HOMOTOPY UNIQUENESS OF CLASSIFYING SPACES OF

COMPACT CONNECTED LIE GROUPS AT PRIMES

DIVIDING THE ORDER OF THE WEYL GROUP

by Dietrich Notbohm

Table of contents

1. Introduction 1 2. The program 6 3. Background 9 4. p–convenient compact connected Lie groups 14 5. Subgroups of maximal rank and the cohomology of Weyl groups 18 6. p–stubborn subgroups of pseudo simply connected Lie groups 21 7. Maximal tori and Weyl groups 28 8. p–adic Weyl group representations for simply connected Lie groups 30 9. p–adic Weyl group representations for compact connected Lie groups 35 10. Mapping spaces 42 11. The normalizer of the maximal torus I 48 12. The normalizer of the maximal torus II 54 13. The map BG⟶X I : G pseudo simply connected 59 14. The map BG⟶X II : The general case 62 15. The integral homotopy type of BG 65 References 67 List of notation 70

1. Introduction.

As a truism, Lie groups - in particular compact connected Lie groups - are very rigid objects. The perhaps best known instance of this rigidity was formulated in Hilbert’s fifth problem and proved by Gleason, Montgomery and Zippin in the early fifties: It requires only very weak assumptions on the topology of a topological group to get a Lie group(for a survey see [H]).

Trying to distinguish two compact connected Lie groups, another kind of rigidity

Typeset by AMS-TEX

occurs. Very often, the rich structure of a Lie group is totally described by little information. For example, simply connected compact Lie groups or compact connected Lie groups up to the local isomorphism type are classified by pure combinatorical data, namely the Dynkin diagram. Semi simple Lie groups are distinguished by their normalizer of the maximal torus [C–W–W].

Similar phenomena seem to occur, if one considers the classifying space BG of a compact connected Lie group G. Surprisingly, pure algebraic data, given by cohomology or complex K–theory, is enough to distinguish BG as a space from other spaces. That is what most of the paper is about.

p–adic completion of spaces makes life a lot easier. Most of the results are about the p–adic completion BG_{p}^{∧}. For a large class of compact connected Lie groups, ‘global’ results are also obtained.

We are concerned with three concepts: The homotopy type, the p–adic type, and the mod–p type of a classifying space. We will explain these concepts in detail in a moment. The last two notions are purely algebraic. Each concept is weaker than the preceeding one. The main theorems will say that, under certain conditions, the first two are equivalent and characterize the homotopy type of BG_{p}^{∧}. That is what we understand by homotopy uniqueness.

We will use the following notation throughout: T_{G}↪G denotes a fixed maximal torus of G, N(T_{G})↪G the normalizer of T_{G}, and W_{G} the Weyl group of G.

We say a p-complete space X has the mod–p type of BG, if there exists an isomorphism

\overline{ϕ} : H^{∗}(X; ℤ/p) ⟶ H^{∗}(BG; ℤ/p)

as algebras over the Steenrod algebra. Under this condition, with an extra assumption for p = 2, Dwyer, Miller, and Wilkerson constructed a ‘maximal torus’ f_{T} : BT_{X}⟶X of X, T_{X} a torus, and an action of W_{G} on BT_{X}. With the trivial action on X the map f_{T} is W_{G}–equivariant up to homotopy. There also exists a map BT_{G}⟶BT_{X} such that

BT_{G} ⟶ BT_{X}
↓  ↓
BG −−→^{\overline{ϕ}} X

commutes and is W_{G}–equivariant in mod–p cohomology [D-M-W 2]. Here, W_{G} acts trivially on BG and X. The dotted arrow indicates that the map only exists in cohomology, but looks like being induced by a topological map. For details and exact definitions, see section 7.

For a space X with the mod–p type of BG, we say that X has the p–adic type of BG, if there exist a W_{G}–equivariant map BT_{G}_{p}^{∧}⟶X, and an isomorphism

ϕ_{p}^{∧} : H^{∗}(X; ℤ_{p}^{∧}) ⟶ H^{∗}(BG; ℤ_{p}^{∧}) ,

compatible with \overline{ϕ} via the long exact sequence in cohomology associated to the sequence ℤ_{p}^{∧}⟶ℤ_{p}^{∧}⟶ℤ/p, such that

[figure: \begin{aligned}& BT_{G} \\ BG & & X \end{aligned}; ϕ_{p}^{∧}]

commutes in p–adic cohomology. Roughly speaking, this means that the W_{G}– actions on BT_{G} and BT_{X} are conjugate over ℤ_{p}^{∧}. On the other hand we can understand this as a topological realization of

H^{∗}(X; ℤ_{p}^{∧})⟶^{ϕ_{p}^{∧}}H^{∗}(BG; ℤ_{p}^{∧})⟶H^{∗}(BT_{G}; ℤ_{p}^{∧})

with W_{G} as Weyl group.

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

K ⟶ G_{s} × T ⟶ G ,

where K is finite abelian, G_{s} is simply connected, and T is a torus. Such coverings we call finite universal. Only the cover G_{s} × T is uniquely determined, but not the map. We could compose the covering with a self covering of T. The factor G_{s} is called the simply connected part of G.

Definition. Let G be a compact connected Lie group.

(1) BG is called p–torsion free, if H^{∗}(BG; ℤ) or, equivalently H^{∗}(G; ℤ), contains no p–torsion.

(2) G is called p–convenient if BG is p–torsion free and if

H^{∗}(BG_{s}; ℤ/p) ≅ H^{∗}(BT_{G_{s}}; ℤ/p)^{W_{G}} .

(3) G is called pseudo simply connected, if G is a product of unitary groups, and simply connected Lie groups which are not isomorphic to SU(n) (i.e. we replace SU(n) by U(n)).

(4) G is called pseudo projective if there exists a homomorphism Z(G_{s})⟶T of the center of G_{s} into a torus T, such that G ≅ (G_{s} × T)/Z(G_{s}).

The center of a pseudo projective Lie group is connected. This condition is also sufficient to characterize pseudo projective Lie groups. Both implications follow by easy arguments from the definition.

If G := (G_{s} × T)/Z(G_{s}) is p–convenient and pseudo projective, then π_{1}(G) ≅ π_{2}(BG) ≅ H_{2}(BG; ℤ) is p–torsion free. This is equivalent to the condition that the kernel of the homomorphism Z(G_{s})⟶T is injective on p–torsion.

Now we are prepared to state our main theorems.

Theorem 1.1. Let G be a compact connected Lie group and let X be a p–complete space with the mod–p type of BG.

(1) If G is p–convenient, there exists a compact connected Lie group H such that BH has the same mod-p type as BG and X and such that X has the p–adic type of BH.

(2) If G is p–convenient and simply connected, pseudo simply connected, or pseudo projective, or if G is a product of unitary groups, then X has the p–adic type of BG.

(3) If p does not divide the order of the Weyl group W_{G}, then X has also the p–adic type of BG.

Theorem 1.2. Let G be a p–convenient compact connected Lie group or a product of unitary groups. If X has the p–adic type of BG, then X and BG_{p}^{∧} are homotopy equivalent.

Theorem 1.1 and 1.2 together imply the following corollary:

Corollary 1.3. Let G satisfy one of the following conditions:

(1) G is p–convenient and simply connected, pseudo simply connected, or pseudo projective.

(2) G is a product of unitary groups.

(3) p does not divide the order of W_{G}.

If X has the mod–p type of BG, then X and BG_{p}^{∧} are homotopy equivalent.

The condition of being p–convenient is essential for our proofs. A more natural condition for theorems of this type would be given by conditions like BG is mod–p polynomial, i.e. H^{∗}(BG; ℤ/p) is a polynomial algebra, or BG is p–torsion free. For odd primes we can weaken our technical assumptions.

1.4 Proposition. Let G be a compact connected Lie group. For p an odd prime, the following conditions are equivalent:

(1) G is p–convenient.

(2) BG is p–torsionfree.

(3) BG is mod–p polynomial.

For p = 2, the spaces BSO(n), n ≥ 3, and BG_{2} are mod–2 polynomial, but not 2–torsion free. BSp(n), n ≥ 1, is 2–torsion free, but Sp(n) not 2–convenient. Our methods do not cover the case G = Sp(n) for p = 2. But we can handle the case of U(2), respectively the case of products of unitary groups, which are not 2–convenient in general.

If p does not divide the order of W_{G}, the Lie group G is always p–convenient. In this case, H^{∗}(BG, ℤ/p) ≅ H^{∗}(BT_{G}, ℤ/p)^{W_{G}} and H^{∗}(BG_{s}×BT, ℤ/p) ≅ H^{∗}(BG, ℤ/p) [Bo 2] [D].

The following table lists up the simple simply connected Lie groups and the primes for which the group is p–convenient, p–torsionfree, or which satisfy the condition (p, |W_{G}|) = 1.

\underline{Lie group}

(\underline{(p, |W_{G}|) = 1})

p–torsion free p–convenient

SU(2)  p odd  all p  p odd
SU(n), n ≥ 3  p > n  all p  all p
Spin(n)  p > n  p odd  p odd
Sp(n)  p > n  p odd  p odd
G_{2}  p ≥ 5  p odd  p odd
F_{4}  p ≥ 5  p ≥ 5  p ≥ 5
E_{6}  p ≥ 7  p ≥ 5  p ≥ 5
E_{7}  p ≥ 11  p ≥ 5  p ≥ 5
E_{8}  p ≥ 11  p ≥ 7  p ≥ 7

For the calculation of the p–torsion see [Bo 3].

Products of p–convenient Lie groups are also p–convenient. We can characterize the p–convenient Lie group in terms of finite universal coverings.

1.5 Proposition. Let K ⟶ G_{s} × T ⟶ G be a finite universal covering of the compact connected Lie group G. Then G is p–convenient if and only if G_{s} is p–convenient and K_{s} := K ∩ G_{s} × {1} has order coprime to p.

After having studied the local problem, it is natural to ask for the analogous global question. In some cases, we can offer a characterization of the integral homotopy type of BG.

Theorem 1.6. Let G be a 2–convenient compact connected Lie group,such that BG is torsion free, or let G be a product of unitary groups. Let X be a simply connected CW–complex of finite type, such that H^{∗}(X; ℤ) is torsion free. If there exists an isomorphism

K(Y) ≅ K(BG)

as λ–rings, then Y and BG are homotopy equivalent.

Questions of this type were first studied by Dwyer, Miller, and Wilkerson. They proved the analogous statement to corollary 1.3 for SU(2) and SO(3) at all primes [D-M-W 1]. They also proved corollary 1.3 under the condition (3) [D-M-W 2]. McClure and Smith got the same result for U(2) at p = 2 [M-S]. For U(2), theorem 1.6 was already proved in [N-S 2].

The problem when proving theorems 1.1 and 1.2 is to pass from more or less pure algebraic data to geometric information. Let X be a space of the mod–p type of BG. Then, as mentioned above, there exists a maximal torus f_{T} : BT_{X}⟶X and Weyl group action of W_{G} on BT_{X} _{p}^{∧}. The associated representation L^{∗}T_{X} _{p}^{∧} := H^{2}(BT_{X}; ℤ_{p}^{∧}) of W_{G} is mod-p isomorphic to L^{∗}T_{G}_{p}^{∧} := H^{2}(BT_{G}; ℤ_{p}^{∧}) (see proposition 7.2). To prove theorem 1.1 we have to study the p–adic liftings of the W_{G}–module L^{∗}T_{G}/p := H^{2}(BT_{G}; ℤ/p).

The construction of a homotopy equivalence BG_{p}^{∧}⟶X (i.e the proof of theorem 1.2), is based on the mod–p decomposition of BG of Jackowski, McClure, and Oliver [J-M-O] and on two ways to calculate map(BV, BG), V an elementary abelian p–group, by Lannes–theory [L 2] and by using the centralizer of V in G [D-Z][L 2]. The structure of BG which has to be taken into account is too rich for constructing a map BG⟶X directly. The mod–p decomposition of BG breaks BG_{p}^{∧} into simpler spaces, namely the classifying spaces of p–toral stubborn subgroups (for the definition of stubborn see section 2). In section 2 we set up a program which reduces the construction of a homotopy equivalence BG_{p}^{∧}⟶X to a series of propositions and lemmas.

This program goes through for pseudo simply connected Lie groups and products of unitary groups. In the case of simply connected Lie groups, all necessary propositions and lemmas should also be true, but the proofs are much harder. The difficulties come from dealing with SU(n) (see proposition 6.7 and 12.1). In particular, N(T_{U(n)}), as a wreath product, is much simpler to handle than N(T_{SU(n)}), respectively the Weyl group action of Σ_{n} = W_{U(n)} = W_{SU(n)} on T_{U(n)},which is induced from the trivial action on S^{1}, is simpler than the one on T_{SU(n)}.

Every compact connected Lie group G = (G_{s} × T)/K is the quotient of the product of a simply connected Lie group G_{s} and a torus T by a finite abelian group K. This classification allows to pass first from pseudo simply connected to simply connected Lie groups, and second to general compact connected Lie groups which completes the proof of theorem 1.2.

The paper is organized as follows: Section 3 contains background including, what is needed from Lannes theory, [D-Z], and [No 1]. In section 4, p–convenient compact connected Lie group are discussed and the propositions 1.4 and 1.5 are proved. For later purpose we need information about the cohomology of Weyl groups in low dimensions. These calculations are done in section 5. In section 6 we study p-toral stubborn subgroups of pseudo simply connected Lie groups.

The construction of a maximal torus for spaces of the mod–p type of BG, as well as the proof of theorem 1.1 is contained in section 7. The necessary calculations about p–adic Weyl group representations are worked out in section 8 and 9.

For A an abelian p–toral group, there is a generalization of the main results of [D-Z] and [No 1] contained in section 10: namely a calculation of the mod–p cohomology of the components of map(BA, X), where X is a space with the mod– p type of BG. The results of this section are independent of the previous ones, but are already qouted earlier, e.g. in section 9. We put section 10 in this place because we feel that the used arguments are similiar to the ones of section 7, but more complicated.

This algebraic replacement of [D-Z] is necessary for constructing a normalizer of the maximal torus of a space X of the p–adic type of BG, i.e. a map BN(T_{G})⟶X which looks cohomologically like BN(T_{G})⟶BG. This is done in section 11 and 12.

After having formulated the classical Lie group theory like maximal tori, Weyl groups, and normalizers for spaces with the p–adic type of BG, we construct a homotopy equivalence f_{G} : BG_{p}^{∧}⟶X in section 13–14, first for pseudo simply connected Lie groups and products of unitary groups, then for simply connected Lie groups, and finally in the general case. This finishes the proof of theorem 1.2.

In the last sections we discuss the integral homotopy type of BG and prove theorems 1.6.

Whenever we deal with completions, we mean completion in the sense of Bousfield and Kan [B-K].

We are grateful to C.Broto, H.W.Henn, and S.Priddy for discussions about parts of this work, and to R.Oliver, who was a great help to detect mistakes in earlier version of some proofs and to get several proofs organized.

2. The program.

For a space X with the p–adic type of BG the construction of a map f : BG⟶X is based on the mod–p approximation of BG of [J-M-O] via p–toral subgroups of G, i.e. extensions of a torus by a finite p–group. A p–toral subgroup P of G is called stubborn if the Weyl group W (P) := N(P)/P, the quotient of the normalizer of P in G by P, is finite and p–reduced, i.e. the intersection of all p–Sylow subgroups of W (P) is trivial.

Let R_{p}(G) ⊂ O(G) be the full subcategory of the orbit category O(G), whose objects are homogenous spaces G/P with P stubborn and p–toral. Let EG denote an acyclic free G–CW–complex.

2.1 Theorem [J-M-O 1]. Let G be a connected Lie group. The canonical map

holim_{\overrightarrow{R_{p}(G)}} EG ×_{G} G/P ⟶ BG

is a p–local equivalence.

Remark. A map X⟶Y is a p–local equivalence, if it induces an isomorphism H^{∗}(Y; ℤ_{(p)})⟶H^{∗}(X; ℤ_{(p)}), where ℤ_{(p)} are the integers localised at p. Analogously we speak of mod–p and p–adic equivalences.

Because every p–toral subgroup P ↪G is conjugate to a subgroup of N(T_{G}) [J-M-O 1], the theorem is still true if we take the full subcategory of those objects G/P of R_{p}(G), where P is in N(T_{G}). This subcategory contains at least one object of every isomorphism class of objects of R_{p}(G). This full subcategory we call R_{p}(G) too.

Let X be a p-complete space with the p–adic type of BG, and let

ϕ_{p}^{∧} : H^{∗}(X; ℤ_{p}^{∧})⟶^{≅}H^{∗}(BG; ℤ_{p}^{∧})

be an isomorphism of algebras such that

[figure: \begin{aligned}& BT_{G} \\ BG & & X \end{aligned}; ϕ_{p}^{∧}]

is W_{G}–equivariant and commutes in p–adic cohomology. We set up a program to construct a realization f_{G} : BG⟶X of ϕ_{p}^{∧}. By theorem 2.1, we have

map(BG, X) ≃ map(holim_{\overrightarrow{R_{p}(G)}} EG ×_{G} G/P, X)

= holim_{\overleftarrow{R_{p}(G)}} map(EG ×_{G} G/P, X) .

To get the desired map f_{G} : BG⟶X, we first construct an extension f_{N} : BN(T_{G})⟶X of f_{T} : BT_{G}⟶X, which looks in mod–p cohomology like the map BN(T_{G})⟶BG. The map f_{N} is the homotopy theoretical version of the normalizer of the maximal torus of X.

2.2 Proposition. Let G be a p–convenient pseudo simply connected Lie group or a product of unitary groups and let X be a p–complete space with the p–adic type of BG.

(1) There exists an extension

[figure: \begin{matrix}BN(T_{G}) \\ BT_{G} & X & . \end{matrix}; f_{N}; f_{T}]

(2) The map f_{N} can be chosen such that the diagram

[figure: \begin{aligned}& BN(T_{G}) \\ BG & & X \end{aligned}; Bi; f_{N}]

commutes in mod–p cohomology.

Proof. (1) is proved in section 11, and (2) in section 12. □

Now we are ready to begin with the construction of the map f_{G} : BG⟶X. Let P be a p–toral subgroup of G contained in N(T_{G}). We define a map f_{P} : BP ⟶X by the composition BP ⟶BN(T_{G})⟶^{f_{N}}X. Denote by {BP }_{R_{p}(G)} the diagram given by the Borel construction applied to the category R_{p}(G). The collection of the maps f_{P} is a map

{BP }_{R_{p}(G)}⟶X

from the diagram to X, i.e. from any object to X. Because of the second part of the last proposition the diagram {BP }_{R_{p}(G)}⟶X commutes in mod–p cohomology. Surprisingly this piece of information is sufficient to prove the following proposition.

2.3 Proposition. Let G be a p–convenient pseudo simply connected Lie group or a product of unitary groups. Then the diagram {BP }_{R_{p}(G)}⟶X commutes up to homotopy.

Proof. See section 13. □

By theorem 2.1, holim_{\overrightarrow{R_{p}(G)}} EG ×_{G} G/P ⟶BG is a mod–p equivalence. The homo-

topy commutativity of the above diagram induces a map on the 1–skeleton of the homotopy colimit to X. The question of extending this map to the total homotopy colimit is decided by the obstruction groups

\underleftarrow{lim}^{i}_{R_{p}(G)} π_{j}(map(BP, X)_{f_{P}}) ,

where lim_{⟵}^{i} is the i–th derived functor of the inverse limit functor [B-K] [Wo]. For X = BG_{p}^{∧} these obstruction groups vanish, if G is simply connected [J-M-O 1], or if G is pseudo simply connected, which is an easy conclusion (see the proof of lemma 13.2). Therefore we compare the two functors

Π_{i}(X), Π_{i}(G) : R_{p}(G)⟶Ab

given by

Π_{i}(X)(G/P) := π_{i}(map(BP, X)_{f_{P}})
Π_{i}(G)(G/P) := π_{i}(map(BP, BG_{p}^{∧})_{(Bi_{P})_{p}^{∧})} .

Ab denotes the category of abelian groups. We remark that Π_{1}(G)(G/P) ≅ π_{1}(BZ_{P}) is an abelian group as well as Π_{1}(X)(G/P) (see section 13).

2.4 Proposition. If G is a p–convenient pseudo simply connected Lie group or a product of unitary groups, there exists a natural transformation

T : Π_{i}(G) ⟶ Π_{i}(X)

which is an equivalence

Proof. See chapter 13. □

Now this proposition and lemma 13.2 imply that

0 = \underleftarrow{lim}^{i}_{R_{p}(G)} π_{j}(map(BP, BG_{p}^{∧})_{(Bi_{P})_{p}^{∧}}) ≅ \underleftarrow{lim}^{i}_{R_{p}(G)} π_{j}(map(BP, X)_{f_{P}}) .

The obstruction groups vanish, and, for a pseudo simply connected Lie group or a product of unitary groups, we get the desired map

f : BG⟶X .

By construction, f realizes the cohomological isomorphism

ϕ_{p}^{∧} : H^{∗}(X; ℤ_{p}^{∧}) ≅ H^{∗}(BT_{G}; ℤ_{p}^{∧})^{W_{G}} ≅ H^{∗}(BG; ℤ/p)

we started with, and hence is a homotopy equivalence.

3. Background.

In this chapter we fix a prime p. The cohomology groups are always taken with coefficients in ℤ/p and H^{∗}( ) always means H^{∗}( ; ℤ/p). We denote by K the category of unstable algebras over the Steenrod algebra A_{p}.

Let V be an elementary abelian p–group. An algebra A over A_{p} is called of finite type if A is finite in each dimension.

3.1 Theorem [L 2]. If X is a p–complete space and H^{∗}(X) of finite type, then the canonical map

[BV, X] ⟶ Hom_{A_{p}}(H^{∗}(X), H^{∗}(BV))

is an isomorphism.

The evaluation map

BV × map(BV, X) ⟶ X

induces a cohomological map

H^{∗}(X) ⟶ H^{∗}(BV) ⊗ H^{∗}(map(BV, X)) .

J. Lannes studied the functor T^{V} : K⟶K which is the left adjoint of the functor H^{∗}(BV)⊗_{ℤ/p} . Taking the adjoint of the evaluation map yields

T^{V} H^{∗}(X) ⟶ H^{∗}(map(BV, X)) .

For any map g : BV ⟶X, there is an associated direct summand T_{g^{∗}}^{V}H^{∗}(X) of T^{V} H^{∗}(X) which corresponds to the summand H^{∗}(map(BV, X)_{g}) of H^{∗}(map(BV, X)). With respect to this splitting the above map is a direct sum of maps with coordinates

T_{g^{∗}}^{V}H^{∗}(X) ⟶ H^{∗}(map(BV, X)_{g}) .

3.2 Theorem [L 2, 3.3.2]. Let X be a space, such that H^{∗}(X) is of finite type. Let g : BV ⟶X be a map. The map

T_{g^{∗}}^{V}H^{∗}(X) ⟶ H^{∗}(map(BV, X_{p}^{∧})_{g})

is an isomorphism if T_{g^{∗}}^{V}H^{∗}(X) is of finite type and one of the following three conditions is satisfied:

(1) T_{g^{∗}}^{V}H^{∗}(X) is zero in degree 1.

(2) map(BV, X_{p}^{∧})_{g} is 1–connected.

(3) There is a connected space Z with the property that H^{∗}(Z) is of finite type

and a map

Z ⟶ map(BV, X)_{g} ,

such that the associated map

T_{g^{∗}}^{V}H^{∗}(X) ⟶ H^{∗}(Z)

is an isomorphism.

3.3 Theorem ([L 2, 3.4.3]). In addition to the assumptions of theorem 2.2 let X be p–complete. Then the following conditions are equivalent:

(1) map(BV, X)_{g} is p–complete.

(2) T_{g^{∗}}^{V}H^{∗}(X)⟶H^{∗}(map(BV, X)_{g}) is an isomorphism.

If we consider a collection S of maps BV ⟶X, then of course we get a direct summand T_{S^{∗}}^{V}H^{∗}(X), where S^{∗} is the collection of the associated cohomological maps. The theorems 3.2 and 3.3 are still true in this situation [L 2].

Let V be an elementary abelian p–group and let BV ⟶^{f}X⟶^{g}Y be a pair of maps, such that the application of the Lannes functor calculates the mod–p cohomology of the mapping spaces map(BV, X)_{f} and map(BV, Y)_{gf}. The following lemma, whose proof is straight forward, describes the induced map

H^{∗}(map(BV, g)) : H^{∗}(map(BV, Y)_{gf}) ⟶ H^{∗}(map(BV, X)_{f})

in terms of the T –functor.

3.4 Lemma. Under the above assumptions, the map H^{∗}(map(BV, g)) is given by the application of the Lannes–functor, i.e.

H^{∗}(map(BV, g)) = T_{f}^{V} (g) : H^{∗}(map(BV, Y)_{gf}) ⟶ H^{∗}(map(BV, X)_{f} .

In the case of X being the classifying space BG of a connected Lie group G, the functor T^{V} can be calculated.

For a p–toral group P we denote by Rep(P, G) the set of representation classes P ⟶G, i.e. G–conjugacy classes of homomorphism P ⟶G. Passing to classifying spaces gives a map Rep(P, G)⟶[BP, BG]. For a particular representation ρ : P ⟶G let C_{G}(ρ) denote the centralizer of ρ in G. The obvious product homomorphism C_{G}(ρ) × P ⟶G has as adjoint on the level of classifying spaces the map BC_{G}(ρ)⟶map(BP, BG)_{Bρ}.

3.5 Theorem ([L 2, 3.4.5]). Let V be an elementary abelian p–group. Then the maps

T^{V} H^{∗}(BG) ⟶ H^{∗}(map(BV, BG)) ⟶ ∐_{ρ∈Rep(V,G)} H^{∗}(BC_{G}(ρ)) ,

and, if we choose a homomorphism ρ : V ⟶G,

T_{Bρ^{∗}}^{V}H^{∗}(BG) ⟶ H^{∗}(map(BV, BG)_{Bρ}) ⟶ H^{∗}(BC_{G}(ρ))

are isomorphisms.

In [D-Z] and [No 1] mapping spaces are calculated, having the classifying space of a finite p–group respectively a p–toral group as source.

3.6 Theorem ([D-Z], [No 1]). For a p–toral group P and a compact Lie group G the natural map

Rep(P, G) ⟶ [BP, BG]

is an isomorphism, and the map

BC_{G}(ρ) ⟶ map(BP, BG)_{Bρ}

a mod–p equivalence.

For a finite extension of a torus as target, there is a stronger result.

3.7 Proposition. Let ρ : H⟶N be a homomorphism from a compact connected Lie group H to a finite extension N of a torus. Then

BC_{N}(ρ) ⟶ map(BH, BN)_{Bρ}

is a homotopy equivalence.

Proof. BN fits into the fibration

BT ⟶ BN ⟶ BW ,

where W := N/T, and where T is the toral part of N, i.e. the component of the unit. Because H is connected, the image of ρ is contained in T. Let Iso(ρ) ⊂ W be the group of those elements, which act trivially on the image of ρ. The Dwyer– Zabrodsky construction induces a commutative diagram

∏_{w∈W/Iso(ρ)} BT  ⟶  BC_{N}(ρ)  ⟶  BW
↓  ↓  ↓
∏_{w∈W/Iso(ρ)} map(BH, BT)_{w∘Bρ} ⟶ map(BH, BN) ⟶ map(BH, BW)_{c} ,

where c denotes the constant map. Since the components of BC_{N}(ρ) are given by Iso(ρ), the upper row is a fibration. The bottom row is a fibration, because every map BH⟶BT is induced by a homomorphism [L-M-S]. The left and right columns are equivalences, which follows from a short calculation respectively the Sullivan conjecture [M]. Therefore, the middle column is an equivalence which proves the statement. □

We work mainly with completed spaces. Hence, it is useful to know if the considered mapping spaces are complete.

3.8 Proposition [B-N]. Let G and H be compact Lie groups, H connected. Then, for any map f : BG⟶BH, the component map(BG, BH_{p}^{∧})_{f_{p}^{∧}} is p–complete, and

(map(BG, BH)_{f})_{p}^{∧} ⟶ map(BG, BH_{p}^{∧})_{f_{p}^{∧}}

is a homotopy equivalence.

This leads to a result slightly stronger than theorem 3.6 (see also [J-M-O 1]).

3.9 Theorem. For any homomorphism ρ : P ⟶G of a p–toral group P into a compact connected Lie group G, the map

BC_{G}(ρ)_{p}^{∧} ⟶ map(BP, BG_{p}^{∧})_{Bρ_{p}^{∧}}

is a homotopy equivalence.

Now let π be a group acting on a space X. Let X^{π} denote the fixed–point set. The homotopy fixed–point set X^{hπ} is defined to be the mapping space map_{π}(Eπ, Y) of equivariant maps. Here Eπ is an acyclic free π–CW–complex of finite type. There is another description of X^{hπ}. Consider the fibration

(∗)  X ⟶ Eπ ×_{π} X ⟶ Bπ .

Note the one to one relation between sections in this bundle and elements in X^{hπ}. Thus, we can also define X^{hπ} as the fiber of the fibration

X^{hπ} ⟶ map(Bπ, Eπ ×_{π} X)_{sec} ⟶ map(Bπ, Bπ)_{id} ,

where map(−, −)_{sec} is the union of the components of the sections in the bundle (∗).

For an elementary abelian p–group V acting on X, the multiplication BV × BV ⟶BV of the H–space BV induces an equivalence

α : BV ⟶ map(BV, BV)_{id} .

This multiplication can also be used to construct a map

β : BV × X^{hV} ⟶ map(BV, EV ×_{V} X)_{sec}

by defining β(b, f) := f ∘ α(b). Here f ∈ X^{hV} is interpreted as a section BV ⟶EV ×_{V} X. The map β fits into the diagram

(∗∗)  \begin{matrix}X^{hV} & ⟶ & BV × X^{hV} & ⟶ & BV \\ ∥ & & ^{β}↓ & & ^{α}↓ \\ X^{hV} & ⟶ & map(BV, EV ×_{V} X)_{sec} & ⟶ & map(BV, BV)_{id} \end{matrix}

This is a trivialization of the fibration in the bottom line.

Let s : EV ⟶X be an equivariant map respectively s : BV ⟶EV ×_{V} X a section. Applying the Lannes functor T_{s^{∗}}^{V} gives a map

(***)  T_{s^{∗}}^{V}H^{∗}(EV ×_{V} X) ⟶ H^{∗}(map(BV, EV ×_{V} X)_{s}) .

The natural isomorphisms

H^{∗}(BV) ⟶^{≅} T_{id^{∗}}^{V}H^{∗}(BV) ⟶^{≅} H^{∗}(map(BV, BV)_{id})

and the canonical map H^{∗}(BV)⟶ℤ/p allow to apply the functor − ⊗_{H^{∗}(BV)} ℤ/p to (∗ ∗ ∗). This yields a map

T_{s^{∗}}^{V}H^{∗}(EV ×_{V} X) ⊗_{H^{∗}(BV)} ℤ/p ⟶ H^{∗}(map(BV, EV ×_{V} X)_{s}) ⊗_{H^{∗}(BV)} ℤ/p .

Because of the equivalences in (∗∗) the right side is isomorphic to H^{∗}(X_{s}^{hV}). Here X_{s}^{hV} denotes the component of s of the homotopy fixed–point set X^{hV}. In [L 2, § 4] the left side is used to define a functor

HF^{V} : T op_{V} ⟶K : X ↦ HF^{V} (X) := T_{sec^{∗}}^{V}H^{∗}(EV ×_{V} X) ⊗_{H^{∗}(BV)} ℤ/p

from the category of spaces with a V –action to the category K. By sec we denote the set of homotopy classes of sections in the bundle EV ×_{V} X ⟶ BV, and by sec^{∗} the set of the associated induced maps in cohomology. For any subset S ⊂ sec, i.e. for any set of homotopy classes of homotopy fixed–points, there is an obvious map

HF_{S^{∗}}^{V}(X) ⟶ H^{∗}(X_{S}^{hV}) .

3.10 Theorem ([L 2, 4.9.3]). Let X be a space with a V –action, such that H^{∗}(X) is of finite type. Let S ⊂ π_{0}(X^{hV}) be a set of homotopy fixed–points. The map

HF_{S^{∗}}^{V}(X) ⟶ H^{∗}((X_{p}^{∧})_{S}^{hV})

is an isomorphism, if H_{S^{∗}}^{∗V} (X) is of finite type and one of the following two conditions is satisfied:

(1) HF_{S^{∗}}^{V}(X) is zero in degree 1.

(2) (X_{p}^{∧})_{s}^{hV} is 1–connected for each s ∈ S.

3.11 Theorem ([L 2, 4.9.2]). In addition to the assumptions of theorem 3.10 let X be p–complete. Then the following conditions are equivalent:

(1) X_{S}^{hV} is p–complete.

(2) HF_{S^{∗}}^{V}(X)⟶H^{∗}(X_{S}^{hV}) is an isomorphism.

3.12 Remark. Let K⟶G⟶H be an exact sequence of compact Lie groups. The ‘extended’ classifying space B̃K := EG/K ≃ BK of K carries a free H action, and B̃K/H = BG. For any space X, which H acts trivially on, we have

map(BG, X) = map(B̃K/H, X)
= map_{H}(B̃K, X)
≃ map_{H}(EH × B̃K, X)
= map_{H}(EH, map(B̃K, X))
= map(B̃K, X)^{hH} .

This implies the following lemma:

3.13 Lemma. Let f_{K} : BK⟶Xbe a map and map(BK, X)_{f_{K}} be simple, i.e the fundamental group acts trivial on the homotopy groups.

(1) The obstructions for extending f_{K} to a map f_{G} : BG⟶X are contained in the groups H^{∗+1}(BH, π_{∗}(map(B̃K, X)_{f_{K}})) (twisted coefficients).

(2) The homotopy classes of extensions BG⟶X are classified by obstructions in H^{∗}(BH, π_{∗}(map(B̃K, X)_{f_{K}})) (twisted coeficients).

Proof. Extensions BG⟶X are given by sections in the fibration

map(B̃K, X)_{f_{K}} ⟶ map(BH, EH×_{H} map(B̃K, X)_{f_{K}}) ⟶ BH . □

We finish this section with a purely algebraic result about the Lannes functor. Let A be an unstable algebra over A_{p}, let W a finite group acting on A, and let ϕ : A⟶H^{∗}(BV) an A_{p}–algebra map. We call

Iso(ϕ) := {w ∈ W |ϕ ∘ w = ϕ}

the isotropy group of ϕ. Because H^{∗}(BV) is an injective object in K, every map ϕ : A^{W} ⟶H^{∗}(BV) can be extended to a map ϕ : A⟶H^{∗}(BV).

3.14 Theorem [L 1].

T^{V} (A^{W}) = ∐_{ϕ∈Hom_{K}(A,H^{∗}(BV))/W} T_{ϕ}^{V} A^{W} ≅ ∐_{ϕ∈Hom_{K}(A,H^{∗}(BV))/W}(T_{ϕ}^{V} A)^{Iso(ϕ)} ,

or, in a compact version, T^{V} (A^{W}) ≅ (T^{V} A)^{W}.

4. p-convenient compact connected Lie groups.

Let K ⟶ G ⟶^{q} H be an exact sequence of connected Lie groups, G and H

connected and K finite. We have associated oriented fibrations BK ⟶ BG ⟶^{Bq} BH

and BK ⟶ BT_{G} ⟶^{Bq} BT_{H}.

4.1 Lemma. If BH is p–torsion free, then BG is p–torsion free.

Proof. By a theorem of Borel [Bo 3] BG is p–torsion free if and only if every elementary abelian p–subgroup of G is conjugate to a subgroup of T_{G}. Let V be an elemetary abelian p–subgroup of G. Then, up to conjugation, q(V) ⊂ T_{H}, and, up to conjugation, V ⊂ q^{−1}(q(V)) ⊂ q^{−1}(T_{H}) = T_{G}. □

For any compact connected Lie group G there exists a finite covering

1 ⟶ K ⟶ G̃ = G_{s} × T ⟶ G ⟶ 1

of compact Lie groups, where G_{s} is simply connected and T a torus (see for example [B-tD]). Such coverings we call finite universal. The cover G̃ is unique up to isomorphism, but the homomorphism G̃ ⟶ G is not unique. We can choose a finite self covering T ⟶ T and can take the composition G_{s} ×T ⟶ G_{s} ×T ⟶ G. Among the finite universal coverings of G there is a minimal one, characterized by the condition that K ⟶ G_{s} × T ⟶G_{s} is an injection. The minimal finite universal covering is a quotient of all the others.

Every finite universal covering establishes two associated commutative diagrams of exact sequences of connected Lie groups:

(*)  \begin{matrix}K_{s} & ⟶ & G_{s} & ⟶ & G_{s}/K_{s} := \overline{G}_{s} \\ ↓ & & ↓ & & ↓ \\ K & ⟶ & G_{s} × T & ⟶ & G \\ ↓ & & ↓ & & ↓ \\ \overline{K} = K/K_{s} & ⟶ & T & ⟶ & T/\overline{K} := \overline{T} \end{matrix}

and

(**)  \begin{matrix}K_{s} & & K_{s} & ⟶ & ∗ \\ ↓ & & ↓ & & ↓ \\ K & ⟶ & G_{s} × T & ⟶ & G \\ ↓ & & ↓ & & ∥ \\ \overline{K} & ⟶ & \overline{G}_{s} × T & ⟶ & G & , \end{matrix}

where K_{s} := K ∩ (G_{s} × \{0\}). Because H^{1}(\overline{G}_{s}; ℤ) = 0, the sequence \overline{G}_{s}⟶G⟶\overline{T} induces isomorphisms H^{1}(\overline{T}; ℤ) ≅ H^{1}(G; ℤ) and H^{2}(B\overline{T}; ℤ) ≅ H^{2}(BG; ℤ). On the other hand these isomorphisms determine the maps G⟶\overline{T} and BG⟶B\overline{T}, and therefore, the fibration B\overline{G}_{s}⟶BG⟶B\overline{T}.

If BG is p–torsion free, then, by lemma 4.1, BG_{s} × BT and B\overline{G}_{s} × BT as well as BG_{s} and B\overline{G}_{s} are p–torsionfree. The order |K_{s}| of K_{s} is coprime to p, because π_{2}(B\overline{G}_{s}) ≅ H_{2}(B\overline{G}_{s}; ℤ) ≅ K_{s}.

4.2 Lemma. Let BG be p–torsion free. If G_{s} or G_{s} × T are p–convenient, then G also is p–convenient, and

H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W_{G}} .

Proof. Denote by E_{∗}^{∗,∗}(T_{G}) and E_{∗}^{∗,∗}(G) the terms of the mod–p cohomology Leray– Serre spectral sequences of the oriented fibrations

BT_{\overline{G}_{s}} ⟶ BT_{G} ⟶ B\overline{T}
↓  ↓  ∥
B\overline{G}_{s} ⟶ BG ⟶ B\overline{T} .

By the above remarks \overline{G}_{s} is p–torsionfree and the spectral sequences collapses, i.e E_{2}^{∗,∗}(G) = E_{∞}^{∗,∗}(G) and E_{2}^{∗,∗}(T_{G}) = E_{∞}^{∗,∗}(T_{G}).

If G_{s} is p–convenient, we have

E_{∞}^{∗,∗}(G) ≅ H^{∗}(B\overline{T}; ℤ/p) ⊗ H^{∗}(B\overline{G}_{s}; ℤ/p)
≅ H^{∗}(B\overline{T}; ℤ/p) ⊗ H^{∗}(BT_{\overline{G}_{s}}; ℤ/p)^{W}
≅ (H^{∗}(B\overline{T}; ℤ/p) ⊗ H^{∗}(B\overline{G}_{s}; ℤ/p))^{W}
≅ E_{∞}^{∗,∗}(T_{G})^{W} .

Using the 5–lemma and an induction over the extension problems of the two associated spectral sequences, one can show that H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G})^{W}. □

In [D] is discussed, whether the canonical map

H^{∗}(BT_{G}; ℤ_{(p)})^{W_{G}} ⊗ ℤ/p⟶H^{∗}(BT_{G}; ℤ/p)^{W_{G}}

is an isomorphism. By the main theorem, proposition 6, and proposition 8 of [D], this is an isomorphism in the following cases:

SU(n) , n ≥ 3  all primes
Spin(n), Sp(n), n ≥ 1, G_{2}  p ≥ 3
F_{4}, E_{6}, E_{7}  p ≥ 5
E_{8}  p ≥ 7 .

If p is odd, the pair (G, p), G simple and simply connected, belongs to the list if and only if BG is p–torsionfree. For p = 2 the space BSp(n) is 2-torsionfree, but Sp(n) is not 2–convenient.

4.3 Proposition. Let p be an odd prime, and let G be a compact connected Lie group. If BG is p–torsion free, then G is p–convenient, and

H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W_{G}} .

Proof. Let K⟶G_{s} ×T ⟶G be a finite universal covering of G, where G_{s} is simply connected and T is a torus. We can split G_{s} = G_{1} ×...×G_{r} into a product of simple simply connected factors G_{i}. By lemma 4.1 G_{s} is p–torsionfree. The arguments of the proof of [Bo 2; 20.3] show that the map

H^{∗}(BH; ℤ_{(p)})⟶H^{∗}(BT_{H}; ℤ_{(p)})^{W_{H}}

is an isomorphism, if BH is p–torsionfree. The pairs (G_{i}, p) belong to the above list. We get

H^{∗}(BG_{s}; ℤ/p) ≅ H^{∗}(BG_{s}; ℤ_{(p)}) ⊗ ℤ/p
≅ H^{∗}(BT_{G_{s}}; ℤ_{(p)})^{W_{Gs}} ⊗ ℤ/p
≅ H^{∗}(BT_{G_{s}}; ℤ/p)^{W_{Gs}} .

Therefore G_{s} and G_{s}×T are p–convenient. By lemma 4.2 the statement follows. □

Proof of proposition 1.4. The equivalence of (1) and (2) follows from proposition 4.3 and the equivalence of (2) and (3) from [Bo 1; proposition 5.2]. □

Proof of proposition 1.5. Let K ⟶ G_{s} × T ⟶ G be a universal finite covering of G. Let W := W_{G_{s}} ≅ W_{G}. The diagram (∗) establishes a fibration B\overline{G}_{s} ⟶ BG ⟶ B\overline{T}.

Let us assume that G_{s} is p–convenient and that the order of K_{s} = K ∩ G_{s} × {1} is coprime to p. Because H^{∗}(B\overline{G}_{s}; ℤ/p) ≅ H^{∗}(BG_{s}; ℤ/p) ≅ H^{∗}(BT_{G_{s}}; ℤ/p)^{W}, the spectral sequence of the fibration collapses because of degree reasons. This implies that H^{∗}(BG; ℤ/p) is concentrated in even degrees and that G is p–convenient.

If G is p–convenientthen G_{s} is p-convenient by definition. If the order of K_{s} is divisible by p, the semi simple part B\overline{G}_{s} has p–torsion. Lemma 4.1 applied to the bottom horizintal row of (∗∗) shows that BG has p–torsion which is a contradiction. □

For later purpose we are interested in centralizers of abelian p–toral subgroups of compact connected Lie groups. The next statement handles this question and has a purely albgraic analogous (see theorem 10.1).

4.4 Proposition. Let A be a p–toral abelian compact Lie group and G a compact connected Lie group such that BG is p–torsion free. Then the following statements are true:

(1) Rep(A, T_{G}) ⟶ Rep(A, G) is onto.

(2) Rep(A, T_{G})/W_{G} ⟶ Rep(A, G) is a bijection.

(3) For any homomorphism ρ : A ⟶ T_{G} the centralizer C_{G}(ρ) is connected, and BC_{G}(ρ) is p–torsion free.

Proof. The first statement follows from [Bo 3; 5.2]. The other conditions we prove by induction on the number of topologically cyclic factors of A.

Let A ≅ S^{1} or A ≅ ℤ/p^{k}. Inparticuliar, be a toplogical cyclic compact p–toral group. Then part (2) follows from the isomorphism between the orbit space T_{G}/W_{G} and the set of conjugacy classes of elements in G (e.g. see [B-tD]).

Let ρ : A ⟶ T_{G} be an homomorphism and let g ∈ C_{G}(ρ) =: C be a p–torsion element or a generator of a torus. Then the group B :=< ρ(A), g >, generated by ρ(A) and g is p–toral and therefore, as a subgroup of G, subconjugated to T_{G}. We choose h ∈ G such that hBh^{−1} ⊂ T_{G}. By (2) the conjugation by h on ρ(A) can be realized by an element n ∈ N(T_{G}). Now n^{−1}h centralizes ρ and conjugates B into T_{G} = T_{C}. That is that every p–torsion element of C is subconjugated to T_{C}. By [J–M–O 1; Appendix] π_{0}(C) is a finite p–group. Therefore the p–torsion elements of G are dense in G. This implies that G is connected which is the first part of (3).

Now let V ⊂ C be an elementary abelian subgroup. As a subgroup of G, the group < ρ(A), V > is subconjugated to T_{G}. We can apply the same trick as above to see that V is subconjugated to T_{C} as a subgroup of C. By [Bo 3] every elementar abelian p–subgroup of C is subconjugated to T_{C} if and only if BC is p–torsion free. This finishes the proof for cyclic groups.

Any p–toral abelian group A splits into a product A ≅ A_{0}×A_{1} such that A_{1} ≅ S^{1} or A_{1} ≅ ℤ/p^{k}. Let ρ, ρ′ : A ⟶ T_{G} be two homorphisms which are conjugate in G. By induction hypothesis ρ_{0} := ρ|_{A_{0}} and ρ′_{0} := ρ′|_{A_{0}} differ on;y by an element in W_{G}. We can assume that ρ_{0} = ρ′_{0}. Then ρ and ρ′ correspond to homomophisms α, α′ : A_{1} ⟶ T_{C_{G}(ρ(A_{0}))} = T_{G} which are conjugated in C := C_{G}(ρ(A_{0}). By induction hypothesis again α and α′ differ only by an element in W_{C}. This implies part (2) in the general case.

The last two statements follow from the identity C_{G}(ρ) = C_{C_{G}(ρ(A_{0}))}(ρ(A_{1})) and the induction hypothesis. □

We finish this section with the following two lemmas.

4.5 Lemma. If G is p–convenient, the order of the cokernel of Z(G)↪T_{G}^{W_{G}} is coprime to p.

Proof. Let i : A↪T_{G}^{W_{G}} be the p–toral part of T_{G}^{W_{G}}. By lemma 4.4 the centralizer C := C_{G}(A) is connected and W_{C} = W_{G}. This implies that C = G and that A ⊂ Z(G). □

4.6 Lemma. Let V_{G} be the maximal elementary abelian p–subgroup of T_{G}. If H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W_{G}}, then C_{G}(V_{G}) = T_{G}.

Proof. By lemma 4.4 C_{G}(V_{G}) is connected and of the same rank as T_{G}. Let W_{C} be the Weylgroup of C_{G}(V_{G}) which is also the isotropy group of the homomorphism V_{G}⟶G. Then W_{C} acts trivially on H^{∗}(BT_{G}; ℤ/p) and H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W_{G}/W_{C}}. This implies that H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p) is an integral Galois extension of degree |W_{G}/W_{C}|. But it is well known that this degree is given by the Euler characteristic of G/T_{G} which is the order of W_{G}. Therefore W_{C} is the trivial group which proves the statement. □

Remark. If p is odd the lemma is true for every compact connected Lie group G [D-M-W 2].

5. Subgroups of maximal rank and the cohomology of Weyl groups. For studying the p-stubborn subgroups of the simple simply connected Lie

groups, which we will do in the next chapter, it is very useful to have a list of subgroups of maximal rank satisfying certain conditions. For each prime p and each p-convenient simple simply connected Lie group G, we choose two compact connected subgroups H and K of G given by the following list. The first is maximal of maximal rank and contains the second, which only is of maximal rank. The number in brackets denotes the Weyl group index.

\underline{G}  \underline{p}  \underline{H}  \underline{K}
Sp(n)  p odd  U(n) [2^{n}]  U(n) [2^{n}]
SO(2n + 1)  p odd  U(n) [2^{n}]  U(n) [2^{n}]
SO(2n)  p odd  U(n) [2^{n−1}]  U(n) [2^{n−1}]
G_{2}  p = 3  SU(3) [2]  SU(3) [2]
G_{2}  p ≥ 5  SU(3) [2]  U(2) [6]
F_{4}  p ≥ 5  SU(3) ×_{ℤ/3} SU(3) [2^{5}]  T_{F_{4}} [2^{7} · 3^{2}]
E_{6}  p ≥ 5  SU(2) ×_{ℤ/2} SU(6) [2^{2} · 3^{2}]  S^{1} ×_{ℤ/2} U(5) [2^{4} · 3^{3}]
E_{7}  p ≥ 5  SU(8)/ℤ/2 [2^{3} · 3^{2}]  U(7)/ℤ/2 [2^{6} · 3^{2}]
E_{8}  p ≥ 7  SU(9)/ℤ/3 [2^{7} · 3 · 5]  U(8)/ℤ/3 [2^{7} · 3^{3} · 5]

This list is taken from [J-M-O, §6], where a complete list of all maximal subgroups of maximal rank is given as well as the information about the Weyl group orders. See also [B–S] where the local isomorphy type of the all maximal subgroups of maximal rank is described. For p odd BSpin(n)_{p}^{∧} ≃ BSO(n)_{p}^{∧}. All the subgroups occuring in the list are quotients of compact connected Lie groups by a finite group of order coprime to p.

5.1 Definition. For a homomorphism ρ : H⟶G between Lie groups we say H is a mod-p subgroup of G, if H^{∗}(BH; ℤ/p) is a finite generated module over H^{∗}(BG; ℤ/p).

If ρ is also a surjection we call ρ a mod–p isomorphism of groups.

Remark. For a homomorphism ρ : H⟶G the group H is a mod–p subgroup iff ker(ρ) is finite of order coprime to p [Q].

We collect the properties of the subgroups of the above list in the following lemma.

5.2 Lemma. For every p-convenient simple simply connected Lie group G, G ≠ SU(n), there exist mod-p subgroups K⟶H⟶G of G satisfying the following conditions:

(1) H is maximal of maximal rank and a unitary group or a product of special unitary groups. The index [W_{G} : W_{H}] of the Weyl groups is coprime to p.

(2) K is of maximal rank and a product of unitary groups or SU(p). The index [W_{G} : W_{K}] is coprime to p. Moreover K⟶C_{G}(V) is a mod–p isomorphism of groups, where V ⊂ Z(K) is the maximal p-elementary abelian subgroup of the center Z(K) of K.

Proof. It is only left to show the last statement, namely K⟶C_{G}(V) is a mod–p isomorphism of groups. All the other facts can be read of from the above list.

First we observe, that for a sequence L ⊂ M ⊂ G of connected subgroups of G of maximal rank the index [W_{G} : W_{M}] divides [W_{G} : W_{L}].

Let p ≥ 5, G = E_{6}, H = SU(2) × SU(6), V = ℤ/p × ℤ/p, \overline{H} = H/(ℤ/2), K = S^{1} × U(5), and \overline{K} = K/(ℤ/2). V is a central subgroup of \overline{H} too. The centralizer C_{E_{6}}(ℤ/p × {0}) is connected ([B-S] or theorem 10.1) and containes S^{1} ×_{ℤ/2} SU(6) by lemma 6.4. Using the complete list of maximal subgroups of maximal rank of [J-M-O 1] and the above condition on the Weyl group orders it follows that \overline{H} is the only one containing S^{1} ×_{ℤ/2} SU(6). Therefore C_{E_{6}}(ℤ/p × \{0\}) = C\overline{H}(ℤ/p × \{0\}) and C_{E_{6}}(V) = C\overline{H}(V) = \overline{K} by lemma 6.4 again.

For F_{4} and p ≥ 5 lemma 4.5 proves the last statement. In all the other cases V = ℤ/p and the argument using the Weyl group index relation can be applied. □

We end this section by listing the calculation of some cohomology groups of Weyl groups in low dimensions. There are canonical W_{G}–actions on LT_{G}_{p}^{∧} := H_{2}(BT_{G}; ℤ_{p}^{∧}), LT_{G}/p^{k} := H_{2}(BT_{G}; ℤ/p^{k}), L^{∗}T_{G}_{p}^{∧} := H^{2}(BT_{G}; ℤ_{p}^{∧}), and L^{∗}T_{G}/p^{k} := H^{2}(BT_{G}; ℤ/p^{k}).

5.3 Lemma. Let G be a p–convenient connected Lie group.

(1) If p is odd all the groups H^{1}(W_{G}; ℤ/p^{k}), H^{2}(W_{G}; ℤ/p^{k}), H^{2}(W_{G}; ℤ_{p}^{∧}),H^{3}(W_{G}; ℤ_{p}^{∧}), H^{1}(W_{G}; L^{∗}T_{G}/p^{k}) and H^{3}(W_{G}; LT_{G}_{p}^{∧}) are trivial.

(2) If p is odd and G pseudo simply connected, H^{1}(W_{G}; LT_{G}/p^{k}) = 0. we have H^{1}(W_{G}; L^{∗}T_{G}/2^{k}) = 0.

(3) H^{3}(Σ_{n}; ℤ_{2}^{∧}) = ℤ/2 for n ≥ 4 and vanishes otherwise.

Proof. We can assume that k = 1. Because L^{∗}T_{U(n)}_{p}^{∧} and LT_{U(n)}_{p}^{∧} are induced by the 1–dimensional trivial Σ_{n−1}–module,

H^{j}(Σ_{n}; L^{∗}T_{U(n)}/p) ≅ H^{j}(Σ_{n−1}; ℤ/p) ≅ H^{j}(Σ_{n}; LT_{U(n)}/p)

≅ 0 for p odd and j ≤ 2

H^{j}(Σ_{n}; LT_{U(n)}_{p}^{∧}) ≅ H^{j}(Σ_{n−1}; ℤ_{p}^{∧})
≅ \{ \begin{matrix}0 , j ≤ 3 and p odd or p = 2, j = 3, and n ≤ 4 \\ ℤ/2 , j = 3, p = 2, and n ≥ 5 \end{matrix}

For the calculation of the mod–p cohomology and homology, see [Na],[K-P]. This proves (1) and (2) for G = U(n) and p odd and also proves (3).

Now let p be odd. The long exact sequences in cohomology, associated to the exact sequences

(*)  \begin{matrix}ℤ/p & ⟶ & L^{∗}T_{U(n)}/p & ⟶ & L^{∗}T_{SU(n)}/p \\ LT_{SU(n)}_{p}^{∧} & ⟶ & LT_{U(n)}_{p}^{∧} & ⟶ & ℤ_{p}^{∧} \end{matrix}

imply (1) for G = SU(n).

If G is p–convenient and simply connected, we choose a mod–p subgroup H⟶G of G such that H is a product of SU(n)’s and U(n)’s (lemma 5.2). Then,

\begin{matrix}H^{∗}(W_{G}; L^{∗}T_{G}/p) & ⟶ & H^{∗}(W_{H}, L^{∗}T_{G}/p) ≅ H^{∗}(W_{H}; L^{∗}T_{H}/p) \\ H^{∗}(W_{G}; LT_{G}_{p}^{∧}) & ⟶ & H^{∗}(W_{H}, LT_{G}_{p}^{∧}) ≅ H^{∗}(W_{H}; LT_{H} _{p}^{∧}) \end{matrix}

are injections. A spectral sequence argument or a Künneth formula argument and an induction over the number of factors of H reduces the calculation to the case H = U(n) or H = SU(n), which we already proved.

For a general p–convenient compact connected Lie group, there are exact sequences

\begin{matrix}L^{∗}T/p & ⟶ & L^{∗}T_{G}/p & ⟶ & L^{∗}T_{\overline{G}_{s}}/p \\ LT_{\overline{G}_{s}} _{p}^{∧} & ⟶ & LT_{G}_{p}^{∧} & ⟶ & LT_{p}^{∧} , \end{matrix}

where G_{s} is the simply connected part of G. The quotient \overline{G}_{s} is p–convenient. Moreover, L^{∗}T_{\overline{G}_{s}}/p ≅ L^{∗}T_{G_{s}}/p and LT_{\overline{G}_{s}} _{p}^{∧} ≅ LT_{G_{s}} _{p}^{∧} because G_{s}⟶\overline{G}_{s} is a mod–p isomorphism of groups. (See section 4). Again, the associated long exact sequences in cohomolgy complete the proof of (1).

Let p be odd and G pseudo simply connected. By lemma 5.2, we choose a mod–p subgroup H⟶G, such that H is a product of unitary groups and SU(3)’s. SU(3) can only occur, if p = 3 and G_{2} is among the factors of G. As already shown H^{1}(W_{U(n)}; LT_{U(n)}/p) = 0, and H^{1}(W_{G_{2}}; LT_{G_{2}}/3) also vanishes. This follows because W_{G_{2}} ≅ Σ_{3} × Σ_{2} and because Σ_{2} acts non trivially on H^{1}(Σ_{3}; LT_{SU(3)}/3) ≅ ℤ/3. This proves (2). □

6. p–stubborn subgroups of pseudo simply connected Lie groups. In [O] is given a complete list of representatives of the conjugacy classes of p–

stubborn groups of the classical Lie groups, i.e. U(n), SU(n), Sp(n), and SO(n) for all primes. We will recall these results for U(n) and SU(n).

Let Σ_{p^{k}} be the symmetric group of order p^{k}!. The permutations σ_{0}, ..., σ_{k−1} are defined by

σ_{r}(i) = \{ \begin{matrix}i + p^{r} & if i ≡ 1, ..., (p − 1)p^{r} mod p^{r+1} \\ i − (p − 1)p^{r} & if i ≡ (p − 1)p^{r} + 1, ..., p^{r+1} mod p^{r+1} . \end{matrix}

A_{0}, ..., A_{k−1} ∈ U(p^{k}) denote diagonal matrices with entries

(A_{r})_{ii} = ζ^{[(i−1)/p^{r}]} ζ = exp(2πi/p) ,

and B_{0}, ..., B_{k−1} are the permutation matrices of σ_{0}, ..., σ_{k−1}.

Definition. For each prime p and k ≥ 0 the subgroups E_{p^{k}} ⊂ Σ_{p^{k}} and Γ_{p^{k}}^{U} are defined by setting

E_{p^{k}} = ⟨σ_{0}, ..., σ_{k−1}⟩ ≅ (ℤ/p)^{k}  and
Γ_{p^{k}}^{U} = ⟨λ · I, A_{r}, B_{r} : λ ∈ S^{1}, 0 ≤ r ≤ k − 1⟩ ⊂ U(p^{k}) .

6.1 Remark. The matrices A_{r} and B_{s} satisfy the commutator relations

[A_{r}, B_{s}] = [A_{r}, A_{s}] = [B_{r}, B_{s}] = 1 for r ≠ s and [B_{r}, A_{r}] = ζ · I .

Also Γ_{p^{k}}^{U} sits in the central extension

1⟶S^{1}⟶Γ_{p^{k}}^{U}⟶(ℤ/p)^{2k}⟶1

and in the splitting extension

1⟶S^{1} × (ℤ/p)^{k}⟶Γ_{p^{k}}^{U}⟶(ℤ/p)^{k}⟶1 ,

where S^{1} × (ℤ/p)^{k} = Γ_{p^{k}}^{U} ∩ T_{U(p^{k})}.

A subgroup H of U(n) is called irreducible, if the associated representation is irreducible.

6.2 Theorem ([O]).

(1) For any n ≥ 1, an irreducible p-toral subgroup P ⊂ U(n) is stubborn iff it is conjugate to an iterated wreath product of the form

P = Γ_{p^{k}}^{U} ≀ E_{q_{1}} ≀ ... ≀ E_{q_{r}} ,

where q_{i} = p^{t_{i}} and n = p^{k+t_{1}+...+t_{r}}.

(2) If P ⊂ U(n) is an arbitrary p–stubborn subgroup then it is conjugate to a subgroup of the form P_{1} ×...×P_{s}, where each P_{i} is an irreducible subgroup of U(n_{i}) and where ∑ n_{i} = n.

(3) A p–stubborn subgroup P ⊂ U(n) is irreducible iff C_{U(n)}(V) = U(n), where V ⊂ Z(P) is the maximal p–elementary subgroup of the center of P.

(4) A p–toral subgroup P ⊂ SU(n) is stubborn iff ⟨P, Z(U(n))⟩ is stubborn in U(n) iff P = P′ ∩ SU(n) for some p–stubborn subgroup P′ ⊂ U(n).

Proof. Everything is proved in [O], but the third statement. That follows from the first two properties. □

Remark. This theorem describes standard models for every conjugacy class of p–stubborn subgroups in U(n) and SU(n).

6.3 Proposition ([J–M–O 1, proposition 1.6]).

(1) For connected Lie groups G and H, R_{p}(G × H) ≅ R_{p}(G) × R_{p}(H).

(2) For a finite covering A⟶G̃⟶G of compact connected Lie groups, we have R_{p}(G̃) ≅ R_{p}(G). In particular, P̃ ⊂ G̃ is p–stubborn iff P = P̃/(P̃ ∩ A) ⊂ G is p–stubborn.

(3) If P ⊂ G is p–stubborn then C_{G}(P) = Z(P).

Information about centralizers of p-toral subgroups of mod-p subgroups is provided by the next lemma.

6.4 Lemma. Let ρ : G⟶\overline{G} be a mod–p isomorphism of compact connected Lie groups.

(1) For any p-toral subgroup \overline{P} ⊂ \overline{G} there exists a p-toral subgroup P ⊂ G,

which is mod–p isomorphic to \overline{P}.
(2) C_{G}(P)⟶C\overline{G}(\overline{P}) is a mod–p isomorphism.
(3) Z(P)⟶Z(\overline{P}) is a mod–p isomorphism.

Proof. Q := ρ^{−1}(\overline{P}) is toral, i.e a finite extension of a torus. The p–toral Sylow subgroup P of Q maps onto \overline{P}.

C_{G}(P)⟶C\overline{G}(\overline{P}) is a mod–p subgroup. Let l be the order of the kernel of ρ, let x ∈ P, and let g ∈ ρ^{−1}(C\overline{G}(\overline{P})). Then gx^{l}g^{−1} = (gxg^{−1})^{l} = (xa)^{l} = x^{l}, where a ∈ ker(ρ). The map P ⟶P : x ↦ x^{l} is a surjection because l is coprime to p. That is C_{G}(P)⟶C\overline{G}(\overline{P}) is a surjection which proves (2).

(3) follows analogously by replacing G by P. □

The following lemma is the key to get control over the p–stubborn subgroups of all p–convenient simple simply connected Lie groups, in particular of the p–convenient exeptional Lie groups.

6.5 Lemma. Let G be a compact connected Lie group and P ⊂ G be p–stubborn. Then P is p-stubborn in C_{G}(Z(P)).

Proof. P is contained in K := C_{G}(Z(P)). The normalizer N_{G}(P) of P in G normalizes Z(P) and K. The intersection N_{G}(P) ∩ K = N_{K}(P) is a normal subgroup in N_{G}(P). Let Q be the intersection of all p–toral Sylow subgroups of N_{K}(P). A conjugation by an element of N_{G}(P) permutes the p–toral Sylow subgroups of N_{K}(P). Thus Q is normal in N_{G}(P). If Q/P is not trivial, i.e P is not stubborn in K, then P is also not stubborn in G. □

Now we are able to describe the p–stubborn subgroups of the simple simply connected Lie groups. By im(ρ) we denote the image of a homomorphism ρ : H⟶G between groups.

6.6 Proposition. Let P ⊂ G be a p–stubborn group of a p–convenient simple simply connected Lie group G.

(1) If G ≠ SU(n), G_{2}, there exist a mod–p subgroup

ρ : H ≅ U(n_{1})×, ..., ×U(n_{k})⟶G

and a p–stubborn subgroup P′ = P_{1}×, ..., ×P_{k}, such that P_{i} is irreducible in U(n_{i}), im(ρ) = C_{G}(Z(P)), and im(ρ|_{P′}) = P.

(2) For G = G_{2} and p = 3, P is stubborn in SU(3), and Z(P) = Z(SU(3)). For G = G_{2} and p ≥ 5 we have P = T_{G_{2}}.

Proof. Let ρ : K⟶G be a mod–p subgroup of G satisfying the conditions of lemma 5.2 (2), and let \overline{K} = im(ρ). Since the Weylgroup index [W_{G} : W_{K}] is coprime to p, the groups G and \overline{K} have isomorphic and G and K mod–p isomorphic p–toral Sylow subgroups. Therefore any p-stubborn subgroup P ⊂ G sits in \overline{K} up to conjugation. By lemma 6.4 there exists a p–toral subgroup P′ ⊂ K with ρ(P′) = P and ρ(Z(P′) = Z(P). In the sequence

Z(P) = C_{G}(P) ⊇ C\overline{K}(P) ⊇ Z(P)

all the groups are equal. The first identity follows from lemma 6.3. We get Z(P) ⊇ Z(\overline{K}) and C_{G}(Z(P)) = C\overline{K}(Z(P)). By lemma 6.5, P is p–stubborn in C\overline{K}(Z(P)), and by proposition 6.3, P′ is p–stubborn in C_{K}(Z(P′)).

For G ≠ SU(n), G_{2}, the group K is a product of unitary groups. The centralizer of any subgroup of such a group always is of the same type. Thus C_{K}(Z(P′)) ≅ U(n_{1}) × ... × U(n_{r}). By theorem 6.2 we have P′ = P_{1}×, ..., ×P_{r} and P_{i} ⊂ U(n_{i}) is irreducible p–stubborn because C_{U(n_{i})}(Z(P_{i})) = U(n_{i}).

For G = G_{2} and p ≥ 5 we have P = T_{G_{2}} because (p, |W_{G_{2}}|) = 1. For G = G_{2} and p = 3 we have K = \overline{K} = SU(3). If C_{G_{2}}(Z(P)) = C_{SU(3)}(Z(P)) ≠ SU(3) this centralizer is isomorphic to U(2) or T_{G_{2}} = T_{SU(3)}. In both cases P = T_{G} = T_{SU(3)}, because |W_{C_{G_{2}} (Z(P))}| ≤ 2. This is a contradiction because W_{G_{2}} has a normal subgroup of order 3. Hence P is p–stubborn in SU(3).

Theorem 6.2 finishes the proof. □

For the proof of theorem 1.2, we need certain properties of p–stubborn groups. With the help of the last proposition we can do this for a special class of compact connected Lie group, namely pseudo simply connected Lie groups.

6.7 Proposition. Let G be a p–convenient pseudo simply connected Lie groupand T_{G}↪G a fixed maximal torus. In the conjugacy class (P) of any p–stubborn subgroup of G exists a representative i : P ↪N(T_{G})↪G satisfying the following conditions:

(1) P_{T} := P ∩ T_{G} = S × V, where S is a torus and V an elementary abelian p–group, and Z(P) ⊂ P_{T}.

(2) C_{G}(P_{T}) = T_{G}.

(3) For any extension α : P ⟶G of i : P_{T} ⟶G, we have C_{G}(α) = i(Z(P)).

(4) The canonical map

π_{0}(map(BP, BG_{p}^{∧})_{Bα|BP_{T} =Bi}) ⟶ Hom(H^{∗}(BG; ℤ/p), H^{∗}(BP; ℤ/p))

is an injection.

Remark. By map(BP, BG_{p}^{∧})_{Bα|_{BP_{T}} =Bi} we denote the components of map(BP, BG_{p}^{∧}) given by maps Bα : BP ⟶BG_{p}^{∧}, such that Bα|_{BP_{T}} ≃ Bi. The homotopy classes of these extensions are classified by obstructions in the groups H^{∗}(BP/P_{T}; π_{∗}(map(BP_{T} , BG_{p}^{∧})_{B}i) (lemma 3.13). By (1) of the above theorem follows that there is only one obstruction group, namely H^{2}(BP/P_{T}; π_{2}(map(BP_{T} , BG_{p}^{∧})_{B}i)) ≅ H^{∗}(BP/P_{T}; π_{2}(BT_{G}_{p}^{∧})).

In the rest of this chapter we will prove this proposition in several steps.

Step 1. Let G = U(p^{k}) and P = Γ_{p^{k}}^{U}.

Proof of (1) and (2). For abbreviation we set Γ := Γ_{p^{k}}^{U}. The condition (1) follows from remark 6.1, and Γ_{T} ≅ (ℤ/p)^{k} × S^{1}. The j-th coordinate is generated by A_{j−1}, and the last coordinate is the the center of U(p^{k}). The inclusion i : Γ_{T} ⟶T_{U(p^{k})} is a splitting in 1–dimensional representations of the p^{k}–dimensional representation Γ⟶U(p^{k}). The j-th summand is given by the homomorphism i_{j} : Γ_{T} ⟶S^{1} defined by

i_{j}(a_{1}, ..., a_{k}, λ) = ∏_{s} ζ^{[(j−1)/p^{s}]·a_{s}} · λ , ζ = exp(2πi/p) .

All these 1–dimensional representation are pairwise nonisomorphic (see the next lemma), which implies that C_{U(p^{k})}(Γ_{T}) = T_{U(p^{k})}. □

6.8 Lemma. The two homorphism i_{j}, i_{l} : Γ_{T} ⟶S^{1} are equal iff j = l.

Proof. If i_{j} = i_{l} we have for all (a_{1}, ..., a_{k}) ∈ (ℤ/p)^{k}

∏_{s=1}^{k} ζ^{[(j−1)/p^{s−1}]·a_{s}} = ∏_{s=1}^{k} ζ^{[(l−1)/p^{s−1}]·a_{s}}

respectively

∑_{s=1}^{k}[(j − 1)/p^{s−1}] · a_{s} ≡ ∑_{s=1}^{k}[(l − 1)/p^{s−1}] · a_{s} mod p .

This condition is satisfied iff [(j − 1)/p^{s−1}] ≡ [(l − 1)/p^{s−1}] mod p for 1 ≤ s ≤ k. This implies j = l, because j, l ≤ k. □

Proof of (3) and (4). Extensions Bα : BΓ⟶BU(p^{k})_{p}^{∧} of Bi : BΓ_{T} ⟶BU(p^{k}) are classified by obstructions in the groups H^{r}(Γ/Γ_{T}; π_{r}(map(BΓ_{T} , BU(p^{k})_{p}^{∧})) (see lemma 3.12). Because of theorem 3.9 and (1), there is only one obstruction group H^{2}(Γ/Γ_{T}; π_{2}(BT_{U(p^{k})})) ≅ H^{2}(Γ/Γ_{T}; (ℤ_{p}^{∧})^{p^{k}}). Here Γ/Γ_{T} ≅ E_{p^{k}} ≅ (ℤ/p)^{p^{k}} acts on ℤ_{p}^{∧}^{p^{k}} via the inclusion E_{p^{k}}↪Σ_{p^{k}} and the permutation representation. In particular Γ/Γ_{T} acts transitively on the standard basis of ℤ_{p}^{∧}^{p^{k}}. The canonical map ℤ_{p}^{∧}[Γ/Γ_{T}]⟶ℤ_{p}^{∧}^{p^{k}} : 1 ↦ e_{1} is an isomorphism, where e_{1} is the first basis vector of the standard basis. Hence the obstruction group H^{2}(Γ/Γ_{T}; (ℤ_{p}^{∧})^{p^{k}}) vanishes and the two extensions α and β are conjugate. This completes the proof of (3) and (4) as well as of step 1. □

Step 2. Let P ⊂ G = SU(p) be a standard p–stubborn subgroup.

Proof. By theorem 6.2 the only p–stubborn subgroups of SU(p) are T_{SU(p)} (for p ≥ 5), P_{1} = Γ_{p}^{U} ∩ SU(p) and P_{2} = (S^{1} ≀ ℤ/p) ∩ SU(p). We have P_{1}_{T} = (ℤ/p)^{2} because Γ_{p}^{U}_{T} = S^{1} × ℤ/p and because C_{SU(p)}(P_{1}_{T}) = T_{SU(p)}. For P_{2} we get P_{2}_{T} = T_{SU(p)} and C_{SU(p)}(P_{2}_{T}) = T_{SU(p)}. To calculate the extensions of BP_{i}_{T} ⟶BSU(p) we have only to consider the obstruction group H^{2}(ℤ/p; ℤ_{p}^{∧}^{p−1}), where ℤ/p acts on ℤ_{p}^{∧}^{p−1} via the action of Σ_{p} on π_{2}(BT_{SU(p)}). The exact sequence ℤ_{p}^{∧}^{p−1}⟶ℤ_{p}^{∧}^{p}⟶ℤ_{p}^{∧} of ℤ/p–modules induces a long exact sequence

... ⟶ H^{1}(ℤ/p; ℤ_{p}^{∧}) ⟶ H^{2}(ℤ/p; ℤ_{p}^{∧}^{p−1}) ⟶ H^{2}(ℤ/p; ℤ_{p}^{∧}^{p}) ⟶ ... .

The left and the right term vanish. Hence there exists only one extension and C_{SU(p)}(P_{i}) = Z(P_{i}). For the p–stubborn subgroup T_{SU(p)} all the statements are obvious. □

For the next step, namely the case of an arbitrary p–stubborn subgroup of U(p^{k}), we need some technical lemmas.

6.9 Lemma. Let P ⊂ U(p^{k}) be a standard irreducible p–stubborn subgroup. Then P_{T} is normal in P, the qoutient P/P_{T} is an iterated wreath product of elementary abelian p–groups, and the homomorphism P ⟶P/P_{T} splits.

Proof. By theorem 6.2, P = Γ_{p^{r_{0}}}^{U} ≀ E_{p^{r_{1}}} ≀ .... ≀ E_{p^{rs}} and P/P_{T} = Γ_{p^{r_{0}}}^{U} /Γ_{p^{r_{0}}}^{U}_{T} ≀ E_{p^{r_{1}}} ≀ .... ≀ E_{p^{rs}} which is an iterated wreath product of elementary abelian p–groups. The homomorphism Γ_{p^{r_{0}}}^{U} ⟶Γ_{p^{r_{0}}}^{U} /Γ_{p^{r_{0}}}^{U}_{T} splits by remark 6.1. Thus P ⟶P/P_{T} splits. □

6.10 Lemma. Let α : Q⟶G be a homomorphism of an iterated wreath product of elementary abelian p–groups into a compact Lie group. If H^{∗}(Bα; ℤ/p) = 0, then α is constant.

Proof. By theorem 3.1 the statement is true for elementary abelian p–groups. Now let Q = Q_{0} ≀ (ℤ/p)^{k}. By induction hypothesis the homomorphism α|_{Q_{0}×...×Q_{0}} : Q_{0} × ... × Q_{0}⟶G is constant restricted to one factor, and hence constant on the product. α splits over \overline{α} : (ℤ/p)^{k}⟶G which is trivial too, because H^{∗}(B\overline{α}; ℤ/p) = 0. □

Step 3. P ⊂ U(p^{k}) is a standard irreducible p–stubborn subgroup.

Proof. By proposition 6.2, P = P′ ≀ E_{p^{r}}, where P′ ⊂ U(p^{k−r}) is also a standard irreducible p–stubborn subgroup. Therefore P_{T} = (P′_{T})^{p^{r}} is a product of a torus and an elementary abelian p–group. Since the splitting of P_{T} into the product comes from the homomorphism U(p^{k−r})^{p^{r}}⟶U(p^{k}), we have C_{U(p^{k})}(P_{T}) = (C_{U(p^{k−r})}(P′_{T}))^{p^{r}} = (T_{U(p^{k−r})})^{p^{r}} = T_{U(p^{k})} by induction hypothesis. This proves the first two parts.

Now let α, β : P ⟶U(p^{k}) be two homomorphisms, such that H^{∗}(Bα; ℤ/p) = H^{∗}(Bβ; ℤ/p) and Bα|_{BP_{T}} = Bβ|_{BP_{T}} = Bi|_{BP_{T}}, where i : P ⟶U(p^{k}) is the standard inclusion. Let

α′, β′ : (P′)^{p^{r}}⟶U(p^{k})

be the restrictions of α and β.

We get Z(P′) = C_{U(p^{k−r})}(P′) = Z(U(p^{k−r})) = S^{1} and (U(p^{k−r}))^{p^{r}} = C_{U(p^{k})}(V), where V ⊂ (S^{1})^{p^{r}} = (Z(P′))^{p^{r}} ⊂ P_{T} is the maximal elementary abelian p– subgroup. The homorphism α′ and β′ split over

α′′, β′′ : P′^{p^{r}}⟶(U(p^{k−r}))^{p^{r}} .

The induced maps in cohomology are determined by the application of the Lannes functor T_{Bi^{∗}}^{V} (remark 3.4), in particular H^{∗}(Bα′′) = H^{∗}(Bβ′′).

The homomorphism α′′ can be described by a (p^{r} × p^{r})–matrix with homomorphisms α_{ij} : P′_{i}⟶U(p^{k−r})_{j} as entries. The indices i, j denote the components in the products. Analogously β′′ is described by a matrix B with entries β_{i,j} : P′_{i}⟶U(p^{k−r})_{j}. By induction hypothesis Bα_{ii} ≃ Bβ_{ii}, i.e α_{ii} and β_{ii} are conjugate (theorem 3.6), and

C_{U(p^{k−r})_{i}}(α_{ii}) = C_{U(p^{k−r})_{i}}(β_{ii}) = Z(P′_{i}) = Z(U(p^{k−r})_{i}) = S_{i}^{1} .

We can assume that α_{ii} = β_{ii}. Because P′_{i} and P′_{j} commute, the homomorphisms α_{ij}, β_{ij} split over the inclusion S_{j}^{1} = Z(U(p^{k−r})_{j})⟶U(p^{k−r})_{j}. Since α′′|_{P_{T}} = β′′|_{P_{T}}, the homomorphisms α_{ij} · β_{ij}^{−1} : P′_{i}⟶S_{j}^{1} factor over homomorphisms ρ_{ij} : (P′/P′_{T})_{i}⟶S_{j}^{1}. and fit together to a map ρ : (P′/P′_{T})^{p^{r}}⟶(S^{1})^{p^{r}}. β′′ can be described by the composition

(P′)^{p^{r}} ⟶^{Δ} (P′)^{p^{r}} × (P′/P′_{T})^{p^{r}} ⟶^{α′′×ρ} U(p^{k−r})^{p^{r}} × (S^{1})^{p^{r}} ⟶^{μ} U(p^{k−r})^{p^{r}},

where Δ is the diagonal composed with taking the quotient, and where μ is given by the canonical map S^{1} × U(p^{k−r})⟶U(p^{k−r}). Because H^{∗}(Bα′′) = H^{∗}(Bβ′′) and because ρ_{ii} is constant, the composition

H^{∗}(BU(p^{k−r})^{p^{r}}) ⟶^{Bρ^{∗}} H^{∗}((BP′/P′_{T})^{p^{r}}) ⟶ H^{∗}(BP′^{p^{r}})

is trivial. By lemma 6.9 and lemma 6.10 the homorphism ρ is constant. That is that α′′ = β′′, and C_{U(p^{k})}(α′) = C_{U(p^{k−r})^{p^{r}}} (α′′) = (Z(U(p^{k−r}))^{p^{r}} = (S^{1})^{p^{r}}.

The extensions Bα : BP = BP′ ≀ E_{p^{r}}⟶BU(p^{k})_{p}^{∧} of Bα′ : BP′^{p^{r}}⟶BU(p^{k})_{p}^{∧} are classified by the obstructions in the groups

H^{∗}(P/(P′^{p^{r}}); π_{∗}(map(BP′^{p^{r}}, BU(p^{k})_{p}^{∧})_{Bα′_{p}^{∧}})

(lemma 3.13). By theorem 3.8

map(BP′^{p^{r}}, BU(p^{k})_{p}^{∧})_{Bα′} = BC_{U(p^{k})}(α′)_{p}^{∧}
= BC_{U(p^{k−r})^{p^{r}}} (α′′)_{p}^{∧}
= (BZ(U(p^{k−r})_{p}^{∧}))^{p^{r}}
= (BS^{1}_{p}^{∧})^{p^{r}} .

The only obstruction group H^{2}(E_{p^{r}}; ℤ_{p}^{∧}^{p^{r}}) vanishes because ℤ_{p}^{∧}^{p^{r}} is a free E_{p^{r}}–module (see the proof of step 1). There exists one extension α of α′, and the centralizer C_{U(p^{k})}(P) is given by the fixed–point set

C_{U(p^{k})}(P) = (C_{U(p^{k−r})^{p^{r}}} (P′^{p^{r}}))^{E_{p^{r}}} = (S^{1}^{p^{r}})^{E_{p^{r}}} = S^{1} = Z(P) . □

Step 4. Let P ⊂ G be a p–stubborn subgroup of a pseudo simply connected Lie group.

Proof. According to proposition 6.6 we choose a mod–p subgroup

ρ : H = H_{1} × ... × H_{s}⟶G

and a p–stubborn subgroup P′ = P_{1} × ... × P_{s}, such that H_{i} is isomorphic to a unitary group or to SU(p), P_{i} ⊂ H_{i} is p–stubborn, P_{i} ⊂ H_{i} is irreducible if H_{i} is a unitary group, ρ(P′) = P, and H⟶\overline{H} := C_{G}(ρ|_{V}) is a mod–p isomorphism of groups, where V ⊂ Z(P) = Z(\overline{H}) is the maximal elementary abelian p–subgroup. Up to conjugacy P_{i} ⊂ H_{i} is the standard model of theorem 6.2. P_{T} is a product of a torus and an elementary abelian p–group, which is the statement of (1).

Because P_{T} ⊃ Z(P) ⊃ V, step 3 induces the sequence

T_{H} = ∏_{i} T_{H_{i}} = ∏_{i} C_{H_{i}}(P_{i}) = C_{H}(P′_{T}) ⟶ C\overline{H}(P_{T}) = C_{G}(P_{T})

which is a mod–p isomorphism by lemma 6.4. Thus, C_{G}(P_{T}) = T_{G} as required in (2).

Let α_{G}, β_{G} : P ⟶G be two homomorphisms such that

H^{∗}(Bα_{G}; ℤ/p) = H^{∗}(Bβ_{G}; ℤ/p) and α_{G}|_{P_{T}} = β_{G}|_{P_{T}} = i|_{P_{T}} .

Both, α_{G} and β_{G}, split over homomorphisms α\overline{H}, β\overline{H} : P ⟶\overline{H} = C_{G}(V). Applying the Lannes functor T_{Bi^{∗}}^{V} we get H^{∗}(Bα\overline{H}) = H^{∗}(Bβ\overline{H}) (remark 3.4). By lemma 6.4 there exist lifts α_{H}, β_{H} : P′⟶H of α\overline{H} and β\overline{H}. Of course H^{∗}(Bα\overline{H}) = H^{∗}(Bα_{H}) and H^{∗}(Bβ\overline{H}) = H^{∗}(Bβ_{H}). Now we are in a position to apply the methods of step 3. This implies Bα_{H} ≃ Bβ_{H}, Bα\overline{H} ≃ Bβ\overline{H}, and hence Bα_{G} ≃ Bβ_{G}. This proves (4). The condition (3) is satisfied, because

Z(P′) = C_{H}(α_{H}) ⟶ C\overline{H}(α\overline{H}) = C_{G}(α_{G})

and

Z(P′) ⟶ Z(P)

are mod–p isomorphisms by lemma 6.4 again and step 3. This finishes the proof of step 4 as well as the proof of proposition 6.7. □

7. Maximal tori and Weyl groups.

In [R] the concept of maximal tori and Weyl groups of a finite loop space is introduced (see also [NS 1,2,3], [R-S]). An algebraic version of this homotopy theoretical definition is given in [D-W]. For our purpose the following definition is convenient.

7.1 Definition. A maximal torus of a space X with the mod–p type of BG is a map

f_{T} : BT_{X} ⟶ X or f_{T} _{p}^{∧} : BT_{X} _{p}^{∧} ⟶ X ,

where T_{X} is a torus, such that (1) rank T_{X} = rank G

(2) H^{∗}(BT_{X}; ℤ/p) is a finite generated module over H^{∗}(X; ℤ/p).

The Weylgroup of X is the group

W_{X} := {[w] : BT_{X} _{p}^{∧}⟶BT_{X} _{p}^{∧} : f_{T} _{p}^{∧} ∘ w ≃ f_{T} _{p}^{∧}}

of homotopy classes of self maps of BT_{X}.

The definition of the Weyl group might depend on the chosen maximal torus f_{T} : BT_{X}⟶X. If p is odd or G is 2–convenient, similar methods as in the proof of [N-S 1; proposition 2.4] can be applied to show that any two maximal tori

f′_{T} , f′′_{T} : BT_{X} ⟶ X

fit into a commutative diagram

BT_{X} _{p}^{∧} ⟶^{g} BT_{X} _{p}^{∧}
f′_{T} ↓  ↓^{f′′_{T}}
X  X ,

where g is an equivalence. Therefore W_{X} is well defined up to ‘conjugation’.

Following ideas of Dwyer, Miller, and Wilkerson [D-M-W 2] we will construct a maximal torus of X. This construction we recall in detail because later we use similar methods in more complicated situations.

Now we assume that G satisfies the condition H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W_{G}}. Let X be a p–complete space with the mod–p type of BG. By theorem 3.1 the composition

H^{∗}(X; ℤ/p)⟶^{ϕ}H^{∗}(BG; ℤ/p)⟶H^{∗}(BT_{G}; ℤ/p)⟶H^{∗}(BV_{G}; ℤ/p) .

has a topological realization f_{V} : BV_{G}⟶X. Applying the Lannes–functor we get the diagram

T_{f_{V}^{∗}}^{V_{G}}H^{∗}(X; ℤ/p)  ⟶^{≅} T_{Bi_{V}^{∗}}^{V_{G}} H^{∗}(BG; ℤ/p)
↓  ↓
H^{∗}(map(BV_{G}, X)_{f_{V}}; ℤ/p))  H^{∗}(map(BV_{G}, BG)_{Bi_{V}}) ⟶^{≅} H^{∗}(BT_{G}; ℤ/p) .

The last isomorphism comes from theorem 3.5 and lemma 4.5. The upper horizontal arrow is an isomorphism. The space BT_{G} is 1–connected. Thus, T_{V_{G}}^{f_{V}^{∗}} H^{∗}(X; ℤ/p) vanishes in degree 1, and the left vertical arrow is also an isomorphism (theorem 3.2). The mapping space map(BV_{G}, X)_{f_{V}} is p–complete by theorem 3.3. Among the p–complete spaces, BT_{G}_{p}^{∧} is determined up to homotopy by its mod–p cohomology. The evaluation map e : BT_{X} _{p}^{∧} := map(BV_{G}, X)_{f_{V}} ⟶X, defined by choosing a suitable base point in BV_{G}, induces a map f_{T} : BT_{X} _{p}^{∧}⟶X which plays the role of a maximal torus of X. Here T_{X} is a torus of the same rank as T_{G}.

The composition BV_{G} ×BV_{G}⟶BV_{G}⟶X of the multiplication of the H–space BV_{G} and f_{V} has as adjoint the map

Bj : BV_{G}⟶map(BV_{G}, X)_{f_{V}} = BT_{X} .

There exists an equivalence BT_{G}_{p}^{∧}⟶BT_{X} _{p}^{∧} which fits into the diagram

BV_{G} = BV_{G}
↓  ↓
BT_{G} ⟶ BT_{X} _{p}^{∧}
↓  ↓
BG −−→ X ,

where the upper square commutes up to homotopy and the lower square in mod–p cohomology. The dotted arrow means a map which only exists in cohomology, in this case in mod–p cohomology.

The Weyl group W_{G} acts on BV_{G} and hence on map(BV_{G}, X). The action on the components of map(BV_{G}, X) is determined by pure cohomological properties, the action on Hom_{A_{p}}(H^{∗}(X; ℤ/p), H^{∗}(BV_{G}; ℤ/p)) (theorem 3.1). Up to homotopy W_{G} fixes the map Bi : BV_{G}⟶BG and therefore the component map(BV_{G}, X)_{f_{V}} of map(BV_{G}, X). We get an action of W_{G} on BT_{X} _{p}^{∧} such that f_{T} ∘ w ≃ f_{T} for every w ∈ W_{G}. All the spaces in the above diagram carry an W_{G}–action ( BG and X the trivial action), and the maps induce equivariant maps in mod–p cohomology.

We collect the results in the following proposition

7.2 Proposition. Let G be a compact connected Lie group, and let X be a p– complete space with the mod–p type of BG. If H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W G} then there exist a maximal torus f_{T} : BT_{X} _{p}^{∧}⟶X, a Weyl group action of W_{G} on BT_{X}, and a mod–p equivalence BT_{G}⟶BT_{X} _{p}^{∧}, equivariant in mod–p cohomology, such that the diagram

BT_{G} ⟶ BT_{X} _{p}^{∧}
↓  ↓
BG −−→ X

commutes in mod–p cohomology. Moreover, H^{∗}(X; ℤ_{p}^{∧}) ≅ H^{∗}(BT_{X}; ℤ_{p}^{∧})^{W_{G}}.

Proof. Only the last statement needs a comment. It follows from the isomorphism H^{∗}(X; ℤ/p) ≅ H^{∗}(BT_{X}; ℤ/p)^{W_{G}} and the Nakayama lemma. □

Remark. All but the last statement is due to Dwyer, Miller and Wilkerson and is true under the weaker assumption C_{G}(V_{G}) = T_{G}, in particular, for all compact connected Lie groups at odd primes [D-M-W 2].

To prove theorem 1.1 we have to study the W_{G}–modules

L^{∗}T_{X} _{p}^{∧} = H^{2}(BT_{X}; ℤ_{p}^{∧}) and L^{∗}T_{G}_{p}^{∧} = H^{2}(BT_{G}; ℤ_{p}^{∧}) .

7.3 Definition. A ℤ_{p}^{∧}[W_{G}]–module L is called p–reducible if

P (L)^{W_{G}} ⊗ ℤ/p ≅ P (L ⊗ ℤ/p)^{W_{G}} ,

where P (−) denotes the symmetric part of the tensor algebra.

Proof of theorem 1.1. The proof is based on results of the sections 8 and 9. By proposition 7.2 the modules L^{∗}T_{X} _{p}^{∧} and L^{∗}T_{G}_{p}^{∧} are p–reducible, and there mod–p reduction are isomorphic as ℤ/p[W_{G}]–modules. Let G be p–convenient.

For p ≠ 2 and G simply connected or pseudo simply connected, for p = 2 and G simply connected, and for (p, |W_{G}|) = 1 we can apply proposition 8.1. If G is pseudo projective we can use proposition 9.5, and if G is a product of unitary groups we can apply proposition 9.9. This proves that in these cases L^{∗}T_{G}_{p}^{∧} ≅ L^{∗}T_{X} _{p}^{∧} as ℤ_{p}^{∧}[W_{G}]–modules.

In the general case there exists a compact connected Lie group H, such that BH has the mod–p type of BG and X, and such that L^{∗}T_{H} _{p}^{∧} ≅ L^{∗}T_{X} _{p}^{∧}. This follows from proposition 9.2.

We can identify BT_{G}_{p}^{∧}, respectively BT_{H} _{p}^{∧}, and BT_{X} _{p}^{∧} via this map. This proves that X has the p–adic type of BG respectively BH. By the Nakayama lemma follows that H^{∗}(X; ℤ_{p}^{∧}) ≅ H^{∗}(BT_{H}; ℤ_{p}^{∧})^{W_{G}}. □.

8. p–adic Weyl group representations, for simply connected Lie groups. The action of the Weyl group W_{G} on the maximal torus BT_{G} of a compact

connected Lie group G gives a ℤ_{p}^{∧}[W_{G}]–module L^{∗}T_{G}_{p}^{∧} := H^{2}(BT_{G}; ℤ_{p}^{∧}). In this chapter we will study the p–adic liftings of the associated mod–p representation H^{2}(BT_{G}; ℤ/p) for a simply connected Lie group G. The general case is discussed in the next chapter.

Definitions and Notation. A ℤ_{p}^{∧}[W_{G}]–module L is called simply connected, if L^{W_{G}} = 0. The module L⊗ℤ/p^{k} is denoted by L/p^{k}, and L/p^{k}^{∗} = Hom(L/p^{k}, ℤ/p^{k}) is the dual module of L/p^{k} as a ℤ/p^{k}[W_{G}]–module. Gl(L/p^{k}) denotes the isomorphisms and M(L/p^{k}) the endomorphisms of L/p^{k}. The symmetric part of the tensor algebra of L is given by P (L).

8.1 Proposition. Let G be a compact connected Lie groupand L be a torsion free ℤ_{p}^{∧}[W_{G}]–module. Let

\overline{α} : L^{∗}T_{G}/p ⟶ L/p

be a ℤ_{p}^{∧}[W_{G}]–isomorphism. Then there exists a ℤ_{p}^{∧}[W_{G}]–isomorphism

α : L^{∗}T_{G}_{p}^{∧} ⟶ L ,

which is a lift of \overline{α}, if one of the following conditions is satisfied:

(1) p is odd, and G is p–convenient and simply connected or pseudo simply connected.

(2) p = 2, L is p–reducible, and G is p–convenient and simply connected.

(3) (p, |W_{G}|) = 1.

For the proof we need several lemmas.

A ℤ_{p}^{∧}[W_{G}]–module L, which is free as ℤ_{p}^{∧}–module of rank n, induces a homomorphism

ρ_{L} : W_{G} ⟶ Gl(L) ≅ Gl(n, ℤ_{p}^{∧}).

We denote by ρ_{L}(k) the homorphism associated to the ℤ/p^{k}[W_{G}]–module L/p^{k}. The kernel of the restriction Gl(n, ℤ/p^{k+r})⟶Gl(n, ℤ/p^{k}) is the group

ker := {id + p^{k}A | A ∈ M(n, ℤ/p^{r})} ,

where M(n, ℤ/p^{r}) ≅ M(L/p^{r}) ≅ Hom(L/p^{r}, L/p^{r}) ≅ L/p^{r}^{∗} ⊗ L/p^{r} is the group of n × n matrices over ℤ/p^{r}. The multiplication in ker is given by

(id + p^{k}A)(id + p^{k}B) = (id + p^{k}(A + B + p^{k}AB)) .

Thus, if r ≤ k, the map A ↦ id+p^{k} ·A induces an isomorphism M(n, ℤ/p^{k}) ≅ ker. The lifts in the diagram

Gl(n, ℤ/p^{k+r})
↓
W_{G} ⟶^{ρ_{L}(k)} Gl(n, ℤ/p^{k})

are classified up to conjugation by the obstruction group

H^{1}(W_{G}; Hom(L/p^{r}, L/p^{r})) ,

where W_{G} acts on Hom(L/p^{r}, L/p^{r}) via conjugation and the homomorphism

W_{G} ⟶^{ρ_{L}(r)} Gl(n, ℤ/p^{r}). To prove lemma 8.1 we have to calculate this obstruction group.

Let G and H be two compact connected Lie groups. Then L^{∗}T_{G}_{p}^{∧} × L^{∗}T_{H} _{p}^{∧} as well as L^{∗}T_{G}_{p}^{∧} and L^{∗}T_{H} _{p}^{∧} are W_{G} × W_{H}–modules, where W_{H} acts trivially on L^{∗}T_{G}_{p}^{∧} and W_{G} trivially on L^{∗}T_{H} _{p}^{∧}.

8.2 Lemma. Let G and H be two connected Lie groups.

H^{1}(W_{G} × W_{H}; Hom(L^{∗}T_{G}/p^{k}, L^{∗}T_{G}/p^{k})) (1)

≅ H^{1}(W_{G}; Hom(L^{∗}T_{G}/p^{k}, L^{∗}T_{G}/p^{k}))
⊕ H^{1}(W_{H}; ℤ/p^{k}) ⊗ Hom(L^{∗}T_{G}/p^{k}, L^{∗}T_{G}/p^{k})^{W_{G}}

If p is odd the second summand vanishes.

H^{1}(W_{G} × W_{H}; Hom(L^{∗}T_{G}/p^{k}, L^{∗}T_{H}/p^{k})) (2)

≅ (LT_{G}/p^{k})^{W_{G}} ⊗ H^{1}(W_{H}; L^{∗}T_{H}/p^{k})
⊕ H^{1}(W_{G}; LT_{G}/p^{k}) ⊗ (L^{∗}T_{H}/p^{k})^{W_{H}} .

If G and H are p–convenient and simply connected or if p is odd and G is p– convenient and pseudo simply connected then both summands vanish.

(3)  H^{1}(Σ_{n}; Hom(L^{∗}T_{U(n)}/p^{k},L^{∗}T_{U(n)}/p^{k}))
≅ H^{1}(Σ_{n−1}; ℤ/p^{k}) ⊕ H^{1}(Σ_{n−2}; ℤ/p^{k})

(4) For n ≥ 3 we have

H^{1}(Σ_{n}; Hom(L^{∗}T_{SU(n)}/p^{k}, L^{∗}T_{SU(n)}/p^{k})) ≅ H^{1}(Σ_{n−2}; ℤ/p^{k}) .

(5) If G is p–convenient and pseudo simply connected or simply connected, and if p is odd,

H^{1}(W_{G}; Hom(L^{∗}T_{G}/p^{k}, L^{∗}T_{G}/p^{k})) = 0 .

(6) If (p, |W_{G}|) = 1, then H^{1}(W_{G}; Hom(L^{∗}T_{G}/p^{k}, L^{∗}T_{G}/p^{k}) = 0.

Proof. The Hochschild–Serre spectral sequence and some easy calculations with the coefficients establish the two isomorphism of (1) and (2) . If p is odd H^{1}(W_{H}, ℤ/p^{k}) as well as H^{1}(W_{H}; L^{∗}T_{H}/p^{k}) vanish, and H^{1}(W_{G}; LT_{G}/p^{k}) vanishes if G is pseudo simply connected. All this follows from lemma 5.3. Because W H is generaterd by elements of order 2 the first group H^{1}(W_{H}, ℤ/p^{k}) vanishes for every compact connected Lie group. If G and H are simply connected then (L^{∗}T_{H}/p^{k})^{W_{H}} = 0 = (L^{∗}T_{H}/p^{k})^{W_{H}}. These facts prove the vanishing of the summands which finishes the proof of (1) and (2) .

The representation L^{∗}T_{U(n)}/p^{k} ≅ ind_{Σ_{n−1}}^{Σ_{n}}ℤ/p^{k} is induced by the 1-dimensional trivial Σ_{n−1}–module ℤ/p^{k}. Therefore

Hom(L^{∗}T_{U(n)}/p^{k}, L^{∗}T_{U(n)}/p^{k}) ≅ ind_{Σ_{n−1}}^{Σ_{n}}Hom(ℤ/p^{k}, L^{∗}T_{U(n)}/p^{k})
≅ ind_{Σ_{n−1}}^{Σ_{n}}(L^{∗}T_{U(n−1)}/p^{k} ⊕ ℤ/p^{k}) ,

and

H^{1}(Σ_{n}; Hom(L^{∗}T_{U(n)}/p^{k},L^{∗}T_{U(n)}/p^{k}))
≅ H^{1}(Σ_{n−1}, L^{∗}T_{U(n−1)}/p^{k}) ⊕ H^{1}(Σ_{n−1}; ℤ/p^{k})
≅ H^{1}(Σ_{n−2}; ℤ/p^{k}) ⊕ H^{1}(Σ_{n−1}; ℤ/p^{k}) .

The exact sequence

0 ⟶ ℤ/p^{k} ⟶ L^{∗}T_{U(n)}/p^{k} ⟶ L^{∗}T_{SU(n)}/p^{k} ⟶ 0

splits as sequence of ℤ/p^{k}–modules and induces an exact sequence

0 ⟶ Hom(L^{∗}T_{SU(n)}/p^{k},L^{∗}T_{SU(n)}/p^{k}) ⟶ Hom(L^{∗}T_{U(n)}/p^{k}, L^{∗}T_{SU(n)}/p^{k})

⟶ Hom(ℤ/p^{k}, L^{∗}T_{SU(n)}/p^{k}) ≅ L^{∗}T_{SU(n)}/p^{k} ⟶ 0 .

Because of lemma 5.3, the associated long exact sequence in cohomology shows that

H^{1}(Σ_{n}; Hom(L^{∗}T_{SU(n)}/p^{k},L^{∗}T_{SU(n)}/p^{k}))
≅ H^{1}(Σ_{n}; Hom(L^{∗}T_{U(n)}/p^{k}, L^{∗}T_{SU(n)}/p^{k}))
≅ H^{1}(Σ_{n−1}; Hom(ℤ/p^{k}, L^{∗}T_{SU(n)}/p^{k}))
≅ H^{1}(Σ_{n−1}, L^{∗}T_{U(n−1)}/p^{k})
≅ H^{1}(Σ_{n−2}; ℤ/p^{k}) .

This proves (3) and (4).

To prove (5) we choose by remark 5.2 a mod–p subgroup H⟶G of G, such that the Weyl group index is coprime to p and such that H is a product of unitary groups and special unitary groups. The restriction

H^{1}(W_{G}; Hom(L^{∗}T_{G}/p^{k}, L^{∗}T_{G}/p^{k})) ⟶ H^{1}(W_{H}; Hom(L^{∗}T_{H}/p^{k}, L^{∗}T_{H}/p^{k})) is an injection. Let H = H_{1} × ... × H_{r} be the splitting into the factors; i.e H_{i} ≅ SU(n_{i}), n_{i} ≥ 3, or H_{i} ≅ U(n_{i}). If p is odd H^{1}(W_{H_{i}}; ℤ/p^{k}) = 0, and, by (1), (2), (3), and (4),

H^{1}(W_{H}; Hom(L^{∗}T_{H}/p^{k},L^{∗}T_{H}/p^{k}))

≅ ⨁_{i} H^{1}(W_{H_{i}}; Hom(L^{∗}T_{H_{i}}/p^{k}, L^{∗}T_{H_{i}}/p^{k}))

= 0

which finishes the proof of (5).

The statement (6) follows because Hom(L^{∗}T_{G}/p^{k}, L^{∗}T_{G}/p^{k}) is a ℤ_{p}^{∧}–modul and (p, |W_{G}|) = 1. □

8.3 Lemma. Let G and H be 2–convenient compact connected Lie groups. If G is simply connected then

H^{1}(W_{G} × W_{H}; Hom(L^{∗}T_{G}/4, L^{∗}T_{G}/4))
⟶ H^{1}(W_{G} × W_{H}; Hom(L^{∗}T_{G}/2, L^{∗}T_{G}/2)

is the trivial map.

Proof. The only 2–convenientcompact connected Lie groups are qoutients of a product of SU(n)’s, n ≥ 3 and a torus. Hence W_{G} and W_{H} are products of symmetric groups. Because of the last lemma we have only to show that

H^{1}(Σ_{n}; ℤ/4)⟶H^{1}(Σ_{n}; ℤ/2)

is the trivial map. This is obvious, because H_{1}(Σ_{n}; ℤ) = ℤ/2. □

Now we are in the position to prove proposition 8.1.

Proof of proposition 8.1.

Proof. The two ℤ_{p}^{∧}[W_{G}]–modules L^{∗}T_{G}_{p}^{∧} and L induce two homomorphisms

ρ_{L^{∗}T_{G}_{p}^{∧}}, ρ_{L} : W_{G} ⟶ Gl(n; ℤ_{p}^{∧}) .

By assumption

ρ_{L^{∗}T_{G}_{p}^{∧}}(1), ρ_{L}(1) : W_{G} ⟶ Gl(n; ℤ/p)

are conjugate. We can assume they are equal. Because the obstruction groups H^{1}(W_{G}; Hom(L^{∗}T_{G}/p, L^{∗}T_{G}/p)) vanish for p odd and for (p, |W_{G}|) = 1 (lemma 8.2), all the lifts

ρ_{L^{∗}T_{G}_{p}^{∧}}(k), ρ_{L}(k) : W_{G} ⟶ Gl(n; ℤ_{p}^{∧}^{k})

are conjugate. Thus, ρ_{L^{∗}T_{G}_{p}^{∧}} and ρ_{L} are conjugate. This proves 8.1 under the assumption (1) and (3).

Next we consider the case G = SU(n), n ≥ 3, and p = 2. We denote by det : Σ_{n}⟶ℤ_{2}^{∧} the composition of ρ_{L^{∗}T_{SU(n)}_{p}^{∧}} and the determinant. For n ≥ 4 the group H^{1}(Σ_{n−2}; ℤ/2) ≅ ℤ/2, and for n = 3 the group vanishes; i.e. for n = 3 there is no obstruction and the lift ρ_{L^{∗}T_{G}_{p}^{∧}}(2) is unique up to conjugation, and for n ≥ 4 there are two lifts of ρ_{L^{∗}T_{G}_{p}^{∧}}(1), given by ρ_{L^{∗}T_{G}_{p}^{∧}}(2) and ρ_{L^{∗}T_{G}_{p}^{∧}}(2)⊗det. But these two lifts can be distinguished by looking at the invariants

P (L^{∗}T_{SU(n)}/4)^{ρ_{L^{∗}T_{SU(n)}^{∧}_{p}} (2)(Σ_{n})} and P (L^{∗}T SU(n)/4)^{(ρ_{L^{∗}T_{SU(n)}^{∧}_{p}} (2)⊗det)(Σ_{n})} ,

as a short calculation of the invariants of degree 6 shows. We assumed that L/4 is p–reducible, and hence, L^{∗}T_{SU(n)}/4 ≅ L/4.

If G is simply connected and 2–convenient G splits into factors G = G_{1} ×...×G_{r}, where G_{i} = SU(n_{i}) and n_{i} ≥ 3. In this case, L/4 ≅ \overline{L}_{1}×...×\overline{L}_{r} by lemma 8.2, where \overline{L}_{i} is a ℤ/4[W_{G_{1}} × ... × W_{G_{r}}]–module. Because L is 2–reducible, \overline{L}_{1} is 2–reducible as W_{G_{1}} × W_{H}–module, where H = G_{2} × ...G_{r}. In the commutative diagram

(∗)  \begin{matrix}P (\overline{L}_{1})^{W_{G_{1}} ×W_{H}} & ⟶^{α} & P (\overline{L}_{1})^{W_{G}} \\ ↓^{β} & & ↓^{γ} \\ P (\overline{L}_{1}/2)^{W_{G_{1}} ×W_{H}} & ⟶^{δ} & P (\overline{L}_{1}/2)^{W_{G}} \end{matrix}

δ and β ⊗ ℤ/2 are isomorphisms, and γ ⊗ ℤ/2 is an injection, hence also an isomorphism. This shows that \overline{L}_{1} is 2–reducible as W_{G}–module. Therefore, by the above argument, L^{∗}T_{G_{1}} _{p}^{∧} ≅ \overline{L}_{1} as W_{G}–modules, and α is an isomorphism by the Nakayama lemma.

The Weyl group W_{H} must act on \overline{L}_{1} via a homorphism

W_{H} ⟶ C_{GL(\overline{L}_{1})}(W_{G}) ≅ C_{GL(L^{∗}T_{G_{1}} /4)}(W_{G}) ≅ ℤ/4 .

That is to say that W_{H} acts via scalar multiplication. A non trivial action of W_{H} on \overline{L}_{1} contradicts the fact that the map α in the diagram (∗) is an isomorphism. Thus L^{∗}T_{G_{1}}/4 ≅ \overline{L}_{1} and L^{∗}T_{G}/4 ≅ L/4 as W_{G}–module.

Now, for an induction argument, we can assume that

L^{∗}T_{G}/2^{k} ≅ L/2^{k}, ρ_{L^{∗}T_{G}_{p}^{∧}}(k) = ρ_{L}(k) , and k ≥ 2 .

The two lifts ρ_{L^{∗}T_{G}_{p}^{∧}}(k + 2) and ρ_{L}(k + 2) differ by an obstruction in H^{1}(W_{G}; Hom(L^{∗}T_{G}/4, L^{∗}T_{G}/4)). Because the map

H^{1}(W_{G}; Hom(L^{∗}T_{G}/4, L^{∗}T_{G}/4)) ⟶ H^{1}(W_{G}; Hom(L^{∗}T_{G}/2, L^{∗}T_{G}/2))

is trivial (lemma 8.3) the two lifts ρ_{L^{∗}T_{G}_{p}^{∧}}(k + 1) and ρ_{L}(k + 1) are conjugate, i.e. L^{∗}T_{G}/2^{k+1} ≅ L/2^{k+1} and L^{∗}T_{G}_{p}^{∧} ≅ L as ℤ_{p}^{∧}[W_{G}]–modules. This proves part (2). □

9. p–adic Weyl group representations, for general compact connected Lie groups.

In this section we use the same notation as in the last one. G denotes in this chapter a compact connected Lie group. Uniqueness results about the p–adic liftings of the Weyl group representation L^{∗}T_{G}/p are not true in general. In remark 9.6 we consrtruct a counterexample. Nevertheless we can describe all torsion free p– reducible p–adic representations L of W_{G} which are lifts of L^{∗}T_{G}/p. The results to be applied in section 7 are the propositions 9.5 and 9.9.

Any compact connected Lie group G fits into an exact sequence \overline{G}_{s}⟶G⟶T, where G_{s} is simply connected and where T is a torus. If G is p– convenient G_{s} is also p–convenient, and G_{s}⟶\overline{G}_{s} is a mod–p isomorphism of groups (see section 4). Moreover, L^{∗}T_{G_{s}} _{p}^{∧} ≅ L^{∗}T_{\overline{G}_{s}} _{p}^{∧} and L^{∗}T_{G_{s}}/p ≅ L^{∗}T_{\overline{G}_{s}}/p.

9.1 Lemma. Let G be a p–convenient compact connected Lie group. Let L be a torsion free p–reducible ℤ_{p}^{∧}[W_{G}]–module. If L/p ≅ L^{∗}T_{G}/p as ℤ_{p}^{∧}[W_{G}]–module then there is an exact sequence

1 ⟶ L^{∗}T_{p}^{∧} ⟶ L ⟶ L^{∗}T_{G_{s}} _{p}^{∧} ⟶ 1

Proof. The qoutient L/L^{W_{G}} is torsion free because L is. Thus, applying the functor ⊗ℤ/p to the exact sequence

0 ⟶ L^{W_{G}} ⟶ L ⟶ L/L^{W_{G}} ⟶ 0 ,

establishes the diagram

0 ⟶ L^{W_{G}}/p ⟶ L/p ⟶  (L/L^{W_{G}})/p  ⟶ 0
↓  ↓  ↓
0 ⟶ (L/p)^{W_{G}} ⟶ L/p ⟶  (L/p)/(L/p)^{W_{G}}  ⟶ 0
↓≅  ↓≅  ↓≅
0 ⟶ (L^{∗}T_{G}/p)^{W_{G}} ⟶ L^{∗}T_{G}/p ⟶ (L^{∗}T_{G}/p)/(L^{∗}T_{G}/p^{W_{G}}) ⟶ 0

of exact rows. The exact sequence \overline{G}_{s}⟶G⟶T establishes the isomorphism (L^{∗}T_{G}/p)^{W_{G}} ≅ L^{∗}T/p. Because G is p–convenient we have

(L^{∗}T_{G}_{p}^{∧}/L^{∗}T_{G}_{p}^{∧}^{W_{G}})/p ≅ (L^{∗}T_{G}/p)/(L^{∗}T_{G}/p^{W_{G}}) ≅ L^{∗}T_{\overline{G}_{s}}/p ≅ L^{∗}T_{G_{s}}/p .

The two lower rows in the diagram are isomorphic by assumption. The two upper rows also are isomorphic, because L is p–reducible. We have L^{W_{G}} ≅ L^{∗}T_{p}^{∧} and, by proposition 8.1, we get L/L^{W_{G}} ≅ L^{∗}T_{G_{s}} _{p}^{∧} as ℤ_{p}^{∧}[W_{G}]–module. That is there exists an exact sequence

0 ⟶ L^{∗}T_{p}^{∧} ⟶ L ⟶ L^{∗}T_{G_{s}} _{p}^{∧} ⟶ 0

of ℤ_{p}^{∧}[W_{G}]–modules as desired. □

Remark. If

Ext_{W_{G}}(L^{∗}T_{G_{s}} _{p}^{∧}, L^{∗}T_{p}^{∧}) ≅ H^{1}(W_{G}; Hom(L^{∗}T_{G_{s}} _{p}^{∧}, L^{∗}T_{p}^{∧}))
≅ H^{1}(W_{G}; LT_{G_{s}} _{p}^{∧}) ⊗ L^{∗}T_{p}^{∧} = 0 ,

then L ≅ L^{∗}T_{G}_{p}^{∧} as ℤ_{p}^{∧}[W_{G}]–modules. Thus, proposition 8.1 is true if (p, |W_{G}|) = 1, as already shown, or if G_{s} is a p–convenient simply connected Lie group , which contains no factor isomorphic to some SU(n), i.e. G_{s} is also pseudo simply connected.

Next we will study the group of extensions Ext_{W_{G}}(L^{∗}T_{G_{s}} _{p}^{∧}, L^{∗}T_{p}^{∧}) or, more generally, the extensions Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧}) for a compact connected Lie group G and a torus T. We denote by GrExt(T, G) the set of exact sequences G⟶G′⟶T of compact connected Lie groups dividing out the usual equivalence relations for extensions. There is a canonical map

\begin{aligned}L : & GrExt(T, G) & ⟶ & Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧}) \\ & (G→G′→T) & ↦ & (L^{∗}T_{p}^{∧}→L^{∗}T_{G}′_{p}^{∧}→L^{∗}T_{G}_{p}^{∧}) \end{aligned}

9.2 Proposition. Let G be a compact connected Lie group. If the inclusion Z(G)⟶T_{G}^{W_{G}} has a cokernel of order coprime to p the map L is a surjection.

Remark. By lemma 4.4 the assumption of 9.2 are satisfied if G is p–convenient.

Proof. Let E : L^{∗}T_{p}^{∧}⟶L⟶L^{∗}T_{G}_{p}^{∧} be an exact sequence. Because Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧}) ≅ H^{1}(W_{G}; Hom(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧})) is a finite group we can choose an injective self map α : L^{∗}T_{p}^{∧}⟶L^{∗}T_{p}^{∧} with finite cokernel K such that α_{∗}(E) splits. Therefore, there exists a commutative diagram

L^{∗}T_{p}^{∧} ⟶  L  ⟶ L^{∗}T_{G}_{p}^{∧}
α↓  ^{β}↓  ∥
L^{∗}T_{p}^{∧} ⟶ L^{∗}T_{p}^{∧} ⊕ L^{∗}T_{G}_{p}^{∧} ⟶ L^{∗}T_{G}_{p}^{∧}
↓  ↓
K  K .

The equivariant maps L^{∗}T_{p}^{∧}⟶K ≅ H^{2}(BBK; ℤ_{p}^{∧}) and L^{∗}T_{p}^{∧} ⊕ L^{∗}T_{G}_{p}^{∧}⟶K ≅ H^{2}(BBK; ℤ_{p}^{∧}) are induced by homomorphisms K⟶T and K⟶T × T_{G}. Because K is a finite abelian p–group and because H^{∗}(BK, ℤ/p) is a finitely generated H^{∗}(BT; ℤ/p)–module, both homomorphisms are injections [Q]. We have K ⊂ T × T_{G}^{W_{G}}, and hence K maps into Z(G) ⊂ T^{W_{G}}. The compact connected Lie group G′ := (T × G)/K fits into an exact sequence G⟶G′⟶T. By construction we get an isomorphism of extensions, namely

L^{∗}T_{p}^{∧} ⟶ L ⟶ L^{∗}T_{G}_{p}^{∧}
∥  ↓  ∥
L^{∗}T_{p}^{∧} ⟶ L^{∗}T_{G′} _{p}^{∧} ⟶ L^{∗}T_{G}_{p}^{∧} . □

Because L^{∗}T_{G}_{p}^{∧} is torsionfree the functor ⊗ℤ/p induces maps

\begin{matrix}α : Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧}) & ⟶ & Ext_{W_{G}}(L^{∗}T_{G}/p, L^{∗}T/p) \\ β : Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T/p) & ⟶ & Ext_{W_{G}}(L^{∗}T_{G}/p, L^{∗}T/p) . \end{matrix}

The exact sequence

L^{∗}T_{p}^{∧} ⟶^{m} L^{∗}T_{p}^{∧} ⟶^{q_{T}} L^{∗}T/p ,

where m is the multiplication with p, and the projection q_{G} : L^{∗}T_{G}_{p}^{∧}⟶L^{∗}T_{G}/p establish the diagram

[figure: \begin{matrix}Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧}) \\ Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧}) \\ Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T/p) & Ext_{W_{G}}(L^{∗}T_{G}/p, L^{∗}T/p) & . \end{matrix}; m_{∗}; q_{T} ∗; α; β; q_{G}^{∗}]

The map m_{∗} is given by multiplication with p in the Ext–group, and β is a left inverse of q_{G}^{∗}, i.e. βq_{G}^{∗} = id. The commutativity of

L^{∗}T_{p}^{∧} ⟶ L ⟶ L^{∗}T_{G}_{p}^{∧}
q_{T} ↓  ↓  ∥
L^{∗}T/p ⟶ L′ ⟶ L^{∗}T_{G}_{p}^{∧}
∥  ↓  ^{q_{G}}↓
L^{∗}T/p ⟶ L/p ⟶ L^{∗}T_{G}/p

shows that βq_{T ∗} = α and q_{G}^{∗}α = q_{T}_{∗}. This implies the following lemma:

9.3 Lemma. ker(α) = ker(q_{T}_{∗}) = im(m_{∗}).

Next we will show that the p–adic lifting of the Weyl group representation L^{∗}T_{G}/p is unique for another class of Lie groups, namely for pseudo projective p–convenient Lie groups. We introduce the following definition. Definition. An extension

E : L^{∗}T_{p}^{∧}⟶L⟶L^{∗}T_{G}_{p}^{∧} ∈ Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧})

is called initial if, for every other extension Ê ∈ Ext_{W_{G}}(L^{∗}T_{G}_{p}^{∧}, L^{∗}T_{p}^{∧}), there exists a self map g : L^{∗}T_{p}^{∧}⟶L^{∗}T_{p}^{∧} such that Ê = g_{∗}E.

The next lemma produces initial extensions. 9.4 Lemma. Let G_{s} be simply connected. If G ∈ GrExt(T, G_{s}) is pseudo projective the extension

E : L^{∗}T_{p}^{∧} ⟶ L^{∗}T_{G}_{p}^{∧} ⟶ L^{∗}T_{G_{s}} _{p}^{∧}

is initial. Proof. By proposition 9.2 every extension E′ : L^{∗}T_{p}^{∧}⟶L⟶L^{∗}T_{G_{s}} _{p}^{∧} is induced by a group extension G_{s}⟶G′⟶T or alternatively by an exact sequence

K ⟶ G_{s} × T ⟶ G′ ,

where the composition K⟶G_{s}×T ⟶T is an injection. The inclusion K⟶Z(G_{s}) can be extended to

K ⟶ Z(G_{s})
↓ ↓
T ⟶ T .

This establishes a finite covering G′⟶G. By construction we get

L^{∗}T_{p}^{∧} ⟶ L^{∗}T_{G}_{p}^{∧} ⟶ L^{∗}T_{G_{s}} _{p}^{∧}
↓ ↓ ∥
L^{∗}T_{p}^{∧} ⟶ L^{∗}T_{G′} _{p}^{∧} = L ⟶ L^{∗}T_{G_{s}} _{p}^{∧}

which shows that E is initial. □

Remember that a compact connected Lie group is pseudo projective if Z(G) is connected. 9.5 Proposition. Let G_{s} be a p–convenient simply connected Lie group, and let G ∈ GrExt(T, G_{s}) be pseudo projective. Let L be a torsion free p–reducible ℤ_{p}^{∧}[W_{G}]–module, and let \overline{α} : L/p⟶L^{∗}T_{G}/p be a ℤ/p[W_{G}]–isomorphism. Then there exists a ℤ_{p}^{∧}[W_{G}]–isomorphism

α : L ⟶ L^{∗}T_{G}_{p}^{∧} , which is a lift of \overline{α}.

Proof. Let E_{G} : L^{∗}T_{p}^{∧}⟶L^{∗}T_{G}_{p}^{∧}⟶L^{∗}T_{G_{s}} _{p}^{∧} be the sequence associated to G, and E : L^{∗}T_{p}^{∧}⟶L⟶L^{∗}T_{G_{s}} _{p}^{∧} the sequence associated to L (lemma 9.1). The difference E_{1} := E −E_{G} is mod–p trivial. Thus, by lemmma 9.3, there is an extension E_{2} such that E_{1} = m_{∗}(E_{2}). The map m : L^{∗}T_{p}^{∧}⟶L^{∗}T_{p}^{∧} is the multiplication with p. The exact sequence E_{G} is initial (lemma 9.4). Hence E_{2} = a_{∗}(E_{G}) and E_{1} = m_{∗}a_{∗}(E_{G}) for a suitable homomorphism a : L^{∗}T_{p}^{∧}⟶L^{∗}T_{p}^{∧}. This yields

E_{G} :  L^{∗}T_{p}^{∧}  ⟶ L^{∗}T_{G}_{p}^{∧} ⟶  L^{∗}T_{G_{s}} _{p}^{∧}
(id,am)↓  ↓  ↓^{Δ}
E_{G} + E_{1} : L^{∗}T_{p}^{∧} ⊕ L^{∗}T_{p}^{∧} ⟶ L^{∗}T_{G}_{p}^{∧} ⊕ L_{1} ⟶ L^{∗}T_{G_{s}} _{p}^{∧} ⊕ L^{∗}T_{G_{s}} _{p}^{∧}
↓  ↓  ∥
L^{∗}T_{p}^{∧}  ⟶  \overline{L}  ⟶ L^{∗}T_{G_{s}} _{p}^{∧} ⊕ L^{∗}T_{G_{s}} _{p}^{∧}
∥  ↑  ↑^{Δ}
E :  L^{∗}T_{p}^{∧}  ⟶  L  ⟶  L^{∗}T_{G_{s}} _{p}^{∧} ,

where the three lower rows represent the sum E_{G} + E_{1} = E and where Δ is the diagonal map. Therefore E = (id + am)_{∗}(E_{G}), and, because id + am ≡ id mod p, the map L^{∗}T_{G}_{p}^{∧}⟶L is an isomorphism. □

9.6 Remark. Let ℤ/p ↪ ℤ/p^{2} = Z(SU(p^{2})) and ℤ/p↪S^{1} be the standard inclusions, and let G = (SU(p^{2}) × S^{1})/ℤ/p. G is a p–fold cover of U(p^{2}). This fits into

Bℤ/p  Bℤ/p
↓  ↓
BSU(p^{2}) ⟶ BG ⟶ BS^{1}
∥  ↓  ↓^{f}
BSU(p^{2}) ⟶ BU(p^{2}) ⟶ BS^{1}

and establishes

L^{∗}S^{1}_{p}^{∧} ⟶ L^{∗}T_{G}_{p}^{∧} ⟶ L^{∗}T_{SU(p^{2})}_{p}^{∧}
f^{∗}↑  ↑  ∥
L^{∗}S^{1}_{p}^{∧} ⟶ L^{∗}T_{U(p^{2})}_{p}^{∧} ⟶ L^{∗}T_{SU(p^{2})}_{p}^{∧} ,

where f^{∗} is multiplication by p. Therefore, by lemma 9.3, the upper row splits after applying the functor ⊗ℤ/p. That is to say that L^{∗}T_{G}/p ≅ L^{∗}T_{SU(p^{2})}/p⊕L^{∗}S^{1}/p as Σ_{p^{2}}–modules. Because G is p–convenient BG and BSU(p^{2}) × BS^{1} have the same mod–p type, but not the same p–adic type. This is the generic example of compact Lie groups having the same mod–p type but different p–adic type, i.e. all examples come from p–fold coverings.

For p = 2, the last proposition only covers the case of products of unitary groups U(n), n ≥ 3, which are all pseudo projective, but not the group U(2) which is not 2–convenient. Nevertheless, with one extra assumption, the same statement is still true for products of all unitary groups at p = 2 as we will see. First we consider the case of a product of U(2)’s.

9.7 Proposition. Let G = U(2)^{r} and let L be a torsion free 2–reducible ℤ_{2}^{∧}[W_{G}]– module. Let

\overline{α} : L^{∗}T_{G}/2 ⟶ L/2

be a ℤ_{2}^{∧}[W_{G}]–isomorphism. Then there exists a ℤ_{2}^{∧}[W_{G}]–isomorphism

α : L^{∗}T_{G} ⟶ L

which is a lift of \overline{α}.

Proof. The two representations L^{∗}T_{G} and L establish homomorphisms ρ_{L}, ρ_{L^{∗}T_{G}} : W_{G} ⟶ Gl(2r, ℤ_{2}^{∧}). Because L is 2–reducible and because P (L)^{W_{G}}/2 ≅ P (L/2)^{W_{G}} is a polynomial algebra, the 2-adic invariants P (L)^{W_{G}} are also a polynomial algebra as well as P (L)^{W_{G}} ⊗ ℚ ≅ P (L ⊗ ℚ)^{W_{G}}. Therefore the rational representation ρ_{L} : W_{G} ⟶ Gl(2r, ℚ_{2}^{∧}) represents W_{G} as a pseudo reflection group. We can choose s_{1}, ..., s_{r} ∈ (W_{G}) ≅ (ℤ/2)^{r} which generate W_{G} and which are pseudo reflections with respect to ρ_{L}. The only pseudo reflections in the image of W_{G} in Gl(2r, ℤ_{2}^{∧} are given by the generators of the Weyl groups of the factors of G. Therefore, after reordering, the element s_{i} generates the Weyl group of the i-th factor.

Let W′ ⊂ W be the subgroup generated by s_{2}, ..., s_{r}. The fixed–points of a pseudo reflection acting on a ℤ_{2}^{∧}–module or ℚ_{2}^{∧}–module has codimension 1 or codimension 0. Therefore, rk(L^{W′}) ≥ 2r − r + 1 = r + 1 and rk(L/L^{W′}) ≤ r − 1. Here rk( ) denotes the rank of free modules. We notice that the qoutient L/L^{W′} is torsion free.

In the commutative diadram of exact rows

0 ⟶ L^{W′} ⟶ L ⟶  L/L^{W′}  ⟶ 0
↓  ↓  ↓
0 ⟶ L^{W′}/2 ⟶ L/2 ⟶  (L/L^{W′})/2  ⟶ 0
↓  ∥  ↓
0 ⟶ L/2^{W′} ⟶ L/2 ⟶  (L/2)/(L/2^{W′})  ⟶ 0
≅↓  ≅↓  ≅↓
0 ⟶ L^{∗}T_{G}/2^{W′} ⟶ L^{∗}T_{G}/2 ⟶ (L^{∗}T_{G}/2)/(L^{∗}T_{G}/2^{W′}) ⟶ 0

the right vertical composition of maps is an epimorphism. Hence, rk_{ℤ_{2}^{∧}}L/L^{W′} = rk_{ℤ/2}(L^{∗}T_{G}/2)/((L^{∗}T_{G}/2)^{W′}) = r − 1. This implies that (L/L^{W′})/2 ⟶ (L/2)/((L/2)^{W′}) is an isomorphism as well as L^{W′}/2 ⟶ (L/2)^{W′}.

According to the splitting G = U(2)^{r} we choose a basis \overline{B} = \{\overline{x}_{1}, \overline{y}_{1}, ..., \overline{x}_{r}, \overline{y}_{r}\} of L/2 characterized by the properties s_{i}(\overline{x}_{j}) = \overline{x}_{j} and s_{i}(\overline{y}_{j}) = \overline{y}_{j} for i ≠ j and the property s_{i}(\overline{x}_{i}) = \overline{y}_{i}. By the above considerations there exist lifts x_{1}, y_{1} ∈ L^{W′} of \overline{x}_{1} and \overline{y}_{1}, such that s_{1}(x_{1}) = y_{1}. Analogously we can choose lifts x_{i}, y_{i} of \overline{x}_{i} and \overline{y}_{i} for all i such that B = \{x_{1}, y_{1}, ..., x_{r}, y_{r}\} is a basis of L, such thats_{i}(x_{i}) = y_{i} and such that, for i ≠ j, s_{i}(x_{j}) = x_{j} and s_{i}(y_{j}) = y_{j}. This implies that there exist a ℤ_{2}^{∧}[W_{G}]–isomorphism L^{∗}T_{G} ⟶ L which is a lift of \overline{α}. □

Let X be a space with the mod–p type of BG × BH. The assumption that G and H are p–convenient is too strong for what follows. We only assume that

(*) H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W_{G}} and H^{∗}(BH; ℤ/p) ≅ H^{∗}(BT_{H}; ℤ/p)^{W_{H}} .

Let BT_{X}⟶X be the maximal torus of X, let V_{G} ⊂ T_{G} and V_{H} ⊂ T_{H} be the maximal elementary abelian p–subgroups of the maximal tori T_{G} and T_{H}, and let g : BV_{H}⟶BV_{H} × BV_{G}⟶X be the obvious map (see section 7). The isotropy group of g is W_{G}. Hence, for Y := map(BV_{H}, X)_{g},

H^{∗}(Y; ℤ/p) ≅ H^{∗}(BT_{X}; ℤ/p)^{W_{G}} ≅ H^{∗}(BT_{G} × BT_{H}; ℤ/p)^{W_{G}}

(theorem 10.1). In particular, H^{∗}(Y; ℤ/p) is concentrated in even degrees, and therefore, H^{∗}(Y; ℤ_{p}^{∧})⟶H^{∗}(Y; ℤ/p) is a surjection.

9.8 Lemma. If G and H are two compact connected Lie groups satifying (∗) then L^{∗}T_{X} _{p}^{∧} is p–reducible as W_{G}–module.

Proof. We have H^{∗}(BT_{X};_{p}^{∧}) ≅ P (L^{∗}T_{X} _{p}^{∧}) which is the symmetric part of the tensor algbra of L^{∗}T_{X} _{p}^{∧}. This induces maps

H^{∗}(Y; ℤ_{p}^{∧})/p ⟶ P (L^{∗}T_{X} _{p}^{∧})^{W_{G}}/p ↪ P (L^{∗}T_{X}/p)^{W_{G}} ≅ H^{∗}(Y; ℤ/p) .

The composition is an isomorphism. This implies the statement. □

Now we are in the position to prove the following result.

9.9 Proposition. Let G be a product of unitary groups. If X is a space with the mod–2 type of BG, then L^{∗}T_{X} _{p}^{∧} ≅ L^{∗}T_{G}_{p}^{∧} as W_{G}–modules.

Proof. The group G = G′ × G′′ splits into a product, where G′ contains all factors isomorphic to U(2) and G′′ all the other factors U(n), n ≥ 3. Then L^{∗}T_{X} _{2}^{∧} is 2–reducible as W_{G′}–module (lemma 9.7), and L^{∗}T_{X} _{2}^{∧}^{W_{G′}} has as W_{G′′}–module the mod–2 type of L^{∗}Z(G′)⊕L^{∗}T_{G′′} _{2}^{∧}. Moreover, Z(G′)×G′′ is pseudo projective, and therefore, L^{∗}T_{X} _{2}^{∧}^{W_{G′}} ≅ L^{∗}Z(G′)_{2}^{∧} ⊕ L^{∗}T_{G′′} _{2}^{∧} as W_{G′′}–modules (proposition 9.5).

Now we consider the diagram of exact sequences

0 ⟶ L^{∗}T_{G′′} _{2}^{∧} ⟶  L^{∗}T_{X} _{2}^{∧}  ⟶ L^{∗}T_{X} _{2}^{∧}/L^{∗}T_{G′′} _{2}^{∧} ⟶ 0
↓  ↓  ↓
0 ⟶ L^{∗}T_{G′′}/2 ⟶  L^{∗}T_{X}/2  ⟶ (L^{∗}T_{X} _{2}^{∧}/L^{∗}T_{G′′} _{2}^{∧})/2 ⟶ 0
∥  ≅↓  ≅↓
0 ⟶ L^{∗}T_{G′′}/2 ⟶ L^{∗}T_{G′}/2 ⊕ L^{∗}T_{G′′}/2 ⟶  L^{∗}T_{G′}/2  ⟶ 0 .

The middle arrow is also an exact sequence because L^{∗}T_{X} _{2}^{∧}/L^{∗}T_{G′′} _{2}^{∧} is torsion free. Moreover, the analogous diagram for the polynomial algebras of the modules shows that L^{∗}T_{X} _{2}^{∧}/L^{∗}T_{G′′} _{2}^{∧} is 2–reducible as W_{G′}–module. By proposition 9.7 this implies that L^{∗}T_{X} _{2}^{∧}/L^{∗}T_{G′′} _{2}^{∧} ≅ L^{∗}T_{G′} _{2}^{∧} as W_{G′}–modules.

L^{∗}T_{X} can be considered as an element in

Ext_{W_{G′} ×W_{G′′}} (L^{∗}T_{G′} _{2}^{∧}, L^{∗}T_{G′′} _{2}^{∧}) ≅ H^{1}(W_{G′} × W_{G′′}; Hom(L^{∗}T_{G′} _{2}^{∧}, L^{∗}T_{G′′} _{2}^{∧}))
≅ LT_{G′} _{2}^{∧}^{W_{G′}} ⊗ H^{1}(W_{G′′}; L^{∗}T_{G′′} _{2}^{∧})
⊕ H^{1}(W_{G′}; LT_{G′} _{2}^{∧}) ⊗ L^{∗}T_{G′′} _{2}^{∧}^{W_{G′′}}
= 0 .

This follows by lemma 8.2 and because H^{1}(W_{G′}; LT_{G′} _{2}^{∧}) = H^{1}(W_{G′′}; L^{∗}T_{G′′} _{2}^{∧}) = 0. Finally we get L^{∗}T_{X} _{2}^{∧} ≅ L^{∗}T_{G′} _{2}^{∧} ⊕ L^{∗}T_{G′′} _{2}^{∧} = L^{∗}T_{G}_{2}^{∧} as W G–modules. □

10. Mapping spaces.

In this section G is a compact connected Lie group satifying the condition H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W_{G}}. Let X be a space with the mod–p type of BG. In section 7 we constructed a maximal torus f_{T} : BT_{X}⟶X, respectively f_{T} : BT_{X} _{p}^{∧}⟶X, with Weyl group W_{G} such that the diagram

BT_{G}_{p}^{∧} ⟶^{≃} BT_{X} _{p}^{∧}
↓  ↓
BG_{p}^{∧} −−→ X

commutes in mod–p cohomology. In this chapter we always work with mod–p cohomology and define H^{∗}( ) := H^{∗}( ; ℤ/p).

Let A be an abelian p–toral group. Then W_{G} acts on map(BA, BT_{X}). This action induces a map

[BA, BT_{X} _{p}^{∧}]/W_{G} ⟶ [BA, X] .

For a map g : BA⟶BT_{X} _{p}^{∧} we define

Iso(g) := {w ∈ W_{G}|w ∘ g ≃ g}

to be the isotropy group of g. The composition of g with the maximal tori of X is also denoted by g.

10.1 Theorem. Let G be a compact connected Lie group such that H^{∗}(BG; ℤ/p) ≅ H^{∗}(BT_{G}; ℤ/p)^{W_{G}}. Let X be a space with the mod–p type of BG, and let A be an abelian p–toral group. Then the following hold:

(1) [BA, BT_{X} _{p}^{∧}] ⟶ [BA, X] is a surjection.

(2) [BA, BT_{X} _{p}^{∧}]/W_{G} ⟶ [BA, X] is a bijection.
(3) For any g : BA⟶BT_{X} _{p}^{∧},
H^{∗}(map(BA, X)_{g}) ≅ H^{∗}(map(BA, BT_{X} _{p}^{∧})_{g})^{Iso(g)}
and map(BA, X)_{g} is p-complete.

10.2 Remark.

(1) By 10.1 (1) there exists for any map g : BA⟶X a lift g′ : BA⟶BT_{X} _{p}^{∧}, which, by 10.1 (2), is unique up to conjugation by elements of the Weyl group. Thus Iso(g′) ⊂ W_{G} is uniquely determined up to conjugation by g.

(2) The isomorphism of 10.1 (3) is induced by the maximal torus BT_{X} _{p}^{∧}⟶X. Of course, BG_{p}^{∧} satisfies the assumptions of the theorem. Let X have the p–adic type of BG. Part (3) and theorem 3.8 induce canonical isomorphisms

H^{∗}(map(BA, X)_{g}) ≅ H^{∗}(map(BA, BG_{p}^{∧})_{g}) ≅ H^{∗}(map(BA, BG)_{g}) .

(3) Let g : BA⟶BT_{G} be a map and g ≃ Bα for a suitable homomorphism α : A⟶T_{G} (theorem 3.6). π_{0}(C_{G}(α)) is a p–group [J-M-O; A.4]. Therefore, by 10.1 (3), the centralizer C_{G}(α) is connected and p–convenient, and W_{C_{G}(α)} = Iso(g).

(4) Theorem 10.1 is true under some weaker assumptions as the proof will show. We only have to assume that X is a complete space with maximal torus f : BT_{X} _{p}^{∧} ⟶ X and Weyl group W_{X} such that H^{∗}(X; ℤ/p) ≅ H^{∗}(BT_{X} _{p}^{∧}; ℤ/p)^{W_{X}}; i.e. it is not necessary that X is of the mod p–type of the classifying space of a compact connected Lie group.

We can reformulate theorem 10.1, using the identity

H^{∗}(map(BA, BT_{X} _{p}^{∧}))^{W_{G}} ≅ ∏_{g∈[BA,BT_{X}]/W_{G}} H^{∗}(map(BA, BT_{X} _{p}^{∧})_{g})^{Iso(g)} .

This identity is in anology to theorem 3.14.

10.3 Theorem. Under the assumption of 10.1, the maximal torus BT_{X} _{p}^{∧}⟶X induces an isomorphism

H^{∗}(map(BA, X)) ≅ H^{∗}(map(BA, BT_{X} _{p}^{∧}))^{W_{G}} ,

and map(BA, X) is p–complete, i.e. every component is p–complete.

For the next statement, we assume that X has the p–adic type of BG and that there exists an extension

BN(T_{G}) ⟶ X ,

of the maximal torus BT_{G}⟶X, such that the diagram

[figure: \begin{aligned}& BN(T_{G}) \\ BG & & X \end{aligned}]

commutes in mod–p cohomology. For any abelian abelian p–toral group A and any map g : BA ⟶ BT_{G}, theorem 10.3 establishes the following sequence of isomorphisms

H^{∗}(map(BA, X)_{g}) ≅ H^{∗}(map(BA, BT_{X} _{p}^{∧})_{g})^{W_{G}}
≅ H^{∗}(map(BA, BT_{G}_{p}^{∧})_{g})^{W_{G}}
≅ H^{∗}(map(BA, BG_{p}^{∧})_{g}) ,

and therefore, a dotted arrow map(BA, BG)_{g} −−−→ map(BA, X)_{g}

10.4 Proposition. For any map g : BA⟶BT_{G}, the diagram

[figure: \begin{aligned}& map(BA, BN(T_{G}))_{g} \\ map(BA, BG)_{g} & & map(BA, X)_{g} \end{aligned}]

commutes in mod–p cohomology.

In the rest of this chapter we will prove theorem 10.3 and proposition 10.4.

10.5 Lemma. It is sufficient to prove 10.3 and 10.4 for finite abelian p–groups.

Proof. Let A be a p–toral abelian group. We denote by A_{k} ⊂ A the elements of order p^{k} and define A_{∞} := ⋃ A_{k}. Then A_{k} is a finite p–group. The map BA_{∞}⟶BA is a mod–p equivalence, which implies that map(BA, X) ≃ holim_{⟵} map(BA_{k}, X) and

map(BA, BT_{X}) ≃ holim_{⟵} map(BA_{k}, BT_{X}). The inclusions BA_{k−1}⟶BA_{k} induce

a commutative diagram

... ⟶ map(BA_{k}, BT_{X} _{p}^{∧}) ⟶ map(BA_{k−1}, BT_{X} _{p}^{∧}) ⟶ ...
↓  ↓
... ⟶ map(BA_{k}, X) ⟶ map(BA_{k−1}, X) ⟶ ....

By theorem 10.3 for finite abelian p–groups we have H^{1}(map(BA_{k}, BT_{X} _{p}^{∧})) = H^{1}(map(BA_{k}, X)) = 0.

The two function spaces map(BA_{r}, BT_{X} _{p}^{∧}) and map(BA_{r}, X) are p–complete. Hence, both mapping spaces are also 1-connected [B-K; I.6]. The Milnor sequence for calculating components of homotopy inverse limits therefore reduces to

[BA, BT_{X} _{p}^{∧}] ⟶^{≅} lim_{⟵} [BA_{k}, BT_{X} _{p}^{∧}]
↓  ↓
[BA, X] ⟶^{≅}  lim_{⟵} [BA_{k}, X] .

The obvious map

[BA, BT_{X} _{p}^{∧}]/W_{G} ⟶ lim_{⟵} ([BA_{k}, BT_{X} _{p}^{∧}]/W_{G})

is a bijection. That is to say that

H^{0}(map(BA, X)) ≅ H^{0}(map(BA, BT_{X} _{p}^{∧}))^{W_{G}} .

Let g : BA⟶BT_{X} _{p}^{∧} be a map and denote by g_{k} the restriction g|_{BA_{k}}. The sequence

... ⊃ Iso(g_{k−1}) ⊃ Iso(g_{k}) ⊃ ...

stabilizes, since W_{G} is a finite group. We can choose k′ (big enough), such that Iso(g) = Iso(g_{k}) for k ≥ k′. Thus

map(BA_{k+1}, X)_{g_{k+1}} ⟶ map(BA_{k}, X)_{g_{k}}

are mod–p equivalences and therefore equivalences for k ≥ k′ because all the spaces are p–complete. This implies

H^{∗}(map(BA, X)_{g}) ≅ H^{∗}(map(BA_{k′}, X)_{g_{k′}})
≅ H^{∗}(map(BA_{k′}, BT_{X} _{p}^{∧})_{g_{k′}})^{Iso(g_{k′})}
≅ H^{∗}(map(BA, BT_{X} _{p}^{∧})_{g})^{Iso(g)} ,

which finishes the proof of first part of the statement.

To prove 10.4, we consider a map g : BA⟶BT_{G}. According to remark 10.2 (3), we can speak of the centralizers C_{G}(g_{k}) and C_{G}(g) = ⋂_{k} C_{G}(g_{k}), which are compact Lie groups. The sequence C_{G}(g_{k+1}) ⊂ C_{G}(g_{k}) stabilizes. Again we can choose k′ big enough such that C_{G}(g) = C_{G}(g_{k′}) and, analogously, C_{N(T_{G})}(g) = C_{N(T_{G})}(g_{k′}). Theorems 10.1 and 3.6 and the above considerations show that the maps

\begin{matrix}map(BA, BN(T_{G}))_{g} & ⟶ & map(BA_{k′}, BN(T_{G}))_{g_{k′}} \\ map(BA, BG)_{g} & ⟶ & map(BA_{k′}, BG)_{g_{k′}} \\ map(BA, X)_{g} & ⟶ & map(BA_{k′}, X)_{g_{k′}} \end{matrix}

are mod–p equivalences which reduces 10.4 to the case of a finite group. □

Let A be a finite abelian p–group. We can choose a subgroup A_{0} ⊂ A of index p and get an exact sequence

1 ⟶ A_{0} ⟶ A ⟶ ℤ/p ⟶ 1 .

We will prove 10.3 and 10.4 for a finite group A by an induction over the order of A. In remark 3.12 we defined B̃A_{0} := EA/A_{0} ≃ BA_{0} which carries a free ℤ/p–action. To this end we introduce the following notation and abbreviations:

MX := map(BA, X)
MX_{0} := map(B̃A_{0}, X)
MT := map(BA, BT_{X} _{p}^{∧})
MT_{0} := map(B̃A_{0}, BT_{X} _{p}^{∧})
W := W_{G} .

Then MX ≅ MX_{0}^{hℤ/p} and MT ≅ MT_{0}^{hℤ/p}. Because the actions of W and ℤ/p on MT_{0} commute the group W acts on the Borel product Eℤ/p ×_{ℤ/p} MT_{0}.

Now let us assume that 10.3 and 10.4 is true for A_{0}.

10.6 Lemma.

(1) The fibration

MT_{0} ⟶ Eℤ/p ×_{ℤ/p} MT_{0} ⟶ Bℤ/p

is fiber homotopic trivial.

(2) H^{∗}(Eℤ/p ×_{ℤ/p} MX_{0}) ≅ H^{∗}(Eℤ/p ×_{ℤ/p} MT_{0})^{W}.

Proof. (1) Every map g_{0} : BA_{0}⟶BT_{X} _{p}^{∧} is, up to homotopy, induced by a homomorphism. For every g_{0} we can choose an extension g : BA⟶BT_{X} _{p}^{∧} such that g|_{BA_{0}} = g_{0}. ℤ/p acts trivially on \overline{MT} := ⋃_{g} map(BA, BT_{X} _{p}^{∧})_{g} and on Eℤ/p ×_{ℤ/p} \overline{MT} = Bℤ/p × \overline{MT}. By theorem 3.9 the restriction from BA to B̃A_{0} induces a homotopy equivalence

\overline{MT} ⟶ MT_{0} .

Moreover, this map is ℤ/p–equivariant, and fits into a commutative diagram

\overline{MT} ⟶ Bℤ/p × \overline{MT} ⟶ Bℤ/p
↓  ↓  ∥
MT_{0} ⟶ Eℤ/p ×_{ℤ/p} MT_{0} ⟶ Bℤ/p

of homotopy equivalent fibrations, which proves (1).

(2) By the assumptions H^{∗}(MX_{0}) ≅ H^{∗}(MT_{0})^{W}. Moreover, this is a direct summand in H^{∗}(MT_{0}) considered as vector spaces. The diagram

MT_{0} ⟶ Eℤ/p ×_{ℤ/p} MT_{0} ⟶ Bℤ/p
↓  ↓  ∥
MX_{0} ⟶ Eℤ/p ×_{ℤ/p} MX_{0} ⟶ Bℤ/p

and (1) show that both fibrations are oriented and that all differentials in the associated Serre spectral sequences are trivial. For every n we have

H^{n}(Bℤ/p; H^{∗}(MX_{0})) ≅ H^{∗}(MX_{0})
≅ H^{∗}(MT_{0})^{W}
≅ H^{n}(Bℤ/p; H^{∗}(MT_{0}))^{W} .

The isomorphism between the first and last group is compatible with the inclusion H^{n}(Bℤ/p; H^{∗}(MX_{0}))⟶H^{n}(Bℤ/p; H^{∗}(MT_{0})) because W acts on the space level. Applying the 5–lemma to the extension problems yields the desired isomorphism

H^{∗}(Eℤ/p ×_{ℤ/p} MX_{0}) ≅ H^{∗}(Eℤ/p ×_{ℤ/p} MT_{0})^{W} . □

Proof of 10.3 for A a finite abelian p–group. By induction hypothesis H^{∗}(MX_{0}) ≅ H^{∗}(MT_{0})^{W}. To calculate MX = MX_{0}^{hℤ/p} we apply the Lannes functor for homotopy fixed–point sets (see section 3). Because MT is p–complete and H^{1}(MT) = 0, the W –equivariant map

HF^{ℤ/p}(MT_{0}) ⟶ H^{∗}(MT)

is an isomorphism (theorem 3.10). Using this fact and theorem 3.14 we get

HF^{ℤ/p}(MX_{0}) ≅ T_{sec^{∗}}^{ℤ/p}H^{∗}(Eℤ/p ×_{ℤ/p} MX_{0}) ⊗_{H^{∗}(Bℤ/p)} ℤ/p
≅ T_{sec^{∗}}^{ℤ/p}(H^{∗}(Eℤ/p ×_{ℤ/p} MT_{0})^{W}) ⊗_{H^{∗}(Bℤ/p)} ℤ/p
≅ T_{sec^{∗}}^{ℤ/p}((H^{∗}(Bℤ/p) ⊗ H^{∗}(MT_{0}))^{W}) ⊗_{H^{∗}(Bℤ/p)} ℤ/p
≅ (T_{id}^{ℤ/p}(H^{∗}(Bℤ/p)) ⊗ T^{ℤ/p}(H^{∗}(MT_{0})^{W}) ⊗_{H^{∗}(Bℤ/p)} ℤ/p
≅ H^{∗}(Bℤ/p) ⊗ T^{ℤ/p}(H^{∗}(MT_{0}))^{W} ⊗_{H^{∗}(Bℤ/p)} ℤ/p
≅ H^{∗}(MT)^{W} ,

where sec^{∗} denotes the cohomological maps induced by sections in the bundle Eℤ/p ×_{ℤ/p} MT_{0}. The third isomorphism follows from lemma 10.6 which says that the bundle Eℤ/p × MT_{0} ⟶ Bℤ/p is fiber homotopic trivial. The fourth isomorphism follows because T commutes with tensor products and because every section in the trivial bundle Bℤ/p × MT_{O} ⟶ Bℤ/p is given by a pair (id_{Bℤ/p}, g) of maps where g : Bℤ/p ⟶ MT_{0}. Hence HF^{ℤ}/p(MX_{0}) vanishes in degree 1. By theorem 3.10 again

H^{∗}(MT)^{W} ≅ HF^{ℤ/p}(MX_{0}) ⟶ H^{∗}(MX)

is an isomorphism, and by theorem 3.11, MX is p–complete which finishes the proof of 10.3. □

Proof of 10.4 for A a finite abelian p–group. Let g ≃ Bα : BA→BT_{G} be a map and denote by S_{p}N(T_{G}) ⊂ N(T_{G}) a p–toral Sylow subgroup. Then C_{G}(α) is connected (remark 10.2) and

C_{S_{p}N(T_{G})}(α) ⊂ C_{N(T_{G})}(α) = N(T_{C_{G}(α)})

is an inclusion of index coprime to p and therefore, also a p–toral Sylow subgroup. In particular,

H^{∗}(BC_{N(T_{G})}(α)) ⟶ H^{∗}(BC_{S_{p}N(T_{G})}(α))

and, by theorem 3.6,

H^{∗}(map(BA, BN(T_{G}))) ⟶ H^{∗}(map(BA, BS_{p}N(T_{G})))

are injections. In order to prove 10.4 it is sufficient to replace N(T_{G}) by S_{p}N(T_{G}). BC_{S_{p}N(T_{G})}(α)_{p}^{∧} ≃ (map(BA, BS_{p}N(T_{G}))_{g})_{p}^{∧} are p–complete (theorem 3.9), and

map(BA, BS_{p}N(T_{G}))_{g}_{p}^{∧} ≃ map(BA, BS_{p}N(T_{G})_{p}^{∧})_{g_{p}^{∧}}
≃ (map(BA_{0}, BS_{p}N(T_{G})_{p}^{∧})_{g_{0}_{p}^{∧}})^{hℤ/p}
≃ (map(BA_{0}, BS_{p}N(T_{G}))_{g_{0}} _{p}^{∧})_{g_{p}^{∧}}^{hℤ/p}

is the homotopy fixed–point set of a p–complete space. Therefore and because of theorem 3.11

HF_{g^{∗}}^{ℤ/p}(map(BA_{0}, BS_{p}N(T_{G}))_{g_{0}}) ≅ HF_{g^{∗}}^{ℤ/p}(map(BA_{0}, BS_{p}N(T_{G})_{p}^{∧})_{g_{0}_{p}^{∧}})
≅ H^{∗}(map(BA, BS_{p}N(T_{G})_{p}^{∧})_{g_{p}^{∧}})
≅ H^{∗}(map(BA, BS_{p}N(T_{G}))_{g}) .

The diagram (*)

[figure: \begin{aligned}& H^{∗}(map(BA, BS_{p}N(T_{G}))_{g}) \\ H^{∗}(map(BA, X)_{g}) & & H^{∗}(map(BA, BG)_{g}) \end{aligned}]

comes up by repeated applications of the functor HF^{ℤ/p} to the commutative diagram

[figure: \begin{aligned}& H^{∗}(BS_{p}N(T_{G})) \\ H^{∗}(X) & & H^{∗}(BG) & . \end{aligned}]

That is that the diagram (∗) also commutes (remark 3.4) which finishes the proof of 10.4. □

11. The normalizer of the maximal torus I.

Let G be a p–convenient compact connected Lie group, and let X be a space of the p–adic type of BG. In this section we construct an extension of the maximal torus

f_{T} : BT_{G} ⟶ X

to a map

f_{N} : BN(T_{G}) ⟶ X

of the classifying space of the the normalizer N(T_{G}) of T_{G}.

The space BN(T_{G}) is a two stage Posnikov system given by the fibration

BT_{G} ⟶ BN(T_{G}) ⟶ BW_{G} .

We have to find a suitable element in map(BN(T_{G}), X) ≃ map(BT_{G}, X)^{hW_{G}} (remark 3.12); i.e. a section in the bundle

EW_{G} ×_{W_{G}} map({ \atop BT_{G}}, X)_{f_{T}} ⟶ BW_{G} .

The obstructions are lying in

H^{∗+1}(BW_{G}; π_{∗}(map({ \atop BT_{G}}, X)_{f_{T}}) ≅ H^{3}(BW_{G}; LT_{G}_{p}^{∧})

(twisted coefficients). The isomorphism follows from theorem 10.1 and the identity π_{2}(BT_{G}_{p}^{∧}) ≅ H_{2}(BT_{G}; ℤ_{p}^{∧}) = LT_{G}_{p}^{∧}. By lemma 5.3 the obstruction group vanishes for odd primes which is sufficient to prove the following statement for odd primes.

11.1 Proposition. Let G be a compact connected Lie group, and let X be a space with the p–adic type of BG. If p is odd and G is p–convenient or if p = 2 and G is a product of unitary groups, there is an extension

[figure: \begin{matrix}BT_{G} & X \\ BN(T_{G}) \end{matrix}; f_{T}; f_{N}]

of the maximal torus BT_{G}⟶X to BN(T_{G})⟶X.

Because the obstruction group does not vanish for p = 2, we need a different approach. The proof in this case is postponed to the end of the section.

Choosing a fixed–point of the W_{G}–action on BT_{G} the evaluation at this fixpoint

map(BT_{G}_{p}^{∧}, X)_{f_{T}} ⟶ X

is equivariant with respect to the trivial action of W_{G} on X. The group W_{G} acts freely on the product BT_{G}_{p}^{∧} ≃ EW_{G} × map(BT_{G}_{p}^{∧}, X)_{f_{T}}. We get a well defined map on the orbit space Y := (EW_{G} × map(BT_{G}_{p}^{∧}, X)_{f_{T}})/W_{G} and an extension

[figure: \begin{matrix}BT_{G}_{p}^{∧} ≃ map(BT_{G}_{p}^{∧}, X)_{f_{T}} & X \\ Y \end{matrix}; f_{T}]

We will show that Y is nothing but the fiberwise completion BN(T_{G})_{p}^{∘} of BN(T_{G}) of the fibration BT_{G}⟶BN(T_{G})⟶BW_{G} (see [B-K; I.8]). By construction Y fits into a fibration

BT_{G}_{p}^{∧} ⟶ Y ⟶ BW_{G} .

We will consider a more general situation. Let W be a finite group, acting on a torus T via homomorphisms. Fibrations of the form

BT_{p}^{∧} ⟶ Y ⟶ BW

can be classified by homotopy classes of maps

BW ⟶BHE(BT _{p}^{∧}) ,

where HE(BT _{p}^{∧}) is the monoid of self equivalences of BT _{p}^{∧} [St]. Denote by SHE(BT _{p}^{∧}) the component of the identity. Then Gl(LT _{p}^{∧}) ≅ π_{0}(HE(BT _{p}^{∧})) is the group of the components. The Dwyer–Zabrodsky map

BT _{p}^{∧} ⟶ SHE(BT _{p}^{∧})

is a homotopy equivalence. Considering BT as a group the map is given by mapping each element to the associated left translation. Therefore, it is a homomorphism of monoids. Moreover, it is W –equivariant because W acts via homomorphisms. We get an W –equivariant equivalence

BBT _{p}^{∧} ⟶BSHE(BT _{p}^{∧}) .

Up to homotopy, the composition BW ⟶BHE(BT _{p}^{∧}) ⟶ BGl(LT _{p}^{∧}) is induced by an homomorphism which is given by the action of W on BT _{p}^{∧} associated to the fibration. The difference between two clasifying maps ϕ, ψ : BW ⟶BHE(BT _{p}^{∧}) with the same action on BT _{p}^{∧} is measured by an obstruction class

d(ϕ, ψ) ∈ H^{3}(BW; π_{3}(BSHE(BT _{p}^{∧}) ≅ H^{3}(BW; π_{2}(BT _{p}^{∧}) ≅ H^{3}(W; LT _{p}^{∧})

for lifting homotopies in the fibration

BSHE(BT _{p}^{∧}) ⟶BHE(BT _{p}^{∧}) ⟶ BGl(LT _{p}^{∧}) .

Notice that H^{3}(W; LT _{p}^{∧}) ≅ H^{2}(W; T)_{p}^{∧} which, forgetting the completion, describes the equivalence classes of group extensions T ⟶ H ⟶ W.

Let s : BW ⟶BHE(BT _{p}^{∧}) be the classifying map of the fibration BT _{p}^{∧}⟶B(T ⋊ W)_{p}^{∘}⟶BW.

Definition. By F ib(BW, BT _{p}^{∧}) we denote the set of fibrations divided out the equivalence relations given by

BT _{p}^{∧} ⟶ Y_{1} ⟶ BW
∥  ↓  ∥
BT _{p}^{∧} ⟶ Y_{2} ⟶ BW .

11.2 Lemma. The map

\begin{matrix}F ib(BW, BT _{p}^{∧}) & ⟶ & H^{3}(W; LT _{p}^{∧}) \\ (ϕ : BW →BHE(BT _{p}^{∧})) & ↦ & d(s, ϕ) \end{matrix}

is a bijection.

Proof. Every cohomology class can be realized as an obstruction. □

11.3 Corollary. The canonical map

\begin{aligned}B : & GrExt(W, T) & ⟶ & F ib(BW, BT _{p}^{∧}) \\ & T →N→W & ↦ & BT _{p}^{∧}→BN_{p}^{∘}→BW \end{aligned}

is a surjection.

Proof. Group extensions are described by cohomology classes in H^{2}(W; T) ≅ H^{3}(W; LT). The canonical map H^{2}(W; T)⟶H^{3}(W; LT _{p}^{∧}) is a surjection and reflects the map B on the cohomological level. □

11.4 Remark. Obviously there is an integral version of corollary 11.3. That is that

GrExt(W, T) ⟶ F ib(BW, BT)

is a bijection.

Now let us assume that G = G_{1} × G_{2} is a product of unitary groups. W_{G} acts trivially on the centers Z(G) and Z_{i} := Z(G_{i}), which are all tori. The canonical map

F ib(BW_{G}, BZ_{1}_{p}^{∧}) ⟶ H^{3}(W_{G}, LZ_{1}_{p}^{∧})

is a bijection and establishes a map

F ib(BW_{G}, BZ_{1}_{p}^{∧}) ⟶ F ib(BW_{G}, BT_{G_{1}} _{p}^{∧})

given by the inclusion Z_{1}⟶T_{G_{1}}.

The inclusion S^{1} = Z(U(n))↪T_{U(n)} induces

H^{3}(Σ_{n}; ℤ_{p}^{∧}) ≅ H^{3}(Σ_{n}; LS^{1}_{p}^{∧})

⟶ H^{3}(Σ_{n}; LT_{U(n)}_{p}^{∧}) ≅ H^{3}(Σ_{n−1}; LS^{1}_{p}^{∧}) ≅ H^{3}(Σ_{n−1}; ℤ_{p}^{∧})

This composition is given by the restriction and therefore an isomorphism for n ≠ 4 and a surjection for n = 4. In dimensions ≤ 2 this restriction induces an isomorphism for n ≠ 2, 4 and a surjection in all cases.

Now a spectral sequence argument shows that

H^{3}(W_{G}; LZ_{1}_{p}^{∧}) ⟶ H^{3}(W_{G}; LT_{G_{1}} _{p}^{∧})

and

F ib(BW_{G}, BZ_{1}_{p}^{∧}) ⟶ F ib(BW_{G}, BT_{G_{1}} _{p}^{∧})

are surjective. They are isomorphisms, if G_{1} has no factor isomorphic to U(4) or U(2).

11.5 Lemma. Let G be a product of unitary groups. Let

BT_{G}_{2}^{∧} ⟶ Y ⟶ BW_{G} ∈ F ib(BW_{G}, BT_{G}_{2}^{∧})

be a fibration such that H^{2}(Y; ℤ/2)⟶H^{2}(BT_{G}_{2}^{∧}; ℤ/2)^{W_{G}} is a surjection. Then there exists an equivalence of fibrations

BT_{G}_{2}^{∧} ⟶  Y  ⟶ BW_{G}
∥  ↓  ∥
BT_{G}_{2}^{∧} ⟶ B(T_{G} ⋊ W_{G})_{2}^{∘} ⟶ BW_{G} .

Proof. Let G = U(n_{1})×...×U(n_{k}). By corollary 11.3 there exists a group extension

E : T_{G} ⟶ N ⟶ W_{G}

such that

BT_{G}_{2}^{∧} ⟶ BN_{2}^{∘} ⟶ BW_{G}
∥  ↓  ↓
BT_{G}_{2}^{∧} ⟶ Y ⟶ BW_{G}

is an equivalence of fibrations. We have to show that BN_{2}^{∘} ≃ B(T_{G} ⋊ W_{G})_{2}^{∘}. The isomorphism

H^{2}(W_{G}; T_{G}) ≅ ⨁_{i} H^{2}(W_{G}; T_{U(n_{i})})

establishes group extensions

E_{i} : T_{U(n_{i})} ⟶ N_{i} ⟶ W_{G} .

E is given by the semi direct product, i.e N ≅ T_{G} ⋊ W_{G}, if and only if E_{i} is isomorphic to the semi direct product for all i. Moreover, we have a map

BT_{G}_{2}^{∧} ⟶ BN_{2}^{∘} ⟶ BW_{G}
↓  ↓  ∥
BT_{U(n_{i})}_{2}^{∧} ⟶ BN_{i}_{2}^{∘} ⟶ BW_{G}

between the two fibrations.

H^{2}(Y; ℤ/2) ≅ H^{2}(BN_{2}^{∘}; ℤ/2)⟶H^{2}(BT_{G}; ℤ/2)^{W_{G}} is an epimorphism. That is to say that the differential

d^{2} : H^{0}(BW_{G}; H^{2}(BT_{G}; ℤ/2)) ≅ H^{2}(BT_{G}; ℤ/2)^{W_{G}} ⟶ H^{3}(BW_{G}; ℤ/2)

of the Leray–Serre spectral sequence is trivial. Because

H^{0}(W_{G}; H^{2}(BT_{U(n_{i})}; ℤ/2)) ⟶ H^{0}(W_{G}; H^{2}(BT_{G}; ℤ/2))

is an injection the differential

d^{2} : H^{2}(BT_{U(n_{i})}; ℤ/2)^{W_{G}} ⟶ H^{3}(BW_{G}; ℤ/2)

is also trivial.

By the above remark there is a commutative diagram

BS^{1}_{2}^{∧} ⟶ Y_{i} ⟶ BW_{G}
↓  ↓  ∥
BT_{U(n_{i})}_{2}^{∧} ⟶ BN_{i}_{2}^{∘} ⟶ BW_{G} .

Because the action of W_{G} on BS^{1} is trivial the classifying map of the top fibration lifts to a map BW_{G}⟶BSHE(BS^{1}_{2}^{∧}) ≃ BBS^{1}_{2}^{∧}. This map is totally determined by the first transgression

d^{2} : H^{2}(BS^{1}_{2}^{∧}; ℤ_{2}^{∧}) ⟶ H^{3}(BW_{G}; ℤ_{2}^{∧})

in the Serre spectral sequence of the 2–adic cohomology and, because H^{3}(BΣ_{n}; ℤ_{2}^{∧}) and H^{3}(W_{G}; ℤ_{2}^{∧}) are annihilated by 2, also by the differential

d^{2} : H^{2}(BS^{1}_{2}^{∧}; ℤ/2) ⟶ H^{3}(BΣ_{n}; ℤ/2) .

Now we first assume that n is odd. Then H^{2}(BT_{U(n)}; ℤ/2)^{W_{G}} ≅ H^{2}(BS^{1}, ℤ/2), and the above transgression is given by the differential

d^{2} : H^{2}(BT_{U(n)}_{2}^{∧}; ℤ_{2}^{∧})^{W_{G}} ⟶ H^{3}(BW_{G}; ℤ_{2}^{∧})

which, by assumption, vanishes. Thus, the fibration BS^{1}_{2}^{∧}⟶Y_{i}⟶BW_{G} is trivial, and the fibration BT_{U(n_{i})}_{2}^{∧} ⟶ BN_{i}_{2}^{∘} ⟶ BW_{G} is given by the fiber wise completion B(T_{U(n_{i})} ⋊ W_{G})_{2}^{∘}.

Now let n_{i} be even, and let G = U(n_{i}) × G_{i}. Let S^{1}⟶T_{U(n_{i})} be the inclusion in the last factor and ℤ/2⟶S^{1} the standard inclusion. Then

C_{N_{i}}(ℤ/2)/C_{T_{U(n_{i})}}(ℤ/2) = C_{N_{i}}(ℤ/2)/T_{U(n_{i})} = Σ_{n_{i}−1} × G_{i} ,

and S^{1}⟶C_{N}(ℤ/2) is central. We define \overline{N}_{i} := C_{N_{i}}(ℤ/2)/S^{1} and get a commutative diagram of fibrations

BT_{U(n_{i}−1)}_{2}^{∧} ⟶ B\overline{N}_{i}_{2}^{∘}  ⟶ BΣ_{n_{i}−1} × BW_{G_{i}}
↑  ↑  ∥
BT_{U(n_{i})}_{2}^{∧} ⟶ BC_{N_{i}}(ℤ/2)_{2}^{∘} ⟶ BΣ_{n_{i}−1} × BW_{G_{i}}
∥  ↓  ↓
BT_{U(n_{i})}_{2}^{∧} ⟶  BN_{i}_{2}^{∘}  ⟶ BΣ_{n_{i}} × BW_{G_{i}} .

A comparison of the two lower rows shows that the transgression

d^{2} : H^{2}(BT_{U(n_{i})}; ℤ/2)^{Σ_{n_{i}−1}×W_{G_{i}}} ⟶ H^{3}(BΣ_{n_{i}−1} × BW_{G_{i}}; ℤ/2)

is trivial, and a comparison between the two upper rows implies that the differential of the top fibration

d^{2} : H^{2}(BT_{U(n_{i}−1)}; ℤ/2)^{Σ_{n_{i}−1}×W_{G_{i}}} ⟶ H^{3}(BΣ_{n_{i}−1} × BW_{G_{i}}; ℤ/2)

is also trivial. Because n_{i} − 1 is odd the top fibration is given by the semi direct product.

On the level of the cohomology groups, describing the group extensions, the above construction goes along with the maps

H^{2}(Σ_{n_{i}} ×W_{G_{i}}; T_{U(n_{i})}) ⟶ H^{2}(Σ_{n_{i}−1}×W_{G_{i}}; T_{U(n_{i})}) ⟶ H^{2}(Σ_{n_{i}} ×W_{G_{i}}; T_{U(n_{i}−1)}) .

The composition

H^{2}(Σ_{n−1}; R) ⟶^{≅} H^{2}(Σ_{n};R^{n}) ⟶ H^{2}(Σ_{n−1}; R^{n})

⟶ H^{2}(Σ_{n−1}; R^{n−1}) ⟶^{≅} H^{2}(Σ_{n−2}; R)

is given by the restriction. This is an isomorphism for n = 4, 2, n ≥ 6, and R = S^{1} or R = ℤ/2 [Na]. Hence, by a spectral sequence argument follows that the above composition is an isomorphism for even n_{i}’s.

Passing to the obstruction groups, describing the associated fibrations, we see that

BT_{U(n_{i})}_{2}^{∧} ⟶BN_{i}_{2}^{∘} ⟶ BΣ_{n_{i}} × W_{G_{i}}

is given by the semi direct product if and only if the fibration

BT_{U(n_{i}−1)}_{2}^{∧} ⟶B\overline{N}_{i}_{2}^{∘} ⟶ BΣ_{n_{i}−1} × W_{G_{i}}

is given by the semi direct product which we already proved. □

Now we are in the position to finish the proof of proposition 11.1.

Proof of proposition 11.1. It is only left to consider one case, namely G is a product of unitary groups and p = 2. The composition of maps

BT_{G}_{2}^{∧} ⟶ Y ⟶ X

shows that H^{∗}(Y; ℤ/2)⟶H^{∗}(BT_{G}_{2}^{∧}; ℤ/2)^{W_{G}} is surjective. By lemma 11.5 the fibration

BT_{G}_{2}^{∧} ⟶Y ⟶ BW_{G}

is given by the fiberwise completion of the classifying spaces of the exact sequence

T_{G} ⟶T_{G} ⋊ W_{G} ⟶W_{G} .

The observation T_{G} ⋊ W_{G} = N(T_{G}) completes the proof. □

12. The normalizer of the maximal torus II.

In this section G is a compact connected Lie group which is p–convenient for p odd and a product of unitary groups for p = 2. Let X be a space with the p–adic type of BG.

In section 11 we constructed an extension f_{N} : BN(T_{G})⟶X of the maximal torus f_{T} : BT_{G}⟶X, which fits into the diagram

[figure: \begin{aligned}BT_{G} & BN(T_{G}) & BG \\ & & X & , \end{aligned}; Bi^{∗}; f_{T}; Bi^{∗}; f_{N}]

where the outer triangle commutes in p–adic cohomology. We consider the question, whether the inner triangle commutes in mod-p cohomology.

The extension BN(T_{G})⟶X induces a lift in

(*)  [figure: \begin{aligned}& H^{∗}(BN(T_{G}); ℤ/p) \\ H^{∗}(BG; ℤ/p) & H^{∗}(BT_{G}; ℤ/p) . \end{aligned}; Bi^{∗}; a; Bi^{∗}]

Here we work in the category of unstable algebras over the Steenrod algebra. The lift a is given by the composition

H^{∗}(BG; ℤ/p) ⟵^{\overline{ϕ}} H^{∗}(X; ℤ/p) ⟶^{f_{N}^{∗}} H^{∗}(BN(T_{G}); ℤ/p) ,

where \overline{ϕ} is an isomorphism. The map Bi^{∗} : H^{∗}(BG; ℤ/p)⟶H^{∗}(BT_{G}; ℤ/p) is induced by the standard inclusion. Another lift comes from the standard inclusion Bi^{∗} : BN(T_{G})⟶BG.

For any homomorphism ψ : W_{G}⟶T_{G}^{W_{G}}, we define j_{ψ} : N(T_{G})⟶N(T_{G}) to be the automorphism

N(T_{G}) ⟶^{Δ} N(T_{G}) × W_{G} ⟶^{id×ψ} N(T_{G}) × T_{G}^{W_{G}} ⟶^{μ} N(T_{G}) ,

where Δ is the diagonal composed with the projection on W_{G} and μ the multiplication. Obviously, j_{ψ}|_{T_{G}} = id|_{T_{G}}, and N(T_{G})⟶^{j_{ψ}}N(T_{G})⟶^{i}G induces another lift in the diagram (∗). We will prove the following statement:

12.1 Proposition. Let G be a p–convenient pseudo simply connected Lie group or a product of unitary groups, and let

a : H^{∗}(BG; ℤ/p) ⟶ H^{∗}(BN(T_{G}); ℤ/p)

be a lift of H^{∗}(BG; ℤ/p)⟶H^{∗}(BT_{G}; ℤ/p).

(1) If p is odd then a = Bi^{∗}; i.e. there is only one algebraic lift, given by the standard inclusion.

(2) If p = 2 there exists a homomorphism ψ : W_{G}⟶T_{G}^{W_{G}} such that a = Bj_{ψ}^{∗}Bi^{∗}; i.e a can be realized by an self automorphism of N(T_{G}) composed with the standard inclusion.

12.2 Corollary. Let G be a p–convenient pseudo simply connected Lie group or a product of unitary groups. Then there exists an extension f_{N} : N(T_{G})⟶X of f_{T} : BT_{G}⟶X such that

[figure: \begin{aligned}BT_{G}Bi^{∗} & BN(T_{G}) & BG \\ & & X & , \end{aligned}; f_{T}; Bi^{∗}; f_{N}]

commutes in mod–p cohomology.

Proof. By proposition 11.1 we choose an extension f′_{N} : BN(T_{G})⟶X of f_{T} : BT_{G}⟶X. If p is odd we can directly apply the above theorem.

If p = 2 we have f′_{N}^{∗}\overline{ϕ}^{−1} = Bj_{ψ}^{∗}Bi^{∗} for a suitable homomorphism ψ : W_{G}⟶T_{G}^{W_{G}}. By construction j_{ψ} is an isomorphism. Hence f_{N} := f′_{N}B(j_{ψ}^{−1}) satisfies the statemant. □

For the following we assume that G = U(n_{1}) × ... × U(n_{r}) is a product of unitary groups. We fix a lift a : H^{∗}(BG; ℤ/p)⟶H^{∗}(BN(T_{G}); ℤ/p) of the standard inclusion into H^{∗}(BT_{G}; ℤ/p).

The next lemma says that, for the proof of proposition 12.1, it suffices to look at compositions

H^{∗}(BG; ℤ/p) ⟶^{a} H^{∗}(BN(T_{G}); ℤ/p) ⟶^{Bj^{∗}} H^{∗}(BV; ℤ/p)

for every elementary abelian p–subgroup j : V ⟶N(T_{G}).

12.3 Lemma. Let G be a product of unitary groups. Then H^{∗}(BN(T_{G}); ℤ/p) is detected by elementary abelian subgroups.

Proof. [G-L-Z]. □

Every element x ∈ N(T_{U(p)}) can be written in the form x = (λ_{1}, ..., λ_{p}; τ), λ_{i} ∈ S^{1} and τ ∈ Σ_{p}. Let σ ∈ Σ_{p} be the permutation represented by the cycle (1, 2, ..., p), let V_{2} ⊂ N(T_{U(p)}) be the group generated by (1, ..., 1; σ), and let V_{1} ⊂ N(T_{U(p)}) be the subgroup generated by diag(ω, ..., ω), ω = exp(2πi/p). Here diag( ) denotes the canonical element in T_{U(p)} ⊂ N(T_{U(p)}). Then V_{1} × V_{2}⟶N(T_{U(p)}) is a subgroup.

12.4 Lemma. The elements

(λ_{1}, ..., λ_{p}; σ), (ωλ_{1}, ..., ωλ_{p}; σ), (λ_{1}, ..., λ_{p}; σ^{l}), (μ, ..., μ; σ)

are conjugate in N(T_{U(p)}) if (l, p) = 1 and μ^{p} = ∏_{i} λ_{i}.

Proof. Choose α_{1} = 1 and α_{i} = α_{i−1}λ_{i−1}μ^{−1}. Then (α_{1}, ..., α_{p}; 1) conjugates the first element into the last one. σ and σ^{l} have the same cycle type. They are conjugate in Σ_{p}. With these operation we can construct all necessary conjugations. □

For abbreviation, we set

H_{n}(k) := U(p)^{k} × (S^{1})^{n−pk} ⊂ U(n) ,
W_{n}(k) := (V_{1} × V_{2})^{k} × (ℤ/p)^{n−pk} ⊂ H_{n}(k) ⊂ U(n) ,
V_{n}(k) := (V_{1})^{k} × (ℤ/p)^{n−pk} = W_{n}(k) ∩ T_{U(n)} ⊂ H_{n}(k) ⊂ U(n) .

12.5 Lemma. Every element x ∈ N(T_{G}) of order p is conjugate to an element in a suitable subgroup

W_{n_{1}}(k_{1}) × ... × W_{n_{r}}(k_{r}) ⊂ N(T_{U(n_{1})}) × ... × N(T_{U(n_{r})}) = N(T_{G}) .

Proof. x can be written in the form ∏_{i}(λ_{i}; τ_{i}), λ_{i} ∈ T_{U(n_{i})} and τ_{i} ∈ Σ_{n_{i}}. τ_{i} is represented by a product of cycles of length p. An application of lemma 12.4 finishes the proof. □

Let j : V ⟶N(T_{G}) be an elementary abelian p–subgroup of N(T_{G}). By the theorems 3.1 and 3.6 the map Bj^{∗}a can be realized by a homomorphism ρ_{V} : V ⟶G which is unique up to conjugation. The character χ_{ρ_{V}} (x) depends on the chosen element x ∈ N(T_{G}) and on a, but not on the subgroup V containing x, nor on the homomorphism ρ_{V}. Therefore χ(x) := χ_{a}(x) := χ_{ρ_{V}} (x) is well defined for all elements x ∈ N(T_{G}) of order p. By T r(x) we denote the trace of the associated matrix, given by N(T_{G})⟶G = U(n_{1}) × ... × U(n_{r}). Now we are prepared for the proof of theorem 12.1.

Proof of proposition 12.1 for G a product of unitary groups.

Let G = U(n_{1}) × ... × U(n_{r}). Let j : V′⟶N(T_{G}) be an elementary abelian p–subgroup of N(T_{G}). Because G is a product of unitary groups, it is a question of characters to determine ρ_{V′}. Thus we have only to consider cyclic subgroups of order p.

Let x ∈ N(T_{G}) be an element of order p. Up to conjugation

x ∈ W_{N,K} := W_{n_{1}}(k_{1}) × ... × W_{n_{r}}(k_{r}) ⟶^{j} N(T_{G})

for suitable k_{1}, ..., k_{r}. We choose k_{1}, ..., k_{r} minimal; i.e x is not contained in a group W_{n_{1}}(l_{1}) × ... × W_{n_{r}}(l_{r}) of the same form such that l_{i} ≤ k_{i} for all i and at least for one i there is strict inequality. Because the cohomological restriction of a : H^{∗}(BG; ℤ/p)⟶H^{∗}(BN(T_{G}); ℤ/p) to H^{∗}(BT_{G}; ℤ/p) is the standard inclusion, we can assume that ρ_{W_{N,K}} |_{V_{N,K}} is the standard inclusion, where V_{N,K} := V_{n_{1}}(k_{1})× ... × V_{n_{r}}(k_{r}).

We set

H_{N,K} := H_{n_{1}}(k_{1}) × ... × H_{n_{r}}(k_{r}) = C_{G}(V) .

Applying the Lannes functor T_{Bi^{∗}}^{V} to

H^{∗}(BG; ℤ/p) ⟶^{a} H^{∗}(BN(T_{G}); ℤ/p) ⟶ H^{∗}(BT_{G}; ℤ/p)

gives

H^{∗}(BH_{N,K}; ℤ/p) ⟶^{a} H^{∗}(BN(T_{H}); ℤ/p) ⟶ H^{∗}(BT_{H}; ℤ/p) = H^{∗}(BT_{G}; ℤ/p)

(theorem 3.5) which shows that ρ_{W_{N,K}} can be taken to have image in H_{N,K}. x ∈ W_{N,K} can be thought of having the form

x = (x_{1}, ..., x_{r})
x_{i} = (\underbrace{(1, ..., 1; σ), ..., (1, ..., 1; σ)}_{k_{i} times}, λ_{1}(i), ..., λ_{n_{i}−pk_{i}}(i)) ∈ W_{n_{i}}(k_{i}) .

We also consider y = (y_{1}, ..., y_{r}) with

y_{i} = (\underbrace{(1, ..., 1; σ), ..., (1, ..., 1; σ)}_{k_{i} times}, 1, ..., 1) ∈ W_{n_{i}}(k_{i}) ,

and write ρ_{W_{N,K}} (y) = (B_{1}, ..., B_{r}) ∈ H_{N,K} with

B_{i} = (A_{1}(i), ..., A_{k_{i}}(i), β_{1}(i), ..., β_{n_{i}−pk_{i}}(i)) ∈ H_{n_{i}}(k_{i}) ,

where A_{j}(i) ∈ U(p) and β_{l}(i) ∈ S^{1}. This implies

ρ_{W_{N,K}} (x) = (D_{1}, ..., D_{r})
D_{i} = (A_{1}(i), ..., A_{k_{i}}(i), β_{1}(i)λ_{1}(i), ..., β_{n_{i}−pk_{i}}(i)λ_{n_{i}−pk_{i}}(i))
= B_{i}diag(1, ..., 1, λ_{1}(i), ..., λ_{n_{i}−pk_{i}}(i)) .

Since y_{i} and y′_{i} := y_{i}diag(ω, ..., ω, 1, ..., 1) (ω pk_{i}–times) are conjugate in N(T_{H_{n_{i}} (k_{i})}) (lemma 12.4), χ(y) = χ(y_{1}, ..., y′_{i}, ..., y_{r}). This shows that T r(A_{1}(i)) = 0. Analogously follows that T r(A_{j}(i)) = 0 for all j and i.

We first assume p is odd. Since y_{i} is conjugate to (y_{i})^{2} in N(T_{H_{n_{i}} (k_{i})}) and y_{i}diag(1, ..., 1, ω) to (y_{i})^{2}diag(1, ..., 1, ω), we get ∑_{j} β_{j}(i) = ∑_{j} β_{j}(i)^{2} and β_{n_{i}−pk_{i}}(i)(ω − 1) = β_{n_{i}−pk_{i}}(i)^{2}(ω − 1). This implies β_{n_{i}−pk_{i}}(i) = 1. An analogous argument shows that β_{j}(i) = 1 for all j and i.

Let π_{l} : G⟶U(n_{l}) be the projection onto the l–th factor. Because ρ_{W_{N,K}} |_{V_{N,K}} is the standard inclusion, it follows immediately that

χ(x) = χ_{π_{l}ρ_{W_{N,K}}} (x) = ∑_{j=1}^{n_{l}} λ_{j}(l) ,

which is the character of the inclusion

V (x)↪W_{N,K} ⟶^{i} G ⟶^{π_{l}} U(n_{l}) ,

where V (x) is the cyclic group of order p generated by x. Thus, π_{l}ρ_{W_{N,K}} j is conjugate to π_{l}i|_{V (x)} and ρ_{W_{N,K}} j = ρ_{V (x)} is conjugate to the standard inclusion.

Now we assume p = 2. We have χ(x) = T r(ρ_{W_{N,K}} (x)) = ∑_{i,j} β_{j}(i)λ_{j}(i). If we permute the λ_{j}(i)’s for a fixed i the new element is conjugate to x in N(T_{G}). We can vary x to realize different values for the λ_{j}(i)’s. Therefore, β_{j}(i) = β_{l}(i) for all j and l. As elements of order 2, β_{j}(i) = ±1. This implies that ρ_{W_{N,K}} (x) = (D_{1}, ..., D_{r}) with

D_{i} = (A_{1}(i), ..., A_{k_{i}}(i), ε_{i}(x)λ_{1}(i), ..., ε_{i}(x)λ_{n_{i}−pk_{i}}(i)) .

The function ε_{i} : W_{G}⟶{±1} is constant on conjugate elements and multiplicative if restricted to an 2–elementary abelian subgroup. We choose for every i a transposition τ_{i} ∈ Σ_{n_{i}} and set

z(τ_{i}) := (z_{1}, ...z_{r}) , z_{i} ∈ N(T_{U(n_{i})})
z_{j} := \{ \begin{matrix}diag(1, ..., 1)  if j ≠ i \\ (1, ..., 1; τ_{i})  if j = i . \end{matrix}

We define a homomorphism

ψ = (ψ_{j,k}) : W_{G} = ∏_{j=1}^{r} Σ_{n_{j}} ⟶ T_{G}^{W_{G}} = ∏_{k=1}^{r} S^{1}

by setting

ψ_{j,k} : Σ_{n_{j}} ⟶ S^{1} : σ ↦ \{ \begin{matrix}sign(σ) if ε_{j}(z(τ_{k})) = −1 \\ +1 if ε_{j}(z(τ_{k})) = +1 . \end{matrix}

Again we denote by π_{l} : G⟶U(n_{l}) the projection on the l–th factor, and by V (x) ⊂ W the subgroup generated by x. Then, by construction, π_{l}ρ_{V (x)} and

the map V (x)↪N(T_{G})⟶^{j_{ψ}}N(T_{G})⟶^{i}G⟶^{π_{l}}U(n_{l}) have the same character for all x ∈ N(T_{G}). That is to say that both homomorphisms are conjugate. To finish the proof we can proceed as for p odd. □

In order to prove theorem 12.1 in the other case, namely for a p–convenient pseudo simply connected Lie group G, we will use the mod–p subgroup H⟶G of G given by proposition 5.2. Unfortunately, for p = 3, H might have a factor SU(p) besides unitary groups. We have to prove the theorem for products of unitary groups and SU(p)’s. For odd primes the p–toral Sylow subgroup of N(T_{SU(p)}) is T_{SU(p)} ⋊ ℤ/p. Thus, lemma 12.4 is still true.

If we define

H′_{p}(0) := T_{SU(p)} ⊂ SU(p)
H′_{p}(1) := SU(p)
W′_{p}(1) := V_{1} × V_{2} ⊂ SU(p)
W′_{p}(0) := T_{SU(p)} =: V′_{P} (0)
V′_{p}(1) := V_{1} ⊂ W′_{p}(1) ∩ T_{SU(p)} ⊂ H′_{p}(1) ⊂ SU(p) ,

few minor changes make the proof work. Only the replacement of lemma 12.3 needs some comments.

12.6 Lemma. For p odd, H^{∗}(BN(T_{SU(p)}); ℤ/p) is detected by elementary abelian subgroups.

Proof. We consider the Gysin sequence

... ⟶ H^{∗−2}(BN(T_{SU(p)}); ℤ/p) ⟶ H^{∗}(BN(T_{U(p)}); ℤ/p)

⟶^{Bi^{∗}} H^{∗}(BN(T_{SU(p)}); ℤ/p) ⟶^{d} H^{∗−1}(BN(T_{U(p)}); ℤ/p) ⟶ ...

of the fibration

S^{1} ⟶ BN(T_{SU(p)}) ⟶^{Bi} BN(T_{U(p)}) .

Let x ∈ H^{∗}(BN(T_{U(p)}); ℤ/p), such that d(x) ≠ 0. We choose an elementary abelian p–subgroup V ⟶BN(T_{U(p)}), detecting d(x). Let ⟨V, Z(U(p))⟩ be the group generated by V and the center Z(U(p)) of U(p). Then W := ⟨V, Z(U(p))⟩ ∩ N(T_{SU(p)}) is an elementary abelian group, fits into the fibration

S^{1} ⟶ BW ⟶ B⟨V, Z(U(p))⟩ ,

and detects x.

If d(x) = 0 then x = Bi^{∗}(y) for y ∈ H^{∗}(BN(T_{U(p)}); ℤ/p). Because S^{1} ≀ ℤ/p is the p–toral Sylow group of N(T_{U(p)}), the spectral sequence of B(S^{1} ≀ ℤ/p) shows that every class in H^{∗}(BN(T_{U(p)}); ℤ/p) is detected by W_{p}(1) which, for p odd, is contained in SU(p), or by (ℤ/p)^{p} ⊂ T_{U(p)}. In the first case W_{p}(1) detects x, and in the second case (ℤ/p)^{p−1}. □

Proof of theorem 12.1 in the p–convenient pseudo simply connected case.

We can assume that p is odd because every 2–convenient pseudo simply connected Lie group is a product of unitary groups. By lemma 5.2, there is a pelementary subgroup i : V ⟶T_{G}⟶G of G, such that H := C_{G}(V) is mod–p isomorphic as group to an product of unitary groups and SU(p)’s. Moreover the index [N(T_{G}) : N(T_{H})] is coprime to p. Applying the Lannes functor T_{Bi^{∗}}^{V} to a lift a : H^{∗}(BG; ℤ/p)⟶H^{∗}(BN(T_{G}); ℤ/p) establishes a commutative diagram

H^{∗}(BG; ℤ/p) ⟶^{a} H^{∗}(BN(T_{G}); ℤ/p) ⟶ H^{∗}(BT_{G}; ℤ/p)
↓  ↓  ∥
H^{∗}(BH; ℤ/p) ⟶^{b} H^{∗}(BN(T_{H}); ℤ/p) ⟶ H^{∗}(BT_{H}; ℤ/p) ,

where b = T_{Bi^{∗}}^{V}(a). The composition in the bottom line is induced by

BT_{H} _{p}^{∧} ⟶ map(BV, BT_{G}_{p}^{∧})_{Bi} ⟶ map(BV, BG_{p}^{∧})_{Bi} ≃ BH_{p}^{∧} ,

which can be considered as the standard inclusion (see remark 3.4). The two left vertical arrows are injections, and b is induced by the standard inclusion. Therefore, a is also given by the standard inclusion. □

13. The map BG ⟶X part I: G pseudo simply connected.

In this section G is a p–convenient pseudo simply connected Lie group for p odd, and a product of unitary groups for p = 2. For p = 2 this includes the case of pseudo simply connected Lie groups. As usual, X is a space with the p–adic type of BG. We will prove theorem 1.2 in these cases, i.e we have to prove the propositions 2.3 and 2.4.

By the propositions 11.1 and 12.1, there exists a triangle

(*)  [figure: \begin{aligned}& BN(T_{G}) \\ BG_{p}^{∧} & & X \end{aligned}; Bi; f_{N}]

commuting in mod–p cohomology. As in section 2 R_{p}(G) is the orbit category of p–stubborn subgroups of G contained in N(T_{G}). For any p–stubborn subgroup

P ⟶^{i_{p}} N(T_{G}) ⟶ G of G, we defined in section 2 a map

f_{P} := f_{N} ∘ Bi_{p} : BP ⟶ BN(T_{G}) ⟶ X .

The collection of these maps gives a diagram

{BP }_{R_{p}(G)} ⟶X

commuting in mod–p cohomology because of the diagram (∗)

As first step we show that this diagram commutes up to homotopy, which is the statement of proposition 2.3.

Proof of proposition 2.3.

Let c_{g} : G/P ⟶G/P′ be a map of R_{p}(G) given by a conjugation. We have to show that

BP ⟶^{Bi_{P}} BN(T_{G}) ⟶^{f_{N}} X
Bc_{g} ↓  ∥
BP′ ⟶^{Bi_{P′}} BN(T_{G}) ⟶^{f_{N}} X

commutes up to homotopy. Without loss of generality we can assume that, according to a fixed chosen maximal torus and normalizer T_{G}⟶N(T_{G})⟶G, the subgroup P is the special representative of the conjugacy class of P; i.e. P satisfies proposition 6.7. Moreover, we can think of α := i_{P′}c_{g} as being a homomorphism such that f_{N}Bα and f_{N}Bi_{P} induce the same map in mod–p cohomology, and i_{P} is conjugate to α in G. Under these assumptions, we will show that f_{N}Bα ≃ f_{N}Bi_{P}.

The intersection P_{T} = P ∩ T_{G} is an abelian group. The restrictions α|_{P_{T}} and i_{P} |_{P_{T}} are conjugate in G, and hence, by theorem 10.1 (i) (see also remark 10.2 (i) ), also conjugate in N(T_{G}). That is that f_{N}Bi_{P} |_{P_{T}} and f_{N}Bα|_{P_{T}} are homotopic. The following proposition, the analogy of proposition 6.7 (iv), shows that the extensions of f_{N}Bi_{P} |_{P_{T}} to maps BP ⟶X are classified by mod-p cohomology. This finishes the proof. □

13.1 Proposition. For any p–toral group P satisfying the hypothesis of proposition 6.7, the canonical map

π_{0}(map(BP, X)_{g|_{BP_{T}} ≃f_{N} Bi_{P} |_{BP_{T}}}) ⟶ Hom(H^{∗}(X; ℤ/p), H^{∗}(BP; ℤ/p))

is an injection.

Proof. We have to relate map(BP, BG) to map(BP, X).

The intersection P_{T} = P ∩ T_{G} is a normal subgroup of P, because P ↪N(T_{G}) normalizes T_{G} and P. The quotient Q := P/P_{T} acts on { \atop BP_{T}} ≃ BP_{T} and on

map({ \atop BP_{T}}, −) ≃ map(BP_{T} , −). By remark 3.12 map(BP, −) ≃ map({ \atop BP_{T}}, −)^{hQ}.

We define i_{P_{T}} := i_{P} |_{P_{T}}. The diagram (∗)

[figure: \begin{aligned}& map({ \atop BP_{T}}, BN(T_{G}))_{Bi_{P_{T}}} \\ map({ \atop BP_{T}}, BG_{p}^{∧})_{Bi_{N} Bi_{P_{T}}} & & map({ \atop BP_{T}}, X)_{f_{N} Bi_{P_{T}}} \end{aligned}]

commutes in mod–p cohomology (proposition 10.4). By the theorems 3.9 and 10.1

map({ \atop BP_{T}}, BG_{p}^{∧})_{Bi_{N} Bi_{P_{T}}} ≃ BT_{G}_{p}^{∧} ≃ map({ \atop BP_{T}}, X)_{f_{N} Bi_{P_{T}}} ,

and by proposition 3.8

map({ \atop BP_{T}}, BN(T_{G}))_{Bi_{P_{T}}} ≃ BT_{G} .

The diagonal maps in (∗) are Q–equivariant, because the components are fixed, and mod–p equivalences. Taking homotopy fixed–points we get (∗∗)

\begin{aligned}& map({ \atop BP_{T}}, BN(T_{G}))_{Bi_{P_{T}}}^{hQ} \\ map({ \atop BP_{T}}, BG_{p}^{∧})_{Bi_{N} Bi_{P_{T}}}^{hQ} & & map({ \atop BP_{T}}, X)_{f_{N} Bi_{P_{T}}}^{hQ} & . \end{aligned} [figure] [figure]

Both diagonal maps are also mod–p equivalences, because an equivariant mod–p equivalence between 1–connected spaces induces a mod–p equivalence between the homotopy fixed–point sets.

The components of map({ \atop BP_{T}}, BG_{p}^{∧})_{Bi_{N} Bi_{P_{T}}}^{hQ} are distinguished by mod–p cohomology (proposition 6.7). The triangle

[figure: \begin{aligned}& BN(T_{G}) \\ BG & & X \end{aligned}]

commutes in mod–p cohomology. Any map in map({ \atop BP_{T}}, X)_{f_{N} Bi_{P_{T}}}^{hQ} has a lift to BN(T_{G}). The obstruction groups which classify the extensions up to homotopy (see remark 3.12 and lemma 3.13)

H^{2}(Q; π_{2}(map(BP_{T} , X)_{f_{N} Bi_{P_{T}}})) ≅ H^{2}(Q; π_{2}(BT_{G}))
≅ H^{2}(Q; π_{2}(map(BP_{T} , BG_{p}^{∧})_{Bi_{N} Bi_{P_{T}}})

are isomorphic. The isomorphisms of the obstruction groups follow from 10.1. All this together implies that the components of

map(BP_{T} , X)_{f_{N} Bi_{P_{T}}}^{hQ} ≃ map(BP, X)_{g|_{BP_{T}} ≃f_{N} Bi_{P_{T}}}

are also distinguished by mod–p cohomology. □

Proposition 2.3 establishes a map from the 1–skeleton of the homotopy colimit holim_{\overrightarrow{R_{p}(G)}} BP to X. In order to extend this map to the homotopy colimit, we have to

calculate the obstruction groups

\underleftarrow{lim}^{i+1}_{R_{p}(G)} π_{i}(map(BP, X)_{f_{P}}) i ≥ 2

for extending this map on the 1–skeleton to the homotopy colimit. \underleftarrow{lim}^{j} is the j–th derived functor of the inverse limit functor. This we will do by comparing the two functors

Π_{i}(X), Π_{i}(G) : R_{p}(G) ⟶ Ab ,

given by

Π_{i}(X)(G/P) := π_{i}(map(BP, X)_{f_{P}})
Π_{i}(G)(G/P) := π_{i}(map(BP, BG_{p}^{∧})_{Bi_{P}}) .

That is we have to contruct a natural transformation between the two functors which is an equivalence. The existence of this transformation is claimed in proposition 2.4.

Proof of proposition 2.4.

The triangle (∗∗) induces for every p–stubborn subgroup P ↪G homotopy equivalences

map(BP, BG_{p}^{∧})_{Bi_{P}} ⟵^{≃} (map(BP, BN(T_{G}))_{Bi_{P}})_{p}^{∧} ⟶^{≃} map(BP, X)_{f_{N} Bα_{N}} ,

which depend on the chosen lift i_{P} : BP ⟶BN(T_{G}) of i_{P} : BP ⟶BG. Two lifts only differ by a conjugation c_{g} (theorem 3.6). We already proved that f_{N}Bi_{P} ≃ f_{N}c_{g}Bi_{P}. Therefore the above equivalence induces well defined isomorphisms

Π_{i}(G)(G/P) ⟶^{≅} Π_{i}(X)(G/P) ,

which are compatible with the maps in R_{p}(G). □

The following lemma fills the last open gap of the program of section 2 and completes the proof of theorem 1.2 for pseudo simply connected Lie groups and products of unitary groups.

13.2 Lemma.

\underleftarrow{lim}^{j}_{R_{p}(G)} Π_{i}(X)(G/P) = 0 i, j ≥ 1 .

Proof. Because of proposition 2.4 we have to calculate \underleftarrow{lim}^{j}_{R_{p}(G)} Π_{i}(G)(G/P).

G = G′ × G′′ splits into two factors; G′′ is simply connected and G′ a product of unitary groups. Since R_{p}(G′ × G′′) ≅ R_{p}(G′) × R_{p}(G′′) ( proposition 6.3), we have Π_{i}(G) ≅ Π_{i}(G′) × Π_{i}(G′′). We have

\underleftarrow{lim}^{j}_{R_{p}(G′)} Π_{1}(G′) = 0

\underleftarrow{lim}^{j}_{R_{p}(G′′)} Π_{1}(G′′) = 0

\underleftarrow{lim}^{j}_{R_{p}(G)} Π_{i}(G) = 0 for i ≥ 2 .

The last two equations follow from [J-M-O 1; theorem 4.8], and the first from the fact that centralizers in unitary groups are always products of unitary groups, and therefore Π_{1}(G′) ≡ 0. □

14. The map BG ⟶ X part II : The general case.

After having proved theorem 1.2 in the pseudo simply connected case we complete the proof of theorem 1.2 in this section. First we assume that G is a p– convenient simply connected Lie group. Then G = G′ × G′′ splits into two factors, where G′ := SU(n_{1}) × ... × SU(n_{l}) is a product of special unitary groups and G′′ a product of the other simply connected Lie groups. There exist exact sequences

K ⟶ G′ × (S^{1})^{l} ⟶ U(n_{1}) × ... × U(n_{l}) =: \overline{G}′

and

K ⟶ G × (S^{1})^{l} ⟶ \overline{G}′ × G′′ =: H ,

where K := ℤ/n_{1} × ... × ℤ/n_{l}. Now let X be a space with the p–adic type of BG, and let f_{T} : BT_{G}⟶X be the maximal torus of X.

We consider the composition

f_{K} := f_{T} ∘ Bi : BK_{p} ⟶ BT_{G} × (BS^{1})^{l} ⟶ X × (BS^{1})^{l} ,

where K_{p} is the p–Sylow subgroup of the abelian group K. Because K_{p} ⊂ G×(S^{1})^{l} is a central subgroup, the evaluation map

map(BK_{p}, X × (BS^{1}_{p}^{∧})^{l})_{f_{K}} ⟶ X × (BS^{1})^{l}_{p}^{∧}

is a mod–p equivalence (theorem 10.1). Moreover, the evaluation map is a homotopy equivalence, because both spaces are p–complete (theorem 10.1).

We can think of BK_{p} as a group which acts freely on

EBK_{p} × map(BK_{p}, X × (BS^{1})^{l}_{p}^{∧}) ≃ X × (BS^{1})^{l}_{p}^{∧} .

We define the p–complete space Y by the fibration

BK_{p} ⟶ X × (BS^{1})^{l}_{p}^{∧} ⟶ (X × (BS^{1})^{l}_{p}^{∧})/BK_{p} =: Y .

We have BT_{H} _{p}^{∧} ≃ B(T_{G} × (S^{1})^{l}/K)_{p}^{∧} ≃ BT_{G}_{p}^{∧} × (BS^{1})^{l}_{p}^{∧}/BK_{p}. Thus, the above fibration fits into the commutative diagram

BK ⟶ BG × (BS^{1})^{l} ⟶ BH
∥  ↑  ↑
BK ⟶ BT_{G} × (BS^{1})^{l} ⟶ BT_{H}
↓  ↓  ↓
BK_{p} ⟶ X × (BS^{1})^{l}_{p}^{∧} ⟶ Y .

BT_{H}⟶Y is a maximal torus. The Weyl group action of W_{G} on BT_{G} × (BS^{1})^{l} commutes with the action of BK_{p}. Therefore W_{G} = W_{H} acts as Weyl group on BT_{H}⟶Y. Analogously to the proof of proposition 4.2 follows that H^{∗}(Y; ℤ_{p}^{∧}) is p–torsionfree, that H^{∗}(Y; ℤ/p) ≅ H^{∗}(BT_{H}; ℤ/p)^{W_{H}} ≅ H^{∗}(BH; ℤ/p), and, by the lemma of Nakayama, that H^{∗}(Y; ℤ_{p}^{∧}) ≅ H^{∗}(BT_{G}; ℤ_{p}^{∧})^{W_{H}} ≅ H^{∗}(BH; ℤ_{p}^{∧}). That is to say that Y has the p–adic type of BH. We collect these considerations into the following proposition.

14.1 Proposition. Let G be a p–convenient simply connected Lie group, and H the associated pseudo simply connected Lie group. Let X have the p–adic type of BG. Then there exists a p-complete space Y with the p–adic type of BH, and a diagram

[figure: \begin{aligned}BG & & X \\ & BT_{G} \\ & BT_{H} \\ BH & & Y & , \end{aligned}]

which commutes in p–adic cohomology.

Now we are ready to prove theorem 1.2. We do this in several steps, first for simply connected Lie groups, then for products of simply connected Lie groups and tori, and finally in the general case.

Proof of theorem 1.2 for simply connected Lie groups.

Let X be a space with the p–adic type of BG, and let Y be the space constructed above. We can think of X as the fibre of the map

Y = (X × (BS^{1})^{l}_{p}^{∧})/BK_{p} ⟶ (BS^{1})^{l}_{p}^{∧}/BK_{p} =: BT_{p}^{∧}

as well as of BG_{p}^{∧} as the fibre of

BH_{p}^{∧} ⟶ BT_{p}^{∧} .

By section 13, there exists a homotopy equivalence BH_{p}^{∧}⟶Y, which can be extended to

BG_{p}^{∧} ⟶ BH_{p}^{∧} ⟶ BT_{p}^{∧}
↓  ^{≃}↓  ^{≃}↓
X ⟶ Y ⟶ BT_{p}^{∧} .

Obviously BG_{p}^{∧}⟶X is a homotopy equivalence. □

Proof of theorem 1.2 for products of simply connected Lie groups and tori.

Let X be a space with the p–adic type of BG_{s} × BT, where G_{s} is a simply connected Lie group and T a torus. The projection BG_{s}_{p}^{∧} × BT_{p}^{∧}⟶BT_{p}^{∧} corresponds to a map X⟶BT_{p}^{∧} which has an canonical left inverse s : BT_{p}^{∧}⟶BT_{G_{s}} _{p}^{∧}× BT_{p}^{∧}⟶X. The fiber X_{s} of X⟶BT_{p}^{∧} has the p–adic type of BG_{s}. Thus we have BG_{s}_{p}^{∧} ≃ X_{s}. By theorem 10.1 map(BT_{p}^{∧}, X)_{s} ≃ X. The adjoint

BG_{s}_{p}^{∧} × BT_{p}^{∧} ⟶ X

of

BG_{s}_{p}^{∧} ≃ X_{s} ⟶ X ≃ map(BT_{p}^{∧}, X)_{s}

is a mod–p isomorphism and hence, a homotopy equivalence. □

Proof of theorem 1.2 in the general case.

Let G be a p–convenient compact connected Lie group and X a space with the p–adic type of BG. The finite covering

K ⟶ G_{s} × T ⟶ G ,

G_{s} simply connected and T a torus, establishes a fibration

BK_{p} ⟶ BG_{s}_{p}^{∧} × BT_{p}^{∧} ⟶ BG_{p}^{∧} ,

where K_{p} is the p–Sylow subgroup of K. This fibration is classified by a map BG_{p}^{∧}⟶BBK_{p}. Since H^{∗}(BG; ℤ/p^{r}) ≅ H^{∗}(X; ℤ/p^{r}) for all r, there is a corresponding map X⟶BBK_{p} classifying the fibration

BK_{p} ⟶ Y ⟶ X .

This fibration fits into

BK_{p} ⟶ BG_{s}_{p}^{∧} × BT_{p}^{∧} ⟶ BG_{p}^{∧}
∥  ↑  ↑
BK_{p} ⟶ BT_{G_{s}} _{p}^{∧} × BT_{p}^{∧} ⟶ BT_{G}_{p}^{∧}
∥  ↓  ↓
BK_{p} ⟶  Y  ⟶ X .

The Weyl group W_{G} = W_{G_{s}} acts as Weyl group on BT G_{s}_{p}^{∧} × BT_{p}^{∧}⟶Y. The space Y is p–complete and of the p–adic type of BG_{s}_{p}^{∧} × BT_{p}^{∧}. This follows from a comparison of the Serre spectral sequences for p–adic cohomology of all three horizontal fibrations and implies that BG_{s} × BT and Y are homotopy equivalent. Moreover, this equivalence fits in a diagram

(*)  \begin{matrix}BK_{p} & ⟶ & BG_{s}_{p}^{∧} × BT_{p}^{∧} & ⟶ & BG_{p}^{∧} \\ ∥ & & ↓^{≃} \\ BK_{p} & ⟶ & Y & ⟶ & X & . \end{matrix}

We have to fill in the arrow in the right collumn.

The canonical G–equivariant homotopy equivalence

X ⟶ map({ \atop BK_{p}}, X)_{c} ,

c : { \atop BK_{p}}⟶X the constant map, induces equivalences

map(BG, X) = X^{hG} ≃ map(B̃K_{p}, X)_{c}^{hG} ≃ map(BG_{s}_{p}^{∧} × BT_{p}^{∧}, X)_{g|_{BKp} ≃c}

(see remark 3.12), where we take in the last mapping space only the components of those maps whose restriction to BK_{p} is trivial up to homotopy. This implies that there exists a map BG_{p}^{∧}⟶X making the diagram (∗) commutative. Moreover, this map is an equivalence. □

15. The integral homotopy type.

In this chapter we prove theorem 1.6.

Proof of theorem 1.6. Let X be a 1–connected CW–complex of finite type. If H^{∗}(X; ℤ) is torsionfree the λ–ring structure of K(X) completely determines the algebra H^{∗}(X; ℤ/p) as well as the Steenrod algebra action on H^{∗}(X; ℤ/p) for all primes p [At]. Therefore, if H^{∗}(X; ℤ) and H^{∗}(BG; ℤ) are torsion free,

H^{∗}(X; ℤ/p) ≅ H^{∗}(BG; ℤ/p)

as algebras over the Steenrod algebra. If in addition G is 2–convenient or G is a product of unitary groups we can construct maximal tori

f_{T} _{p}^{∧} : BT_{X} _{p}^{∧} ⟶ X_{p}^{∧}

and Weyl group actions of W_{G} on BT X_{p}^{∧} for all p. Moreover, H^{∗}(X; ℤ_{p}^{∧}) ≅ H^{∗}(BT_{X} _{p}^{∧}; ℤ_{p}^{∧})^{W_{G}}. All this follows from proposition 7.2.

Using the results of [At] a Chern character argument shows that

K(X; ℤ_{p}^{∧}) ≅ K(BT_{X} _{p}^{∧}; ℤ_{p}^{∧})^{W_{G}} .

K(−; ℤ_{p}^{∧}) denotes complex K–theory with p-adic coefficients. By [Wil] there exist λ–ring maps

ϕ : K(BT_{X} _{p}^{∧}; ℤ_{p}^{∧}) ⟶ K(BT_{G}_{p}^{∧}; ℤ_{p}^{∧})
ψ : K(BT_{G}_{p}^{∧}; ℤ_{p}^{∧}) ⟶ K(BT_{X} _{p}^{∧}; ℤ_{p}^{∧})

such that

K(X_{p}^{∧}; ℤ_{p}^{∧}) ≅ K(BT_{X} _{p}^{∧}; ℤ_{p}^{∧})^{W_{G}} ⟶ K(BT_{X} _{p}^{∧}; ℤ_{p}^{∧})
≅↓ ≅↓ ϕ↓ ↑ψ
K(BG_{p}^{∧}; ℤ_{p}^{∧}) ≅ K(BT_{G}_{p}^{∧}; ℤ_{p}^{∧})^{W_{G}} ⟶ K(BT_{G}_{p}^{∧}; ℤ_{p}^{∧}) .

By an argument analogous to [A-M; 1.7] ϕψ and ψϕ differ from the identity only by an element of the Weyl group. Thus, ϕ and ψ are isomorphisms. Because K–theory classifies maps BT_{G}_{p}^{∧}⟶BG_{p}^{∧} up to homotopy [No 1] [N-S 1], the representations of W_{G}, given by the action on BT_{G} and BT_{X} _{p}^{∧}, are conjugate over ℤ_{p}^{∧}. That is to say that X_{p}^{∧} has the p–adic type of BG. By theorem 1.2 we get X_{p}^{∧} ≃ BG_{p}^{∧}.

The rationalizations X_{ℚ} and BG_{ℚ} are homotopy equivalent, because BG_{ℚ} is a product of rational Eilenberg–McLane spaces and because

H^{∗}(X; ℚ) ≅ K(X) ⊗ ℚ ≅ K(BG) ⊗ ℚ ≅ H^{∗}(BG; ℚ) .

Hence X is in the p–adic genus of BG. The finite universal cover G_{s} × T ⟶G does not contain any Sp(n) or Spin(n), n ≥ 3, as factor. Therefore, in this case complex K–theory classifies the adic genus [No 2; remark 5.4, theorem 1.8 and the proof]. Since K(X) ≅ K(BG) this implies that BG ≃ X. □

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List of notation.

ℤ/p  group of order p
ℤ_{(p)}  integers localised at p
ℤ_{p}^{∧}  p–adic integers
X_{p}^{∧}  mod–p completion in the sense of Bousfield and Kan
X_{p}^{∘}  fiberwise completion
map(X, Y)  mapping space
map(X, Y)_{f}  the component of map(X, Y) associated to the map f
F ib(B.F)  set of equivalence classes of fibrations F ⟶E⟶B
BG  classifying space of the group G
im(ρ), ker(ρ)  image and kernel of a homomorphism between groups
Rep(G, H)  set of representations from G to H,i.e. conjugacy classes

of homomorphisms G⟶H

S_{p}G  p–toral Sylow subgroup of a compact Lie group G
Z(G)  center of the group G
C_{G}(H), C_{G}(ρ)  centralizer of a subgroup H or of the image of a

homorphism ρ in G

N_{G}(H)  normalizer of H in G
GrExt(G, H)  set of equivalence classes of exact sequences

H⟶K⟶G of groups

L/p  L ⊗ ℤ/p
L_{p}^{∧}  p–adic completion of a module L, or L ⊗ ℤ_{p}^{∧}
X−−→Y  map which exists only in cohomology as H^{∗}(Y) ⟶ H^{∗}(X)

and which is waiting for a topological realization

For a compact connected Lie group G, we denote by

T_{G}  the maximal torus of G
W_{G}  the Weyl group of G
N(T_{G})  the normalizer of T_{G}
V_{G}  the maximal elementary abelian p–subgroup of T_{G}
LT_{G}  H_{2}(BT_{G}; ℤ)
L^{∗}T_{G}  H^{2}(BT_{G}; ℤ)