THE GLUING PROBLEM AND Σ_{p^{2}}-EQUIVARIANT COHOMOLOGY.

ASSAF LIBMAN

Abstract. Given a block b in kG where k is an algebraically closed field of characteristic p, there are classes α_{Q} ∈ H^{2}(Aut_{F} (Q); k^{×}), constructed by Külshammer and Puig, where F is the fusion system associated to b and Q is an F-centric subgroup. The gluing problem in F has a solution if these classes are restriction of a class α ∈ H^{2}(F^{c}; k^{×}). Linckelmann showed that a solution to the gluing problem gives rise to a refrmulation of Alperin’s weight conjecture. He then shows that the gluing problem has a solution if for every finite group G, the equivariant Bredon cohomology group H_{G}^{1}(|Δ_{p}(G)|; A^{1}) vanishes, where |Δ_{p}(G)| is the simplicial complex of the non-trivial p-subgroups of G and A^{1} is the coefficient functor G/H ↦ Hom(H, k^{×}). The purpose of this note is to show that if G = Σ_{p^{2}} then this group does vanish.

1. Introduction

Given a functor M: C → Ab, we will write H^{∗}(C; M) for the groups \underleftarrow{lim}_{C}^{∗} M. When C has one object with a group G of automorphisms, a functor M: C → Ab is the same thing as as a G-modue and H^{∗}(G; M) ≅ \underleftarrow{lim}_{C}^{∗} M.

Let us now fix a prime p and let F be the fusion system of a block b of a finite group G. As usual, we will write F^{c} for the full subcategory generated by the F-centric subgroups in F. Let k be an algebraically closed field of charateristic p. In [4] Külshammer and Puig show that for every F-centric subgroup Q there is a canonically chosen class α_{Q} ∈ H^{2}(Aut_{F}(Q); k^{×}). We say that the gluing problem can be solved in F is there exists a class α ∈ H^{2}(F^{c}; k^{×}) where k^{×} is the constant functor, such that the restriction α|_{Aut_{F} (Q)} is equal to α_{Q} for all Q ∈ F^{c}.

Linckelmann shown in [6] that if the gluing problem can be solved for all fusion systems of block then Alperin’s weight conjecture is logically equivalent to a relation between the number k(b) of complex representations of G associated to b by Knörr and Robinson [3] and the Euler characteristic of a certain cochain complex built from the fusion system of b.

Let G a finite group and C a finite G-poset. The (combinatorial) simplicial complex associated to C, see [7, Chap. 3], is denoted S(C). The n-simplices are squences c = c_{0} ⪵ ⋯ ⪵ c_{n} in C and face maps are inclusion of simplices. Clearly G acts on S(C) whose space [S(C)] is also a simplicial complex and its simplices form, as usual, a small category.

As a special case consider the poset Δ_{p}(G) of the non-trivial p-subgroups of G. Note that the isotropy group of an n-simplex P = (P_{0} < ⋯ < P_{n}) in S(Δ_{p}(G)) is

N_{G}(P) = ∩_{i=0}^{n}N_{G}(P_{i}).

Date: May 26, 2009.

Key words and phrases. Equivariant cohomology, gluing problem, Alperin’s conjecture.

There is a functor N_{G}: [S(Δ_{p}(G))] → Ab defined by Linckelmann in [5]

N_{G}([P]) = Hom(N_{G}(P), k^{×}) = Hom(N_{G}(P)_{ab}, k^{×}).

Theorem ??? of [5] implies that the gluing problem in F has a solution if we can prove that H^{1}([S(Δ_{p}(K))]; N_{K}) = 0 for all K = Aut_{F}(Q)/Inn(Q) where Q is an F-centric subgroup. Thus, if we can prove that H^{1}([S(Δ_{p}(G))]; N_{G}) = 0 for all finite groups G, then the gluing problem has a solution for all fusion systems. The purpose of this note is to show that this programme is not feasable.

1.1. Theorem. Set G = Σ_{p^{2}}. If p ≥ 5 then H^{1}([S(Δ_{p}(G))]; N_{G}) ≠ 0.

2. Subdivision categories and higher limits

Let G be a finite group. As in the introduction, if C is a G-poset, we let S(C) denote the associated G-simplicial complex and let [S(C)] denote its orbit space. We will denote the set of n-simplices of S(C) and [S(C)] by S(C)_{n} and [S(C)]_{n}. Given an n-simplex c = (c_{0} ⪵ ⋯ ⪵ c_{n}) where n ≥ 1, we will write ∂_{i}c for the n − 1-simplex obtained by removing c_{i} where 0 ≤ i ≤ n. In this way we obtain face maps

∂_{i}: S(C)_{n} → S(C)_{n−1}  and  [∂_{i}]: [S(C)]_{n} → [S(C)]_{n−1},  (0 ≤ i ≤ n)

where ∂_{i} is G-equavariant and an n-simplex [c] give rise to a map of transitive G-

sets [∂_{i}c] ⟶^{[∂_{i}]} [c]. The simplices of [S(C)] form a small category with one morphism [c′] → [c] if [c′] is a face of [c].

2.1. Definition. Fix a commutative ring R. Let C be a finite G-poset and consider a functor A: [S(C)] → R -mod. Define a cochain complex C^{∗}(A) as follows.

C^{n}(A) = ∏_{[c]∈[S(C)_{n}]} A([c]), d: C^{n}(A) \underrightarrow{∑_{j=0}^{n}(−1)^{j} ∏_{[c]} A([∂_{j}])} C^{n−1}(A)

2.2. Lemma. With the notation of Definition 2.1, the cohmology groups of C^{∗}(A) are isomorphic to H^{∗}([S(C)]; A).

Proof. For every n ≥ 0 define functors P_{n}: [S(C)] → R -mod

P_{n} = ⨁_{[c]∈[S(C)]_{n}} R ⊗ Mor_{[S(C)]}([c], −)

where R ⊗ Mor_{S(C)}([c], −) is the functor that sends [x] ∈ [S(C)] to the free Rmodule generated by the set Mor_{[S(C)]}([c], [x]). Thus, P_{n} are projective functors and by Yoneda’s lemma we have a natural isomorphism

Hom_{[S(C)]}(P_{n}, A) = ∏_{[c]∈[S(C)]_{n}} A([c]) = C^{n}(A).

For every j = 0, …, n, the map [∂_{j}]: [S(C)]_{n} → [S(C)]_{n−1} yields, by representability, a natural transformations ∂_{j}^{∗}: P_{n} → P_{n−1}. Define natural transformations

P_{n} ⟶^{∂} P_{n−1} as usual by ∂ = ∑_{j=0}^{n}(−1)^{j}∂_{j}^{∗}. We now claim that

⋯ → P_{n} ⟶^{∂} P_{n−1} ⟶^{∂} … ⟶^{∂} P_{1} ⟶^{∂} P_{0} → \underline{R}

where \underline{R} is the constant functor, is a projective resolution for \underline{R}. We only need to show that evaluation at any object [c] ∈ [S(C)] gives an acyclic cochain complex. Fix [c] ∈ [S(C)]_{n} and let [S(C)]^{≤[c]} denote the subcomplex of the faces of [c]. It is clear that it is isomprohic to the standard n-simplex and it is therefre contractible. For every j we observe that P_{j}([c]) is the free R-module generated by the j-dimensional faces of [c]. Thus P_{•}([c]) → R is isomorphic to the augmented cochain complex C_{∗}(Δ^{n}; R) → R and it is therefore acyclic. The Lemma now follows since H^{∗}([S(C)], A) ≅ H^{∗}Hom_{[S(C)]}(P_{•}, A) ≅ H(C^{∗}(A)). □

3. Bredon cohomology

Throughout this section a space means a simplicial set. Let G be a finite group. A coefficient functor M for G is a contravariant functor {G-sets} → Ab which turns coproducts of G-sets into products of abelian groups. By applying M to the sets of simplices of a G-space X, one obtains a cosimplicial abelian group M(X). The associated (normalised) cohain complexes, which are chain homotopic, are denoted

C_{G}^{∗}(X; M)  and  NC_{G}^{∗}(X; M).

Their homology groups are called the Bredon cohomology groups H_{G}^{∗}(X; M). If Y ⊆ X is a G-subspace, there results an obvious short exact sequence of cochain complexes

0 → C_{G}^{∗}(X, Y; M) → C_{G}^{∗}(X; M) → C_{G}^{∗}(Y; M) → 0

which defines the relative cohomology groups H_{G}^{∗}(X, Y; M) together with the usual long exact sequences. Bredon cohomology is an equivariant cohomology theory. In particular it turns G-homotopy equivalences into isomorphisms and if X is a union of subspaces Y_{1} ∪ Y_{2}, one has the usual Mayer Vietoris sequence.

Recall that the Borel construction of a G-space U is U ×_{G} EG where EG is a contractible space on which G acts freely. If U = G/K then U ×_{G} EG = BK is the classifying space of K.

3.1. Definition. Fix a finite group G, an abelian group A and an integer n ≥ 0. Define a coefficient functor H^{n} for G by H^{n}(U) = H^{n}(U ×_{G} EG; A). Observe that H^{n}(G/K) = H^{n}(K; A) where A has the trivial action of K.

3.2. Definition. Let C be a finite G-poset and let M be a coefficient system. Define a functor A_{M}: [S(C)] → Ab by A_{M}([c]) = M([c]). If [c′] is a face of [c], we define A_{M}([c′] → [c]) by applying M to the map [c] → [c′] of transitive G-sets.

Bredon cohomology gives a convenient interpretation of the derived functors of A_{M}.

3.3. Lemma. Let C be a finite G-poset and let M be a coefficient functor for G. With the notation of Definition 3.2 we have an isomorphism

H^{∗}([S(C)]; A_{M}) ≅ H_{G}^{∗}(|C|; M).

Proof. By inspection the cochain complex C^{∗}(A_{M}) defined in 2.1 is equal to the normalised cochain complex NC_{G}^{∗}(|C|; M). Now apply Lemma 2.2. □

4. Proof of Theorem 1.1

Set G = Σ_{p^{2}}. First we observe that Hom(K, A) = H^{1}(K; A) for any finite group K and any abelian group A. Thus, the functor N_{G} defined in the introduction is canonically isomorphic to A_{H^{1}} as defined in 3.2 and in 3.1 with A = k^{×}. In light of Lemma 3.3 we need to prove that H_{G}^{1}(|Δ_{p}(G)|; H^{1}) ≠ 0. Consider the G-subposet B_{p}(G) of the radical p-subgroups of G, namely the non-trivial p-subgroup P ≤ G such that N_{G}(P)/P contains no non-trivial normal p-subgroup. It is well known that the inclusion |B_{p}(G)| ⊆ |Δ_{p}(G)| is a G-homotopy equivalence, see e.g. [2, Proposition 6.6.5]. It therefore remains to prove that

(4.1)  H_{G}^{1}(|B_{p}(G)|; H^{1}) ≠ 0.

The radical p-subgroups of the symmetric groups were determined by Alperin and Fong in [1]. In G = Σ_{p^{2}} they form the following conjugacy classes:

(a) The Sylow p-subgroup of Σ_{p^{2}}. This is C_{p} ≀ C_{p} which, following Alperin and Fong, we denote by V_{1,1}. Its normaliser is V_{1,1} ⋊ (GL_{1}(p) × GL_{1}(p)) with the diagonal action on C_{p}.

(b) The subgroup V_{2} = C_{p} × C_{p} embedded in Σ_{p^{2}} via its action on itself by translation. Its normaliser is V_{2} ⋊ GL_{2}(p).

(c) For every k = 1, …, p the subgroup V_{1}^{×k} which is isomorphic to C_{p}^{×k} as a subgroup of Σ_{p}^{×k} ≤ Σ_{p^{2}}. The normaliser of V_{1}^{×k} is (GL_{1}(p) ≀ Σ_{k}) × Σ_{p^{2}−kp}.

4.2. Definition. Consider the following subposets of B_{p}(G).

(1) Let D_{1} be the subposet generated by the conjugacy classes of V_{1,1} and V_{1}, V_{1}^{×2}, …, V_{1}^{×p}.

(2) The subposet V_{1} is generated by the conjugacy classes of V_{1}, V_{1}^{×2}, …, V_{1}^{×p}.

(3) Let D_{2} be the subposet generated by the conjugacy classes of V_{1,1} and V_{2}.

(4) Let D_{3} be the subposet generated by the conjugacy class of V_{1,1}.

Observe that V_{2} acts transitively so it cannot be conjugate to a subgroup of V_{1}^{×k} whose orbits have cardinality p. Since V_{2} acts freely it cannot contain a conjugate of V_{1}^{×k} since the latter does not act freely. We see that up to conjugacy B_{p}(G) has the form

[V_{1}] < [V_{1}^{×2}] < … [V_{1}^{×p}] < [V_{1,1}] > [V_{2}]

and it follows that

(4.3)  |B_{p}(G)| = |D_{1}| ∪ |D_{2}|,  |D_{3}| = |D_{1}| ∩ |D_{2}|.

Mayer Vietoris sequence gives an exact sequence (4.4) ⋯ → H_{G}^{0}(|D_{1}|; H^{1}) ⊕ H_{G}^{0}(|D_{2}|; H^{1}) → H_{G}^{0}(|D_{3}|; H^{1}) → H_{G}^{1}(|B_{p}(G)|; H^{1}) → …

Throughout, let L denote Hom(GL_{1}(p), k^{×}) ≅ 𝔽_{p}^{×}. We will prove

4.5. Proposition. H_{G}^{∗}(|D_{3}|; H^{1}) ≅ L × L.

4.6. Proposition. H_{G}^{∗}(|D_{2}|; H^{1}) ≅ L.

4.7. Proposition. H_{G}^{∗}(|D_{1}|; H^{1}) ≅ C_{2}.

Proposirtions 4.5-4.7 together with the exact sequence (4.4) immediately imply (4.1) because |L| ≥ 4. It therefore remains to prove these propositions.

Proof of Proposition 4.5. Since D_{3} is a single orbit with isotropy group N_{G}(V_{1,1}) we see that

H_{G}^{∗}(|D_{3}|; H^{1}) = H^{1}([V_{1,1}]) = Hom(V_{1,1} ⋊ (GL_{1}(p) × GL_{1}(p)); k^{×}) = L × L.

□

Proof of Proposition 4.6. Since |B_{p}(G)| is G-equivalent to |Δ_{p}(G)|, Symond’s resolution of Webb’s conjecture [8] shows that the orbit space |B_{p}(G)|_{/G} is contractible. But (4.3) shows that |B_{p}(G)|_{/G} = |D_{1}|_{/G} ∨ |D_{2}|_{/G}. It follows that |D_{2}|_{/G} is contractible and since it is a 1-dimensional simplicial complex, the poset of its simplices must have the form

[V_{2}] → [V_{2} < V_{1,1}] ← [V_{1,1}].

Since V_{2} ≤ V_{1,1} = C_{p} ≀ C_{p} is generated by the C_{p} at the top and the diagonal C_{p} in the base group C_{p} × ⋯ × C_{p}, we see that N_{G}(V_{2} < V_{1,1})/V_{2} is isomorphic to the normaliser of a Sylow p-subgroup of GL_{2}(p), that is to the group T_{2}(p) of upper triangular matrices. With the notation of Definition 3.2 we have

A_{H^{1}}([V_{2} < V_{1,1}]) = Hom(T_{2}(p); k^{×}) ≅ Hom(GL_{1}(p) × GL_{1}(p); k^{×}) ≅ A_{H^{1}}([V_{1,1}]),

and A_{H^{1}}([V_{2}]) = Hom(GL_{2}(p), k^{×}) ≅ L because the Sylow p-subgroup of GL_{2}(p) generate SL_{2}(p). In light of Lemma 3.3, the groups H_{G}^{∗}(|D_{2}|; H^{1}) are isomorphic to H^{∗}([S(D_{2})]; A_{H^{1}}), namely to the derived functors of the pullback diagram L ⟶^{Δ} L × L ⟵^{id} L × L. This completes the proof. □

4.8. Lemma. The inclusion V_{1} ⊆ D_{1} induces a G-homotopy equivalence on nerves.

Proof. Given a subgroup P of G let δ_{1}(P) denote the subgroup of P generated by all the permutations g ∈ P whose support contains at most p elements. Observe that δ_{1} is invariant under conjugation, namely δ_{1}(gP g^{−1}) = gδ_{1}(P)g^{−1}. By inspection δ_{1}(V_{1}^{×k}) = V_{1}^{×k} and δ_{1}(V_{1,1}) = V_{1}^{×p}. We obtain a G-equivariant morphism of posets δ_{1} : D_{1} → V_{1}. Clearly, δ_{1} ∘ incl_{V_{1}}^{D_{1}} = Id_{V_{1}}. The inclusion δ_{1}(P) ≤ P gives a G-equivariant natural transformation incl_{V_{1}}^{D_{1}} ∘ δ_{1} → Id_{D_{1}}. The result follows. □

We leave the following result as an easy exercise for the reader.

4.9. Lemma. Let K be a finite group. The abeliabisation of G = K ≀ Σ_{n} is isomorphic to K_{ab} × (Σ_{n})_{ab}. The restriction of G → G_{ab} to any one of the factors K in K^{n} ≤ G is the canonical projection K → K_{ab} and the restriction of G → G_{ab} to Σ_{n} is the projection onto (Σ_{n})_{ab}.

If n, m ≥ 1, the inclusion (K ≀ Σ_{n}) × (K ≀ Σ_{m}) ≤ (K ≀ Σ_{n+m}) induces on abelianisation the fold map K_{ab} × K_{ab} → K_{ab} and (Σ_{n})_{ab} × (Σ_{m})_{ab} → (Σ_{n+m})_{ab}.

4.10. Notation. The following non-standard description of the (n − 1)-simplex Δ^{n−1} will be used throughout. The r-simplices of Δ^{n−1} are sequences i_{0} < ⋯ < i_{r} where 1 ≤ i_{0}, …, i_{r} ≤ n. Face maps are obtained by inclusion of sequences. (The usual convention is 0 ≤ i_{0}, …, i_{r} ≤ n − 1.)

Proof of Proposition 4.7. In light of Lemma 4.8, we must prove that H_{G}^{∗}(|V_{1}|; H^{1}) ≅ C_{2}. The hight transitivity of G = Σ_{p^{2}} implies that every r-simlex of S(V_{1}) is Gconjugate to a simplex of the form V_{1}^{×i_{0}} < ⋯ < V_{1}^{×i_{r}} where 1 ≤ i_{0} < ⋯ < i_{r} ≤ p. With the notation of 4.10 we see that [S(V_{1})] = Δ^{p−1}.

For any group K let K̂ denote the abelian group Hom(K, k^{×}). Let N denote the normaliser of C_{p} in Σ_{p}. Thus N = C_{p} ⋊ GL_{1}(p) and observe that GL_{1}(p) is generated by an odd cycle since p is odd. Set

L = N̂ = Hom(N, k^{×}) = Hom(GL_{1}(p), k^{×}) ≅ C_{p−1}

Consider the following functor Φ: (Δ^{p−1})^{op} → {Groups}. On objects

Φ(i_{0} < ⋯ < i_{r}) = ( ∏_{t=0}^{r} N ≀ Σ_{i_{t}−i_{t−1}}) × Σ_{p^{2}−i_{r}p}, (i_{−1} = 0)

On face maps ∂_{j}: (Δ^{p−1})_{r} → (Δ^{p−1})_{r−1} the effect of Φ is induced by the inclusions

(N ≀ Σ_{a}) × (N ≀ Σ_{b}) ≤ (N ≀ Σ_{a+b}),  and for j = r by  (N ≀ Σ_{a}) × Σ_{c} ≤ Σ_{ap+c}.

By inspection,A_{H^{1}} = Φ̂ and in light of Lemmas 3.3 and 2.2 it remains to prove that

(4.11)  H^{∗}(Δ^{p−1}; Φ̂) ≅ C_{2}.

Consider the following subfunctor Ψ of Φ defined by

Ψ(i_{0} < ⋯ < i_{r}) = ( ∏_{t=0}^{r} N ≀ Σ_{i_{t}−i_{t−1}}) × (N ≀ Σ_{p−i_{r}})

via the inclusion N ≀ Σ_{p−i_{r}} ≤ Σ_{p^{2}−i_{r}p}. We obtain a morphism of functors Φ̂ → Ψ̂ of abelian groups. Our goal now is to prove that it is a monomorphism. Fix an r-simplex i = (i_{0} < ⋯ < i_{r}) and consider Φ̂(i) → Ψ̂(i). Case (a). If i_{r} = p then Σ_{p^{2}−i_{r}p} and N ≀ Σ_{p−i_{r}} are trivial groups and therefore Φ̂(i) → Ψ̂(i) is injective Case (b). If i_{r} = p − 1 then N ≀ Σ_{p−i_{r}} = N and Σ_{p(p−i_{r})} = Σ_{p} and since N contains an odd permutaion, by applying Hom(−, k^{×}) to the inclusion N ≤ Σ_{p} induces the iinclusion C_{2} → L and therefore Φ̂(i) → Ψ̂(i) is injective. Case (c). Assume that i_{r} ≤ p − 2. The inclusion of N^{p−i_{r}} ≤ Σ_{p(p−i_{r})} contains odd permutations. Since p is odd, also the inclusion Σ_{p−i_{r}} ≤ Σ_{p(p−i_{r})} contains odd permutations. Therefore the induced map { \atop Σ_{p(p−i_{r})}} → { \atop N ≀ Σ_{p−i_{r}}} is the diagonal inclusion C_{2} → L ⊕ C_{2}. It follows that Φ̂(i) → Ψ̂(i) is injective.

We obtain a short exact sequence of functor (Δ^{p−1})^{op} → Ab

0 → Φ̂ → Ψ̂ → Γ → 0

where the functor Γ has the form

Γ(i) = \{ \begin{matrix}0 & if i_{r} = p \\ L/C_{2} & if i_{r} − p − 1 \\ L & if i_{r} ≤ p − 2 \end{matrix}

with all maps induced by the identity map and the projection L → L/C_{2}.

Let Γ′ be the functor with Γ′(i) = 0 if i_{r} = p and constant Γ′(i) = L for i_{r} ≤ p−1. Let Γ′′ be the functor with Γ′′(i) = C_{2} if i_{r} = p − 1 and Γ′′(i) = 0 otherwise. We get a shortr exact sequence of functors

0 → Γ′′ → Γ′ → Γ → 0.

We view Δ^{p−2} as a the (p − 1)st face of Δ^{p−1}, that is Δ^{p−2} consist of the simplices i with i_{r} < p. Similarly Δ^{p−3} is the (p − 2)nd face of Δ^{p−2}. By inspection C^{∗}(Δ^{p−1}; Γ′′) is equal to the cochain complex C^{∗}(Δ^{p−2}, Δ^{p−3}; C_{2}) and since p ≥ 5 the contractibility of the standard simplices implies that

H^{∗}(Δ^{p−1}; Γ′′) ≅ H^{∗}(Δ^{p−2}, Δ^{p−3}; C_{2}) = 0.

By inspection C^{∗}(Δ^{p−1}; Γ′) = C^{∗}(Δ^{p−2}; L), hence Γ′ → Γ induces an isomorphism

(4.12)  H^{∗}(Δ^{p−1}; Γ) ≅ H^{∗}(Δ^{p−1}; Γ′) ≅ L.

We now observe that Ψ̂ has the following form

Ψ̂(i) = ( ∏_{t=0}^{r} L × { \atop Σ_{i_{t}−i_{t−1}}}) × \{ \begin{matrix}0 & if i_{r} = p \\ L × { \atop Σ_{p−i_{r}}} & if i_{r} < p. \end{matrix}

We obtain a constant subfunctor Ψ′(i) = L via the diagonal inclusion. By inspection the following square commutes

\xymatrix{Ψ′ \ar[r] \ar[d] & Ψ̂ \ar[d] \\ Γ′ \ar[r] & Γ.}

The map Ψ′ → Γ′ induces the canonical projection

C^{∗}(Ψ) = C^{∗}(Δ^{p−1}; L) → C^{∗}(Δ^{p−2}; L) = C^{∗}(Γ′)

which induces an isomorphis in homology. It follows that Ψ′ → Γ′ induces an isomorphism in H^{∗}(Δ^{p−1}; −). Taking (4.12) into account, we now see that Ψ̂ → Γ induces an epimorphism in H^{∗}(Δ^{p−1}; −). In light of (4.12) again, the proof of this proposition will be complete if we prove that H^{∗}(Δ^{p−1}; Ψ̂) = L ⊕ C_{2}.

Set K = N ≀ Σ_{p} and let it act on the poset Ω of the non-empty subsets of {1, …, p} via the projection onto Σ_{p}. One easily checks that [S(Ω)] = Δ^{p−1} and that, by choosing appropriate representatives,

Iso_{K}(i_{0} < ⋯ < i_{r}) = Ψ(i_{0} < ⋯ < i_{r}).

Thus, if H_{K}^{1} is the coefficient functor for K defined in 3.1 with A = k^{×}, we see that C^{∗}(Ψ̂) = C^{∗}(A_{H_{K}^{1}} ), whence by Lemma 3.3,

H^{∗}(Δ^{p−1}; Ψ̂) ≅ H^{∗}(Δ^{p−1}; A_{H_{K}^{1}} ) ≅ H_{K}^{∗}(|Ω|; H_{K}^{1}).

Now, |Ω| is K-equivalent to a point because {1, …, p} is a maximal element in Ω fixed by the action of K. Therefore H_{K}^{∗}(|Ω|; H_{K}^{1}) ≅ H_{K}^{1}(pt) = { \atop N ≀ Σ_{p}} = L ⊕ C_{2}. This completes the proof. □

References

[1] Alperin, J. L.; Fong, P. Weights for symmetric and general linear groups. J. Algebra 131 (1990), no. 1, 2–22.

[2] Benson, D. J. Representations and cohomology. II. Cohomology of groups and modules. Cambridge Studies in Advanced Mathematics, 31. Cambridge University Press, Cambridge, 1991.

[3] Knörr, Reinhard; Robinson, Geoffrey R. Some remarks on a conjecture of Alperin. J. London Math. Soc. (2) 39 (1989), no. 1, 48–60.

[4] Külshammer, Burkhard; Puig, Llus Extensions of nilpotent blocks. Invent. Math. 102 (1990), no. 1, 17–71.

[5] Linckelmann Markus, On H^{∗}(C; k^{×}) for fusion systems. To appear in HHA.

[6] Linckelmann Markus, Alperin’s weight conjecture in terms of equivariant Bredon cohomology. Math. Z. 250 (2005), no. 3, 495–513.

[7] Spanier, Edwin H. Algebraic topology. McGraw-Hill

[8] Symonds, Peter. The orbit space of the p-subgroup complex is contractible. Comment. Math. Helv. 73 (1998), no. 3, 400–405.

Department of Mathematical Sciences, King’s College, University of Aberdeen, Aberdeen AB24 3UE , Scotland, U.K.

   E-mail address: assaf@maths.abdn.ac.uk