STABLE SPLITTINGS FOR CLASSIFYING SPACES OF ALTERNATING, SPECIAL ORTHOGONAL AND SPECIAL UNITARY GROUPS

by

Hans–Werner Henn and Huỳnh Mui

Abstract

Let G(n) denote either the symmetric group Σ(n), the orthogonal group O(n) or the unitary group U(n) and let SG(n) denote either the alternating group A(n), the special orthogonal group SO(n) or the special unitary group SU(n). The classifying spaces BG(n) are known to split stably as BG(n) ≃ ⋁_{ℓ=1}^{n} BG(ℓ)/BG(ℓ − 1). We con-

sider the case of BSG(n) and prove that, after localizing at any prime p, there are similar although somewhat coarser splittings. E.g. we get a stable 2-local splitting

BA(4n) ≃ BA(4n)/BA(4n − 2) ∨ ⋁_{ℓ=1}^{n−1} BA(4ℓ + 2)/BA(4ℓ − 2) . A crucial ingredient in

our proof is a careful study, for finite p- groups P, of the morphism sets mor_{A}(P, SG(n)) in the “Burnside category” A , and in particular the effect of transfers on these sets.

Introduction

Stable splittings of classifying spaces have received a lot of attention in the last decade. A good introductory exposition is given in [P]. Perhaps the first known

splittings were those for the symmetric groups, BΣ(n) ≃ ⋁_{ℓ=1}^{n} BΣ(ℓ)/BΣ(ℓ − 1),

which can be obtained via a geometric version of Dold’s proof [D] of Nakaoka’s [Na] decomposition theorem for the homology of symmetric groups (cf. [KP]). (As usual here and in the following quotients like BΣ(ℓ)/BΣ(ℓ − 1) are to be interpreted as cofibres of the maps BΣ(ℓ − 1) → BΣ(ℓ) induced by the inclusion Σ(ℓ − 1) → Σ(ℓ).) In [MP] Mitchell and Priddy show how to extend these

1991 Mathematics Subject Classification Primary 55R35, Secondary 55P42, 55R12 This paper is in final form and no version of it will be submitted for publication elsewhere

2 HANS–WERNER HENN and HUỲNH MUI

methods and they produce stable splittings BG(n) ≃ ⋁_{ℓ=1}^{n} BG(ℓ)/BG(ℓ − 1) with G(ℓ) = O(ℓ), U(ℓ), Sp(ℓ) or GL(ℓ, _{q}). In the last case one has to invert p if q = p^{k}. In this paper we prove the following

THEOREM 0. For each natural number n there are p–local stable splittings of the form

BSG(n) ≃_{(p)} ⋁_{ℓ=1}^{n} BSG(ℓ)/BSG(ℓ − δ(ℓ))

in each of the following cases: (0.1.) p = 2, SG(n) = A(n), the alternating group on n letters, and

δ(ℓ) = \{ \begin{matrix}4 & if ℓ ≡ 2 mod 4 \\ 0 & if ℓ ≢ 2 mod 4 and ℓ ≠ n \\ k & if ℓ ≡ 2 + k mod 4, k ∈ \{1, 2, 3\} and ℓ = n \end{matrix}

(0.2.) p > 2, SG(n) = A(n), and

δ(ℓ) = \{ \begin{matrix}p & if ℓ ≡ 2 mod p \\ 0 & if ℓ ≢ 2 mod p and ℓ ≠ n, \\ k & if ℓ ≡ 2 + k mod p, k ∈ \{1, 2, …, p − 1\} and ℓ = n \end{matrix}

(0.3.) p = 2, SG(n) = SO(n), the special orthogonal group of real n × n–matrices, and

δ(ℓ) = \{ \begin{matrix}2 & if ℓ ≡ 1 mod 2 , \\ 0 & if ℓ ≡ 0 mod 2 and ℓ ≠ n, \\ 1 & if ℓ ≡ 0 mod 2 and ℓ = n \end{matrix}

(0.4.) p any prime, SG(n) = SU(n), the special unitary group of complex n × n–matrices, and

δ(ℓ) = \{ \begin{matrix}2 & if ℓ ≡ 1 mod p , \\ 0 & if ℓ ≡ 0 mod p and ℓ ≠ n, \\ 1 & if ℓ ≢ 0, 1 mod p; or ℓ ≡ 0 mod p and ℓ = n \end{matrix}

(By convention SG(m) will be the trivial group if m ≤ 0.)

EXAMPLES

1) BA(4n) ≃ BA(4n)/BA(4n − 2) ∨ ⋁_{ℓ=1}^{n−1} BA(4ℓ + 2)/BA(4ℓ − 2), after localization at 2.

2) BSO(2n + 1) ≃ ⋁_{ℓ=1}^{n} BSO(2ℓ + 1)/BSO(2ℓ − 1), after localization at 2. After inverting 2, Snaith [Sn] and Mitchell and Priddy [MP] constructed also stable splittings BSO(2n + 1) ≃ ⋁_{ℓ=1}^{n} BSO(2ℓ + 1)/BSO(2ℓ − 1). Is there a global splitting of this form?

REMARKS

a) The proof of Theorem 0 will actually show that the natural “filtration” of BSG(n) by the BSG(ℓ) with δ(ℓ) ≠ 0 splits (stably and after localization) and hence we may include the limiting case n = ∞ in Theorem 0.

b) The splittings of Theorem 0 are coarser than those for Σ(n), O(n) and U(n), where δ(ℓ) was always 1. The following observations show that this is to be expected, at least for A(n) and SO(n).

The restriction maps H^{∗}(BA(4n + 2); _{2}) → H^{∗}(BA(4n); _{2}) and H^{∗}(BA(pn + 2); _{p}) → H^{∗}(BA(pn); _{p}) are not epi if n ≥ 1 (cf. Remark 4.2.(b) below).

The restriction map H^{∗}(BSO(2n + 1); _{2}) → H^{∗}(BSO(2n); _{2}) is not split epi as a homomorphism of modules over the Steenrod algebra (for this consider the action of Sq^{1} on the Stiefel–Whitney class w_{2n}).

c) It can be shown (cf. [B]) that the inclusion A(n) → A(n + 1) induces an isomorphism in mod p cohomology if n ≢ 1, 3 mod 4 and p = 2, or n ≢ 1, p − 1 mod p and p > 2. Then BA(n) and BA(n + 1) are stably homotopy equivalent at p and hence there are other choices of δ(ℓ) which still describe the same splittings.

d) If n ≢ 0, 1 mod p, p > 2, then our splitting of BA(n) is an easy consequence of the splitting of BΣ(n) and the fact that the inclusion A(n) → Σ(n) induces a p–local stable equivalence in this case (cf. [B]).

The paper is organized as follows. In Section 1 we construct the splitting maps h : BSG(n) → ⋁_{ℓ=1}^{n} BSG(ℓ)/BSG(ℓ − δ(ℓ)) by a modification of the construction

in the case of BG(n). Our proof of Theorem 0 then depends on studying the composition of h with (suitable stable) maps BP → BSG(n) where P runs through the class of finite p–groups. The splitting map itself is a Section 2 we will describe a convenient algebraic set up for transfer calculations (cf. [AGM], [Ma], [Ni] and [HLS]) and state an algebraic analogue (Theorem 2.7) of Theorem 0 in this set up. In Section 3 we show how this leads to a proof of Theorem 0 and in Section 4 we give a proof of Theorem 2.7.

The authors would like to thank the Forschungsschwerpunkt Geometrie at the University of Heidelberg for enabling them to work together. The second author is particularly grateful to Prof. D.Puppe for making his visit to Heidelberg (in late 1988) possible.

A preliminary version of this work was available in preprint form in mid 1989. Its final publication has been delayed by the difficulties the authors experienced in trying to exchange information since then.

1. Construction of the splitting maps

In the following all spaces and maps are to be considered in the stable category of CW –spectra. We denote X with a disjoint base point by X^{+}. Stably X^{+} and X∨S^{0} are equivalent. Let (G(n), SG(n)) be either (Σ(n), A(n)) or (O(n), SO(n)) or (U(n), SU(n)), and let i_{n} denote the canonical inclusion SG(n) → G(n) and Bi_{n} the induced map BSG(n)^{+} → BG(n)^{+}.

Let det : G(n) → G(ϵ) denote either the signum or the determinant map, i.e. ϵ = 2 in case G(n) = Σ(n) and ϵ = 1 in the other two cases. Then there is another canonical inclusion ĩ : G(n) × G(ϵ) → G(n + ϵ) given by juxtaposition and the homomorphism

G(n)⟶^{(id,det^{−1})} G(n) × G(ϵ) \underrightarrow{_{i}} G(n + ϵ)

has image in SG(n+ϵ). We denote the resulting homomorphism G(n) → SG(n+ ϵ) and the corresponding stable map by d_{n+ϵ} resp. Bd_{n+ϵ}.

Furthermore we write tr_{n,m} for the transfer belonging to the inclusion G(n) × G(m) → G(n + m) given again by juxtaposition; tr_{n,m} is a stable map BG(n + m)^{+} → (BG(n) × BG(m))^{+}. Finally, the projection map G(n) × G(m) → G(n) resp. (BG(n) × BG(m))^{+} → BG(n)^{+} will be denoted by π_{n,m} resp. Bπ_{n,m} or simply by π resp. Bπ.

We will now define the components of a splitting map

h : BSG(n) → ⋁_{ℓ=1}^{n} BSG(ℓ)/BSG(ℓ − δ(ℓ))

(with δ(ℓ) as in Theorem 0) as the composition of a map h_{n,ℓ} : BSG(n)^{+} → BSG(ℓ)^{+} with the inclusion BSG(n) → BSG(n)^{+} and the canonical projection q_{ℓ} of BSG(ℓ)^{+} to the quotient BSG(ℓ)^{+}/BSG(ℓ − δ(ℓ))^{+} = BSG(ℓ)/BSG(ℓ − δ(ℓ)). The map h_{n,ℓ} is defined as

— identity if n = ℓ,

— as the following composition if ϵ < ℓ < n.

BSG(n)^{+}⟶^{Bi_{n}}BG(n)^{+}⟶^{tr_{ℓ−ϵ,n+ϵ−ℓ}}(BG(ℓ − ϵ) × BG(n + ϵ − ℓ))^{+}

⟶^{Bπ}BG(ℓ − ϵ)^{+}⟶^{Bd_{ℓ}}BSG(ℓ)^{+}.

— any map if ℓ ≤ ϵ. (Then SG(ℓ) and the corresponding wedge summand are trivial.)

It will be convenient to consider the map

h^{+} : BSG(n)^{+} → (⋁_{ℓ=1}^{n} BSG(ℓ)/BSG(ℓ − δ(ℓ))) ∨ S^{0}

with components q_{ℓ}h_{n,ℓ}, 1 ≤ ℓ ≤ n, and BSG(n)^{+} → B{e}^{+} ≃ S^{0}, induced by the trivial homomorphism SG(n) → {e}. Clearly, h^{+} is a p–local equivalence if and only if h is one.

2. A(P, G)

We will study h^{+} by composing it with maps BP^{+} → BSG(n)^{+} of the form BP^{+}⟶^{tr}BQ^{+}⟶^{Bφ}BSG(n)^{+}, where Q is a subgroup of P, tr = tr_{P}^{Q} the transfer associated to the inclusion Q ⊂ P and φ : Q → SG(n) a group homomorphism.

In this section we will describe the appropriate algebraic set up for dealing with such compositions (cf. [AGM], [Ma], [Ni] and [HLS]). The crucial input is the definition of a transfer homomorphism which is modelled after the double coset formula.

2.1. For a fixed finite group P and a compact Lie group G let A(P, G) be the free abelian group with basis the equivalence classes of pairs (Q, φ) consisting of a subgroup Q of P and a homomorphisms φ : Q → G. Here the pairs (Q, φ) and (Q′, φ′) are called equivalent if and only if the groups Δφ = {(q, φq) | q ∈ Q} and Δφ′ = {(q′, φq′) | q′ ∈ Q′} are conjugate as subgroups of P ×G. The equivalence class of (Q, φ) will be denoted by [Q, φ] (or, more precisely, by [Q, φ]_{(P,G)}). It is easy to see, say by a minor modification of the argument given in [Q2, Lemma 6.3], that the set of these equivalence classes is finite. Therefore, G ↦ A(P, G) defines a functor from the category G_{cL} of compact Lie groups to the category Ab_{f} of finitely generated free abelian groups. Furthermore, for a closed subgroup H of G, there is a transfer homomorphism τ = τ_{G}^{H} : A(P, G) → A(P, H) which is defined on a basis element [Q, φ] as follows.

Q acts via φ on the homogeneous space G/H, and the double coset space Q\G/H may be decomposed as a disjoint union of “orbit type manifold components” M_{i} ([F]). More precisely, if Q′ runs through a set of representatives of conjugacy classes of subgroups of Q and if G/H is broken up into the disjoint union of the subspaces (G/H)_{(Q′)}, consisting of those points in G/H whose Q–orbits are isomorphic to Q/Q′, then the M_{i} are the connected components of the corresponding orbit spaces Q\(G/H)_{(Q′)}. Let χ^{#}(M_{i}) be the “internal Euler characteristic” of M_{i}, i.e. χ^{#}(M_{i}) = χ(\overline{M}_{i}) − χ(\overline{M}_{i} − M_{i}) with \overline{M}_{i} denoting the closure of M_{i} in Q\G/H. Finally choose a representative g_{i} ∈ G of any element in M_{i} so that the isotropy subgroup of g_{i}H is Q_{i}. Then we define

(2.2.)  τ_{G}^{H}([Q, φ]) = ∑_{i} χ^{#}(M_{i})[Q_{i}, g_{i}^{−1}φg_{i}].

(Note that g_{i}^{−1}φg_{i} maps Q_{i} into H !) It is easy to see that the right hand side depends neither on the particular choice of g_{i} (e.g. because the space of equivalence classes is finite, in particular discrete) nor on the chosen representative of the equivalence class [Q, φ]. We remark that for G a finite group A(P, Q) agrees with the set of morphisms from P to G in the “Burnside category” as defined in [AGM] .

2.3. This formula for the transfer is fairly complicated. We will now focus on the “leading term” of the transfer and show that it has a more manageable description.

Subconjugation defines a partial order on the set of conjugacy classes (Q) of subgroups of P. From this partial order we get a filtration of A(P, G) by defining F_{(Q)}A(P, G) as the subgroup generated by the classes [Q′, φ] with (Q′) ≤ (Q). We let \overline{F}_{(Q)}A(P, Q) denote the quotient of F_{(Q)}A(P, Q) by the subgroups of lower filtration. Obviously G → \overline{F}_{(Q)}A(P, G) is also a functor from G_{cL} to Ab_{f} and (2.2) shows that the transfer for A(P,?) induces one for \overline{F}_{(Q)}A(P,?) which will be denoted by \overline{τ}.

Next we will give a different description of \overline{F}_{(Q)}. Let Rep(Q, G) denote the set of G–conjugacy classes of homomorphisms from Q to G. The class of a homomorphism φ will be denoted by [φ] (or more precisely by [φ]_{G}). The normalizer N_{P} (Q) of Q in P acts on Rep(Q, G) via conjugation. We denote the quotient of Rep(Q, G) with respect to this action by Rep_{P} (Q, G).

Then the map Rep(Q, G) → A(P, G), [φ] ↦ [Q, φ] induces a natural isomorphism between [Rep_{P} (Q, G)] and \overline{F}_{(Q)}A(P, G). (Here and in the following the free abelian group on a set S will be denoted by [S]). Thus we get a transfer \overline{τ} on [Rep_{P} (Q,?)] which we will now describe.

For homomorphisms φ : Q → G and ψ : Q → H let (G/H)_{ψ}^{φ} = {gH | [g^{−1}φg]_{H} = [ψ]_{H}} consist of those cosets gH which conjugate [φ]_{G} into [ψ]_{H} and let [φ : ψ] denote the Euler characteristic of (G/H)_{ψ}^{φ}. Define τ = τ_{G}^{H} : [Rep(Q, G)] → [Rep(Q, H)] via

(2.4.)  [φ] ⟼ ∑_{[ψ]∈Rep(Q,H)} [φ : ψ][ψ].

(cf. [HLS]). Clearly, τ_{G}^{H} induces a homomorphism τ_{G}^{′H} : [Rep_{P} (Q, G)] → [Rep_{P} (Q, H)].

PROPOSITION 2.5. \overline{τ}_{G}^{H} and τ_{G}^{′H} agree.

\underline{Proof:} Consider \overline{τ}_{G}^{H}. By (2.2) we only have to consider the contributions coming from the components of the fixed points (G/H)^{Q}. These components are closed, hence χ^{#}(M_{i}) and χ(M_{i}) agree. Now the proposition follows from the observation

that (G/H)_{ψ}^{φ} = ∐_{[g_{i}^{−1}φg_{i}]=[ψ]} M_{i}.

The following Lemma will be useful in Section 4 for computing the coefficients [φ : ψ] in concrete cases. The proof is straightforward and left to the reader.

LEMMA 2.6. Suppose (G/H)_{ψ}^{φ} is not empty and let g_{0} ∈ G be any element satisfying g_{0}^{−1}φg_{0} = ψ. Then the map

C_{G}(ψ)/C_{G}(ψ) ∩ H ⟶ (G/H)_{ψ}^{φ}
induced by  g ⟼ g_{0}gH

is a homeomorphism. (Here C_{G}(ψ) denotes the centralizer of the image of ψ in G, i.e. C_{G}(ψ) = {g ∈ G | gψ(q) = ψ(q)g for all q ∈ Q}.)

We will finish this section by stating the algebraic analogue of Theorem 0. For this let P be a finite p–group and p, SG(ℓ) and δ(ℓ) be as in Theorem 0.

We will see in 4.1 that for such P the homomorphism A(P, SG(ℓ − δ(ℓ)) → A(P, SG(ℓ)) induced by inclusion is mono. In Theorem 2.7 below we will identify A(P, SG(ℓ − δ(ℓ)) with its image in A(P, SG(ℓ)).

As in Section 1 we introduce homomorphisms h_{n,ℓ} : A(P, SG(n)) → A(P, SG(ℓ)). For ℓ = n we define h_{n,n} = id. For 0 < ℓ ≤ ϵ we let h_{n,ℓ} be trivial, and for ϵ < ℓ < n we define h_{n,ℓ} as composition d_{ℓ}πτ_{ℓ−ϵ,n+ϵ−ℓ}i_{n} where d_{ℓ}, π and i_{n} are as in Section 1 and τ_{ℓ−ϵ,n+ϵ−ℓ} denotes the transfer τ_{G(n)}^{G(ℓ−ϵ)×G(n+ϵ−ℓ)}.

THEOREM 2.7. For each natural number n and each finite p–group P the homomorphism

h^{+} : A(P, SG(n)) → (⨁_{ℓ=1}^{n} A(P, SG(ℓ))/A(P, SG(ℓ − δ(ℓ)))) ⊕ A(P, {e}) ,

with components induced by h_{n,ℓ} respectively the trivial homomorphism P → {e}, becomes an isomorphism after tensoring with /p.

The proof of Theorem 2.7 will be given in Section 4. In the following the target of h^{+} will be denoted by C(P, SG(n)), and the component A(P, SG(n)) → A(P, {e}) = A(P, SG(0)) by h_{n,0}.

3. Proof of Theorem 0

The following folklore proposition gives the justification for Section 2. As there P will be a finite group, G a compact Lie group and H a closed subgroup of G.

PROPOSITION 3.1. There are homomorphisms λ = λ_{P,G} from A(P, G) to the group {BP^{+}, BG^{+}} of homotopy classes of stable maps from BP^{+} to BG^{+}, defined by λ_{P,G}[Q, φ] = Bφ ∘ tr_{P}^{Q} : BP^{+} → BQ^{+} → BG^{+}.

These homomorphisms are natural in P and Q and commute with transfers, i.e. tr_{G}^{H} ∘ λ_{P,G} = λ_{P,H} ∘ τ_{G}^{H}.

\underline{Proof:} Inner automorphisms induce selfmaps on BP^{+} and BG^{+} which are homotopic to the identity maps; hence λ_{P,Q} is well defined. Naturality is trivial and compatibility with the transfer follows from the double coset formula [F]. (For the existence of transfers BG^{+} → BH^{+} in the compact Lie group case we refer to [C].)

We remark that Lewis, May and McClure [LMM] have shown that λ_{P,Q} induces an isomorphism between a suitable completion of A(P, G) and {BP^{+}, BG^{+}} if G is finite. However, we will not need this in the sequel.

We begin with the proof of Theorem 0, assuming Theorem 2.7. For the remainder of this section let p, SG(ℓ) and δ(ℓ) be as in Theorem 0.

3.2. If P_{ℓ} is any finite p–group and φ_{ℓ} : P_{ℓ} → SG(ℓ), 0 ≤ ℓ ≤ n, are any homomorphisms (recall that SG(0) = {e} by convention), then Theorem 2.7 implies that there are elements x_{ℓ} ∈ A(P_{ℓ}, SG(n)) and x′_{ℓ} ∈ C(P_{ℓ}, SG(n)) such that

\overline{[P_{ℓ}, φ_{ℓ}]} = h^{+}(x_{ℓ}) + px′_{ℓ},

where \overline{[P_{ℓ}, φ_{ℓ}]} denotes the class of [P_{ℓ}, φ_{ℓ}] ∈ A(P_{ℓ}, SG(ℓ)) in C(P_{ℓ}, SG(n)). Then Proposition 3.1 gives a diagram

(3.3.)  \begin{aligned}BSG(n)^{+} & ⟶^{h^{+}} & ⋁_{ℓ=1}^{n} BSG(ℓ)/BSG(ℓ − δ(ℓ)) ∨ S^{0} \\ ⋁_{ℓ=0}^{n} λ(x_{ℓ}) & ↑ ⋁_{ℓ=0}^{n} q_{ℓ}∘Bφ_{ℓ} \\ & ⋁_{ℓ=0}^{n} BP_{ℓ}^{+} \end{aligned}

which commutes after passing to mod p–cohomology. Now Lemma 3.4 below implies easily that h^{+} induces an isomorphism in mod p–cohomology and Theorem 0 follows.

LEMMA 3.4.

a) The mod p cohomology of BSG(n)^{+} and ⋁_{ℓ=1}^{n} BSG(ℓ)/BSG(ℓ − δ(ℓ)) ∨ S^{0}

are of finite type and isomorphic as graded vector spaces.

b) There are finite p–groups P_{ℓ} and homomorphisms φ_{ℓ} such that ⋁_{ℓ=0}^{n} q_{ℓ} ∘ Bφ_{ℓ}

induces a monomorphism in mod p–cohomology.

\underline{Proof:} We claim that q_{ℓ} : BSG(ℓ) → BSG(ℓ)/BSG(ℓ−δ(ℓ)) induces a monomorphism in reduced mod p–cohomology or, equivalently, that the map BSG(ℓ − δ(ℓ))⟶^{Bi}BSG(ℓ) with i denoting the inclusion SG(ℓ − δ(ℓ)) → SG(ℓ) induces an epimorphism in mod p–cohomology. This is clear for SO(ℓ) and SU(ℓ) and will be proved below for A(ℓ). Now part a) follows immediately.

For b) we let P_{ℓ} be either a p–Sylow subgroup of SG(ℓ), if SG(ℓ) = A(ℓ), or the maximal p–torus of SG(ℓ), if SG(ℓ) = SO(ℓ) or SU(ℓ), consisting of all diagonal matrices of order p and determinant 1. For φ_{ℓ} we take the inclusion of P_{ℓ} into SG(ℓ). Part b) follows from the well–known fact that these φ_{ℓ} induce monomorphisms in mod p–cohomology.

It remains to prove the following

LEMMA 3.5. For m ≤ ℓ the restriction map H^{∗}(BA(ℓ); _{p}) → H^{∗}(BA(m); _{p}) is onto provided that m ≢ 0, 1 mod 4 if p = 2 resp. m ≢ 0, 1 mod p if p > 2.

\underline{Proof:} If we replace A(ℓ) and A(m) by Σ(ℓ) and Σ(m) then the restriction map is onto for all m ≤ ℓ by Nakaoka [Na]. Therefore it suffices to show that the restriction map H^{∗}(BΣ(m); _{p}) → H^{∗}(BA(m); _{p}) is onto if m ≢ 0, 1 mod 4 resp. m ≢ 0, 1 mod p.

For odd primes a p–Sylow subgroup of A(m) is also one of Σ(m) and an easy argument (essentially the same as in the proof of Lemma 4.1 below) with stable elements [CE] shows that H^{∗}(BΣ(m); _{p}) → H^{∗}(BA(m); _{p}) is an isomorphism if m ≢ 0, 1 mod p. For another proof we refer to [B].

For p = 2 the Gysin sequence of the S^{0}–bundle BA(m) → BΣ(m) tells us that it is enough to show that multiplication with the mod 2 Euler class e : H^{∗}(BΣ(m); _{2}) → H^{∗}(BΣ(m); _{2}) is mono. Now H^{∗}(BΣ(m); _{2}) is detected by the mod 2–cohomology of its maximal elementary abelian 2–groups E [Q1] and hence it suffices to show that e restricts nontrivially to each H^{∗}(BE; _{2}), E maximal. Finally, e ∈ H^{1}(BΣ(m); _{2}) corresponds to the signum map Σ(m) → /2, and for m ≢ 0, 1 mod 4 each maximal E contains a single transposition (cf. [Mu,p.346f]), hence e restricts nontrivially to H^{∗}(BE; _{2}).

4. Proof of Theorem 2.7

We start by showing that the inclusion j = j_{m,n} : SG(m) → SG(n) induces an injection j = j_{m,n} : A(P, SG(m)) → A(P, SG(n)), at least for those m with δ(m) ≠ 0. Then we will identify A(P, SG(m)) with its image and show that the filtration of A(P, SG(n)) by the subspaces A(P, SG(m)) (for those m with δ(m) ≠ 0) splits via h^{+} after tensoring with /p (cf. Lemma 4.4 and Proposition 4.5 below). This will prove Theorem 2.7 and gives, via the argument of Section 3, the strengthening of Theorem 0 mentioned in Remark a) of the introduction.

Throughout this section P will denote a finite p–group, Q a subgroup of P and p, SG(m) and δ(m) are as in Theorem 0.

LEMMA 4.1 The inclusion SG(m) ⟶^{j} SG(n) induces injections Rep(Q, SG(m)) ⟶^{j} Rep(Q, SG(n)), Rep_{P} (Q, SG(m)) ⟶^{j} Rep_{P} (Q, SG(n)) and A(P, SG(m)) ⟶^{j} A(P, SG(n)) provided δ(m) ≠ 0 (or m = 0).

\underline{Proof:} The cases m = 0 and m = n are trivial, so we may assume n > m > 0 . It suffices to prove the statement for Rep(Q,?). The other cases are immediate consequences.

It is clear that the inclusions from G(m) to G(n) induce injections because elements in Rep(Q, G(m)) correspond to isomorphism classes of Q–sets of cardinality m (if G(m) = Σ(m)) resp. m–dimensional real or complex representations of Q (if G(m) = O(m) or U(m)) and inclusion corresponds to adding a trivial Q–set of cardinality n − m resp. a trivial representation of dimension n − m.

Therefore it suffices to show that the inclusion i_{m} : SG(m) → G(m) induces an injection if δ(m) ≠ 0. For this suppose that φ_{1} and φ_{2} represent elements in Rep(Q, SG(m)) and that there is an element g ∈ G(m) \ SG(m) such that i_{m}φ_{1} = g(i_{m}φ_{2})g^{−1}.

If SG(m) = A(m) we think of i_{m}φ_{k}, k = 1, 2, as Q–set of cardinality m. Then m < n and δ(m) ≠ 0 imply that m ≡ 2 mod 4 (if p = 2) or m ≡ 2 mod p (if p > 2), hence these Q–sets contain either two fixed points or an orbit of length 2. Hence there is a transposition τ which commutes with i_{m}φ_{1} and then we have i_{m}φ_{1} = (τg)(i_{m}φ_{2})(τg)^{−1} with τg ∈ A(m), i.e. φ_{1} and φ_{2} represent the same element in Rep(Q, A(m)).

For SG(m) = SO(m), m < n and δ(m) ≠ 0, we have that m is odd and then the matrix (−id) commutes with i_{m}φ_{1} and has determinant −1. If SG(m) = SU(m), then λ · id commutes with i_{ℓ}φ_{1} for each complex number λ and we can choose λ such that det(λ · id) = det(g)^{−1} . In both cases we can continue as in the case of A(m) and conclude that φ_{1} and φ_{2} agree in Rep(Q, SG(m)).

REMARKS 4.2.

a) The proof shows that the restriction δ(m) ≠ 0 in 4.1 is actually unnecessary if SG(m) = SU(m).

b) On the other hand one can show that there are even elementary abelian p– groups Q for which the map Rep(Q, A(m)) → Rep(Q, Σ(m)) is not injective if m > 0, m ≡ 0 mod 4, p = 2, or m > 0, m ≡ 0 mod p, p odd. This and the proof of 4.1 imply that for such m and all ℓ ≥ m + 2 the map Rep(Q, A(m)) → Rep(Q, A(ℓ)) is not injective and therefore (cf. [HLS, Section 4.2]) the restriction map H^{∗}(BA(ℓ); _{p}) → H^{∗}(BA(m); _{p}) is not even an F - epimorphism in the sense of [Q2], in particular not an epimorphism (cf. Remark b) after Theorem 0).

4.3. Now we introduce filtrations on A(P, SG(n)) and C(P, SG(n)) by defining

E_{m}A(P, SG(n)) = Im(A(P, SG(m))) ⟶^{j} A(P, SG(n))

and

E_{m}C(P, SG(n)) = (⨁_{ℓ=1}^{m} A(P, SG(ℓ))/A(P, SG(ℓ − δ(ℓ)))) ⊕ A(P, SG(0)).

By 4.1 we may identify E_{m}A(P, SG(n)) with A(P, SG(m)) whenever δ(m) ≠ 0. Furthermore, in the definition of E_{m}C, we have used 4.1 in order to identify A(P, SG(ℓ − δ(ℓ))) with Im(A(P, SG(ℓ − δ(ℓ))) ⟶^{j} A(P, SG(ℓ))) for all ℓ. (By convention we take E_{m} equal to A(P, SG(0)) in both filtrations whenever m ≤ 0.)

The maps A(P, SG(m)) ⟶^{j} A(P, SG(n)) map basis elements [Q, φ] to basis elements [Q, jφ] and hence j ⊗ /p is mono whenever j is mono. Therefore Theorem 2.7 is clearly implied by the following two results.

LEMMA 4.4. The homomorphism h^{+} : A(P, SG(n)) → C(P, SG(n)) maps E_{m}A(P, SG(n)) to E_{m}C(P, SG(n)) for all m ≤ n.

PROPOSITION 4.5. The map h^{+} ⊗ /p induces isomorphisms on E_{0} and also on filtration quotients E_{m} ⊗ _{p}/E_{m−δ(m)} ⊗ /p for all m with 0 < m ≤ n and δ(m) ≠ 0.

In the proofs of 4.4 and 4.5 we will interpret homomorphisms Q ⟶^{ρ} G(m) as Q–sets of cardinality m (if G(m) = Σ(m)) resp. as real or complex Q– representations of dimension m (if G(m) = O(m) or U(m)) and we will write ρ ≅ ∑_{λ} c_{λ}λ with nonnegative numbers c_{λ} and λ running through a set of represen-

tatives of the isomorphism classes of irreducible Q–sets resp. Q–representations. If we denote the cardinality of λ resp. the dimension of λ by |λ|, then clearly ∑_{λ} c_{λ}|λ| = m. The trivial Q–set resp. representation will be denoted by λ = 1.

We will abbreviate ∑_{λ≠1} c_{λ}|λ| by ∥ρ∥.

4.6. \underline{Proof of Lemma 4.4:} The case m = 0 is trivial. For the other cases it suffices to show that the homomorphisms h_{n,ℓ} : A(P, SG(n)) → A(P, SG(ℓ)) (cf. Section 2) map E_{m}A(P, SG(n)) into A(P, SG(m) (considered as subspace of A(P, SG(ℓ)) via 4.1) for all 0 ≤ m < ℓ ≤ n. This is trivial for ℓ = n and for ℓ ≤ ϵ.

In the other cases we get from (2.2) that h_{n,ℓ} is given on a class [Q, φ] by a linear combination of the form

h_{n,ℓ}[Q, φ] = ∑_{i} χ_{i}[Q_{i}, d_{ℓ}πψ_{i}].

Here the χ_{i} are suitable integers, the Q_{i} suitable subgroups of Q and there are elements g_{i} ∈ G(n) such that ψ_{i} := g_{i}^{−1}(i_{n}φ)g_{i} maps Q_{i} to G(ℓ−ϵ)×G(n+ϵ−ℓ). We interpret ψ_{i} as a pair of Q_{i}–sets resp. Q_{i}–representations and write ψ_{i} ≅

(∑_{λ} a_{λ}^{i}λ, ∑_{λ} b_{λ}^{i}λ).

If [Q, φ] is in E_{m}A(P, SG(n)), then ∑_{λ≠1}(a_{λ}^{i} + b_{λ}^{i})|λ| ≤ m. It is clear that

[Q_{i}, d_{ℓ}πψ_{i}] is in E_{k+ϵ}A(P, SG(ℓ)), if ∥πψ_{i}∥ = ∑_{λ≠1} a_{λ}^{i}|λ| ≤ k, so we may con-

centrate on the case that m − ϵ < ∥πψ_{i}∥ ≤ m. But then we find b_{λ}^{i} = 0 for all λ ≠ 1 (note that |λ| ≥ ϵ for all λ ≠ 1) and therefore πψ_{i} factors through SG(m). Finally, d_{ℓ} maps SG(m) clearly to itself and this finishes the proof.

4.7. For the proof of Proposition 4.5 we need some preparations. First we observe that the statement about E_{0} is trivial.

Then we define filtrations Ẽ_{m} on [Rep(Q, SG(n))] as in 4.3, i.e.

Ẽ_{m} [Rep(Q, SG(n))] = Im( [Rep(Q, SG(m))] → [Rep(Q, SG(n))]) .

The proof of 4.4 shows also that the maps

h_{n,ℓ} : [Rep(Q, SG(n))] → [Rep(Q, SG(ℓ))]

preserve Ẽ_{m} for all 0 ≤ m ≤ ℓ ≤ n. (Of course, the definition of these maps is as in section 2.)

Now fix ℓ with 0 < ℓ ≤ n and δ(ℓ) ≠ 0. For the proof of 4.5 it suffices now by 4.4 and 2.3–2.5 to show that h_{n,ℓ} induces an isomorphism on the filtration quotient Ẽ_{ℓ} ⊗ /p/Ẽ_{ℓ−δ(ℓ)} ⊗ /p for all subgroups Q of P. We will show that h_{n,ℓ} induces an isomorphism on filtration quotients Ẽ_{m} ⊗ /p/Ẽ_{m−1} ⊗ /p for ℓ − δ(ℓ) < m ≤ ℓ and this is clearly enough.

Again the cases ℓ = n or ℓ ≤ ϵ are trivial, so we will assume ϵ < ℓ < n from now on.

In these cases we get for a basis element [φ] ∈ Rep(Q, SG(n)) by 2.4

h_{n,ℓ}[φ] = ∑_{[ψ]}[i_{n}φ : ψ][d_{ℓ}πψ]

with [ψ] running through Rep(Q, G(ℓ − ϵ) × G(n + ϵ − ℓ)). Now we write i_{n}φ ≅ ∑ c_{λ}λ, ψ ≅ (∑ a_{λ}λ, ∑ b_{λ}λ). Then [i_{n}φ : ψ] = 0 unless a_{λ} + b_{λ} = c_{λ} for all λ. In particular, if we write ∥φ∥ instead of ∥i_{n}φ∥ we have either ∥φ∥ = ∥πψ∥ or ∥πψ∥ ≤ ∥φ∥ − ϵ for such ψ. If ∥φ∥ = ∥πψ∥ then this [ψ] is unique and we will also denote it by [ψ_{0}].

Together with the fact that d_{ℓ} maps terms [πψ] with ∥πψ∥ ≤ k to Ẽ_{k+ϵ} (cf. proof of 4.4) this implies

(4.8) h_{n,ℓ}[φ] ≡ [i_{n}φ : ψ_{0}][d_{ℓ}πψ_{0}] + ∑_{{[ψ] \atop ∥πψ∥=∥φ∥−ϵ}}[i_{n}φ : ψ][d_{ℓ}πψ] mod Ẽ_{∥φ∥−1}.

(By convention the term involving ψ_{0} will be dropped if ∥φ∥ ≠ ∥πψ∥ for all [ψ].)

It will be enough to show that h_{n,ℓ}[φ] ≡ C[φ] mod Ẽ_{∥φ∥−1} for all φ with ℓ−δ(ℓ) < ∥φ∥ ≤ ℓ where C is an integer which is not divisible by p and [φ] on the right hand side is regarded as element in Rep(Q, SG(ℓ)) (cf. 4.1).

In order to evaluate (4.8) we need the following Lemma. But first note that for those [ψ] with a_{λ} + b_{λ} = c_{λ} and ∥πψ∥ = ∥φ∥ − ϵ there is a unique λ′ ≠ 1 with ∥λ′∥ = ϵ and a_{λ′} = c_{λ′} − 1, a_{λ} = c_{λ} for all other λ ≠ 1.

LEMMA 4.9. For [φ] ∈ Rep(Q, SG(n)) and [ψ] ∈ Rep(Q, G(ℓ − ϵ) × G(n + ϵ − ℓ)) with δ(ℓ) ≠ 0 and ℓ − δ(ℓ) < ∥φ∥ ≤ ℓ we get

a) If ∥πψ∥ = ∥φ∥ or ∥πψ∥ = ∥φ∥ − ϵ, and a_{λ} + b_{λ} = c_{λ} for all λ, then [d_{ℓ}πψ] = [φ]. (Here φ is considered as element in Rep(Q, SG(ℓ)) by using 4.1.)

b) If ∥πψ∥ = ∥φ∥ and a_{λ} + b_{λ} = c_{λ} for all λ (i.e. ψ = ψ_{0}), then

[i_{n}φ : ψ] = χ (\frac{G(n − ∥φ∥)}{G(ℓ − ∥φ∥ − ϵ) × G(n + ϵ − ℓ)}) .

c) If ∥πψ∥ = ∥φ∥ − ϵ, and if a_{λ} + b_{λ} = c_{λ} for all λ and λ′ is as above, then

[i_{n}φ : ψ] = χ (\frac{G(c_{λ′})}{G(c_{λ′} − 1) × G(1)} × \frac{G(n − ∥φ∥)}{G(ℓ − ∥φ∥) × G(n − ℓ)}) .

We postpone the proof of 4.9 and give now the proof of 4.5.

4.10. \underline{Proof of Proposition 4.5:} We evaluate (4.8) in the different cases separately.

\underline{Case SG(n) = A(n) and p odd:} We write ℓ = ps + 2 (recall that we discuss the case δ(ℓ) ≠ 0). Now ℓ − δ(ℓ) < ∥φ∥ ≤ ℓ implies ∥φ∥ = ps. Furthermore there are no terms with ∥πψ∥ = ∥φ∥ − ϵ because ∥πψ∥ is divisible by p. Therefore 4.9 implies

h_{n,ℓ}[φ] ≡ ( \begin{matrix}n − ∥φ∥ \\ n + 2 − ℓ \end{matrix} ) [φ] ≡ [φ] mod Ẽ_{∥φ∥−1}.

\underline{Case SG(n) = A(n) and p = 2:} We write ℓ = 4s + 2 (Recall that δ(ℓ) ≠ 0). Now ℓ − δ(ℓ) < ∥φ∥ ≤ ℓ implies ∥φ∥ = 4s + 2 or ∥φ∥ = 4s. If ∥φ∥ = 4s + 2 then there are only terms with ∥πψ∥ = ∥φ∥ − ϵ and we get from 4.9

h_{n,ℓ}[φ] ≡ ∑_{∥λ∥=2} c_{λ} ( \begin{matrix}n − ∥φ∥ \\ n − ℓ \end{matrix} ) [φ] mod Ẽ_{∥φ∥−1}.

Again the binomial coefficient is 1 and ∥φ∥ = 4s + 2 implies that ∑_{∥λ∥=2} c_{λ} is odd.

If ∥φ∥ = 4s we find

h_{n,ℓ}[φ] ≡ \{( \begin{matrix}n − ∥φ∥ \\ n + 2 − ℓ \end{matrix} ) + ∑_{∥λ∥=2} c_{λ} ( \begin{matrix}n − ∥φ∥ \\ n − ℓ \end{matrix} )\} [φ] mod Ẽ_{∥φ∥−1}.

Because of ∥φ∥ = 4s we get ∑_{∥λ∥=2} c_{λ} ≡ 0 mod 2 and ( \begin{matrix}n − ∥φ∥ \\ n + 2 − ℓ \end{matrix} ) = ( \begin{matrix}n − ∥φ∥ \\ n − ∥φ∥ \end{matrix} )

= 1 and we are done again.

\underline{The cases SG(n) = SO(n) or SU(n):} If ∥φ∥ = ℓ, then there are only summands with ∥πψ∥ = ∥φ∥ − ϵ and 4.9 gives

(4.11.)  h_{n,ℓ}[φ] ≡ ∑_{{∥λ∥=1 \atop λ≠1}} χ(G(c_{λ})/G(c_{λ} − 1) × G(1))[φ] mod Ẽ_{∥φ∥−1}.

The space G(c_{λ})/G(c_{λ} − 1) × G(1) may be identified with the projective space KP^{c_{λ}−1} where K denotes in the case of SO(n) and in the case of SU(n). For the Euler characteristic we get χ( P^{c_{λ}−1}) ≡ c_{λ} mod 2 and χ( P^{c_{λ}−1}) = c_{λ}. Therefore we find that the coefficient of [φ] in (4.11) is ∑_{{∥λ∥=1 \atop λ≠1}} c_{λ} for SU(n) and congruent to ∑_{{∥λ∥=1 \atop λ≠1}} c_{λ} mod 2 for SO(n).

Now Q is a p–group and the dimension of an irreducible Q–representation is either 1 or divisible by p (cf. [Se, Chap. 8.1] for the case of complex representations; the case of real representations and p = 2 is easily reduced to the complex case.). Therefore we find ∑_{{∥λ∥=1 \atop λ≠1}} c_{λ} ≡ ∥φ∥ ≡ ℓ mod p, and hence the coefficient of [φ] is

not divisible by p (we have assumed that δ(ℓ) ≠ 0 and ℓ < n).

It remains to consider the case ∥φ∥ ≠ ℓ. Then ℓ − δ(ℓ) < ∥φ∥ < ℓ implies ∥φ∥ = ℓ − 1 and ℓ ≡ 1 mod p. Here 4.9 yields

h_{n,ℓ}[φ] ≡ [φ] + ∑_{{∥λ∥=1 \atop λ≠1}} χ(KP^{c_{λ}−1}) · χ(KP^{n−ℓ})[φ] mod Ẽ_{∥φ∥−1}.

The coefficients in the sum are equal to c_{λ}·(n−ℓ+1), at least mod p, and as above

we see ∑_{{∥λ∥=1 \atop λ≠1}} c_{λ} ≡ ∥φ∥ ≡ ℓ − 1 ≡ 0 mod p. Therefore h_{n,ℓ}[φ] ≡ C[φ] mod Ẽ_{∥φ∥−1}

with C ≡ 1 mod p.

4.12. It remains to give the \underline{Proof of 4.9:} Via 4.1 we consider [φ] as element in Rep(Q, SG(ℓ)). If ∥πψ∥ = ∥φ∥ and a_{λ} + b_{λ} = c_{λ} for all λ (i.e. ψ = ψ_{0}) then πψ factors through SG(ℓ − ϵ) and hence i_{ℓ}d_{ℓ} adds only trivial Q–orbits resp. representations to πψ. Therefore i_{ℓ}[d_{ℓ}πψ] = i_{ℓ}[φ] where i_{ℓ} denotes the homomorphism induced by the inclusion SG(ℓ) → G(ℓ).

If ∥πψ∥ = ∥φ∥ − ϵ and a_{λ} + b_{λ} = c_{λ} for all λ, then πψ does not factor through SG(ℓ − ϵ) and i_{ℓ}d_{ℓ} adds the Q–set resp. Q–representation corresponding to the homomorphism Q⟶^{πψ}G(ℓ−ϵ)⟶^{(det)^{−1}}G(ϵ). It is easy to see that this corresponds just to the unique λ′ with λ′ ≠ 1, ∥λ′∥ = ϵ, a_{λ′} = c_{λ′} − 1 and a_{λ} = b_{λ} for all other λ ≠ 1. Therefore i_{ℓ}[d_{ℓ}πψ] = i_{ℓ}[φ].

Now i_{ℓ} is mono if δ(ℓ) ≠ 0 (cf. proof of 4.1) and hence a) follows.

b) and c): The proofs of b) and c) are similar and are just an application of Lemma 2.6. If we abbreviate G(n) by G, G(ℓ − ϵ) × G(n + ϵ − ℓ) by H and write as before i_{n}φ ≅ ∑ c_{λ}λ, ψ ≅ (∑ a_{λ}λ, ∑ b_{λ}λ), then (G/H)_{ψ}^{i_{n}φ} is nonempty if and only if a_{λ} + b_{λ} = c_{λ} for all λ.

Furthermore, it is not hard to see that

C_{G}(ψ) ≅ ∏_{λ} Aut_{Q}(c_{λ}λ)

with Aut_{Q}(c_{λ}λ) ⊂ G(c_{λ}|λ|) denoting the automorphism group of the Q–set respectively Q–representation c_{λ}λ; similarly we have

C_{G}(ψ) ∩ H ≅ ∏_{λ} Aut_{Q}(a_{λ}λ) × Aut_{Q}(b_{λ}λ).

Hence (G/H)_{ψ}^{i_{n}φ} is homeomorphic to ∏_{λ} Aut_{Q}(c_{λ}λ)/Aut_{Q}(a_{λ}λ)×Aut_{Q}(b_{λ}λ) and

b) and c) follow easily.

For example, in case c) we have c_{λ} = a_{λ} for all λ except for λ = λ′ as above and λ = 1. Now Aut_{Q}(c_{λ}λ) is isomorphic to G(c_{λ}) if |λ| = 1, respectively to the wreath product Aut_{Q}(λ) ≀ G(c_{λ}) if G(n) = Σ(n) and λ is any irreducible Q–set. Hence we get in this case

C_{G}(ψ)/C_{G}(ψ) ∩ H ≅
≅ (G(c_{λ′})/G(c_{λ′} − 1) × G(1)) × (G(n − ∥φ∥)/G(ℓ − ∥φ∥) × G(n − ℓ)) .

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Hans–Werner Henn  Huỳnh Mui
Mathematisches Institut  Department of Mathematics
der Universität  University of Hanoi
Im Neuenheimer Feld 288  Dai Hoc Tong Hop
D–6900 Heidelberg  Hanoi
Federal Republic of Germany  Vietnam