorem 1.1 (1), Ishiguro proved a similiar result for simple Lie groups. [7, 8]. We remark that we have to take the completion of BH. Moreover, as the proof shows, the group H in the theorem is not connected in general. This statement says that, at least, we can divide out by the kernel to make the map injective. Moreover, the homotopy fiber of BG_{p}^{∧} ⟶^{q} BH_{p}^{∧} is closely related to ker(f). One might think of this homotopy fiber as being the kernel of f and of q as a surjection. In general, the homotopy fiber of q might not be the completion of the classifying space of a compact Lie group.

The paper is organized as follows: In the next section we proof Theorem 1.1, the third section contains a proof of Theorem 1.2, and in the last section O_{p}(G)–normal subgroups are discussed to prove Theorem 1.5.

Completion are always meant in the sense of [2].

It is pleasure to thank K. Ishiguro for several discussions on this subject.

2. Prekernels and kernels.

We start with the following observation:

2.1 Lemma. Let X_{p}^{∧} be a p–complete almost Bℤ/p–local space. Then, X_{p}^{∧} is almost BG–local for every compact Lie group G.

Proof. A p–complete space X_{p}^{∧} is always Bℤ/p′–local for any prime p′ ≠ p. If X_{p}^{∧} is also Bℤ/p–local, then, by [11, §9], follows that X_{p}^{∧} is Bπ–local for any locally finite group π. For every compact Lie group G, there exists a mod–p equivalence Bπ⟶BG, where π is locally finite [6]. So, X_{p}^{∧} is BG–local. Because for a pcomplete space X_{p}^{∧}, the loop space Ω(X_{p}^{∧}) is also p–complete, the same arguments apply to show that every p–complete almost Bℤ/p–local space is almost BG–local for every compact Lie group G. □

Let π be a finite group. For x ∈ π, we define ν(x) to be the smallest subgroup of π, which is normal in π and contains x. This is welldefined because the intersection of two normal subgroups is also normal.

2.2 Lemma. If π is a finite p–group and noncyclic, then, for every x ∈ π, ν(x) ⊂ π is a proper subgroup and ν(x) = ⟨yxy^{−1}: y ∈ π⟩.

For a set S of elements of π, we denote by ⟨S⟩ the subgroup generated by the elements of S.

Proof. The center of π is nontrivial and contains ℤ/p as subgroup. Hence, there

exists a central extension ℤ/p ⟶ π ⟶^{q} \overline{π} := π/ℤ/p. If \overline{π} ≅ ℤ/p^{k} is cyclic, all central group extensions are given by abelian groups. Thus , as a noncyclic group, π ≅ ℤ/p ⊕ ℤ/p^{k} and obviously satisfies the statement.

If \overline{π} is noncyclic, we can use an induction over the order of π. Let \overline{x} := q(x). Then, q(ν(x)) ⊂ ν(\overline{x}) ≠ \overline{π} by induction hypothesis. This shows that ν(x) ≠ π.

To prove the second part of the statement, we observe that the group ν′(x) := ⟨yxy^{−1}: y ∈ π⟩, generated by all the conjugates of x, is normal in π. Hence, ν(x) ⊂ ν′(x). On the other hand, yxy^{−1} ∈ ν(x) for all y ∈ π, which shows that ν′(x) ⊂ ν(x). □

To prove Theorem 1.1 we need the following result, which may be found in [12], or [7].

2.3 Lemma. Let K ⟶ G ⟶^{q} H be an exact sequence of groups, and let X be an almost BK–local space. Then,

q^{∗}: map(BH, X) ⟶ ⋃_{f|_{BK}≃const} map(BG, X)_{f}

is an equivalence.

2.4 Proposition. Let f: Bπ ⟶ X_{p}^{∧} be a map, where π is a finite p–group and X_{p}^{∧} a p–complete and almost Bℤ/p–local space. Let {x_{1}, ⋯, x_{r}} be a set of generators. If f|_{B⟨x_{i}⟩} ≃ const for all i, then f is homotopically trivial.

Proof. We pove the statement by an induction over the order of π. If π ≅ ℤ/p^{k} is cyclic, there is nothing to show because one of the elements must generate π. If π is noncyclic, by Lemma 2.1, there exists an exact sequence

ν(x_{1}) ⟶ π ⟶ \overline{π} := π/ν(x_{1}) .

ν(x_{1}) is generated by elements of the form yx_{1}y^{−1}, and f|_{B⟨yx_{1}y^{−1}⟩} ≃ const. The order of ν(x_{1}) is smaller than the order of π. By induction hypothesis, f|_{Bν(x_{1})} ≃ const. By Lemma 2.1, X_{p}^{∧} is Bν(x_{1})–local, and Lemma 2.2 establishes an equivalence map(B\overline{π}, X_{p}^{∧}) ≃ ⋃_{g|_{Bν(x_{1})}≃const} map(Bπ, X_{p}^{∧})_{g}. In particuliar, f

factors over a map \overline{f}: B\overline{π}⟶X_{p}^{∧}.

The quotient \overline{π} is generated by the elements \overline{x}_{i} := q(x_{i}). The exact sequence

ν(x_{1}) ∩ ⟨x_{i}⟩ ⟶ ⟨x_{i}⟩ ⟶ ⟨\overline{x}_{i}⟩

and another application of Lemma 2.2 show that \overline{f}|_{B⟨\overline{x}_{i}⟩} ≃ const. Thus we can again apply the induction hypothesis, which shows that \overline{f} ≃ const. This finishes the proof. □

Now, we are prepared to prove Theorem 1.1.

Proof of Theorem 1.1. Let x, y ∈ preker(f). We want to show that xy ∈ preker(f) or, more generally, that f|_{B⟨x,y⟩} ≃ const.

S_{p∞}G is a locally finite p group. In particuliar, there exists a sequence

Γ_{1} ⊂ Γ_{2} ⊂ ⋯ ⊂ Γ_{r} ⊂ ⋯ S_{p^{∞}}G

of finite p groups such that S_{p^{∞}}G = ⋃_{r} Γ_{r} [5]. Therefore, ⟨x, y⟩ is a finite p–group, and, by the last proposition, f|_{B⟨x,y⟩} ≃ const. This proves (1).

Let Γ′_{r} := Γ_{r} ∩ preker(f). Then, preker(f) = ⋃_{r} Γ′_{r}. By Proposition 2.3, f|_{BΓ′_{r}} is nullhomotopic. Because X_{p}^{∧} is almost Bℤ/p–local and hence almost BΓ′_{r}–local (Lemma 2.1), lim_{⟵}^{1}π_{1}(map(BΓ′_{r}, X_{p}^{∧})_{const} ≅ lim_{⟵}^{1}π_{1}(X_{p}^{∧}) vanishes. The Milnor sequence for calculating the homotopy groups of inverse homotopy limits proves that f|_{Bpreker(f)} is nullhomotopic.

Lemma 2.5 below shows that Bpreker(f)⟶Bker(f) is a mod–p equivalence. For the p–complete space X_{p}^{∧}, the map [B(ker(f), X_{p}^{∧}] ⟶ [B(preker(f), X_{p}^{∧}] between homotopy classes of maps is a bijection [2]. This implies that f|_{Bker(f)} is nullhomotopic and proves part (3).

ker(f) is the closure of preker(f) in S_{p}G. Thus, the group of the components of ker(f) is a finite p–group, and ker(f) is a p–toral group which is part (2). This finishes the proof. □

2.5 Lemma. Let f: BG⟶X_{p}^{∧} be a map as in (S). Then,the map

B(preker(f)) ⟶ B(ker(f))

is a mod–p equivalence.

Proof. Let T (f) denote the component of the unit of ker(f), T_{∞}(f) the intersection of T (f) and S_{p^{∞}}G, and let π := π_{0}(ker(f). These groups fit into the commutative diagram

T_{∞}(f) ⟶ preker(f) ⟶ π
↓  ↓  ∥
T (f) ⟶ ker(f) ⟶ π .

Both rows are exact.

As a locally finite abelian p–group, T_{∞}(f) ≅ (ℤ/p^{∞})^{r} × A, where A is a finite abelian p–group. Because the closure of T_{∞}(f) is T (f), A is trivial, and T (f) ≅ (S^{1})^{r}. So, BT_{∞}(f)⟶BT (f) is a mod–p equivalence. The Serre spectral sequence for mod–p cohomology for the fibrations in the diagram

BT_{∞}(f) ⟶ Bpreker(f) ⟶ Bπ
↓  ↓  ∥
BT (f) ⟶ Bker(f) ⟶ Bπ

proves the statement. □

3. Injective and monic maps.

In this section we proof Theorem 1.2. Let f: BG⟶X_{p}^{∧} be a map as in (S). Let A_{p}(G) denote the Quillen category. The objects are given by elementary abelian p– subgroups and the morphisms by conjugation in G [13]. To get a finite category, we take only one object for every isomorphism class of objects, i.e. for every conjugacy class of a group. The Quillen map

ϕ: H^{∗}(BG; 𝔽_{p}) ⟶ lim_{\overleftarrow{V ∈A_{p}(G)}} H^{∗}(BV; 𝔽_{p})

is H^{∗}(BG; 𝔽_{p})–linear and an F –isomorphism; i.e. kernel and cokernel are nilpotent [13, Theorem 7.2]. Let

B := im(ϕ ∘ f^{∗}) ⊂ lim_{\overleftarrow{V ∈A_{p}(G)}} H^{∗}(BV; 𝔽_{p}) ⊂ ∏_{V ∈A_{p}(G)} H^{∗}(BV; 𝔽_{p})

be the image of H^{∗}(X_{p}^{∧}; 𝔽_{p}).

Proof of Theorem 1.2. First, we assume that f is injective and show that f is monic. For any elementary abelian p–subgroup V ⊂ G, the restriction f|_{BV} is also injective. By [4, Proposition 4.4], which is the analogous statement of Theorem 1.2 for elementary abelian p–groups, this implies that H^{∗}(BV; 𝔽_{p}) is a finitely generated module over H^{∗}(X_{p}^{∧}; 𝔽_{p}) and over B. Because A_{p}(G) is a finite category, ∏_{V ∈A_{p}(G)} H^{∗}(BV; 𝔽_{p}) is a finitely generated module over B and a finitely generated algebra over 𝔽_{p}. Therefore, B is also a finitely generated algebra over 𝔽_{p} [1, Proposition 7.8] and hence noetherian. This implies that lim_{\overleftarrow{V ∈A_{p}(G)}} H^{∗}(BV; 𝔽_{p}), as

a submodule of ∏_{V ∈A_{p}(G)} H^{∗}(BV; 𝔽_{p}), is a finitely generated module over B and over H^{∗}(X_{p}^{∧}; 𝔽_{p}).

If f is not monic, i.e the finitely generated 𝔽_{p}–algebra H^{∗}(BG; 𝔽_{p}) is not a finitely generated module over H^{∗}(X_{p}^{∧}; 𝔽_{p}), there exists an element y ∈ H^{∗}(BG; 𝔽_{p}) such that {y^{i}: i ∈ ℕ} is a set of linearly independent elements over H^{∗}(X; 𝔽_{p}). By the above considerations, for r ∈ ℕ big enough, there exists a relation ϕ(y^{r}) = ∑_{i=0}^{r−1} x_{i}ϕ(y^{i}) with x_{i} ∈ H^{∗}(X_{p}^{∧}; 𝔽_{p}). That is that y^{r} − ∑_{i=0}^{r−1} x_{i}y^{i} is in the kernel of ϕ and hence nilpotent. Thus, for s ∈ ℕ big enough,

0 = (y^{r} − ∑_{i=0}^{r−1} x_{i}y^{i})^{s} = y^{rs} − ∑_{j=0}^{r−1} x′_{j}y^{j}

for suitable x′_{i} ∈ H^{∗}(X_{p}^{∧}; 𝔽_{p}). This is a contradiction and proves that f is monic.

Now, we assume that f is monic. Let ℤ/p ⊂ G be a subgroup of G. Up to conjugation, ℤ/p is contained in S_{p}G. By [13], H^{∗}(Bℤ/p; 𝔽_{p}) is finitley generated over H^{∗}(BG; 𝔽_{p}), and therefore, also over H^{∗}(X_{p}^{∧}; 𝔽_{p}). That is that the map

Bℤ/p ⟶ BG ⟶^{f} X_{p}^{∧} is homotopically nontrivial. This implies that ker(f) = {1} and that f is injective. □

4. O_{p}(G)–normal subgroups.

For every compact connected Lie group G, there exists a finite covering

K ⟶ G̃ ⟶^{α} G ,

where G̃ ≅ G_{s} × T is a product of a simply connected Lie group G_{s} and a torus T. G_{s} ≅ ∏ G_{i} is a product of simply connected simple Lie groups. K is a finite central subgroup of G̃. The group G̃ is unique up to isomorphisms. Such coverings we call universal finite.

4.1 Lemma. Let K ⟶ G̃ ⟶^{α} G be an exact sequence of compact Lie groups, K finite and G̃ and G connected. Let f: BG ⟶ X_{p}^{∧} be a map as in (S).

(1) The sequence

S_{p}K ⟶ preker(f ∘ Bα) ⟶ preker(f)

is exact.

(2) ker(f ∘Bα) ⟶ ker(f) is an epimorphism, and ker(f ∘Bα) = S_{p}α^{−1}ker(f).

Proof. Obviously, α^{−1}(preker(f))∩S_{p^{∞}} G̃ is contained in preker(f ∘Bα). In particuliar, S_{p}K ⊂ preker(f ∘ Bα). Let Γ := preker(f ∘ Bα)/S_{p}K ⊃ preker(f). Then, by Lemma 2.3, map(BΓ, X_{p}^{∧}) ≃ ⋃_{g|_{BSpK}≃const} map(Bpreker(f ∘ Bα), X_{p}^{∧})_{g}. This implies that f|_{BΓ} is homotopically trivial, and hence, that Γ ⊂ preker(f), which establishes the desired sequence of (1).

To prove the second statement we first observe that in this case epimorphism are maintained under taking closures. The second part in (2) follows from the facts that, as a p–toral group, ker(f ∘ Bα) ⊂ S_{p}α^{−1}ker(f), and that f|_{BS_{p}α^{−1}ker(f)} is homotopically trivial. □

4.2 Lemma. Let K ⟶ G̃ ⟶^{α} G be an exact sequence of compact Lie groups, K finite and G̃ and G connected. Let P ⊂ S_{p}G be a p–toral subgroup. Then, P is O_{p}(G)–normal if and only if S_{p}α^{−1}(P) ⊂ S_{p}G̃ is O_{p}(G̃)–normal.

Proof. Let Q̃ := α^{−1}(P) and P̃ := S_{p}Q̃. The composition P̃ ⟶ Q̃ ⟶ P is an epimorphism. This follows, because P̃ and Q̃ have identical components of the unit, and because, passing to the components, the composition π_{0}(P̃) = S_{p}π_{0}(Q̃) ⟶ π_{0}(P) is an epimorphism of finite groups.

K ⊂ G̃ is a central subgroup. The multiplication μ: (K ∩ Q̃) × P̃ ⟶ Q̃ fits into the pull back diagram

(K ∩ Q̃) × P̃ ⟶^{μ} Q̃
↓  ↓^{α}
P̃  ⟶^{α} P .

Thus, μ is an epimorphism, and P̃ ⊂ Q̃ is a normal subgroup. That is that P̃ is the only p–toral Sylow subgroup of Q̃.

Every element x ∈ P of order a power of p has a lift x̃ ∈ P̃, also of order a power of p. Let g̃ ∈ G̃ and g := α(g̃). Then, g̃x̃g̃^{−1} ∈ S_{p}G̃ if and only if gxg^{−1} ∈ S_{p}G, and, because P̃ ⊂ Q̃ is the only p–toral Sylow subgroup, g̃x̃g̃^{−1} ∈ P̃ if and only if gxg^{−1} ∈ P. This proves the statement. □

Lemma 4.1 and Lemma 4.2 reduce the calculation of kernels and O_{p}(G)–normal subgroups, G connected, to the case of products of simply connected Lie groups and tori. Let G̃ ≅ G_{s} × T be such a product. In order to describe O_{p}(G̃)–normal subgroups, we associate for every prime p a subgroup H(G, p) to each simply connected simple Lie group G . We define

H(G, p) := \{ \begin{matrix}NT_{G} & if (p, |W_{G}|) = 1 \\ SU(2) ⋊ ℤ/2 & if G = G_{2} and p = 3 \\ G & else . \end{matrix}

We define H(G_{s}, p) := ∏ H(G_{i}, p) for a product G_{s} = ∏ G_{i} of simply connected simple Lie groups. Then, BH(G_{s}, p)↪BG is a mod–p equivalence. If p = 3 and G = G_{2}, this follows from the isomorphism H^{∗}(BG_{2}; ℤ/p) ≅ H^{∗}(BSU(2); ℤ/p)^{ℤ/2}, and if (p, |W_{G_{s}}|) = 1 from the isomorphism H^{∗}(BG_{s}; ℤ/p) ≅ H^{∗}(BT_{G_{s}}; ℤ/p)^{W_{Gs}}.

4.3 Proposition. Let G̃ = ∏ G_{i} × T be a product of simply connected simple Lie groups G_{i} and a torus T. Let Γ ⊂ N_{p}TG̃ be a O_{p}(G̃)–normal subgroup. Then, we can split G_{s} = G′ × G′′ such that Γ ≅ N_{p}T_{G′} × Γ̂ and Γ̂ ⊂ T_{G′′} × T. Moreover, Γ̂ is normal in H(G′′, p) × T and the image of Γ̂ in G′′ is finite.

We postpone the proof. This result enables us to prove Theorem 1.5.

Proof of Theorem 1.5. Let f: BG ⟶ X_{p}^{∧} be a map as in (S), and let

K ⟶ G̃ ⟶ G

be a universal finite covering, where G̃ ≅ G_{s} × T. By the last proposition and Proposition 1.3, G_{s} ≅ G′ × G′′ and ker(f ∘ Bα) ≅ N_{p}T_{G′} × Γ̂. Now, we define H = (H(G′′, p)×T)/Γ̂. The classical kernel of the projection G′ ×H(G′′, p)×T ⟶ H is given by G′ ×Γ̂, which contains K by construction. We get a commutative diagram

B(G′ × H(G′′, p) × T) ⟶^{Bi} BG̃ ⟶^{f∘Bα} X_{p}^{∧}
Bα̂↓  ^{Bα}↓  ∥
B(G′ × H(G′′, p) × T)/K ⟶^{B\overline{i}} BG ⟶^{f} X_{p}^{∧}
q̂↓  ^{q}↓  ∥
BH_{p}^{∧}  BH_{p}^{∧} ⟶^{\overline{f}} X_{p}^{∧} .

Bi and B\overline{i} are mod–p equivalences. BH_{p}^{∧} is p–complete, because π_{1}(BH) is a finite group [2]. This establishes the map q: BG⟶BH_{p}^{∧}. The quotient (G′ × Γ̂)/K is a normal subgroup of (G′ × H(G′′, p) × T)/K, and X_{p}^{∧} is almost B(G′ × Γ̂)/K–local (Lemma 2.1). Therefore, the map f ∘ B\overline{i} factors over \overline{f}: BH_{p}^{∧}⟶X_{p}^{∧} (Lemma 2.3). Moreover, \overline{f} ∘ q ≃ f, because \overline{f} ∘ q̂ ≃ f ∘ B\overline{i}. This proves the first half of Theorem 1.5.

By Lemma 4.1 , ker(f ∘ Bα ∘ Bi) ⟶^{β∘α̂} ker(\overline{f}) is an epimorphism. This shows that \overline{f} is injective. □

In the rest of this section, we prove Proposition 4.3.

Proof of Proposition 4.3. The subgroup Γ_{T} = Γ ∩ TG̃ is invariant under the Weyl group action. In particuliar, WG̃ acts on the component Γ_{e} of the unit of Γ, and H^{2}(BΓ_{e}; ℚ) is a WG̃–submodule of

H^{2}(BTG̃; ℚ) ≅ ⨁ H^{2}(BG_{i}; ℚ) ⊕ H^{2}(BT; ℚ) .

The first summands are irreducible. Thus, Γ_{T} ∩ T_{G_{i}} = T_{G_{i}} or the intersection is trivial.

Let G′ be the product of all factors G_{i} with T_{G_{i}} ⊂ Γ, and G′′ the product of the other factors of G̃. Let x ∈ N_{p}T_{G′} but x ∉ T_{G′}. Then x is conjugated to an element in T_{G′} and therefore, x ∈ Γ. This implies that N_{p}T_{G′} ⊂ Γ. Moreover, N_{p}T_{G′} is a normal subgroup of Γ. In the commutative diagram

N_{p}T_{G′} ⟶^{i}  Γ  ⟶ Γ̂ := Γ/N_{p}T_{G′}
∥  ↓  ↓
N_{p}T_{G′} ⟶^{j} N_{p}T_{G′} × N_{p}T_{G′′} × T ⟶ N_{p}T_{G′′} × T

both rows are central extensions. The map j has a section, which establishes also a section for i. Therefore, the upper sequence is the trivial extension, Γ ≅ N_{p}T_{G′} ×Γ̂, and Γ̂ ⊂ N_{p}T_{G′′} × T. This is the first part of the statement.

Because Γ_{e} ⊂ N_{p}T_{G′} × T, the image of Γ̂ in G′′ is a finite group and O_{p}(G′′)– normal. We have to investigate finite O_{p}(G)–normal subgroups of simply connected simple Lie groups. The following proposition finishes the proof. □

4.4 Proposition. Let G be a simply connected simple Lie group, and Γ ⊂ N_{p}T_{G} a O_{p}(G)–normal finite p–subgroup.

(1) If p divides |W_{G}|, and if G ≠ G_{2} or p ≠ 3, then Γ is central in G.

(2) If G = G_{2} and p = 3, then Γ is central in SU(3) and hence normal in H(G_{2}, 3).

(3) If (p, |W_{G}|) = 1, then Γ is normal in H(G, p).

Proof. If x ∈ Γ then txt^{−1}x^{−1} ∈ Γ for all t ∈ T_{G}. Because Γ is finite, all the commutators are trivial. Thus, x centralizes T_{G}. Because C_{G}(T_{G}) = T_{G}, Γ = Γ_{T} ⊂ T_{G}.

If (p, |W_{G}|) = 1, Γ is normal in NT_{G} = H(G, p). If p divides |W_{G}|, we have to prove that Γ is central in G or, for G = G_{2} and p = 3, central in SU(3). Let x ∈ Γ \ Z(G) and x^{p^{k}} = 1. Using this element, we construct an element Γ \ T_{G}, which gives a contradiction. This is done by a case by case checking and very much in the flavour of the proof of [7, Theorem 2’].

Before we start, we make two observations. First, if (1) is true for two groups

G and H, then it is obviously true for the product G × H and, Second, if G ⟶^{q} \overline{G} is a covering of connected Lie groups, then (1) is true for G if and only if \overline{G} satisfies condition (1). To see this, we consider a finite O_{p}(G)–normal p–subgroup Γ ⊂ N_{p}T_{G}. Then, Γ′ := ⟨Γ, S_{p}Z(G)⟩ is also a finite O_{p}(G)–normal p–subgroup. Because Γ′ = S_{p}(q^{−1}(q(Γ′))), the group q(Γ′) is O_{p}(\overline{G})–normal if and only if Γ′ is O_{p}(G)–normal (Lemma 4.2), and q(γ′) ⊂ \overline{G} is central if and only if Γ′ ⊂ G is central.

Let G = SU(n), n ≥ p. Up to conjugation, x can be represented by a diagonal matrix D = D(a_{1}, ⋯, a_{n}), where a_{i} is a p^{k}–th root of unity, and a_{1} ≠ a_{2}. The element y = D(a_{2}, a_{1}, a_{3}, ⋯, a_{n}) is conjugate to x and thus, xy^{−1} = D(a_{1}a_{2}^{−1}, a_{1}^{−1}a_{2}, 1, ⋯, 1) ∈ Γ. Because a_{1}a_{2}^{−1} ≠ 1, conjugates of xy^{−1} generate a subgroup of T_{G}, which contains the maximal elementary abelian p–subgroup V_{G} of T_{G}. Every element of order p is conjugate to an element in V_{G}. This gives a contradiction. For G = U(n), n ≥ p, the same proof works.

Let G = Sp(n), n ≥ p. Then, U(n) and SU(2)^{n} are subgroups of maximal rank. Therefore, Γ ⊂ T_{Sp(n)} is central in U(n) and SU(2)^{n}. But Z(U(n)) ∩ Z(SU(2)^{n}) = ℤ/2 = Z(Sp(n)).

By the above observation, the case of G = Spin(n), n ≥ 2p and n ≥ 5, can be reduced to the case of SO(n). Let n = 2k or n = 2k + 1. Then, U(k) ⊂ SO(2k) ⊂ SO(2k + 1) is a subgroup of maximal rank, and Γ is central in U(k). For n ≥ 5, the only W_{SO(2k)} or W_{SO(2k+1)}–invariant subgroup of S^{1} = Z(U(k)) is ℤ/2. For SO(2k), ℤ/2 ⊂ U(k) ⊂ SO(2k) is central, and for SO(2k + 1), ℤ/2 ⊂ U(k) ⊂ SO(2k + 1) is conjugate to a subgroup in NT_{SO(2k+1)} \ T_{SO(2k+1)}, which proves the statement for SO(n).

Let G be an exceptional Lie group and p a divisor of |W_{G}|. In this case, we choose subgroups of maximal rank, given by the following list:

G_{2} :  p = 2, 3  H = SU(3)
F_{4} :  p = 2  H = SU(3) ×_{ℤ/3} SU(3)  p = 3  Spin(9)
E_{6} :  p = 2, 5  H = SU(3)^{3}/ℤ/3  p = 3  H = SU(2) ×_{ℤ/2} SU(6)
E_{7} :  p = 3, 5, 7  H = SU(8)/ℤ/2  p = 2  H = SU(3) ×_{ℤ/3} SU(6)
E_{8} :  p = 3, 5, 7  H = SU(2) ×_{ℤ/2} E_{7}  p = 2  H = SU(9)/ℤ/3 .

Beside the case G = G_{2} and p = 3, the inclusion H ⊂ G always induces an isomorphism S_{p}Z(H) ≅ S_{p}Z(G) between the p–Sylow subgroups of the centers. The data may be obtained from [9], where one can find a complete list of maximal subgroups of maximal rank of the exeptional Lie groups, and from [14].

Now, we can argue as follows: Let Γ ⊂ N_{p}T_{G} be an O_{p}(G)–normal subgroup. Then, by induction over the rank, by the above observations, and by the already calculated cases, Γ is central in H and hence, central in G.

For G = G_{2} and p = 3, the argument only shows that Γ is central in SU(3). That is Γ = ℤ/3 or Γ is the trivial group. In both cases, Γ is normal in SU(3) ⋊ ℤ/2 = H(G_{2}, 3). This finishes the last open case and the proof of the proposition. □

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