THE GLUING PROBLEM IN THE FUSION SYSTEMS OF THE SYMMETRIC AND THE ALTERNATING GROUPS.

ASSAF LIBMAN

Abstract. A solution to Linckelmann’s gluing problem in all fusion systems of blocks yields and alternative formulation of Alperin’s weight conjecture. Using Bredon equivariant cohomology techniques we solve the gluing problem in fusion systems that are isomorphic to the fusion systems of the symmetric groups or, for odd primes, the alternating groups.

1. Introduction

Fix a prime p and an algebraically closed field k of characteristic p. Let F be a saturated fusion system on a p-group S and F^{c} the full subcategory of the F-centric subgroups, see [5]. An s-simplex in F^{c} is a sequence P_{0} ⪇ ⋯ ⪇ P_{s} of F-centric subgroups of S. It is F-conjugate to P′_{0} ⪇ ⋯ ⪇ P′_{s} if there exists an F-isomorphism P_{s} → P′_{s} which carries every P_{i} onto P′_{i}. The equivalence classes of s-simplices in F^{c} form a poset [S(F^{c})] where [Q_{•}] → [P_{•}] if Q_{•} is F-conjugate to a subsimplex, namely a face, of P_{•}. For every i ≥ 0 there is a functor

A_{F}^{i}: [S(F^{c})] → Ab, [P_{•}] ↦ H^{i}(Aut_{F}(P_{•}); k^{×})

where Aut_{F}(P_{•}) is the subgroup of Aut_{F}(P_{s}) that normalises P_{•} and k^{×} is the group of the invertible elements in k. See [12] and [10, §1] for more details. A family of classes α_{[P_{•}]} ∈ A_{F}^{i}([P_{•}]) is called compatible if every [Q_{•}] → [P_{•}] in [S(F^{c})] carries α_{[Q_{•}]} to α_{[P_{•}]}. In other words, it is an element in \underleftarrow{lim}_{[S(F^{c})]} A_{F}^{i}.

The inclusion of the categories Aut_{F}(P_{•}) with a single object P_{s} in F^{c} yields

H^{i}(F^{c}; k^{×}) ⟶^{γ} \underleftarrow{lim}_{[S(F^{c})]} A_{F}^{i}

where the term on the left is the ith derived functor of the constant funtor F^{c} → Ab with value k^{×}. We say that the gluing problem for the family (α_{[P_{•}]}) has a solution if it is in the image of γ. See [12], [11] and [10, §4].

Suppose that F is the fusion system of a block b. Külshammer and Puig’s construction in [8, 1.8, 1.12], together with Dade’s splitting theorem yields a compatible family of classes (α_{[P_{•}]}) in A_{F}^{2}. Linckelmann showed in [11, 4.2], see also [10, §4], that if a solution to the gluing problem for this family of classes exists for all the fusion systems of blocks, then Alperin’s weight conjeture has an equivalent formulation in terms of the Euler characteristic of certain cochain complexes.

Little is known about the solution of the gluing problem in general. In this paper we will prove the following results.

Date: September 18, 2009.

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

Theorem 1.1. The gluing problem for any family of compatible classes in A_{F}^{2} has a solution in the fusion systems of the symmetric groups.

Theorem 1.2. If p > 2 then the gluing problem for any family of compatible classes in A_{F}^{2} has a solution in the fusion systems of the alternating groups.

The proof of Theorem 1.2 is a slightly more involved variant of the proof of Theorem 1.1 but introduces no new ideas. We therefore prove in an Appendix.

A collection C in F^{c} is a set of F-centric subgroups of S which is closed to Fconjugacy. We let F^{C} denote the full subcategory generated by the object set C. Given a compatible family of classes (α_{[P_{•}]}) in F^{c}, we may restrict it to a family in F^{C}. A natural question to ask is if the gluing problem can be reduced to F^{C}.

Assume that F has an associated centric linking system L, see [5]. We can equip L with morphisms ι: P → Q which are lifts of inclusions P ≤ Q in F in such a way that the composite P ⟶^{ι} Q ⟶^{ι} R is P ⟶^{ι} R. See [5, Proposition 1.11]. By [9, Theorem A] there are functors

A_{L}^{i}: [S(F^{c})] → Ab, [P_{•}] ↦ H^{i}(Aut_{L}(P_{•}); k^{×})

where Aut_{L}(P_{•}) is the preimage of Aut_{F}(P_{•}) under the projection Aut_{L}(P_{s}) ⟶^{π} Aut_{F}(P_{s}) whose kernel is Z(P_{s}). There results a nartural isomorphism of functors

A_{F}^{i} ⟶^{[}≅]π^{∗}A_{L}^{i}. By abuse of notation we denote the image of classes α_{[P_{•}]} ∈ A_{F}^{i} under π^{∗} by the same letter α_{[P_{•}]} and may seek for a solution to the gluing problem in L. Namely, we want to find a class α ∈ H^{i}(L; k^{×}) which restricts to α_{[P_{•}]} ∈ A_{L}^{i}([P_{•}]) for all [P_{•}] ∈ [S(L)]. The upshot of the next resultis that in the presence of a linking system, the gluing problem can be reduced to the collection of the F-centric F-radical subgroups.

Theorem 1.3. Consider a saturated fusion system F on S and assume that it has an associated centric linking system L. Assume that (α_{[P_{•}]}) is a family of compatible classes in A_{F}^{2}. Let C be a collection in F^{c} which contain all the F-centric F-radical subgroups. Then the following are equivalent

(a) The gluing problem for the family (α_{[P_{•}]}) has a solution in F^{c}.

(b) The gluing problem for the family (α_{[P_{•}]}) has a solution in L.

(c) The gluing problem for the family (α_{[P_{•}]})_{[P_{•}]∈[S(F^{C})]} has a solution in L^{C}.

(d) The gluing problem for the family (α_{[P_{•}]})_{[P_{•}]∈[S(F^{C})]} has a solution in F^{C}. In addition, if H^{2}([S(F^{C})]; A_{L}^{1}) = 0 then there is a solution to the gluing problem for any collection of compatible classes in A_{F^{c}}^{2}.

Organization of the paper. The constructions in section 2 are used throughout the paper, particularly important are Examples 2.2 and 2.5. The proof of Theorems 1.1 and 1.2 depends on Theorem 1.3 and the the calculations in Propositions 3.5 and 4.1 whose details are quite technical and the reader may find it a good idea to avoid them at first, as well as the details of Lemma 2.11.

Throughout we fix a prime p and an algebraically closed field k of characteristic p. The word “space” means a simplicial set. A G-space is a simplicial G-set. All groups are assumed to be finite, albeit not necessary for most of the arguments.

2. Bredon Cohomology

Let G be a finite group and O_{G} the category of transitive G-sets; it is equivalent to the subcategory spanned by the coset spaces G/H. A coefficient functor M for G is a contravariant functor {G-sets} → Ab which turns coproducts of G-sets into products of abelian groups. Equivalently, it is an object in the category O_{G} -mod of the contravariant functor O_{G} → Ab. See [2].

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 are denoted

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

They are chain equivalent and their homology groups are the Bredon cohomology groups H_{G}^{∗}(X; M). Note that C_{G}^{n}(X; M) = ∏_{[σ]} M([σ]) where [σ] runs through all the orbits of the n-simplices of X. If Y ⊆ X then there is a short exact sequence

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

giving rise to the ususal long exact sequence in cohomology. The assignment G/H ↦ H_{j}(X^{H}; ℤ) defines an object \underline{H}_{j}(X; ℤ) in O_{G} -mod. There is a spectral sequence, [2, Chap. I, (10.4)]

E_{2}^{i,j} = Ext_{O_{G} -mod}^{i}(\underline{H}_{j}(X; ℤ), M) ⇒ H_{G}^{i+j}(X; M)

which is natural in both X and M. From this spectral sequence it is clear that H_{G}^{∗}(−; M) is homotopy invariant because X → Y is a G-equivalence if and only if X^{H} → Y^{H} are homotopy equivalences. In addition, from Künneth’s theorem, we obtain the following result.

Lemma 2.1. For any G-space X and any integral homology isomorphism of spaces f: A → B, the map A × X ⟶^{f×X} B × X induces an H_{G}^{∗}(−; M)-isomorphism for any coefficient functor M for G.

2.1. Important examples of coefficient systems. Throughout, the orbit space of a G-space X is denoted by either X_{G} or X_{/G}. Its fixed point subspace is denoted as usual by X^{G}.

Example 2.2. Fix an abelian group A. The constant coefficient functor K_{A} has the effect G/H ↦ A. By inspection of the definitions, for every G-space X

(2.2.1)  C_{G}^{∗}(X; K_{A}) = C^{∗}(X_{G}; A).

More generally, the coefficient functor H^{n} for G is defined by H^{n}(G/K) = H^{n}(K; A) where A has the trivial action of G. In this paper the case n = 1 plays a special role. We will define H_{G} as the coefficient functor

(2.2.2)  H_{G}: G/H ↦ Hom(H, k^{×})

where k^{×} is the group of units in an algebraically closed field of characteristic p.

Example 2.3. Consider G = G_{1} × G_{2} and coefficient functors M_{1} and M_{2} for G_{1} and G_{2}. Define a coefficient functor M = M_{1} ⊕ M_{2} for G by

(M_{1} ⊕ M_{2})(U) = M_{1}(U_{/G_{2}}) ⊕ M_{2}(U_{/G_{1}}).

Clearly C_{G}^{∗}(X; M) = C_{G_{1}}^{∗}(X_{G_{2}}; M_{1}) ⊕ C_{G_{2}}^{∗}(X_{G_{1}}; M_{2}) for any G-space X, whence

(2.3.1)  H_{G_{1}×G_{2}}^{∗}(X; M_{1} ⊕ M_{2}) = H_{G_{1}}^{∗}(X_{G_{2}}; M_{1}) ⊕ H_{G_{2}}^{∗}(X_{G_{1}}; M_{2}).

If M = H_{G_{1}×G_{2}}, see (2.2.2), then by inspection M(U_{1} ×U_{2}) = H_{G_{1}}(U_{1})⊕H_{G_{2}}(U_{2}) for any transitive G_{1}- and G_{2}-sets. Therefore

(2.3.2)  H_{G_{1}×G_{2}}^{∗}(X_{1} × X_{2}; H_{G_{1}×G_{2}}) ≅ H_{G_{1}×G_{2}}^{∗}(X_{1} × X_{2}; H_{G_{1}} ⊕ H_{G_{2}})

Example 2.4. Let U^{≤(K)} denote the subset of a G-set U consisting of the orbits whose isotropy group is conjugate to a subgroup of K ≤ G and let U^{≰(K)} denote its complement. The assignment U ↦ U^{≰(K)} is a functor on the category of G-sets. Given a coefficient functor M for G define new coefficient functors

M^{≰(K)}(U) =^{def} M(U^{≰(K)})  and  M^{≤(K)}(U) =^{def} M(U^{≤(K)}).

The inclusions U^{≰(K)} ⊆ U yield a canonical short exact sequence of functors

0 → M^{≤(K)} → M → M^{≰(K)} → 0.

Example 2.5. Let M be a coefficient functor for G and set Γ = G≀Σ_{r}. It contains G × Σ_{r} via the diagonal inclusion G ≤ G^{r}. Let G_{i} ≅ G^{r−1} denote the subgroup of G^{r} whose ith factor is the trivial group. If U is a Γ-set then U_{/G_{i}} is a G-set via the diagonal inclusion G ≤ G^{r}. Thus, ∐_{i=1}^{r} U_{/G_{i}} becomes a G × Σ_{r}-set and M(∐_{i=1}^{r} U_{/G_{i}}) is a Σ_{r}-module. We define coefficient functors for Γ

M^{⊕r}(U) = M(∐_{i=1}^{r} U_{/G_{i}}),  and  (M ≀ Σ_{r})(U) = (M^{⊕r}(U))^{Σ_{r}}

If U is a G-set with orbits {U_{i}}_{i∈I} then U^{r} is naturally a Γ-set. A Γ-orbit W ⊂ U^{r} is the Σ_{r}-orbit of some G^{r}-set of the form (U_{i_{1}})^{e_{1}} ×⋯×(U_{i_{t}})^{e_{t}} where e_{1} +⋯+e_{t} = r, via permutation of the factors. By inspection (M ≀ Σ_{r})(W) is the Σ_{e_{1}} × ⋯ × Σ_{e_{r}}- invariants of M(U_{i_{1}})^{⊕e_{1}} ⊕ ⋯ ⊕ M(U_{i_{r}})^{⊕e_{r}}, whence

(2.5.1)  (M ≀ Σ_{r})(W) ≅ M(U_{i_{1}}) ⊕ ⋯ ⊕ M(U_{i_{r}}).

The next result follows by direct inspection of the cochain complexes and (2.5.1).

Proposition 2.6. Let M_{1}, M_{2} and M_{3} be coefficient functiors for G_{1}, G_{2}, G_{3}. Let X_{1}, X_{2}, X_{3} be G_{1}, G_{2} and G_{3}-spaces whose orbit spaces are equal to a space X̄ with quotinet maps X_{i} ⟶^{π_{i}} X̄. Suppose that for every simplex σ ∈ X̄ we are equipped with a short exact sequence 0 → M_{1}(π_{1}^{−1}(σ)) ⟶^{φ_{σ}} M_{2}(π_{2}^{−1}(σ)) ⟶^{ψ_{σ}} M_{3}(π_{3}^{−1}(σ)) → 0 such that the homomorphisms φ_{σ} and ψ_{σ} commute with the face and degenercay maps of X̄.

Fix a Σ_{r}-subspace Ȳ of X̄^{r} and for every i = 1, 2, 3 set Γ_{i} = G_{i} ≀ Σ_{r}. Let Y_{i} be the Γ_{i}-subspace of (X_{i})^{r} which is the preimage of Ȳ. Then there is a short exact sequence of cochain complexes

0 → C_{Γ_{1}}^{∗}(Y_{1}; M_{1} ≀ Σ_{r}) → C_{Γ_{2}}^{∗}(Y_{2}; M_{2} ≀ Σ_{r}) → C_{Γ_{3}}^{∗}(Y_{3}; M_{3} ≀ Σ_{r}) → 0.

The most useful case is when G_{1} = G_{2} = G_{3} and X_{1} = X_{2} = X_{3} and 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of coefficient functors. Also, if K_{A} is the constant coefficient functor, see Example 2.2, then K_{A}(U_{G}) ⟶^{≅} K_{A}(U) for every G-set U. Hence, if Y is a G ≀ Σ_{r}-subspace of X^{r} then

(2.6.1)  C_{G≀Σ_{r}}^{∗}(Y; K_{A} ≀ Σ_{r}) ≅ C_{G≀Σ_{r}}^{∗}(Y_{G^{r}}; K_{A} ≀ Σ_{r})

Example 2.7. Let K be a subgroup of G. Any K-set U can be induced up to a G-set U ↑_{K}^{G} =^{def} G ×_{K} U. This construction is left adjoint to the restriction of the action from G to K.

(i) Note that K/H ↑_{K}^{G}= G/H. Thus, if Y is a K-subset of a G-set X such that Y_{K} → X_{G} is bijective and such that the G-isotropy group of every y ∈ Y is contained in K, then X ≅ Y ↑_{K}^{G}.

(ii) If M is a coefficient fucntor for G we obtain a coefficient functor for K defined by M ↓_{K}^{G} (U) = M(U ↑_{K}^{G}). Therefore, for any K-space X,

H_{G}^{∗}(X ↑_{K}^{G}; M) = H_{K}^{∗}(X; M ↓_{K}^{G}).

We note that for the coefficient functors H^{n} of Example 2.2 we have

H_{G}^{n} ↓_{K}^{G}= H_{K}^{n}.

The homotopy orbit space of a G-space X is the Borel construction EG ×_{G} X where EG is a contractible space with a free right action of G and we are taking the orbit space of EG × X with the diagonal action of G. Different choices of EG give rise to homotopy equivalent spaces and EG ×_{G} G/K ≅ BK where BK is the classifying space of K. Thus, H^{n} of Example 2.2 has the form U ↦ H^{n}(EG ×_{G} U; A).

There is a standard model for EG obtained as the nerve of the category EG whose object set is G and there is one isomorphism between any two objects. Clearly, if G = N ⋊ K then K acts by conjugation on N, whence on EN. It is then not hard to show that EG = EK × EN where N acts on the right by translation on EN and trivially on EK and K acts by conjugation on EN and by translation on EK. Also note that E(G^{n}) = (EG)^{n}.

Proposition 2.8. Let X be a G-space and set Γ = G ≀ Σ_{r}. Consider the coefficient functors H_{G}, H_{Σ_{r}}, H_{Γ} defined in Example 2.2. If Z is a Γ-subspace of X^{r} then there is a short exact sequence of cochain complexes

0 → C_{Σ_{r}}^{∗}(Z_{G^{r}}; H_{Σ_{r}}) → C_{Γ}^{∗}(Z; H_{Γ}) → C_{Γ}^{∗}(Z; H_{G} ≀ Σ_{r}) → 0.

Proof. As explained above EΓ = EΣ_{r} × EG^{r} where Σ_{r} acts by permuting the factors of EG^{r} and by translation on EΣ_{r} and G^{r} act on EG^{r} by translation. We will now construct several coefficient functors M_{1}, M_{2}, M_{3} and M_{4} for Γ.

Fix a Γ-set W. The map EG^{r} → pt gives rise to a Γ-map EΓ → EΣ_{r} and note that pt ×_{G^{r}}W = W_{G^{r}}. We therefore obtain maps

EΓ ×_{G^{r}} W → EΓ ×_{Γ} W → EΣ_{r} ×_{Γ} W = EΣ_{r} ×_{Σ_{r}} (W_{G^{r}}).

The first space has a natural action of Σ_{r} and the first map is Σ_{r}-equivariant. We therefore obtain coefficient functors

M_{1}(W) = H^{1}(EΣ_{r} ×_{Σ_{r}} W_{G^{r}}; k^{×}) = H_{Σ_{r}}(W_{G^{r}}),
M_{2}(W) = H^{1}(EΓ ×_{Γ} W; k^{×}) = H_{Γ}(W),
M_{3}(W) = (H^{1}(EΓ ×_{G^{r}} W; k^{×}))^{Σ_{r}}

together with natural transformations M_{1} → M_{2} → M_{3}.

For every 1 ≤ i ≤ r we consider the projection EG^{r} ⟶^{π_{i}} EG onto the ith factor. It is clearly a G^{r}-maps and by applying EΓ ×_{G^{r}} − we obtain

EΣ_{r} × EG^{r} ×_{G^{r}} W ⟶^{EΣ_{r}×π_{i}} EΣ_{r} × EG ×_{G} (W_{G_{i}})

where G_{i} ≅ G^{r−1} is the subgroup of G^{r} with the trivial group in the ith factor; see Example 2.5. We obtain coefficient functors

N_{i}(W) = H^{1}(EΣ_{r} × EG ×_{G} (W_{G_{i}}); k^{×}) = H_{G}(W_{G_{i}})

together with natural maps N_{i}(W) → H^{1}(EΓ ×_{G^{r}} W; k^{×}). The action of Σ_{r} on W induces an action on ∐_{i=1}^{r} W_{G_{i}} and we define a coefficient functor M_{4}

M_{4}(W) = ( ⊕_{i=1}^{r} N_{i}(W))^{Σ_{r}} = (H_{G} ≀ Σ_{r})(W).

Observe that for every i and every σ ∈ Σ_{r} the following diagram commutes

\xymatrix{EΣ_{r} × EG^{r} ×_{G^{r}} W \ar[r]^{π_{i}} \ar[d]_{σ} & EΣ_{r} × EG ×_{G} (W_{G_{i}}) \ar[d]^{σ} \\ EΣ_{r} × EG^{r} ×_{G^{r}} W \ar[r]^{π_{σ(i)}} & EΣ_{r} × EG ×_{G} (W_{G_{σ(i)}})}

Thus, ⊕_{i=1}^{r}N_{i}(W) → H^{1}(EΓ×_{G^{r}} W; k^{×}) is Σ_{r}-equivariant and we obtain a natural transformation M_{4} → M_{3}.

Now consider a Γ-orbit in X^{r}. It is the Σ_{r}-orbit via permutation of the factors of U_{1}^{e_{1}} × ⋯ × U_{t}^{e_{t}} where U_{i} are G-orbits of X isomorphic to some G/K_{i} and e_{1} + ⋯ + e_{t} = r. Its isotropy group is easily seen to be ∏_{j=1}^{t} K_{j} ≀ Σ_{e_{j}}. We note that M_{1}(W) → M_{2}(W) → M_{3}(W) are the homomorphisms

∏_{j=1}^{r} H^{1}(BΣ_{e_{j}}; k^{×}) → ∏_{j=1}^{r} H^{1}(BK_{j} ≀ Σ_{e_{j}}; k^{×}) → ∏_{j=1}^{r} (H^{1}(BK_{j}^{e_{j}}; k^{×}))^{Σ_{e_{j}}}

induced by K_{j}^{e_{j}} → K_{j} ≀ Σ_{e_{j}} → Σ_{e_{j}}. We now leave it to the reader to check that since H^{1}(K; A) = Hom(K, A) the folowing sequence is exact for any finite group K

0 → H^{1}(Σ_{s}; A) → H^{1}(K ≀ Σ_{s}; A) → H^{1}(K^{s}; A)^{Σ_{s}} → 0.

Thus, 0 → M_{1}(W) → M_{2}(W) → M_{3}(W) → 0 is exact. Since W_{G_{i}} = U_{j} for a suitable j one easily sees that M_{4}(W) = ⊕_{j=1}^{t}(H^{1}(BK_{j}^{e_{j}}; k^{×}))^{Σ_{e_{j}}} = M_{3}(W). The result now follows from Proposition 2.6. □

2.2. Calculation of some Bredon cohomology groups.

Definition 2.9. Let X be a G-poset. A G-subposet Y ⊆ X is called left closed if for any y ∈ Y and x ∈ X we have x ⪯ y ⇒ x ∈ Y. Similarly Y is right closed if for any y ∈ Y and x ∈ X we have y ⪯ x ⇒ x ∈ Y.

For example (−∞, a] is a left closed subposet of ℝ and [a, ∞) is right closed.

Lemma 2.10. Suppose that (X_{1}, ⪯) and (X_{2}, ⪯) are G_{1}- and G_{2}-posets. Let X′_{i} ⊆ X_{i} be left closed G_{i}-subposets. Set Y_{i} = X_{i} − X′_{i} and Z = X_{1} × X_{2} − (X′_{1} × X′_{2}) and consider Z as G = G_{1} × G_{2}-posets. Then the inclusions Y_{i} ⊆ X_{i} induce an isomorphism of relative G_{1} × G_{2}-spaces

(|Y_{1} × X_{2}|, |Y_{1} × Y_{2}|) ⟶^{≅} (|Z|, |X_{1} × Y_{2}|).

Proof. First, note that |X_{1} × Y_{2}| ∩ |Y_{1} × X_{2}| = |Y_{1} × Y_{2}|. Consider a k-simplex

x = (x_{1}(0), x_{2}(0)) ⪯ (x_{1}(1), x_{2}(1)) ⪯ ⋯ ⪯ (x_{1}(k), x_{2}(k))

in |Z| which is not contained in |X_{1}×Y_{2}|. Then there is some i such that x_{2}(i) ∈ X′_{2}. Since X′_{2} is left closed, x_{2}(0) ∈ X′_{2} and by the definition of Z it follows that x_{1}(0) ∈ Y_{1}. Now, Y_{1} is right closed so x_{1}(j) ∈ Y_{1} for all j, namely x ∈ |Y_{1}×X_{2}|. □

The next technical result is fundamental to this paper.

Lemma 2.11. Let X be a G-poset, where G ≠ 1, with a minimum Z and set Y = X^{G}. Let A be a finite group and let A denote the constant coefficient functor for G, and consider the coefficient functor A^{≤(e)} where e ≤ G is the trivial subgroup; see Examples 2.2 and 2.4. Assume that (a) Y is left closed in X and X − Y has a free action of G, i.e. all its isotropy

groups are trivial.

(b) The orbit poset X̄ = X_{/G} has a maximum x̄^{max} which is distinct from Z. Also assume that Y has a maximum y^{max}.

(c) The natual map |X|_{G} → |X̄| is an isomorphism of simplicial sets. Set Γ = G ≀ Σ_{r} with its natural action on X^{r}. Then

(i) |X^{r} − Z^{r}|_{G^{r}} is Σ_{r}-equivalent to a point and

(ii) H_{Γ}^{∗≥1}(|X^{r} − Z^{r}|; A^{≤(e)} ≀ Σ_{r}) = 0.

Remark: In fact, we will show that H_{Γ}^{∗}(|X^{r} − Z^{r}|; A^{≤(e)} ≀ Σ_{r}) = 0 unless r = 1 and y^{max} = Z, in which case this graded group is isomorphic to A.

Proof. (i) By hypothesis (c) and since Z is fixed by G we have |X^{r} − Z^{r}|_{G^{r}} = |X̄^{r} − Z^{r}| which is Σ_{r}-contractible becauase (x̄^{max}, …, x̄^{max}) is a maximum of X̄^{r} − Z^{r} by hypothesis (b).

(ii) For every integer q ≥ 0 let Γ_{q} denote G ≀ Σ_{q}. For s, t ≥ 0 let Γ_{s,t} denote Γ_{s} × Γ_{t} as a subgroup of Γ_{s+t}. Consider the following Γ_{s,t}-subposets of Y^{s} × X^{t}.

X^{s,t}(q) = {(y, x) ∈ Y^{s} × X^{t} : at least q entries of x belong to Y },
X_{s,t}(q) = = X^{s,t} − {Z}^{s+t}.

Clearly X_{s,t}(q + 1) ⊆ X_{s,t}(q). Also, X_{0,r}(0) = X^{r} − Z^{r} and X_{0,r}(r) = Y^{r} − Z^{r}. In addition, X_{s,t}(q) ⊆ X_{0,s+t}(s + q). We now claim that for 1 ≤ q ≤ r − 1

(2.11.1)  (|X_{0,r}(q)|, |X_{0,r}(q + 1)|) ≅ (|X_{q,r−q}(0)|, |X_{q,r−q}(1)|) ↑_{Γ_{q,r−q}}^{Γ_{r}} .

Indeed, any elelent of X_{0,r}(q) is in the Σ_{r}-orbit of some u ∈ X_{q,r−q}(0). Consider v ∈ X_{0,r}(q) such that v ≤ u. Then v ∈ X_{q,r−q}(0) becuause Y is left closed by hypothesis (a). If in addition v ∉ X_{0,r}(q + 1) then its first q entries belong to Y and the last r − q entries belong to X − Y, hence v ∈ X_{q,r−q}(0) − X_{q,r−q}(1). If w ≤ u and v is in the same Γ_{r}-orbit of v then w and v must be in the same orbit of Γ_{q,r−q} because w, v ∈ Y^{q} × (X − Y)^{r−q}. This also shows that the elements of Γ_{r} that fix a simplex σ in (|X_{q,r−q}(0)|, |X_{q,r−q}(1)|) are contained in Γ_{q,r−q}. Now (2.11.1) follows from Example 2.7. In addition, no non-trivial element of G^{r} fixes a simplex σ in (|X_{0,r}(0)|, |X_{0,r}(1)|) becuase by hypothesis (a) G^{r} acts freely on X_{0,r}(0) − X_{0,r}(1)| = (X − Y)^{r}. We deduce from the definition in Example 2.5 that

(2.11.2)  C_{Γ_{r}}^{∗}(|X_{0,r}(0)|, |X_{0,r}(1)|; A^{≰(e)} ≀ Σ_{r}) = 0.

Note that X^{s,t}(q) has a minimum Z^{s+t} provided q ≤ t. Also, if y^{max} ≠ Z then X_{s,t}(t) = Y^{s+t} − Z^{s+t} has a maximum by hypothesis (b). Therefore

(2.11.3) |X^{s,t}(q)| is Γ_{s,t}-equivalent to a point if q ≤ t and |X_{s,t}(t)| is Γ-equivalent to a point if y^{max} ≠ Z.

We are now ready to prove point (ii). Set X_{r} =^{def} X^{r} − Z^{r} = X_{0,r}(0). First, observe that if y^{max} = Z and r = 1 then X_{1} = X − Y which has a free action of Γ_{1} = G, hence C_{Γ_{1}}^{∗}(|X_{1}|; A^{≤(e)}) = C_{Γ_{1}}^{∗}(|X_{1}|; A). Example (2.2.1) and point (i) now imply H_{Γ_{1}}^{∗}(|X_{1}|; A^{≤(e)}) = A.

We will now prove that if either y^{max} ≠ Z or if y^{max} = Z and r ≥ 2 then H_{Γ_{r}}^{∗}(|X_{r}|; A^{≤(e)} ≀ Σ_{r}) = 0. To do this we recall from Examples 2.4 and Proposition 2.6 that there is a short exact sequence of cochain complexes

0 → C_{Γ_{r}}^{∗}(|X_{r}|; A^{≰(e)} ≀ Σ_{r}) → C_{Γ_{r}}^{∗}(|X_{r}|; A ≀ Σ_{r}) → C_{Γ_{r}}^{∗}(|X_{r}|; A^{≤(e)} ≀ Σ_{r}) → 0.

Example (2.6.1) and point (i) now show that |X_{r}| → pt induces an isomorphism in H_{Γ_{r}}^{∗}(−; A ≀ Σ_{r}). Also note that H_{Γ_{r}}^{∗}(pt; A^{≰(e)} ≀ Σ_{r}) = H_{Γ_{r}}^{∗}(pt; A ≀ Σ_{r}) = A because G ≠ 1. We now see that to prove that H_{Γ_{r}}^{∗}(|X_{r}|; A^{≤(e)} ≀ Σ_{r}) = 0, it only remains to prove that |X_{r}| → pt induces an isomrphism in H_{Γ_{r}}^{∗}(−; A^{≰(e)} ≀ Σ_{r}). Case A: y^{max} ≠ Z. If s ≥ 1 then Y^{s} and Y_{s} = Y^{s} − Z^{s} have a maximum by hypothesis (b), whence their nerves are Γ_{s}-equivalent to a point. Lemma 2.10 implies that for any coefficient functor N for Γ_{s,t}

H_{Γ_{s,t}}^{∗}(|X_{s,t}(q)|, |Y_{s} × X^{0,t}(q)|; N) ≅ H_{Γ_{s,t}}^{∗}(Y^{s} × X_{0,t}(q)|, |Y_{s} × X_{0,t}(q)|; N) ≅

H_{Γ_{s,t}}^{∗}(|X_{0,t}(q)|, |X_{0,t}(q)|; N) = 0.

Now (2.11.1) and Example 2.7 and (2.11.3) imply that if N is a coefficient functor for Γ_{r} and if 1 ≤ q ≤ r − 1 then

H_{Γ_{r}}^{∗}(|X_{0,r}(q)|, |X_{0,r}(q + 1)|; N) ≅ H_{Γ_{q,r−q}}^{∗}(|X^{0,r}(0)|, |X^{0,r}(1)|; N ↓_{Γ_{q,r−q}}^{Γ_{r}}) = 0.

Together with (2.11.2) and (2.11.3) we deduce that

|X_{r}| = |X_{0,r}(0)| ⊇ |X_{0,r}(1)| ⊇ ⋯ ⊇ |X_{0,r}(r − 1)| ⊇ |X_{0,r}(r)| → pt

induce isomorphisms in H_{Γ}^{∗}(−; A^{≰(e)} ≀ Σ_{r}). This completes the proof in this case. Case B: y^{max} = Z and r ≥ 2. Note that Y = {Z} so

X_{s,t}(q) ≅ Y^{s} × X_{0,t}(q).

Also observe that X_{0,r}(r − 1) ≅ ∐_{r}(X − Y) where Σ_{r} acts by permuting the components. Claim. |X_{0,r}(q)|_{Γ_{r}} is acyclic for all 0 ≤ q ≤ r − 1. Proof. If q = 0 then we apply point (i). If q = r − 1 then |X_{0,r}(r − 1)|_{Γ_{r}} ≅ |X − Y |_{G} = |X̄ − Z| is contractible becuase X̄ − Z has a maximum. We will now prove the claim by induction on r. For r = 2 note that q = 0 or q = 1 = r − 1. Thus, we fix some r ≥ 3 and we will use downward induction on 1 ≤ q ≤ r − 1, the base of induction q = r − 1 has already been settled. If 1 ≤ q ≤ r − 2 then we apply (2.11.1) and the induction hypothesis on r to deduce that

H_{∗}(|X_{0,r}(q)|_{Γ_{r}}, |X_{0,r}(q + 1)|_{Γ_{r}}) ≅ H_{∗}(|X_{0,r−q}(0)|_{Γ_{r−q}}, |X_{0,r−q}(1)|_{Γ_{r−q}}) = 0.

From the induction hypothesis on q + 1 we deduce that |X_{0,r}(q)|_{Γ_{r}} is acyclic and this completes the induction step for q, hence for r and completes the proof of the claim.

Let M denote A^{≰(e)}≀Σ_{r}. Since r ≥ 2, hypothesis (a) and the fact that G ≠ 1 and that X_{0,r}(r −1) = ∐_{r}(X −Y)×pt, imply that M([σ]) = A for all σ ∈ |X_{0,r}(r −1)|. We deduce from the claim and the fact that G ≠ 1 that |X_{0,r}(r − 1)| → pt induces an isomorphism in H_{Γ_{r}}^{∗}(−; M) because

H_{Γ_{r}}^{∗}(|X_{0,r}(r − 1)|; M) ≅ H^{∗}(|X_{0,r}(r − 1)|_{Γ_{r}}; A) ≅ H^{∗}(pt; A) ≅ H_{Γ_{r}}^{∗}(pt; M).

Now fix 1 ≤ q ≤ r − 2 and consider a k simplex σ of the form x^{0} ≤ ⋯ ≤ x^{k} in the relative space (|X_{0,r}(q)|, |X_{0,r}(q + 1)|). Then x^{k} must have exactly q of its entries in Y = {Z} because if q + 1 entries are in Y then this is the case for all x^{i} since Y is left closed and this is a contradiction because σ ∉ |X_{0,r}(q + 1)|. By hypothesis (a) we see that σ is fixed by G ≀ Σ_{q} and the remaining r − q factors of G^{r} ≤ Γ_{r} act freely on σ. Since G ≠ 1 and q ≥ 1 we deduce that M([σ]) = A. We now deduce from the claim that

H_{Γ_{r}}^{∗}(|X_{0,r}(q)|, |X_{0,r}(q + 1)|; M) ≅ H^{∗}(|X_{0,r}(q)|_{Γ_{r}}, |X_{0,r}(q + 1)|_{Γ_{r}}; A) = 0.

We finally see that |X_{r}| = |X_{0,r}(0| ⊃ |X_{0,r}(1)| ⊃ ⋯ ⊃ |X_{0,r}(r − 1)| → pt induce isomorphisms in H_{Γ_{r}}^{∗}(−; M). This completes the proof. □

3. The poset of partitions into intervals

Definition 3.1. Let C be a poset and let subs(G) be the poset of all the subgroups of some group G. Given a map of posets F: C^{op} → subs(G), define a G-poset C̃ whose objects are pairs (c, ḡ) where c ∈ C and ḡ is an element in the coset space G/F (c). The partial order is defined by (c, ḡ) ≤ (c′, ḡ′) if c ≤ c′ and if the quotient map G/F (c′) → G/F (c) carries ḡ′ to ḡ. The action of G is via left translation.

There is an inclusion of posets C ⊆ C̃ via c ↦ (c, ē) where e ∈ G is the unit. The orbit of an n-simplex σ of |C| ⊆ |C̃| is denoted [σ]_{G}.

The following result is an easy consequence of the definitions and we leave its proof to the reader.

Lemma 3.2. With the notation of Definition 3.1, (i) |C| ⊆ |C̃| → |C̃|_{G} is an isomorphism of simplicial sets. (ii) The isotropy group of an n-simplex σ ∈ |C| ⊆ |C̃| of the form c_{0} ≤ ⋯ ≤ c_{n}

is F (c_{n}).

Definition 3.3. (i) An interval in \underline{d} = \{1, …, d\} is a non-empty subset of the form {x : a ≤ x ≤ b} for some a, b. Let P(d) denote the poset of all the partitions c of \underline{d} into intervals where c ≤ c′ if c′ is a refinement of c. We write partitions in the form c = (c_{1}, …, c_{t}) where c_{i} is the length of the ith interval.

Define the following partitions consisting of d, 1 and 2 intervals,

λ^{max} = (1, 1, …, 1),  λ^{min} = (d),  λ_{i} = (i, d − i) for 1 ≤ i ≤ d − 1.

Given r ≥ 1 let P(d)_{r} denote the Σ_{r}-poset P(d)^{r} − {(λ^{min}, …, λ^{min})}.

(ii) Fix a prime number p, an ineteger d ≥ 1 and set G = GL_{d}(p). For c = (c_{1}, …, c_{t}) in P(d) let R_{c} ≤ G be the isotropy group of the flag in 𝔽_{p}^{d}

𝔽_{p}^{c_{1}} < 𝔽_{p}^{c_{1}+c_{2}} < ⋯ < 𝔽_{p}^{c_{1}+···+c_{t}} = 𝔽_{p}^{d}

where 𝔽_{p}^{s} is spanned by the first s elements of the standard basis of 𝔽_{p}^{d}.

(iii) Let P̃(d) be the poset defined in 3.1 by the the assignment c ↦ R_{c} yielding P(d)^{op} → subs(G). The orbit of a simplex σ ∈ |P(d)| ⊆ |P̃(d)| is denoted [σ]_{G}.

If c ≤ c′ in P(d) then there is a surjective function of sets c′ → c which maps an interval i′ ∈ c′ to the unique interval i ∈ c which contains it. If A is an abelian group and X is a set, let A^{X} denote the group of all the functions X → A. we obtain a functor P(d) → Ab via the assignment c ↦ A^{c}.

Proposition 3.4. Fix G = GL_{d}(p) and P̃(d) as in Definition 3.3 and consider the inclusion P(d) ⊂ P̃(d). Then for every n-simplex c_{0} ≤ ⋯ ≤ c_{n} of P(d) the following isomorphism holds where H_{G} is defined in Example (2.2.2) and L ≅ 𝔽_{p}^{×}.

H_{G}([c_{0} ≤ ⋯ ≤ c_{n}]_{G}) ≅ L^{c_{n}}.

Moreover, H_{G}([c_{0} ≤ … ĉ_{i} ⋯ ≤ c_{n}]_{G}) → H_{G}([c_{0} ≤ ⋯ ≤ c_{n}]_{G}) is the identity on L^{c_{n}} if i < n and it is the map L^{c_{n−1}} → L^{c_{n}} induced by c_{n} → c_{n−1} if i = n.

Proof. The Sylow p-subgroups of GL_{n}(p) generate SL_{n}(p) so Hom(GL_{n}(p), k^{×}) ≅ L becuase k is algebraically closed of characteristic p. For every c ∈ P(d) where c = (c_{1}, …, c_{t}), the group R_{c} contains GL_{c}(p) := ∏_{i=1}^{t} GL_{c_{i}}(p) as a complement to a normal p-subgroup of R_{c}, hence Hom(R_{c}, k^{×}) ≅ L^{c}. If c ≤ c′ then R_{c′} ≤ R_{c} and GL_{c′}(p) ≤ GL_{c}(p) as an inclusion of block matrices, hence applying Hom(−, k^{×}) to R_{c′} ≤ R_{c} yields L^{c} → L^{c′}. The result now follows by Lemma 3.2(ii). □

The main result of this section is the next proposition.

Proposition 3.5. Let U be a K-poset for some K ≠ 1 and let M be a coefficient functor for K. Assume that U contains P(d) for some d ≥ 2 as a subposet and write [c_{0} ≤ … c_{n}]_{K} for the K-orbit in |U| of an n-simplex of |P(d)|. Assume further that

(a) the composite |P(d)| ⊆ |U| → |U|_{K} is an isomorphism of simplicial sets.

(b) Every simplex σ of |P(d)| is equipped with an isomorphism λ_{σ}: M([σ]_{K}) ⟶^{≅} H_{G}([σ]_{G}) where G = GL_{d}(p) and P̃(d) and [σ]_{G} are defined in 3.3(iii). Assume that these isomorphism are compatible with the face and degeneracy maps of |P(d)|.

Then the following holds. Fix some r ≥ 1 and let U_{r} denote the K ≀ Σ_{r}-subposet of U^{r} which is generated by P(d)_{r}, see Definition 3.3(i). Then

(i) H_{K≀Σ_{r}}^{∗≥1}(|U_{r}|; M ≀ Σ_{r}) = 0  and  (ii) H_{K≀Σ_{r}}^{∗≥1}(|U^{r}|; M ≀ Σ_{r}) = 0.

Proof. First, |U^{r}|_{K^{r}} = |P(d)^{r}| = |P̃(d)^{r}|_{G^{r}} and |U_{r}|_{K^{r}} = |P(d)_{r}| = |P̃(d)_{r}|_{G^{r}} by hypothesis (a) and Proposition 3.2(i) where P̃(d)_{r} is the G ≀ Σ_{r}-subposet of P̃(d)^{r} generated by P(d)_{r}. We deduce from hypothesis (b) and Proposition 2.6 that

H_{K≀Σ_{r}}^{∗}(|U^{r}|; M ≀ Σ_{r}) ≅ H_{G≀Σ_{r}}^{∗}(|P̃(d)^{r}|; H_{G} ≀ Σ_{r}),
H_{K≀Σ_{r}}^{∗}(|U_{r}|; M ≀ Σ_{r}) ≅ H_{G≀Σ_{r}}^{∗}(|P̃(d)_{r}|; H_{G} ≀ Σ_{r}).

Set Γ = G ≀ Σ_{r} and note that P̃(d) has a minimum λ^{min} ∈ P(d) ⊆ P̃(d). Thus, P̃(d)^{r} has a minimum, whence its nerve is Γ-equivalent to a point and in particular H_{Γ}^{∗≥1}(|P̃(d)^{r}|; H_{G} ≀ Σ_{r}) = 0. This proves point (i) of the proposition.

Set L = H_{G}(pt) and note that L = Hom(GL_{d}(p), k^{×}) ≅ 𝔽_{p}^{×}. The G-maps Ω → Ω_{G} for every G-set Ω gives rise to a natural transformation K_{L} → H_{G} where K_{L} is the constant coefficient functor, see Example 2.2. Let H̄_{G} denote the cokernel of this map. Since λ^{min} is fixed by G, Proposition 3.4 implies that K_{L}([σ]_{G}) → H_{G}([σ]_{G}) is the diagonal inclusion L ≤ L^{c_{n}} for any simplex σ of |P(d)| of the form c_{0} ≤ ⋯ ≤ c_{n}. We obtain from Proposition 2.6 a short exact sequence of cochain complexes

0 → C_{Γ}^{∗}(|P̃(d)_{r}|; K_{L} ≀ Σ_{r}) → C_{Γ}^{∗}(|P̃(d)_{r}|; H_{G} ≀ Σ_{r}) → C_{Γ}^{∗}(|P̃(d)_{r}|; H̄_{G} ≀ Σ_{r}) → 0.

First, we claim that H_{Γ}^{∗≥1}(|P̃(d)_{r}|; K_{L} ≀ Σ_{r}) = 0. This follows from Example (2.6.1) because |P̃(d)_{r}|_{G^{r}} = |P(d)_{r}| by Lemma 3.2(i) and P(d)_{r} has a maximum since d ≥ 2 so λ^{min} ≠ λ^{max}, see Definition 3.3. Thus, |P(d)_{r}| is Γ-equivalent to a point.

To prove point (ii) it remains to prove that H_{Γ}^{∗≥1}(|P̃(d)_{r}|; H̄_{G} ≀ Σ_{r}) = 0. Set G_{i} = R_{λ_{i}} where 1 ≤ i ≤ d − 1, see Definitions 3.3 (i) and (ii). Consider the following coefficient functors for G; see Definition 2.2 and 2.4.

N_{i} = (K_{L×L})^{≤(G_{i})},  N̄_{i} = (K_{L})^{≤(G_{i})},  (1 ≤ i ≤ d − 1)

Fix some 1 ≤ i ≤ d − 1. By Lemma 3.2 the isotropy group of c = (c_{1}, …, c_{t}) in P(d) ⊆ P̃(d) is R_{c}. We note that the subspaces of 𝔽_{p}^{d} which are invariant under R_{c} are 𝔽_{p}^{c_{1}+···+c_{i}} for 0 ≤ i ≤ t. We deduce that the isotropy group of (c, ḡ) ∈ P̃(d) is conjugate to a subgroup of R_{λ_{i}} if and only if one of the invariant subspaces of R_{c} has dimension i, if and only if λ_{i} ≤ c. Thus, for every n-simplex σ of the form c_{0} ≤ ⋯ ≤ c_{n} in |P(d)| we obtain a homomorphism

N_{i}([σ]_{G}) → H_{G}([σ]_{G})

which is trivial if λ_{i} ≰ c_{n} and it is L × L = L^{λ_{i}} → L^{c_{n}} if λ_{i} ≤ c; see Proposition 3.4. These homomorphimsms are easily seen to be compatible with face and degeneracy maps in |P(d)|. The diagonal L ≤ L^{2} also induces a natural transformation K_{L}^{≤(G_{i})} → N_{i} whose cokernel is N̄_{i}. It is also easy to check that the composite

K_{L}^{≤(G_{i})}([σ]_{G}) → N_{i}([σ]_{G}) → H_{G}([σ]_{G})

factors through K_{L}([σ]_{G}) → H_{G}([σ]_{G}). By taking cokernels we now obtain maps N̄_{i}([σ]_{G}) → H̄_{G}([σ]_{G}) which commute with the face and degeneracy maps of |P(d)|.

The images of N_{i}([σ]_{G}) → H_{G}([σ]_{G}) where i = 1, …, d − 1 and of K_{L}([σ]_{G}) → H_{G}([σ]_{G}) generate H_{G}([σ]_{G}) because L^{c_{n}} is generated by the diagonal L ≤ L^{c_{n}} and the diagonal inclusions L × L ≤ L^{j} × L^{|c_{n}|−j}. It follows that ⊕_{i=1}^{d−1}N̄_{i}([σ]_{G}) → H̄_{G}([σ]_{G}) is surjective. This is, in fact, an isomorphism because N̄_{i}([σ]_{G}) = L if and only if 𝔽_{p}^{i} is fixed by R_{c_{n}}, hence both sides are isomorphic to L^{|c_{n}|−1}. We deduce from Proposition 2.6 that

C_{Γ}^{∗}(|P̃(d)_{r}|; H̄_{G} ≀ Σ_{r}) ≅ ⨁_{i=1}^{d−1} C_{Γ}^{∗}(|P̃(d)_{r}|; N̄_{i} ≀ Σ_{r}).

It therefore remains to prove that H_{Γ}^{∗≥1}(|P̃(d)_{r}|; N̄_{i} ≀ Σ_{r}) = 0 for all 1 ≤ i ≤ d − 1.

Fix some i = 1, …, d − 1 and let P̃(d, i) be the G_{i}-poset definded in 3.1 by the map of posets P(d)^{op} → subs(G_{i}) which has the effect c ↦ {e} if λ_{i} ≤ c and c ↦ G_{i} if λ_{i} ≰ c. We have seen in Lemma 3.2(i) that the inclusion P(d) ⊆ P̃(d, i) induces an isomorphism of simplicial sets |P(d)| ≅ |P̃(d, i)|_{G_{i}}. Let A be the constant coefficient functor for G_{i} with value L, see Example 2.2. Note that the isotropy group of an n-simplex σ in |P(d)| ⊂ |P̃(d, i)| of the form c_{0} ≤ ⋯ ≤ c_{n} is trivial if λ_{i} ≤ c_{n} and it is equal to G_{i} if λ_{i} ≰ c_{n}. Therefore, since G_{i} ≠ 1 (because d ≥ 2),

A^{≤(e)}([c_{0} ≤ ⋯ ≤ c_{n}]_{G_{i}}) = \{ \begin{matrix}L & if λ_{i} ≤ c_{n} \\ 0 & if λ_{i} ≰ c_{n} \end{matrix} = N̄_{i}([c_{0} ≤ ⋯ ≤ c_{n}]_{G})

because Iso_{G}(c_{n}) ≤ R_{λ_{i}} if and only if λ_{i} ≤ c_{n} as we have seen above. These isomorphisms clearly commute with the face and degeneracy maps in |P(d)| so we deduce from Proposition 2.6 that

C_{Γ}^{∗}(|P̃(d)_{r}|; N̄_{i} ≀ Σ_{r}) ≅ C_{G_{i}≀Σ_{r}}^{∗}(|P̃(d, i)_{r}|; A^{≤(e)} ≀ Σ_{r}),

where P̃(d, i)_{r} is the subposet of P̃(d, i)^{r} generated by P(d)_{r}. Set X = P̃(d, i) and note that X has a minimum λ^{min} and that X_{G_{i}} = P(d) contains a maximum λ^{max}. Now, Y = X^{G_{i}} is isomorphic to the poset {c ∈ P(d): λ_{i} ≰ c} and it is therefore left closed in X and it contains a maximum

λ̂_{i} = (\underbrace{1, 1, …, 1}_{i−1 times}, 2, \underbrace{1, …, 1}_{d−i−1 times})

which is different from λ^{max} ∈ P(d). We can now apply Lemma 2.11 to X, Y and Z = {λ^{min}} to deduce that H_{G_{i}≀Σ_{r}}^{∗≥1}(|P̃(d, i)_{r}|; A^{≤(e)} ≀ Σ_{r}) = 0. □

4. Radical p-subgroups of the symmetric group

4.1. Goals. Fix a prine p. A p-subgroup of a finite group G is p-radical if the largest normal p-subgroup of N_{G}(P)/P denoted O_{p}(N_{G}(P)/P) is trivial. It is pcentric if Z(P) is a Sylow p-subgroup of C_{G}(P). Let D(n) denote the collection of the p-centric p-radical subgroups of Σ_{n}. The goal of this section is to prove the next result. Throughout we fix an algebraically closed filed k of characteristic p.

Proposition 4.1. Let H_{Σ_{n}} be the coefficient functor H ↦ Hom(H, k^{×}), see Example 2.2. Then H_{Σ_{n}}^{∗}(|D(n)|; H_{Σ_{n}}) = 0 for all ∗ ≥ 2.

Let (ℤ/p)^{c} act on itself by left translations and let V_{c} denote its image in Σ_{p^{c}} via this action.

Definition 4.2 (Alperin-Fong, [1]). A p-subgroup P ≤ Σ_{n} is called basic if it is conjugate to

V_{c_{1},...,c_{t}} =^{def} V_{c_{1}} ≀ V_{c_{2}} ≀ ⋯ ≀ V_{c_{t}} (≤ Σ_{p^{c_{1}+···+c_{t}}} .)

Its degree is deg(P) = c_{1} + ⋯ + c_{t}.

Alperin and Fong proved in [1] that P is a p-radical p-subgroup of Σ_{n} if and only if it is a products of basic subgroups. The next result is fundamental to our analysis. Since it is interesting in its own right, we decided to state it at this point.

Lemma 4.3. Let V = V_{c_{1},...,c_{t}} be a basic subgroup of degree d in Σ_{p^{d}}. If U is a basic subgroups of V of degree d which is conjugate to some V_{m_{1},...,m_{s}} then

(a) There are integers q_{1}, …, q_{s−1} such that

(4.3.1) \underbrace{c_{1} + ⋯ + c_{q_{1}}}_{m_{1}}, \underbrace{c_{q_{1}+1} + ⋯ + c_{q_{2}}}_{m_{2}}, \underbrace{c_{q_{2}+1} + ⋯ + c_{q_{3}}}_{m_{3}}, …, \underbrace{c_{q_{s−1}+1} ⋯ + c_{t}}_{m_{s}}

Thus, in the notation of 3.3, the partition (c_{1}, …, c_{t}) refines (m_{1}, …, m_{s}).

(b) If W ≤ V is conjugate to U in the ambient group Σ_{p^{d}}, then U is conjugate to W in V.

4.2. Operations on radical p−subgroups of Σ_{n}.

Definition 4.4. For K ≤ Σ_{n}, let Δ_{r}(K) be the set of the permutations g ∈ K whose support contains at most r elements. Let δ_{r}(K) be the subgroup generated by Δ_{r}(K).

Lemma 4.5. For any K ≤ Σ_{n} we have δ_{r}(K) ≤ K and moreover δ_{r}(K)◁N_{Σ_{n}}(K). In addition δ_{r}(K_{1} × ⋯ × K_{s}) = δ_{r}(K_{1}) × ⋯ × δ_{r}(K_{s}).

Proof. Immediate consequence of the definitions. □

Lemma 4.6. Consider a basic subgroup P = V_{c_{1},...,c_{t}} and some r ≥ 1. Let q be the largest integer ≤ t such that s := p^{c_{1}+···+c_{q}} ≤ r and set e = p^{c_{q+1}+···+c_{t}}. Then δ_{r}(P) = (V_{c_{1},...,c_{q}})^{e} ◁ P. Moreover, δ_{r}(P) contains every basic subgroup of degree ≤ log_{p} r which is contained in P.

Proof. If q = t then the result is a triviality so we assume that q < t. From the choice of q and Lemma 4.5 it is clear that (V_{c_{1},...,c_{q}})^{e} is a subgroup of δ_{r}(V).

Consider now an element g ∈ P − (V_{c_{1},...,c_{q}})^{e}. By the structure of the wreath product and the fact that V_{c} acts freely on p^{c} elements, the support of every nontrivial element of V_{c} ≀K contains at least p^{c} elements. Now, P = V_{c_{1},...,c_{q}} ≀V_{c_{q+1},...,c_{t}} and the image of g under the quotient P → V_{c_{q+1},...,c_{t}} is a non-trivial element ḡ whose support, therefore, contains at least p^{c_{q+1}} elements. Thus, g permutes nontrivially at least p^{c_{q+1}} of the orbits of (V_{c_{1},...,c_{q}})^{e}, so its support contains at least s · p^{c_{q+1}} = p^{c_{1}+···+c_{q+1}} > r elements, thus g ∉ Δ_{r}(P). It follows that δ_{r}(P) = (V_{c_{1},...,c_{q}})^{e}. Finally, if U ≤ V has degree ≤ log_{p} r then U = δ_{r}(U) ≤ δ_{r}(V). □

Proof of Lemma 4.3. Set n = p^{d}. We use induction on s.

Base of induction s = 1. In this case (a) holds trivially because deg(V) = deg(U). We will prove (b) by induction on t. If t = 1 then U = V = W. Assume now that t > 1. Set e = p^{d−c_{1}} and note that V ≤ Σ_{p^{c_{1}}} ≀ Σ_{e}. Set U′ = U ∩ (Σ_{p^{c_{1}}} )^{e}. It acts freely on the underlying set because U, which is conjugate to V_{d}, acts freely. By construction the size of the orbits of U′ is at most p^{c_{1}} and since it acts freely |U′| ≤ p^{c_{1}}. Now, C_{Σ_{n}}(U) is the group of automorphisms of the U-set {1, …, n} which must be isomorphic to U because U acts freely and transitively. Since U is abelian we now deduce that U = C_{Σ_{n}}(U) ≥ C_{Σ_{n}}(V) ≥ Z(V). Now, V_{c_{2},...,c_{t}} centralises precisely the diagonal copy of V_{c_{1}} in (V_{c_{1}})^{e}, whence Z(V) ≅ V_{c_{1}}. Since Z(V) ≤ (Σ_{p^{c_{1}}} )^{e} and |U′| ≤ p^{c_{1}}, we deduce that

U′ = U ∩ (Σ_{p^{c_{1}}} )^{e} = Z(V)

By the same argument W′ = Z(V) where W′ = W ∩ (Σ_{p^{c_{1}}} )^{e}.

Let V̄ , Ū and W̄ be the images of V, U and W under the projection Σ_{p^{c_{1}}} ≀Σ_{e} → Σ_{e}. Clearly V̄ = V_{c_{2},...,c_{t}}. Now, Ū acts transitively on {1, …, e} because U is transitive. Also |Ū| = |U/U′| = p^{d−c_{1}} = e and therefore Ū acts freely. It is also clearly an elementary abelian p-group and therefore Ū is conjugate to V_{d−c_{1}} in Σ_{e}. By the same argument W̄ is conjugate to V_{d−c_{1}}. By induction hypothesis on t there exists g ∈ V̄ which conjugates Ū to W̄. We may regard g as an element of V since V = V_{c_{1}} ≀ V̄ and since g centralises U′ = W′ = Z(V), we see that g conjugates U to W. This completes the induction step for t and the proof when s = 1.

Induction step s > 1. Set r = m_{1} + ⋯ + m_{s−1} and consider V′ = δ_{p^{r}}(V) and U′ = δ_{p^{r}}(U). By Lemma 4.6, U′ = U′_{(1)} × ⋯ × U′_{(p^{ms})} where every U′_{(i)} is conjugate to V_{m_{1},...,m_{s−1}}. Similarly V′ = V′_{(1)} × ⋯ × V′_{(k)} where V′_{(i)} are conjugate to V_{c_{1},...,c_{q}} where q ≤ t is the largest integer such that c_{1} + ⋯ + c_{q} ≤ r and k = p^{c_{q+1}+···+c_{t}}. Thus, the orbits of V′ have length p^{c_{1}+···+c_{q}} ≤ p^{r} and since it contains U′ whose orbits have length p^{r}, we see that c_{1} + ⋯ + c_{q} = r and therefore c_{q+1} + ⋯ + c_{t} = m_{s}. Moroever, by looking at the orbits of the U′_{(i)}’s and the V′_{(i)}’s we deduce that U′_{(i)} ≤ V′_{(i)}. The induction hypothesis on s implies that c_{1}, …, c_{q} and m_{1}, …, m_{s−1} satisfy (4.3.1) with d replaced by r. This completes the induction step for (a).

Set W′ = δ_{p^{r}}(W) and observe that W′ = W′_{(1)}×⋯×W′_{(p^{ms})} and that U′_{(i)}, W′_{(i)} ≤ V′_{(i)} with U′_{(i)}, W′_{(i)} conjugate to V_{m_{1},...,m_{s−1}}. By induction hypothesis on s, the groups U′_{(i)} and W′_{(i)} are conjugate in V′_{(i)} and we may therefore assume that they are all equal to one particular copy of V_{m_{1},...,m_{s−1}} in V_{c_{1},...,c_{q}} ≤ Σ_{p^{r}}. Thus,

U ∩ (Σ_{p^{r}})^{p^{ms}} = U′ = W′ = W ∩ (Σ_{p^{r}})^{p^{ms}} .

Let V̄, Ū and W̄ be the images of V, U and W under the quotient Σ_{p^{r}} ≀ Σ_{p^{ms}} → Σ_{p^{ms}}. Then V̄ = V_{c_{q+1},...,c_{t}} and Ū and W̄ are elementary abelian p-groups of rank m_{s}. They act transitively on the set {1, …, p^{m_{s}}} because U and W are transitive and therefore they must act freely. Thus Ū and W̄ are conjugate to V_{m_{s}} in Σ_{p^{ms}} and by induction hypthesis they are conjugate via some g ∈ V̄ which we may think of as a subgroup of V = V_{c_{1},...,c_{q}} ≀V̄. Now, g normalises U′ = W′ = (V_{m_{1},...,m_{s−1}})^{p^{ms}} since it acts by permuting the factors of (Σ_{p^{r}})^{p^{ms}}. Thus, g conjugates U to W and this completes the induction step. □

4.3. The poset of the p-centric p-radical subgroups.

Definition 4.7. Let B(d) denote the poset of the basic subgroups of degree d in Σ_{p^{d}}. Let B̄(d) denote the quotient poset by the action of Σ_{p^{d}}.

Proposition 4.8. Set n = p^{d} and G = GL_{d}(p). There is a function V : P(d) → B(d) with the following properties.

(i) V is order preserving and if c = (c_{1}, …, c_{t}) then V (c) is a basic subgroup conjugate to V_{c_{1},...,c_{t}}. Thus, P(d) ⊆ B(d).

(ii) The composite |P(d)| → |B(d)| → |B(d)|_{Σ_{n}} is an isomorphism of simplicial

sets. (iii) There are isomorphisms λ_{σ}: H_{Σ_{n}}([σ]_{Σ_{n}}) ⟶^{≅} H_{G}([σ]_{G}) for every simplex

σ ∈ |P(d)| where [σ]_{Σ_{n}} and [σ]_{G} are the orbits of σ in |B(d)| and in |P̃(d)|, see Definitions (2.2.2) and 3.3(iii). These isomorphisms commute with the face and degeneracy maps of |P(d)|.

Proof. Let Sym(X) denote the symmetric group of a set X. If Y ⊆ X then Sym(Y) is a subgroup of Sym(X) which fixes X − Y.

We view the elements of P(d) as partitions into intervals of the standard basis of the vector space U = 𝔽_{p}^{d}. We will identify Σ_{n} with Sym(U).

Fix some c = (c_{1}, …, c_{t}) in P(d) and let W_{j} denote the subspace of U generated by the basis elements in the jth interval; clearly dim W_{j} = c_{j}. For every 1 ≤ j ≤ t set F_{j} = ∑_{i=1}^{j} W_{i} and let W_{j} act on F_{j} via left translation. Now, F_{j} is a subset of U and let W̃_{j} denote the image of the composite W_{j} → Sym(F_{j}) ≤ Sym(U). It is clear that if w̃ is the image of w ∈ W_{j} in W̃_{j} and if u ∈ U then w̃(u) = w + u if u ∈ F_{j} and w̃(u) = u if u ∉ F_{j}.

Since F_{j} = W_{j} ⊕ F_{j−1}, we see that W̃_{j} acts on F_{j} ≅ ∐_{W_{j}} F_{j−1} by permuting the components. Therefore, if K is any subgroup of Sym(F_{j−1}) then W̃_{j} and K generate K ≀ W_{j}. We define

V (c) =^{def} the subgroup of Sym(U) generated by W̃_{1}, …, W̃_{t}.

It is immediate from Definition 4.2 that V (c) is the vasic subgroup V_{c_{1},...,c_{t}}. Also, since W_{j−1} acts diagonally on F_{j} = ∐_{W_{j}} F_{j−1} via its action on F_{j−1}, the image of

W_{j−1} ⊕ W_{j} → Sym(F_{j}) ≤ Sym(U)

is generated by W̃_{j} and the diagonal inclusion of W_{j−1} in W_{j−1} ≀ W_{j} ≅ ⟨W̃_{j−1}, W̃_{j}⟩. In particualr its image is contained in ⟨W̃_{j−1}, W̃_{j}⟩. It now follows that if c′ ≤ c then V (c′) ≤ V (c) because the partition c′ is obtained from c by successively joining adjacent intervals. This completes the proof of point (i). Point (ii) follows immediately from Lemma 4.3 and we now prove point (iii).

For c ∈ P(d) of the form (c_{1}, …, c_{t}) let GL_{c}(p) denote ∏_{j=1}^{t} GL_{c_{j}}(p) as a subgroup of GL_{d}(p) = GL(U) which is itself a subgroup of Sym(U). If g ∈ GL_{c_{j}}(p) and w ∈ W_{i} and u ∈ U then g ∘ w̃ ∘ g^{−1}(u) = g(w) + u if u ∈ F_{i} and g ∘ w̃ ∘ g^{−1}(u) = u if u ∉ F_{i} because g ∈ R_{c} so g(F_{i}) = F_{i}; see Definition 3.3(ii). Thus, g ∘w̃∘g^{−1} = { \atop g(w)} and we deduce that g normalised W̃_{j} and centralises W̃_{i} if i ≠ j. Therefore GL_{c}(p) normalises V (c) which is generated by the W̃_{j}’s.

Next, we argue that no non-trivial element of V (c) acts on U as a homomorphism. We prove this by induction on t for all U. If t = 1 then V (c) = Ũ which acts on U by left translation so if g ∈ Ũ acts as a homomorphism then g = 1. Assume that t > 1 and observe that since W̃_{t} acts freely, any element of V_{c_{1},...,c_{t}} − (V_{c_{1},...,c_{t−1}})^{p^{c_{t}}} acts on U without any fixed point and it therefore cannot act as a homomorphism. Thus, if g ∈ V_{c_{1},...,c_{t}} acts as a homomorphism, then it must act on U leaving the cosets of F_{t−1} invariant. Therefore, g = ∏_{w∈W_{t}} w̃hw̃^{−1} where h ∈ V_{c_{1},...,c_{t−1}} acts as a homomorphism on F_{t−1} = 𝔽_{p}^{d−c_{t}}. By induction hypothesis h = 1, hence g = 1 and this completes the induction step.

We deduce that GL_{c}(p) ∩ V (c′) = 1 for any c, c′ ∈ P(d). Also, N_{Σ_{n}}(V (c)) = V (c) ⋊ GL_{c}(p) by [1, Section 2]. More generally, if σ is an s-simplex in |P(d)| of the form c_{0} ≤ ⋯ ≤ c_{s}, then

V (c_{s})·N_{Σ_{n}}(V (c_{0}) ≤ ⋯ ≤ V (c_{s})) = V (c_{s})·(∩_{i=0}^{s} N_{Σ_{n}}(V (c_{i}))) = V (c_{s})⋊GL_{c_{s}}(p).

Since k is algebraically closed of characteristic p, by applying Hom(−, k^{×}) to the following sequence of inclusions of groups

Iso_{G}(σ) = R_{c_{s}} ≥ GL_{c_{s}}(p) ≤ V_{c_{s}} · ( ∩_{i=0}^{s} N_{Σ_{n}}(V (c_{i}))) ≥ Iso_{Σ_{n}}(σ)

we obtain isomorphisms of groups because all these groups are extenstions of GL_{c_{s}}(p) by a finite p-group. This gives the isomorphism H_{G}([σ]_{G}) ⟶^{≅} H_{Σ_{n}}([σ]_{Σ_{n}}) which is clearly compatible with face and degeneracy maps. □

Definition 4.9. Fix d ≥ 2, r ≥ 1 and set n = rp^{d}. Define the following Σ_{n}-posets

(a) B(d, r) is the poset of all the radical p-subgroups of Σ_{n} which are products of r basic subgroups of degree d.

(b) V(d, r) is the conjugacy class of the radical p-subgroups which are product of r copies of V_{d}.

(c) B^{0}(d, r) = B(d, r) − V(d, r).

Proposition 4.10. The following holds if d ≥ 2. Fix r ≥ 1 and set n = rp^{d}. Let H_{Σ_{n}} be the coefficient functor H ↦ Hom(H, k^{×}) defined in Example 2.2. Then

(i) The spaces |B(d, r)|_{Σ_{n}} and |B^{0}(d, r)|_{Σ_{n}} are contractible.

(ii) H_{Σ_{n}}^{∗}(|B(d, r)|, |B^{0}(d, r)|; H_{Σ_{n}}) = 0 for all ∗ ≥ 2.

Proof. Set K = Σ_{p^{d}} and Γ = K ≀ Σ_{r}. Let V(d) be the subposet of B(d) consisting of the conjugacy class of V_{d}. Denote B(d)_{r} = B(d)^{r} − V(d)^{r} as a Γ-poset.

The high transitivity of Σ_{n} clearly implies that any P ∈ B(d, r) is conjugate to some Q ∈ B(d)^{r} ⊆ B(d, r). It is also clear, by looking at the orbits of the basic subgroups, that if Q, Q′ ∈ B(d)^{r} are conjugate via some g ∈ Σ_{n} then g ∈ Γ. In particular N_{Σ_{n}}(Q) ≤ Γ. We deduce from Example 2.7(i) that

B(d, r) = B(d)^{r} ↑_{Γ}^{Σ_{n}},  and  B^{0}(d, r) = B(d)_{r} ↑_{Γ}^{Σ_{n}} .

Note that P(d)^{r} and P(d)_{r}, see Definition 3.3(i), have Σ_{r}-contractible nerves because d ≥ 2 so they have a maximum. We deduce from Proposition 4.8(ii) that

|B(d, r)|_{Σ_{n}} = |B(d)^{r}|_{Γ} = |P(d)^{r}|_{Σ_{r}} ≃ pt .

Similarly |B^{0}(d, r)|_{Σ_{n}} = |B(d)_{r}|_{Γ} = |P(d)_{r}|_{Σ_{r}} is contractible, whence point (i). Example 2.7(ii) implies

H_{Σ_{n}}^{∗}(|B(d, r)|; H_{Σ_{n}}) = H_{Γ}^{∗}(|B(d)^{r}|; H_{Γ}) and
H_{Σ_{n}}^{∗}(|B^{0}(d, r)|; H_{Σ_{n}}) = H_{Γ}^{∗}(|B(d)_{r}|; H_{Γ}).

In light of Proposition 2.8 and the Σ_{r}-contractibility of |P(d)^{r}| and |P(d)_{r}|, we see that in order to prove point (ii) it remains to prove that

H_{Γ}^{∗≥1}(|B(d)^{r}|; H_{K} ≀ Σ_{r}) = H_{Γ}^{∗≥1}(|B(d)_{r}|; H_{K} ≀ Σ_{r}) = 0.

This follows from Propositions 3.5 and Proposition 4.8. □

The components of a p-radical p-subgroup P of Σ_{n} are the basic subgroups whose product is P; see [1] and Definition 4.2.

Definition 4.11. (a) Let D(n) denote the poset of the p-centreic p-radical

p-subgroups of Σ_{n}.

(b) The degree of P ∈ D(n) is the maximum degree of its components.Let D_{d}(n) be the subposet of P ∈ D(n) such that deg(P) ≤ d.

(c) For r ≥ 0 let E_{d,r}(n) be the subposet of D_{d}(n) of those subgroups P such that at most r of its components are conjugate to V_{d}.

Remark 4.12. (a) Every basic subgroup P of degree d in Σ_{p^{d}} conatins a conjugate of V_{d} which is self centralising because it is abelian and it acts freely and transitively, hence C_{Σ_{p^{d}}} (P) ≤ P. Therefore, if the components of P ∈ D(n) have degrees d_{1}, …, d_{r} and if m = n − p^{d_{1}+···+d_{r}} then C_{Σ_{n}}(P) = Z(P) × Σ_{m}. We deduce that P is p-centric if and only if m < p.

(b) If rp^{d} ≥ n then E_{d,r}(n) = D_{d}(n).

Proposition 4.13. For every d ≥ 2 the inclusion i: D_{d−1}(n) ⊆ E_{d,0}(n) induces a Σ_{n}-equivalence on nerves.

Proof. If Q = V_{c_{1},...,c_{t}} of degree d is different from V_{d} then t > 1 then δ_{p^{d−1}}(Q) = (V_{c_{1},...,c_{t−1}})^{p^{c_{t}}} is a non-trivial radical p-subgroup of degree ≤ d − 1 by Lemma 4.6.

By Proposition 4.6 and Remark 4.12(a), δ_{p^{d−1}} gives an order preserving Σ_{n}-

equivariant map E_{d,0}(n) ⟶^{δ_{p^{d−1}}} D_{d−1}(n). Since δ_{p^{d−1}}(P) ≤ P, we obtain a natural transformation i∘δ_{p^{d−1}} → Id and also δ_{p^{d−1}}∘i = Id. Therefore |i| is a Σ_{n}-equivariant homotpy equivalence. □

Proof of Proposition 4.1. We will prove by induction on d that if n ≥ p then H_{Σ_{n}}^{∗≥2}(|D_{d}(n)|; H_{Σ_{n}}) = 0 and |D_{d}(n)|_{Σ_{n}} is acyclic. This will complete the proof because D_{d}(n) = D(n) if p^{d} > n.

The base of induction d = 1 follows from Remark 4.12(a) because D_{1}(n) is the conjugacy class of (V_{1})^{e} ≤ Σ_{n} where n − ep < p so H^{∗≥1}(|D_{1}(n)|; H_{Σ_{n}}) = 0 and |D_{1}(n)|_{Σ_{n}} = pt.

Fix d ≥ 2 and assume inductively that H_{Σ_{n}}^{∗≥2}(|D_{d−1}(n)|; H_{Σ_{n}}) = 0 and that |D_{d−1}(n)|_{Σ_{n}} is acyclic for all n. In light of Remark 4.12(b), to prove the inductiuon step for d, it suffices to prove that H_{Σ_{n}}^{∗≥2}(|E_{d,r}(n)|; H_{Σ_{n}}) = 0 and that |E_{d,r}(n)|_{Σ_{n}} is acyclic for all r. We do this by induction on r ≥ 0.

The base of induction r = 0 is immediate from Lemma 4.13 and the induction hypothesis on d. We therefore fix some r ≥ 1. If rp^{d} > n then E_{d,r}(n) = E_{d,r−1}(n) and we can use the induction hypothesis directly. If rp^{d} ≤ n then an s-simplex σ of the form P_{0} ≤ ⋯ ≤ P_{s} in the realtive space (|E_{d,r}(n)|, ||E_{d,r−1}(n)|) has the following properties. First, each P_{i} has at most r components which are conjugate to V_{d}. Second, there exists some i such that P_{i} ∉ E_{d,r−1}(n), thus, the nunber of component of P_{i} that are conjugate to V_{d} is exactly r. Set n′ = rp^{d} and m = n − rp^{d}. By Lemma 4.6, V_{d} contains no proper subgroup which is p-radical so P_{0} has exactly r factors conjugate to V_{d}. In particular every P_{i} has at least r components of degree d. Thus, up to conjugation in Σ_{n}, we may assume that σ belongs to the relative Σ_{n′} × Σ_{m}-space (|B(d, r) × E_{d,0}(m)|, |B^{0}(d, r) × E_{d,0}(m)|) and, by looking at P_{0}, if σ′ in this relative space is conjugate to σ then they are conjugate in Σ_{n′} × Σ_{m}. It follows from Example 2.7(i) that

(|E_{d,r}(n)|, |E_{d,r−1}(n)|) ≅ (|B(d, r) × E_{d,0}(m)|, |B^{0}(d, r) × E_{d,0}(m)|) ↑_{Σ_{n′} ×Σ_{m}}^{Σ_{n}} .

By Proposition 4.10(i), H_{∗}(|E_{d,r}(n)|_{Σ_{n}}, |E_{d,r−1}(n)|_{Σ_{n}}) = 0. Also, from Example 2.7(ii), (2.3.2), (2.3.1), Lemma 2.1, Propositon 4.13, the induction hypothesis on d − 1 and Proposition 4.10(ii) we see that for all ∗ ≥ 2,

H_{Σ_{n}}^{∗}(|E_{d,r}(n)|, |E_{d,r−1}(n)|, H_{Σ_{n}}) ≅

H_{Σ_{n′} ×Σ_{m}}^{∗}(|B(d, r) × E_{d,0}(m)|, |B^{0}(d, r) × E_{d,0}(m)|; H_{Σ_{n′} ×Σ_{m}}) =
H_{Σ_{n′} ×Σ_{m}}^{∗}(|B(d, r) × E_{d,0}(m)|, |B^{0}(d, r) × E_{d,0}(m)|; H_{Σ_{n′}} ⊕ H_{Σ_{m}}) ≅

H_{Σ_{m}}^{∗}(|E_{d,0}(m)|, |E_{d,0}(m)|; H_{Σ_{m}}) ⊕ H_{Σ_{n′}}^{∗} (|B(d, r)|, |B^{0}(d, r)|; H_{Σ_{n′}}) = 0.

The induction step for r follows from the induction hypothesis on r − 1 and the long exact sequence in homology and in Bredon cohomology. □

5. The gluing problem

Proof of Theorem 1.3. Throughout [P_{•}] denotes the equivalence class of an s-simplex P_{0} < ⋯ < P_{s} in F^{c}. The kernel of the epimorphism Aut_{L}(P_{•}) → Aut_{F}(P_{•}) is Z(P_{s}) and since H̃^{∗}(Z(P_{s}); k^{×}) = 0, the Leray-Serre spectral sequence implies that A_{F}^{i}([P_{•}]) = A_{L}^{i}([P_{•}]) for all i. In particular \underleftarrow{lim}^{∗} A_{F}^{i} ≅ \underleftarrow{lim}^{∗} A_{L}^{i}. Since k^{×} is a divisible group with no p-torsion it follows from [4, Lemma 1.3] by applying it to every prime q ≠ p at a time, that H^{∗}(L^{C}; k^{×}) ≅ H^{∗}(F^{C}; k^{×}). Here we used H^{∗}(C; k^{×}) ≅ H^{∗}(C^{op}; k^{×}) for every category C. This proves the equivalence of (a) and (b) and of (c) and (d).

Given the collection C we obtain the following diagram in which the square commutes and the bottom row is exact by [12, Theorem 1.1].

\xymatrix{H^{2}(L; k^{×}) \ar[r] \ar[d]_{res} & \underleftarrow{lim}_{[S(F^{c})]} A_{L}^{2} \ar[d]^{res} \\ H^{2}(L^{C}; k^{×}) \ar[r] & \underleftarrow{lim}_{[S(F^{C})]} A_{L}^{2} \ar[r] & H^{2}([S(F^{C})]; A_{L}^{1})}

We will prove below that the first vertical arrow is an isomorphism and that the second is a monomorphism. The equivalence of (b) and (c) then follows by a simple diagram chasing. If the third group in the second row vanishes, then the first arrow is an epimorphism which implies that the gluing problem has a solution in L^{C}.

Since C contains the F-centric F-radical subgroup of S, the inclusion induces a homotopy equivalence |L^{C}| ≃ |L| by [3, Theorem 3.5]. It follows that H^{2}(L; k^{×}) ⟶^{res} H^{2}(L^{C}; k^{×}) is an isomorphism becuase H^{∗}(L; k^{×}) ≅ H^{∗}(|L|; k^{×}).

Consider an element (κ_{[P_{•}]}) in the kernel of

\underleftarrow{lim}_{[S(F^{c})]} A_{L}^{2} ⟶^{res} \underleftarrow{lim}_{[S(F^{C})]} A_{L}^{2}.

Thus, (κ_{[P_{•}]})_{[P_{•}]∈[S(F^{c})]} is a compatible family of classes in H^{2}(Aut_{L}(P_{•}); k^{×}) such that κ_{[P_{•}]} = 0 if P_{0}, …, P_{s} ∈ C. Our goal is to prove that κ_{[P_{•}]} = 0 for all P_{•}. The morphism [P_{0}] → [P_{•}] in [S(F^{c})] carries κ_{[P_{0}]} to κ_{[P_{•}]} since the family is compatible so it suffices to prove that κ_{[P]} = 0 for all P ∈ L.

Let E denote the collection of all P ∈ L such that κ_{[P]} ≠ 0. Clearly E must be disjoint from C. We want to prove that E is empty. If this is not the case, choose P ∈ E which has maximal order. We may assume that P is fully normalised. Note that P ∉ C so P is not F-radical, hence Q = O_{p}(Aut_{L}(P)) ⪈ P. Since P is maximal, Q ∉ E whence κ_{[Q]} = 0. We have inclusion of groups, see [9, Prop. 1.5],

Aut_{L}(P) ≥ Aut_{L}(P < Q) ≤ Aut_{L}(Q).

Since Q is the preimage of O_{p}(Aut_{F}(P)) in S, the map Aut_{F}(P < Q) → Aut_{F}(P) is surjective by [6, Lemma 2.4]. It follows that Aut_{L}(P < Q) → Aut_{L}(P) is surjective, see [5, Lemma 1.10(a)], and therefore it is an isomorphism. Thus, the morphisms [P] ⟶^{φ} [P < Q] ⟵^{ψ} [Q] in [S(F^{c})] induce

A_{L}^{i}([P]) ⟶_{≅}^{φ_{∗}} A_{L}^{i}([P < Q]) ⟵^{ψ_{∗}} A_{L}^{i}([Q]).

Compatibility implies that κ_{[P <Q]} = ψ_{∗}(κ_{[Q]}) = 0, whence κ_{[P]} = 0. This is a contradiction to the choice of P. □

Proof of Theorem 1.1. Write n = rp + m where m < p. Then Σ_{n} and Σ_{rp} have the same fusion system and we may therefore assume that n = rp.

Fix a Sylow p-subgroup S ≤ Σ_{n} and consider the poset D(n), see Definition 4.11. Let F be the fusion system of Σ_{n} on S. Let C denote the collection of subgroups P ∈ D(n) such that P ≤ S. Note that C contains all the F-centric F-radical subgroups^{1}. It is also clear that C is closed to F-conjugacy. From Remark 4.12(a) we see that C_{Σ_{n}}(P) ≤ P for all P ∈ C because p|n. Therefore, Aut_{L}(P) = N_{Σ_{n}}(P), see [5, Remarks after Definition 1.8]. Thus, A_{L}^{1}([P_{•}]) = H_{Σ_{n}}([P_{•}]_{Σ_{n}}) where H_{Σ_{n}} is the coefficient functor H ↦ Hom(H, k^{×}), see Exaple 2.2, and [P_{•}]_{Σ_{n}} is the Σ_{n}-orbit of an s-simplex in L^{C}.

By Theorem 1.3, it suffices to prove that H^{2}([S(F^{C})]; A_{L}^{1}) = 0. By [10, Prop. 3.2], the groups H^{∗}([S(F^{C})]; A_{L}^{1}) are isomorphic to the homology groups of the cochain complex E^{∗} where

E^{n} = ∏_{[P_{0}<···<P_{n}]∈[S(F^{C})]} A_{L}^{1}([P_{0} < ⋯ < P_{n}]),

and coface maps are the usual alternating sums. Note that [S(F^{C})] is the simplicial complex of the non-degenerate simplices of |D(n)|_{Σ_{n}}, hence E^{∗} is isomorphic to the normalised cochain complex NC_{Σ_{n}}^{∗}(|D(n)|; H_{Σ_{n}}). Finally, H^{∗}([S(F^{C})]; A_{L}^{1}) ≅ H_{Σ_{n}}^{∗}(|D(n)|; H_{Σ_{n}}) vanishes for all ∗ ≥ 2 by Proposition 4.1. □

1If Q/P = O_{p}(N_{G}(P)/P) ≠ 1 then QC′_{G}(P)/P C′_{G}(P) ≤ O_{p}(N_{G}(P)/P C′_{G}(P).

Appendix A. The alternating groups

In this section we will prove Theorem 1.2. We will be brief since the proof builds on the proof of Theorem 1.1 so the reader needs less details to be filled in.

Throughout we fix a prime p > 2. If K is a subgroup of Σ_{n} we let K^{ev} denote the subgroup K ∩ A_{n} of the even permutations in K. We let K^{odd} denote the set of odd permutations in K. The commutator subgroup of any group G is denoted Γ^{2}(G). The commutator of x, y ∈ G is denoted [x, y].

We let O^{p′}(G) denote the subgroup of a finite group G generated by all the elements of p-power order. It is clearly a characteristic subgroup of G which is contained in the kernel of any homomorphism G → k^{×} where k is an algebraically closed field of characteristic p.

Lemma A.1. Fix p > 2, a subgroup K ≤ Σ_{n} and assume that one the following holds.

(a) K contains an odd permutation and O^{p′}(K) ⊇ Γ^{2}(K).

(b) K = K_{1} × K_{2} where both K_{1} and K_{2} contain odd permutations.

(c) K = H ≀ Σ_{r} where r ≥ 3 and H ≤ Σ_{m} contains odd permutations. Then the inclusion K^{ev} → K induces a short exact sequence

0 → C_{2} → Hom(K, k^{×}) → Hom(K^{ev}, k^{×}) → 0.

The kernel C_{2} is generated by the signature map K ≤ Σ_{n} ⟶^{sign} C_{2} ≤ k^{×}.

Proof. (a) It is clear that O^{p′}(K^{ev}) = O^{p′}(K) because p > 2 = |K: K^{ev}|. We obtain a short exact seuence of abelian groups 0 → \frac{K^{ev}}{O^{p′} (K)} → \frac{K}{O^{p′} (K)} → C_{2} → 0. The results follows by applying the exact functor Hom(−, k^{×}).

(b) Clearly Γ^{2}(K) = Γ^{2}(K_{1}) × Γ^{2}(K_{2}). Fix some τ ∈ K_{2}^{odd} and observe that [x, y] = [xτ^{i}, yτ^{j}] for all x, y ∈ K_{1}. Choose i, j so that xτ^{i}, yτ^{j} ∈ K^{ev} to deduce that Γ^{2}(K_{1}) ≤ Γ^{2}(K^{ev}) and smilarly Γ^{2}(K_{2}) ≤ Γ^{2}(K^{ev}). We deduce that Γ^{2}(K^{ev}) = Γ^{2}(K) and we obtain a short exact sequence 0 → (K^{ev})_{ab} → K_{ab} → C_{2} → 0 to which we apply Hom(−, k^{×}).

(c) For x ∈ H let x_{i} denote the image of x in the ith factor of H^{r} ◁ K. We will show that Γ^{2}(K^{ev}) = Γ^{2}(K) and deduce the result as in part (b).

The same argument we used in (b) shows that Γ^{2}(H)^{r} is contained in Γ^{2}(K^{ev}). If x ∈ H^{ev} then one easily checks that the commutator of x_{1} and (1ij) ∈ A_{r} where 1 < i < j is x_{1}^{−1}x_{i}. If x ∈ H^{odd} then the commutator of x_{1} · (1i) and x_{j} · (1i) is also x_{1}^{−1}x_{i}. We deduce that Γ^{2}(K^{ev}) contains all the elements of the form x_{i}x_{j}^{−1} for all x ∈ H. In addition Γ^{2}(K^{ev}) contains all the 3-cycles (1ij) ∈ A_{r} because if τ ∈ H^{odd} then the commutator of (1i) · τ_{i} and (1j) · τ_{j} is (1ij). Therefore Γ^{2}(K^{ev}) also contains A_{r}. The quotient of K by the subgroup generated by A_{r}, by Γ^{2}(H) and the elements x_{i}x_{j}^{−1} where x ∈ H, is easily seen to be abelian; in fact the quotient is H_{ab} × C_{2}. In particular Γ^{2}(K^{ev}) ≥ Γ^{2}(K) and therefore equality holds. □

Lemma A.2. Consider H ≤ Σ_{n} that contains odd permutations and set K = H≀Σ_{2} as a subgroup of Σ_{2n}. Then the inclusion i: K^{ev} → K induces a short exact sequence of abelian groups

(A.2.1)  0 → C_{2} → Hom(K, k^{×}) ⟶^{i^{∗}} Hom(K^{ev}, k^{×}) → C_{2} → 0.

Set K_{0} = H_{0} ≀Σ_{2} and K_{1} = H_{1} ≀Σ_{2} where H_{0} ≤ H_{1} ≤ Σ_{n} contain odd permutations. Then the inclusion j: K_{0} → K_{1} induces the commutative diagram below whose rows are exact

(A.2.2) [figure: 0; C_{2}; j^{∗}; Hom(K_{1}, k^{×}); (i_{1})^{∗}; j^{∗}; Hom(K_{1}^{ev}, k^{×}); j^{∗}; C_{2}; j^{∗}; 0; 0; C_{2}; Hom(K_{0}, k^{×}); (i_{0})^{∗}; Hom(K_{0}^{ev}, k^{×}); C_{2}; 0]

Proof. Since K^{ev} → K → C_{2} → 0 is exact there is a short exact sequence

0 → Ker(i_{ab}) → (K^{ev})_{ab} ⟶^{i_{ab}} K_{ab} → C_{2} → 0.

Since Hom(−, k^{×}) is exact the exactness of (A.2.1) follows once we prove that |(K^{ev})_{ab}| = |K_{ab}|. It is easy to see that K_{ab} ≅ H_{ab}×C_{2}, see e.g. the proof of Lemma A.1(c). The argument in the proof of Lemma A.1(b) shows that Γ^{2}(H) ≤ Γ^{2}(K^{ev}).

As in the proof of Lemma A.1(c) let x_{i} denote the image of x ∈ H in the ith factor of H^{2} ◁ H ≀ Σ_{2}. Let δ(H) denote the subgroup of H^{2} consisting of the elements x_{1}x_{2}^{−1}. If y ∈ H^{ev} and x ∈ H^{odd} then the commutator of y_{1} and (12) · x_{2} is y_{1}y_{2}^{−1}. This shows that Γ^{2}(K^{ev}) contains the group L generated by Γ^{2}(H)^{2} and δ(H^{ev}). It is clear that L ◁ K. It is easy to check that K^{ev}/L is abelian. To do this, observe that the elements of K^{ev} have the form (12)x_{1}y_{2} or (12)y_{1}x_{2} where x ∈ H^{odd} and y ∈ H^{ev}, or the form u_{1}v_{2} where u, v ∈ H^{odd} or the form y_{1}z_{2} where y, z ∈ H^{ev}. One then check that these elements commute modulo L. We now deduce that L = Γ^{2}(K^{ev}).

Clearly (H^{ev})^{2} ◁ K and K/(H^{ev})^{2} ≅ C_{2} ≀ C_{2}. Also K^{ev}/(H^{ev})^{2} ≅ ℤ/4 and (H^{ev})^{2}/L ≅ H^{ev}/Γ^{2}(H). Therefore

|(K^{ev})_{ab}| = \frac{|K^{ev}|}{|L|} = \frac{|K^{ev}|}{|H^{ev}|^{2}} · \frac{|H^{ev}|^{2}}{|L|} = 4 · \frac{|H^{ev}|}{|Γ^{2}(H)|} = 2 · \frac{|H|}{|Γ^{2}(H)|} = 2|H_{ab}| = |K_{ab}|.

This completes the proof of the exactness of (A.2.1).

Now consider subgroups H_{0} ≤ H_{1} of Σ_{n} both contain odd permutations. Any odd permutation in K_{0} maps to a generator of K_{0}/K_{0}^{ev} ≅ C_{2} and of K_{1}/K_{1}^{ev} ≅ C_{2}. Therefore Coker((i_{1})_{ab}) → Coker((i_{0})_{ab}) is an isomorphism and this shows that the left arrow j^{∗} in (A.2.2) is an isomorphism.

Recall that L_{1} = Γ^{2}(K_{1}^{ev}) is generated by Γ^{2}(H_{1})^{2} and δ(H_{1}^{ev}) and that Γ^{2}(K_{1}) is generated by Γ^{2}(H_{1})^{2} and δ(H_{1}). We have seen above that |(K_{1}^{ev})_{ab}| = |(K_{1})_{ab}|, whence there exists x ∈ H_{1}^{odd} such that x_{1}x_{2}^{−1} ∉ L_{1}. Since x_{1}x_{2}^{−1} = [x_{1}, (12)], the cosets x_{1}x_{2}^{−1}L_{1} and x_{1}x_{2}^{−1}L_{0} generate the kernels of (i_{1})_{ab} : (K_{1}^{ev})_{ab} → (K_{1})_{ab} and (i_{0})_{ab} : (K_{0}^{ev})_{ab} → (K_{0})_{ab}, both of which are isomorphic to C_{2}. Thus, j_{ab} : (K_{0})_{ab} → (K_{1})_{ab} induces an isomorphism Ker((i_{0})_{ab}) → Ker((i_{1})_{ab}). Therefore the arrow j^{∗} on the right of (A.2.2) is an isomorphism. □

Lemma A.3. If P_{0} ≤ ⋯ ≤ P_{k} is a k-simplex in D(n), see Defintion 4.11, then N_{Σ_{n}}(P_{•}) contains an odd permutation.

Proof. We have seen in the proof of Proposition 4.8 that if P is a basic subgroup V_{d_{1},...,d_{r}} of degree d then N_{Σ_{n}}(P) = P ⋊ GL_{d}(p) where GL_{d}(p) = ∏_{i=1}^{r} GL_{d_{i}}(p) as a subgroup of Σ_{p^{d}} ≅ Sym(𝔽_{p}^{d}). Note that GL_{d_{1}}(p) contains a copy of 𝔽_{p}^{×} via the matrices diag(𝔽_{p}^{×}, 1, …, 1). A generator of 𝔽_{p}^{×} is a product of p^{d−d_{1}} cycles of length p − 1 in GL_{d}(p) and it is therefore an odd permutation.

Since P_{k} is a product of basic subgroups and P_{•} becomes a product of k-simplices, it suffices to assume that P_{k} is a basic subgroup V_{d_{1},...,d_{r}} of degree d which we denote V_{d}. We will now prove by induction on d that there exists some g ∈ P_{k} such that GL_{d_{1}}(p) normalises g · P_{•} · g^{−1}. This implies the result.

If d = 1 then P_{i} = V_{1} for all i so N_{Σ_{n}}(P_{•})/V_{1} = GL_{1}(p). Assume that d ≥ 2 and we will use induction on k to prove that, up to conjugation by an element of P_{k}, GL_{d_{1}}(p) normalises P_{•}. When k = 0 then GL_{d}(p) normalises P_{•}. So we assume that k ≥ 1.

If deg(P_{k−1}) = d then by Lemma 4.3(b) we may assume that P_{k−1} = V_{e} where d is a refinement of e = (e_{1}, …, e_{m}). By induction hypothesis on k, up to conjugation by an element of P_{k−1}, we may assume that GL_{e_{1}}(p) normalises P_{0} ≤ ⋯ ≤ P_{k−1} and therefore GL_{d_{1}}(p) ≤ GL_{e_{1}}(p) normalises P_{•}.

If deg(P_{k−1}) ≤ d − 1 then set P′_{k} = δ_{p^{d−1}}(P_{k}). It is clear from Lemma 4.6 that GL_{d_{1}}(p) normlises P′_{k} and that P′_{k} contains P_{k−1}. Now, the k-simplex P_{0} ≤ ⋯ ≤ P_{k−1} ≤ P′_{k} which we denote P′_{•} is a product of k-simplices of the form Q_{•} where Q_{k} = V_{d_{1},...,d_{r−1}}. By induction hypothesis on d, up to conjugation by an element of Q_{k} we may assume that GL_{d_{1}}(p) normalises Q_{•}. Doing it for each factor Q_{•} of P′_{•} we see that GL_{d_{1}}(p) normalises a conjugate of P_{•} by an element of P_{k}. □

Proof of Theorem 1.2. As in the proof of Theorem 1.1 we may assume that p|n. If P ≤ A_{n} is p-radical then it must be p-radical in Σ_{n} and in light of Remark 4.12(a), if P is also p-centric in A_{n} then it must be p-centric in Σ_{n}. Thus, p-centric p-radical p-subgroups in A_{n} contain their centraliser. Also, we now see that the collection D(n) contains all the p-centric p-radical p-subgroups of A_{n}. Following the argument in the proof of Theorem 1.1, it is enough to prove that H_{A_{n}}^{2}(|D(n)|; H_{A_{n}}) = 0.

We may think of H_{A_{n}} as a coefficient functor for Σ_{n} by restricting the action of Σ_{n}-sets to A_{n}. Specifically, H_{A_{n}}(X) = H^{1}(EΣ_{n} ×_{A_{n}} X; k^{×}). The projection EΣ_{n} ×_{A_{n}} X → EΣ_{n} ×_{Σ_{n}} X gives a natural transformation α: H_{Σ_{n}} → H_{A_{n}} with kernel N and cokernel M and short exact sequences of coefficient functors

(†)  0 → N → H_{Σ_{n}} ⟶^{α} M′ → 0  and  0 → M′ → H_{A_{n}} → M → 0.

By Lemma A.3, the isotropy group of any simplex σ ∈ |D(n)| contains an odd permutation and this implies that the A_{n}-orbit of σ is equal to its Σ_{n}-orbit. Thus, |D(n)|_{Σ_{n}} = |D(n)|_{A_{n}} and

C_{Σ_{n}}^{∗}(|D(n)|; H_{A_{n}}) = C_{A_{n}}^{∗}(|D(n)|; H_{A_{n}}).

It therefore remains to prove that H_{Σ_{n}}^{∗≥2}(|D(n)|; H_{A_{n}}) = 0. The idea is to prove by induction on d ≥ 1 that

(⋆) H_{Σ_{n}}^{∗≥2}(|D_{d}(n)|; H_{A_{n}}) = 0

If d = 1 then |D_{1}(n)| consists of a single orbit and (⋆) is immediate. We now fix d ≥ 2 and use induction on r ≥ 0 to show that H_{Σ_{n}}^{∗≥2}(|E_{d,r}(n)|; H_{A_{n}}) = 0. The base of induction r = 0 is taken care of by the induction hypothesis on d − 1 and Proposition 4.13. So we fix r ≥ 1 and our goal is to prove that

H_{Σ_{n}}^{∗≥2}(|E_{d,r}(n)|, |E_{d,r−1}(n)|; H_{A_{n}}) = 0.

Consider a k-simplex P_{•} in the relative space (|E_{d,r}(n)|, |E_{d,r−1}(n)|). As in the proof of Proposition 4.1, it is conjugate to P′_{•} × P′′_{•} where P′_{•} is a simplex in the relative space (|B_{d,r}(n)|, |B_{d,r}^{0}(n)|) and P′′_{•} ∈ |E_{d,0}(m)| where m = n − rp^{d}. By Lemmas A.3 and A.1 we see that N ([P_{•}]) = C_{2} and M([P_{•}]) = 0 unless m = 0 and P′_{•} = (Q_{•})^{r} for some simplex Q_{•} in the relative space (|B_{d,1}(p^{d})|, |B_{d,1^{0}}(p^{d})|), whence Iso_{Σ_{n}}(P_{•}) = N(Q_{•}) ≀ Σ_{r}. If r ≥ 3 or if r = 1 then N ([P_{•}]) = C_{2} and M([P_{•}]) = 0 by parts (c) and (a) of Lemma A.1, respectively, because in the latter case N(Q_{•}) = Q_{k}⋊GL_{d}(p). If r = 2 then Lemma A.2 shows that N ([P_{•}]) = C_{2} and M([P_{•}]) = C_{2}. Thus, the restriction of N to (|E_{d,r}(n)|, |E_{d,r−1}(n)|) is the constant coefficient functor with value C_{2} and since in the proof of Proposition 4.1 we show that |E_{d,r}(n)|_{Σ_{n}} are acyclic, we now deduce that

H_{Σ_{n}}^{∗}(|E_{d,r}(n)|, |E_{d,r−1}(n)|; N) = 0.

The analysis above also shows that M vanishes on (|E_{d,r}(n)|, |E_{d,r−1}(n)|) unless n = 2p^{d} and in that case it vanishes on all the simplices except those of the form Q_{•} ×Q_{•} where Q_{•} ∈ (|B(d, 1)|, |B^{0}(d, 1)|). Thus, if n = 2p^{d} we see from Proposition 4.10(i) that

H_{Σ_{n}}^{∗}(|E_{d,r}(n)|, |E_{d,r−1}(n)|; M) = H^{∗}(|B(d, 1)|_{Σ_{p^{d}}} , |B^{0}(d, 1)|_{Σ_{p^{d}}}; C_{2}) = 0.

We deduce that in all cases H_{Σ_{n}}^{∗}(|E_{d,r}(n)|, |E_{d,r−1}(n)|; M) = 0. The short exact sequences in (†) and the reults in the proof of Proposition 4.1 imply

H_{Σ_{n}}^{∗≥2}(|E_{d,r}(n)|, |E_{d,r−1}(n)|; H_{A_{n}}) ≅ H_{Σ_{n}}^{∗≥2}(|E_{d,r}(n)|, |E_{d,r−1}(n)|; H_{Σ_{n}}) = 0.

This completes the induction step for r. □

References

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

[2] Bredon, G. E. Equivariant cohomology theories. Lecture Notes in Mathematics, No. 34 Springer-Verlag, Berlin-New York 1967

[3] Broto, Carles; Castellana, Natlia; Grodal, Jesper; Levi, Ran; Oliver, Bob. Subgroup families controlling p-local finite groups. Proc. London Math. Soc. (3) 91 (2005), no. 2, 325–354.

[4] Broto, Carles; Levi, Ran; Oliver, Bob. Homotopy equivalences of p-completed classifying spaces of finite groups. Invent. Math. 151 (2003), no. 3, 611–664.

[5] Broto, Carles; Levi, Ran; Oliver, Bob. The homotopy theory of fusion systems. J. Amer. Math. Soc. 16 (2003), no. 4, 779–856

[6] Kessar, Radha; Linckelmann, Markus ZJ-theorems for fusion systems. Trans. Amer. Math. Soc. 360 (2008), no. 6, 3093–3106.

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

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

[9] Libman, Assaf. The normaliser decomposition for p-local finite groups. Algebr. Geom. Topol. 6 (2006), 1267–1288

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

[11] Linckelmann, Markus, Fusion category algebras. J. Algebra 277 (2004), no. 1, 222–235.

[12] Linckelmann Markus, On H^{∗}(C; k^{×}) for fusion systems. Homology, Homotopy and Applications 11(1), 203–218.

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

E-mail address: a.libman@abdn.ac.uk