A SPECTRAL SEQUENCE FOR FUSION SYSTEMS

ANTONIO DÍAZ RAMOS

Abstract. We build a spectral sequence converging to the cohomology of a fusion system with a strongly closed subgroup. This spectral sequence is related to the Lyndon-Hochschild-Serre spectral sequence and coincides with it for the case of an extension of groups. Nevertheless, the new spectral sequence applies to more general situations like finite simple groups with a strongly closed subgroup and exotic fusion systems with a strongly closed subgroup. We prove Tate’s p-nilpotency criterion following its original proof and using this new spectral sequence.

1. Introduction.

Let K ⊴G be a normal subgroup of the finite group G and consider the extension

K → G → G/K.

The Lyndon-Hochschild-Serre spectral sequence of this short exact sequence is an important tool to analyze the cohomology of G with coefficients in the ℤGmodule M. It has second page E_{2}^{n,m} = H^{n}(G/K; H^{m}(K; M)) with G/K acting on H^{m}(K; M) and converges to H^{n+m}(G; M).

Our aim in this work is to construct a related spectral sequence in the context of fusion systems. This concept was originally introduced by Puig and developed by Broto, Levi and Oliver in [2], to where we refer the reader for notation. It consists of a category F with objects the subgroups of a finite p-group S and morphisms bounded by axioms that mimic properties of conjugation morphisms.

In the setup of fusion systems the concept of short exact sequence is an evasive one: Let F be a fusion system over the p-group S. For a strongly F-closed subgroup T of S there is a quotient fusion system F/T [7, 5.10]. Nevertheless, in general there is no normal fusion subsystem of F that would play the role of the kernel of the morphism of fusion systems F → F/T [1, 8.11 ff.]. So the answer to [13, Conjecture 11] is negative and one cannot expect to construct a Lyndon-Hochschild-Serre spectral sequence for fusion systems. Here we are able to construct a spectral sequence that converges to the cohomology of F, H^{∗}(F; M), where M is a ℤ_{(p)}- module with trivial action of S.

1.1. Theorem. Let F be a fusion system over the p-group S, T a strongly F-closed subgroup of S and M a ℤ_{(p)}-module with trivial S-action. Then there is a first quadrant cohomological spectral sequence with second page

E_{2}^{n,m} = H^{n}(S/T; H^{m}(T; M))^{F}

and converging to H^{n+m}(F; M).

Date: November 15, 2012.

Partially supported by FEDER-MCI grant MTM2010-18089, Junta de Andalucía grant FQM-213 and P07-FQM-2863.

The notation^{F} will be fully described in Section 2 and must be thought as taking F-stable elements: Consider for each subgroup P of S the Lyndon-Hochschild-Serre spectral sequence of the extension

P ∩ T → P → P/P ∩ T ≅ P T/T

converging to H^{∗}(P; M). A morphism φ ∈ Hom_{F}(P, Q) induces a morphism φ^{∗} between the spectral sequences corresponding to Q and P. Hence we have a contravariant functor from F to spectral sequences. The inverse limit spectral sequence or spectral sequence of F-stable elements has E_{2}^{n,m} entry equal to H^{n}(S/T; H^{m}(T; M))^{F}, i.e., the elements z from

H^{n}(S/T; H^{m}(T; M))

such that φ^{∗}(z) = res(z), where φ ∈ Hom_{F}(P, S) and res = ι^{∗} is restriction in

cohomology for the inclusion P ≤^{ι} S. Hence H^{∗}(S/T; H^{∗}(T; M))^{F} is a differential graded subalgebra of the differential graded algebra H^{∗}(S/T; H^{∗}(T; M)) and its differential is just restriction of the differential of the latter. This should be useful in computations. The theorem states that the abutment of this spectral sequence is H^{∗}(F; M).

For the case of a normal subgroup K ⊴ G and F = F_{S}(G) with S ∈ Syl_{p}(G) we have two spectral sequences converging to H^{∗}(G; M). Here, M is a ℤ_{(p)}-module with trivial G-action (and hence trivial S-action). On the one hand, we have the Lyndon-Hochschld-Serre spectral sequence associated to K → G → G/K. On the other hand, we have the spectral sequence associated to F and the strongly F-closed subgroup T = K ∩ S ∈ Syl_{p}(K). In Section 5 we prove that the two spectral sequences coincide. Note that, in particular, this shows that the Lyndon-Hochschild-Serre spectral sequence of the extension K → G → G/K depends only on the intersection of K with a Sylow p-subgroup of G.

As an application of the spectral sequence in Theorem 1.1 we prove Tate’s criterion for p-nilpotency of finite groups translated to the setup of fusion systems:

1.2. Corollary ([14]). Let F be a fusion system over the p-group S. If the restriction map H^{1}(F; 𝔽_{p}) → H^{1}(S; 𝔽_{p}) is an isomorphism then F = F_{S}(S).

This result was already proven in [8] using transfer for fusion systems and in [5] by topological methods. Here we are able to mimic Tate’s cohomological original proof that relies on the five terms exact sequence associated to the Lyndon-Hochschild-Serre spectral sequence. We use instead the spectral sequence of Theorem 1.1, showing that this new spectral sequence fits in the p-local setup of fusion systems.

There are situations where the Lyndon-Hochschild-Serre spectral sequence is not applicable while the spectral sequence from Theorem 1.1 can be utilized. For instance, a classical drawback of the Lyndon-Hochschild-Serre spectral sequence is that it cannot be applied to finite simple groups. Nevertheless there are finite simple groups that do have a strongly closed p-subgroup. In [10], Flores and Foote classified all finite groups with a strongly closed p-subgroup. In particular, they stated which finite simple groups have a stronly closed p-subgroup. This result together with Theorem 1.1 gives the following

1.3. Corollary. If G is any of the following simple finite groups:

- U_{3}(2^{n}), Sz(2^{n}) (p = 2)

- Groups of Lie type in characteristic ≠ p whose Sylow p-subgroup is abelian but not elementary abelian or U_{3}(p^{n}) (p odd),

- Re(3^{2n+1}) or G_{2}(q) with (q, 3) = 1 (p = 3),

- Sporadic groups: J_{2}, J_{3} (p = 3), Co_{2}, Co_{3}, HS, Mc (p = 5) or J_{4} (p = 11). then there is a spectral sequence converging to H^{∗}(G; 𝔽_{p}) where p is the given

prime. The E_{2}-page is a bigraded differential subalgebra of the E_{2}-page of the Lyndon-Hochschild-Serre spectral sequence of T → S → S/T, where S ∈ Syl_{p}(G) and 1 < T < S is a strongly closed p-subgroup of G.

More information on the strongly closed p-subgroup T is given in [10, 2.7]. Notice that even if F is induced from a non-simple finite group F = F_{S}(G) not every strongly closed F-subgroup T of S is of the form T = K ∩ S for some normal subgroup K ⊴ G [1, Example 6.4]. This describes another circumstances where Lyndon-Hochschild-Serre does not apply but Theorem 1.1 does. As final example of this situation consider exotic fusion system with a strongly closed p-subgroup. A family of such exotic fusion systems is described in [9], where the authors classified all the fusion systems over p-groups of p-rank 2 (p odd). From [9, Table 6] and Theorem 1.1 we obtain the following

1.4. Corollary. Let F be any of the following exotic fusion systems

3.F(3^{2k}, 1), 3.F(3^{2k}, 2), 3.F(3^{2k}, 1).2 or 3.F(3^{2k}, 2).2

defined over the maximal nilpotency class group B(3, 2k + 1) with k ≥ 2. Then there is a spectral sequence converging to H^{∗}(F; 𝔽_{3}). The E_{2}-page is a bigraded differential subalgebra of the Lyndon-Hochschild-Serre spectral sequence of ℤ_{3} → B(3, 2k + 1) → B(3, 2k) where ℤ_{3} is strongly closed in F.

1.5. Remark. Theorem 1.1 holds for the wider class of F-stable ℤ_{(p)}S-modules, i.e., for ℤ_{(p)}S-modules M such that for any morphism φ: P → S in F and any p ∈ P we have φ(p) · m = p · m. Also, the Lyndon-Hochschild-Serre spectral sequence of K ⊴G and the spectral sequence from Theorem 1.1 for F = F_{S}(G) and T = S ∩ K coincide for G-stable ℤ_{(p)}G-modules, i.e, for ℤ_{(p)}G-modules M such that g^{−1}hg · m = h · m for any h, g ∈ G.

Organization of the paper: In Section 2, F-stable elements and Mackey functors are defined and some related results introduced. In Section 3, we describe a particular cohomological Mackey functor that will play a central role in the construction of the spectral sequence. In Section 4, the spectral sequence is built and Theorem 1.1 is proven as Theorem 4.1. In Section 5 we compare the spectral sequence from Theorem 1.1 to the Lyndon-Hochschild-Serre spectral sequence. In Section 6 we prove Tate’s Theorem.

Acknowledgments: I would like to thank A. Viruel for several fruitful conversations when developing this paper. Also, I am grateful to P. Symonds for showing me how to prove that in the normal subgroup case the two spectral sequences coincide (Theorem 5.1).

2. Cohomology and F-stable elements.

Throughout this section F denotes a fusion system over the p-group S. We start introducing some notation: If A : F → C is a contravariant functor and C is any category then by φ^{∗} we denote the value A(φ) for φ a morphism in F. For φ = ι_{P}^{S}, the inclusion of P into S, we write res := ι_{P}^{S}^{∗}. If C is a complete category then we denote by A^{F} the inverse limit over F of this functor:

A^{F} := \underleftarrow{lim}_{F} A.

If there is a functor U : C → Sets that creates (inverse) limits then there is favourable description of A^{F}:

2.1. Lemma. Let A : F → C be a contravariant functor with C complete and such that U : C → Sets creates limits. Then

A^{F} = A(S)^{F} := {z ∈ A(S)|res(z) = φ^{∗}(z) for each φ ∈ Hom_{F}(P, S)} ⊆ A(S).

We call the elements in A(S)^{F} the F-stable elements in A(S). The category C = Ab of abelian groups is complete and the forgetful functor U : C → Sets creates limits. Hence the lemma applies. For the complete category C = CCh(Ab) of (unbounded) cochain complexes limits are constructed dimension-wise and the conclusion of the lemma also holds, i.e., for any functor A : F → CCh(Ab) we have that

A^{F} = A(S)^{F}.

For such a functor we can consider the cohomology H^{∗}(A^{F}) = H^{∗}(A(S)^{F}) of A(S)^{F} ∈ CCh(Ab). Notice that we also have functors H^{n}(A) : F → Ab obtained by taking cohomology at degree n. Hence we may also consider the inverse limits H^{∗}(A)^{F} = H^{∗}(A(S))^{F}. We are interested in functors A for which taking F-stable elements and cohomology commute. We prove in this section (Proposition 2.10) that being a cohomological Mackey functor (Definition 2.2) with values in ℤ_{(p)}- modules is sufficient.

2.2. Definition. Let F be a saturated fusion system over the p-group S and let A be an abelian category. A cohomological Mackey functor for F over A is a pair of functors (A, B): F → A with A : F → A contravariant and B : F → A covariant such that:

(1) A(P) = B(P) and A(φ) = B(φ^{−1}) for each P ≤ S and φ ∈ Hom_{F}(P, φ(P)).

(2) (Identity) A(c_{p}), B(c_{p}): A(P) → A(P) are the identity morphisms for every p ∈ P ≤ S, where c_{p} : P → P, x ↦ pxp^{−1} is conjugation by p.

(3) (Double coset formula) A(ι_{Q}^{P}) ∘ B(ι_{R}^{P}) = ∑_{x∈Q\P/R} B(ι_{Q∩^{x}R}^{Q}) ∘ A(ι_{Q∩^{x}R}^{^{x}R}) ∘ A(c_{x^{−1}}_{|^{x}R}) for Q, R ≤ P ≤ S, where Q\P/R are the double cosets.

(4) (Cohomological) B(ι_{P}^{Q}) ∘ A(ι_{P}^{Q}) : A(Q) → A(Q) is multiplication by |Q : P | for every P ≤ Q ≤ S.

See [15] for the classical definition of Mackey functor and of cohomological Mackey functor for finite groups.

2.3. Remark. In Definition 2.2 we have omitted the familiar conditions

• (Transitivity) B(ι_{Q}^{R}) ∘ B(ι_{P}^{Q}) = B(ι_{P}^{R}), A(ι_{P}^{Q}) ∘ A(ι_{Q}^{R}) = A(ι_{P}^{R}) for P ≤ Q ≤ R ≤ S and

• (Conjugation)B(ι_{P}^{Q})∘A(φ_{|P}) = A(φ)∘B(ι\begin{matrix}φ(Q) \\ φ(P) \end{matrix}), B(φ_{|P})∘A(ι_{P}^{Q}) = A(ι\begin{matrix}φ(Q) \\ φ(P) \end{matrix})∘

B(φ) for P ≤ Q ≤ S, φ ∈ Hom_{F}(Q, φ(Q)). In fact, they are consequence of the functoriality of A and B and of condition (1).

We will use several times along the paper that cohomology of finite groups is a cohomological Mackey functor. For a proof of this fact see [4] for example.

2.4. Remark. The map B(ι_{P}^{Q}) for the inclusion ι_{P}^{Q} : P → Q is called the transfer from P into Q. Given the contravariant functor A, the transfers are all what is needed to define the covariant part B: For any morphism φ ∈ Hom_{F}(P, Q) define B(φ) : A(P) → A(Q) by B(φ) = B(ι_{φ(P)}^{Q}) ∘ A(φ̃^{−1}) with φ̃ : P →^{≅} φ(P). Such a B

becomes functorial if for any P ≤ Q ≤ R we have B(ι_{Q}^{R}) ∘ B(ι_{P}^{Q}) = B(ι_{P}^{R}) and for

any P ≤ Q →^{φ} φ(Q) we have B(ι_{P}^{Q}) ∘ A(φ_{|P}) = A(φ) ∘ B(ι_{φ(P)}^{φ(Q)}).

Before proving the main result of this section we need to introduce (G, H)-bisets: sets with commuting free right G-action and free left H-action. Every (G, H)-biset Ω can be decomposed into a disjoint union of transitive (G, H)-bisets of the form

H ×_{φ} G = H × G/ ∼,

with K ≤ G, φ : K → H a monomorphism and

(h, kg) ∼ (hφ(k), g)

for h ∈ H, g ∈ G and k ∈ K. A saturated fusion system gives rise to a special type of biset:

2.5. Proposition ([2, Proposition 5.5]). For any saturated fusion system F over a p-group S, there is an (S, S)-biset Ω with the following properties:

(a) Each transitive component of Ω is of the form S ×_{φ} S for some P ≤ S and φ ∈ Hom_{F}(P, S).

(b) For each P ≤ S and each φ ∈ Hom_{F}(P, S), the (P, S)-biset Ω_{P} obtained by restricting the right action from S to P and the (P, S)-biset Ω_{φ} obtained by restricting the right action from S to P via φ are isomorphic as (P, S)-bisets.

(c) |Ω|/|S| = 1(modp).

We call such an (S, S)-biset an F-stable (S, S)-biset. Now let (A, B): F → A be a cohomological Mackey functor for F over the abelian category A. For each transitive (Q, R)-biset R ×_{φ} Q with φ ∈ Hom_{F}(P, R), P ≤ Q ≤ S, R ≤ S, we have the composition

(2.6)  A(R) →^{A(φ)} A(P) = B(P) →^{B(ι_{P}^{Q})} B(Q) = A(Q).

For each (Q, R)-biset Ω with

Ω = ∐ R ×_{φ} Q

we can define a map A(Ω) : A(R) → A(Q) by

(2.7)  A(Ω) := ∑ B(ι) ∘ A(φ).

2.8. Lemma. Let (A, B) : F → A be a cohomological Mackey functor. Then

(1) For each transitive (Q, R)-biset R ×_{φ} Q the morphism (2.6) depends only on the isomorphism class of R ×_{φ} Q as (Q, R)-biset.

(2) For any (Q, R)-biset Ω the morphism (2.7) depends only on the isomorphism class of Ω as (Q, R)-biset.

(3) For any (Q, R)-biset Ω and any monomorphism ψ : P → Q we have

A(ψ) ∘ A(Ω) = A(Ω_{ψ}),

where Ω_{ψ} is the (P, R)-biset obtained by restricting the right action of Ω from Q to P via ψ.

(4) If A = ℤ_{(p)} - mod and Ω is an F-stable (S, S)-biset then
A(S)^{F} = Im(A(Ω) : A(S) → A(S)).

Proof. Proof of (1): The transitive (Q, R)-bisets R ×_{φ_{1}} Q and R ×_{φ_{2}} Q with φ_{1} : P_{1} → R, φ_{2} : P_{2} → R, P_{1}, P_{2} ≤ Q are isomorphic as (Q, R)-bisets if and only if there exist elements q ∈ Q and r ∈ R such that the following diagram commutes:

\xymatrix{P_{1} \ar[d]^{c_{q}} \ar[r]^{φ_{1}} & R \ar[d]^{c_{r}} \\ P_{2} \ar[r]^{φ_{2}} & R.}

Hence both squares in the following diagram commute:

\xymatrix{A(R) \ar[r]^{A(φ_{1})} & A(P_{1}) \ar[r]^{B(ι_{P_{1}}^{Q})} \ar[d]^{B(c_{q})} & A(Q) \ar[d]^{B(c_{q})} \\ A(R) \ar[u]^{A(c_{r})} \ar[r]^{A(φ_{2})} & A(P_{2}) \ar[u]^{A(c_{q})} \ar[r]^{B(ι_{P_{2}}^{Q})} & A(Q).}

Using (1) and (2) from Definition 2.2 one finds out that

B(ι_{P_{1}}^{Q}) ∘ A(φ_{1}) = B(ι_{P_{2}}^{Q}) ∘ A(φ_{2}).

Proof of (2): any automorphism of Ω permutes its transitive components via isomorphisms. So we may apply (1) from the Lemma to each component.

Proof of (3): write Ω as a disjoint union of transitive (Q, R)-bisets Ω = ∐ R×_{φ}Q. The transitive (Q, R) biset R×_{φ}Q with φ : K → R, K ≤ Q decomposes as a (P, R)- biset with P acting via ψ as follows:

R ×_{φ} Q = ⋃_{q∈ψ(P)\Q/K} R ×_{φ∘c_{q^{−1}} ∘ψ} P,

with P ≥ P ∩ ψ^{−1}(^{q}K) →^{ψ_{|}} ψ(P) ∩^{q}K ⟶^{c_{q^{−1}}} K →^{φ} R. Hence,

A(Ω_{ψ}) = ∑_{φ} ∑_{q∈ψ(P)\Q/K} B(ι_{P ∩ψ^{−1}(^{q}K)}^{P}) ∘ A(φ ∘ c_{q^{−1}} ∘ ψ_{|}).

Using functoriality of A and B we get

A(Ω_{ψ}) = A(ψ̃) ∘ (∑_{φ} ∑_{q∈ψ(P)\Q/K} B(ι_{ψ(P)∩^{q}K}^{ψ(P)}) ∘ A(c_{q^{−1}}) ∘ A(φ)),

with ψ̃ : P →^{≅} ψ(P). Now the Mackey decomposition (3) from Definition 2.2 gives

A(Ω_{ψ}) = ∑_{φ} A(ψ̃) ∘ A(ι_{ψ(P)}^{Q}) ∘ B(ι_{K}^{Q}) ∘ A(φ) = A(ψ) ∘ A(Ω).

Proof of (4): Let z ∈ A(S). We want to see that A(Ω)(z) ∈ A(S)^{F}. So let ψ be a morphism in Hom_{F}(P, S). Then

A(ψ)(A(Ω)(z)) = (A(ψ) ∘ A(Ω))(z) = A(Ω_{ψ})(z)

by (3) of this lemma. By 2.5(b), the (P, S)-bisets Ω_{ψ} and Ω_{ι_{P}^{S}} = Ω_{P} are isomorphic as (P, S)-bisets. Then by (2) of this lemma we have A(Ω_{ψ}) = A(Ω_{ι_{P}^{S}}). Hence,

A(ψ)(A(Ω)(z)) = A(Ω_{ψ})(z) = A(Ω_{ι_{P}^{S}})(z) = A(ι_{P}^{S})(A(Ω)(z))

by (3) of this lemma. Thus A(Ω)(z) ∈ A(S)^{F}.

Now let z ∈ A(S)^{F}. Then

A(Ω)(z) = ∑ B(ι)(A(φ)(z)) = ∑ B(ι)(A(ι)(z))

as z is F-stable. Now by (4) of Definition 2.2 we get

A(Ω)(z) = (∑ |S : P |) · z

and by 2.5(c) the number q = (∑ |S : P |) = |Ω|/|S| is a p′-number. So A(Ω)(\frac{z}{q}) = z and hence z ∈ Im A(Ω). □

For a fusion system F over the p-group S denote by CohMack_{ℤ_{(p)}}(F) the category with objects the cohomological Mackey functors with values in ℤ_{(p)} - mod and morphisms those natural transformations commuting with both the contravariant and covariant parts.

2.9. Lemma. Let F be a fusion system over the p-group S. If the sequence

0 ⇒ (A_{1}, B_{1}) ⇒ (A_{2}, B_{2}) ⇒^{η} (A_{3}, B_{3}) ⇒ 0

is exact in CohMack_{ℤ_{(p)}}(F) then the sequence

0 → A_{1}^{F} → A_{2}^{F} →^{η^{F}} A_{3}^{F} → 0

is exact in ℤ_{(p)} - mod.

Proof. The non-trivial equality to prove is that the arrow A_{2}^{F} → A_{3}^{F} is an epimorphism. So let z be an F-stable element in A_{3}(S). Fix an (S, S)-biset Ω satisfying the properties of Proposition 2.5. (Such a biset exists by the same proposition.) By Lemma 2.8(4) there exists an element z′ ∈ A_{3}(S) with z = A_{3}(Ω)(z′). By hypothesis, the map

A_{2}(S) →^{η_{S}} A_{3}(S)

is an epimorphism and hence there exists an element y′ ∈ A_{2}(S) with η_{S}(y′) = z′. By Lemma 2.8(4) again we have that y =^{def} A_{2}(Ω)(y′) belongs to A_{2}^{F}. Because η commutes with the covariant and contravariant parts of (A_{2}, B_{2}) and (A_{3}, B_{3}) is easy to see that

η^{F}(y) = η^{F}(A_{2}(Ω)(y′)) = A_{3}(Ω)(η^{F}(y′)) = A_{3}(Ω)(z′) = z.
□

2.10. Proposition. Let F be a fusion system over S and let (A, B) : F → CCh(ℤ_{(p)}) be a cohomological Mackey functor. Then

H^{∗}(A(S)^{F}) ≅ H^{∗}(A(S))^{F}.

Proof. Let Ω be an F-stable (S, S)-biset (see 2.5). For any n ∈ ℤ write (A^{n}, B^{n}) : F → ℤ_{(p)} - mod for the degree n component of (A, B). Then (A^{n}, B^{n}) is a cohomological Mackey functor. The subfunctors (Ker A^{n}, Ker B^{n}) and (Im A^{n}, Im B^{n}) of (A^{n}, B^{n}) defined as the kernel and image of the differential on dimension n are also cohomological Mackey functors. The quotient functor (H^{n}(A), H^{n}(B)) =^{def} (Ker A^{n}, Ker B^{n})/(Im A^{n}, Im B^{n}) is also a cohomological Mackey functor and we have a short exact sequence in CohMack_{ℤ_{(p)}}(F)

0 ⇒ (Im A^{n}, Im B^{n}) ⇒ (Ker A^{n}, Ker B^{n}) ⇒ (H^{n}(A), H^{n}(B)) ⇒ 0.

By the previous Lemma 2.9 we have a short exact sequence of ℤ_{(p)}-modules

0 → (Im A^{n})^{F} → (Ker A^{n})^{F} → H^{n}(A)^{F} → 0.

Recall that (Ker A^{n})^{F} = (Ker A^{n}(S))^{F} is exactly

{z ∈ Ker A^{n}(S)|A^{n}(φ)(z) = A^{n}(ι_{P}^{S})(z) for each φ ∈ Hom_{F}(P, S)}

and that (Im A^{n})^{F} = (Im A^{n}(S))^{F} is exactly

{d^{n−1}(z)|z ∈ A^{n−1}(S), A^{n}(φ)(d^{n−1}(z)) = A^{n}(ι_{P}^{S})(d^{n−1}(z)) for each φ ∈ Hom_{F}(P, S)}.

On the other hand,

Ker(A^{n}^{F}) = Ker A^{n}(S)^{F} = {z ∈ A^{n}(S)^{F}|d^{n}(z) = 0}

and a straightforward check shows that

(Ker A^{n}(S))^{F} = Ker A^{n}(S)^{F} = A^{n}(S)^{F} ∩ Ker d^{n}.

For the image functor we have

Im(A^{n}^{F}) = Im A^{n}(S)^{F} = {d^{n−1}(z)|z ∈ A^{n−1}(S)^{F}},

which equals

{d^{n−1}(z)|z ∈ A^{n−1}(S), A^{n−1}(φ)(z) = A^{n−1}(ι_{P}^{S})(z) for each φ ∈ Hom_{F}(P, S)}.

Clearly we have

Im A^{n}(S)^{F} ≤ (Im A^{n}(S))^{F} = A^{n}(S)^{F} ∩ Im d^{n−1}.

To check the reverse inclusion choose d^{n−1}(z) ∈ (Im A^{n}(S))^{F}. We want to find z′ ∈ A^{n−1}(S)^{F} such that d^{n−1}(z) = d^{n−1}(z′). Set z′ = A^{n−1}(Ω)(z). Then z′ ∈ A^{n−1}(S)^{F} by (4) of Lemma 2.8. Now we compute the differential of z′:

d^{n−1}(z′) = d^{n−1}(A^{n−1}(Ω)(z)) = d^{n−1}(∑ B^{n−1}(ι)(A^{n−1}(φ)(z))),

which equals

∑ B^{n}(ι)(A^{n}(φ)(d^{n−1}(z))) = A^{n}(Ω)(d^{n−1}(z)).

By hypothesis d^{n−1}(z) ∈ (Im A^{n}(S))^{F} and hence by the proof of (4) of Lemma 2.8

A^{n}(Ω)(d^{n−1}(z)) = q · d^{n−1}(z),

where q is a p′-number. Hence \frac{z′}{q} ∈ A^{n−1}(S)^{F} and d^{n−1}(\frac{z′}{q}) = d^{n−1}(z). □

2.11. Remark. Let F be a fusion system over the p-group S and let M be a trivial ℤ_{(p)}S-module. By [2, Section 5] the cohomology of F is defined as H^{∗}(F; M) = H^{∗}(S; M)^{F} where H^{∗}(·; M) : F → ℤ_{(p)}-modules is the cohomological Mackey functor with values H^{∗}(P; M). If one could choose (contravariant) cochains C^{∗}(·; M) : F → CCh(ℤ_{(p)}) such that C^{∗}(·; M) was the contravariant part of a cohomological Mackey functor then Proposition 2.10 would give the computationalpurposes formula

H^{∗}(F; M) = H^{∗}(C^{∗}(S; M)^{F}).

In the next section some problems related to the functoriality of cochains will become apparent.

3. A Mackey functor.

Let F be a fusion system over the p-group S, T a strongly F-closed subgroup of S and M a ℤ_{(p)}-module with trivial S-action. In this section we prove that for every n, m ≥ 0 the functor H^{n,m} : F → ℤ_{(p)} - mod sending the subgroup P ≤ S to H^{n}(P/P ∩ T; H^{m}(P ∩ T; M)) is the contravariant part of a cohomological Mackey functor F → CCh^{2}(ℤ_{(p)}) with values in double complexes (Definition 2.2).

For P ≤ S denote by \overline{P} the group P/P ∩T. The bar resolutions B_{P}^{∗} and B∗ ; \overline{P} for P and \overline{P} respectively are projective resolutions of the trivial module ℤ_{(p)} over ℤ_{(p)}P and ℤ_{(p)}\overline{P} respectively. Recall that the bar resolution is functorial (covariant) over finite groups and homomorphisms. Define A^{∗,∗}(P) as the double complex associated to the short exact sequence

0 → P ∩ T → P → \overline{P} ≅ P T/T → 0.

More precisely, for n ≥ 0 and m ≥ 0, we define

A^{n,m}(P) = Hom_{P} (Bn ; \overline{P} ⊗ B_{P}^{m}, M)

where P acts on Bn ; \overline{P} ⊗ B_{P}^{m} by p(y ⊗ x) = \overline{p}y ⊗ px for y ∈ Bn ; \overline{P} and x ∈ B_{P}^{m}.

3.1. Remark. By [11, Equation (9.9), page 350] we could have defined A^{n,m}(P) by Hom\overline{P} (Bn ; \overline{P} , Hom_{P ∩T} (B_{P}^{m}, M)) instead. We have chosen the former definition above for simplicity.

As the action of P on M is trivial the cochains in A^{n,m}(P) are the homomorphisms f ∈ Hom(Bn ; \overline{P} ⊗ B_{P}^{m}, M) such that

f(\overline{p}y ⊗ px) = f(y ⊗ x)

for all y ∈ Bn ; \overline{P}, x ∈ B_{P}^{m} and p ∈ P.

To obtain a double complex we consider the following horizontal and vertical differentials for f ∈ A^{n,m}(P)

d^{h}(f)(y ⊗ x) = (−1)^{n+m+1}f(d(y) ⊗ x), y ∈ Bn+1 ; \overline{P}, x ∈ B_{P}^{m}

and

d^{v}(f)(y ⊗ x) = (−1)^{m+1}f(y ⊗ d(x)), y ∈ Bn ; \overline{P}, x ∈ B_{P}^{m+1},

where we are using the differential d of the complexes B∗ ; \overline{P} and B_{P}^{∗}. We choose the signs as in MacLane’s book [11, XI.10.1] to ensure that d^{h}d^{v} + d^{v}d^{h} = 0. We will obtain the functor H^{n,m} by taking vertical cohomology followed by horizontal cohomology in A^{n,m}.

To define A on morphisms notice that any morphism φ ∈ Hom_{F}(P, Q) takes P ∩ T to Q ∩ T as T is strongly F-closed. Hence it induces a homomorphism

\overline{φ} : \overline{P} → \overline{Q}.

Thus for any φ ∈ Hom_{F}(P, Q) we may define

A^{n,m}(Q) →^{A^{n,m}(φ)} A^{n,m}(P)

mapping the cochain f ∈ A^{n,m}(Q) to the cochain in A^{n,m}(P) that takes y ∈ Bn ; \overline{P} and x ∈ B_{P}^{m} to

f(B^{n}(\overline{φ})(y) ⊗ B^{m}(φ)(x)),

where B^{n}(\overline{φ}) and B^{m}(φ) are part of our choice of functorial resolutions for groups, i.e., the bar resolution. They commute with differentials and satisfy

B^{n}(\overline{φ})(\overline{p} · y) = \overline{φ}(\overline{p}) · B^{n}(\overline{φ})(y)

for every y ∈ Bn ; \overline{P} and every \overline{p} ∈ \overline{P} and

B^{m}(φ)(p · x) = φ(p) · B^{m}(φ)(x)

for every x ∈ B_{P}^{m} and p ∈ P. It is straightforward that A^{n,m}(φ)(f) ∈ A^{n,m}(P) and that A^{n,m} commutes with both the horizontal and vertical differentials.

3.2. Remark. By definition the fusion system F/T is defined over the p-group S/T. For T ≤ P, Q ≤ S the morphisms in Hom_{F/T} (P/T, Q/T) are those homomorphisms \overline{ψ} : P/T → Q/T induced on the quotient from ψ ∈ Hom_{F}(P, Q).

For P, Q ≤ S and φ ∈ Hom_{F}(P, Q) we have a morphism \overline{φ}: \overline{P} → \overline{Q}. Then we have a commutative diagram

\xymatrix{\overline{P} \ar[d]_{≅} \ar[r]^{\overline{φ}} & \overline{Q} \ar[d]^{≅} \\ P T/T \ar[r]^{\overline{φ}} & QT/T}

where \overline{φ}’s are induced by φ and where the vertical arrows are the natural isomorphisms. According to [7, 5.10] bottom morphism \overline{φ} belongs to F/T, i.e., there exists ψ ∈ Hom_{F}(P T, QT) such that the induced map \overline{ψ} : P T/T → QT/T coincides with the given one.

3.3. Remark. The construction of A^{n,m} is clearly functorial and hence so far we have a contravariant functor A^{∗,∗} : F → CCh^{2}(ℤ_{(p)}) with values in double complexes.

Now we define B^{n,m}(P) = A^{n,m}(P) for every P ≤ S and n, m ≥ 0. For each morphism φ ∈ Hom_{F}(P, Q) we will define a morphism of double complexes B^{n,m}(φ) : A^{n,m}(P) → A^{n,m}(Q). This will not make B into a covariant functor F → CCh^{2}(ℤ_{(p)}) as the definition depends on a choice of representatives. Nevertheless, B will become functorial once we pass to cohomology.

To define B^{∗,∗}(ψ) on a generic morphism ψ : P → Q in F write ψ = ι ∘ ψ̃, where ψ̃ : P → ψ(P) is an isomorphism and ι is the inclusion ψ(P) ≤ Q, and set

(3.4)  B^{∗,∗}(ψ) = B^{∗,∗}(ι) ∘ A^{∗,∗}(ψ̃^{−1}).

So we just need to define B on inclusions.

So let ι be the inclusion between subgroups P ≤ Q of S. There are maps of ℤ_{(p)}P -chain complexes and of ℤ_{(p)}\overline{P} -chain complexes respectively

τ_{∗}^{Q,P} : B_{Q}^{∗} → B_{P}^{∗} , \overline{τ}_{∗}^{Q,P} : B∗ ; \overline{Q} → B∗ ; \overline{P}

built as in [4, (D), page 82]. More precisely, the map τ_{∗}^{Q,P} is induced by a map of left P -sets Q →^{ρ} P defined as follows: fix a set of representatives for the right cosets P \Q, then ρ(q) = q\overline{q}^{−1} where \overline{q} is the representative with P q = P \overline{q}. The map \overline{τ}_{∗}^{Q,P} is defined analogously choosing representatives for the right cosets \overline{P} \\overline{Q}. These choices of representatives prevent B^{p,q} from being functorial.

We define the map

B^{n,m}(ι): Hom_{P} (Bn ; \overline{P} ⊗ B_{P}^{m}, M) → Hom_{Q}(Bn ; \overline{Q} ⊗ B_{Q}^{m}, M)

by

(3.5)  B^{n,m}(ι)(f)(y ⊗ x) = ∑_{w∈Q/P} f(\overline{τ}_{n}^{Q,P} (\overline{w}^{−1}y) ⊗ τ_{m}^{Q,P} (w^{−1}x)),

where w runs over a set of representatives of the left cosets Q/P. This formula can be thought as a relative transfer formula for twisted coefficients. Clearly its definition does not depend on the representatives w chosen and B^{n,m}(ι)(f) ∈ A^{n,m}(Q). Moreover, B^{n,m}(ι) commutes with both the horizontal and vertical differentials as τ_{∗} and \overline{τ}_{∗} do and so it is a map of double complexes.

3.6. Remark. By [12] there are finite groups G and \overline{G} such that S is a p-subgroup of G (not necessarily a Sylow p-subgroup), \overline{S} = S/T is a p-subgroup of \overline{G} (not necessarily a Sylow p-subgroup) and with F = F_{S}(G) and F/T = F\overline{S}(\overline{G}). Let B_{G}^{∗} and B∗ ; \overline{G} be the bar resolutions of G and \overline{G} respectively. Then we could have defined for P ≤ S

A^{n,m}(P) = Hom_{P} (Bn ; \overline{G} ⊗ B_{G}^{m}, M),

where P acts on Bn ; \overline{G} ⊗ B_{G}^{m} by restricting the actions of G on B_{G}^{∗} and of \overline{G} on B∗ ; \overline{G}. This means that p(y ⊗x) = \overline{p}y ⊗px for p ∈ P. In this setup clearly one can define a functorial B^{n,m} on inclusions. On the other hand, to realize a morphism φ : P → Q we need to choose g ∈ N_{G}(P, Q) with φ = c_{g} and \overline{g} ∈ N\overline{G}(\overline{P} , \overline{Q}) with \overline{φ} = c\overline{g} and then define

A^{n,m}(φ)(y ⊗ x) = (\overline{g}y ⊗ gx).

It is clear that in general A^{n,m} defined this way will not be functorial on morphisms.

On each double complex A^{∗,∗}(P) with P ≤ S we may take vertical cohomology followed by horizontal cohomology to obtain H^{∗}(\overline{P}; H^{∗}(P ∩ T; M)) [11, Equation (10.2), page 352]. For any homomorphism φ ∈ Hom_{F}(P, Q) the maps A^{∗,∗}(φ) and B^{∗,∗}(φ) are maps of double complexes and hence they induce maps

H^{n,m}(A)(φ) : H^{n}(\overline{Q}; H^{m}(Q ∩ T; M)) → H^{n}(\overline{P}; H^{m}(P ∩ T; M))

and H^{n,m}(B)(φ) : H^{n}(\overline{P}; H^{m}(P ∩ T; M)) → H^{n}(\overline{Q}; H^{m}(Q ∩ T; M)).

3.7. Lemma. For φ : P → Q the map H^{n,m}(A)(φ) factors as

\xymatrix{H^{n}(\overline{Q}; H^{m}(Q ∩ T; M)) \ar[r]^{H^{n}(\overline{φ})} & H^{n}(\overline{P}; H^{m}(Q ∩ T; M)) \ar[r]^{H^{m}(φ)} & H^{n}(\overline{P}; H^{m}(P ∩ T; M))}

where • H^{n}(\overline{φ}) is the map induced by \overline{φ} in cohomology with H^{m}(Q∩T; M)-coefficients,

• H^{m}(φ) is the map induced by the change of coefficients

H^{m}(φ) : H^{m}(Q ∩ T; M) → H^{m}(P ∩ T; M).

This map is a map of ℤ_{(p)}\overline{P} -modules where \overline{P} acts on H^{m}(Q ∩ T; M) via

\overline{P} →^{\overline{φ}} \overline{φ(P)} ≤ \overline{Q}.

Proof. By construction. □

3.8. Lemma. For the inclusion ι : P → Q the map H^{n,m}(B)(ι) factors as

\xymatrix{H^{n}(\overline{P}; H^{m}(P ∩ T; M)) \ar[r]^{H^{m}(tr′)} & H^{n}(\overline{P}; H^{m}(Q ∩ T; M)) \ar[r]^{H^{n}(tr)} & H^{n}(\overline{Q}; H^{m}(Q ∩ T; M))}

where • H^{n}(tr) is the transfer map in cohomology with H^{m}(Q ∩ T; M)-coefficients,

• H^{m}(tr′) is the map induced by the change of coefficients given by the transfer map in cohomology

H^{m}(tr′) : H^{m}(P ∩ T; M) → H^{m}(Q ∩ T; M).

This map is a map of ℤ_{(p)}\overline{P} -modules where \overline{P} acts on H^{m}(Q ∩ T; M) via \overline{P} ≤ \overline{Q}.

Proof. Choose representatives z_{i} ∈ \overline{Q} of the left cosets \overline{Q}/\overline{P} and representatives t_{j} ∈ Q ∩ T of the left cosets (Q ∩ T)/(P ∩ T). Choose also representatives q_{k} ∈ Q of the left cosets \overline{Q} = Q/(Q ∩ T). Then each z_{i} ∈ \overline{Q} is represented as z_{i} = \overline{q_{k_{i}}} for a unique k_{i}. It is an exercise to prove that the set of elements of Q q_{k_{i}}t_{j} for all i and j is a set of representatives of Q/P. Then we can rewrite Equation (3.5) as

∑_{z_{i}∈\overline{Q}/\overline{P}} ∑_{t_{j}∈(Q∩T)/(P ∩T)} f(\overline{τ}_{n}^{Q,P} (\overline{q_{k_{i}}t_{j}}^{−1}y) ⊗ τ_{m}^{Q,P} ((q_{k_{i}}t_{j})^{−1}x)),

Because t_{j} ∈ Q ∩ T then \overline{q_{k_{i}}t_{j}} = \overline{q_{k_{i}}} and the formula simplifies to

∑_{z_{i}∈\overline{Q}/\overline{P}} ∑_{t_{j}∈(Q∩T)/(P ∩T)} f(\overline{τ}_{n}^{Q,P} (\overline{q_{k_{i}}}^{−1}y) ⊗ τ_{m}^{Q,P} (t_{j}^{−1}q_{k_{i}}^{−1}x)).

This coincides with the composition in the statement of the lemma. □

This last lemma proves in particular that the definition of H^{n,m}(B)(ι) does not depend on the representatives chosen to construct the maps τ_{∗}^{Q,P} and \overline{τ}_{∗}^{Q,P}. (Although B^{n,m}(ι) do depends on them.)

3.9. Corollary. For n, m ≥ 0 the assignment

H^{n,m}(B) : F → ℤ_{(p)} - mod

taking P to H^{n}(\overline{P}; H^{m}(P ∩ T; M)) and taking φ ∈ Hom_{F}(P, Q) to H^{n,m}(B)(φ) is a functor.

Proof. By Remarks 2.4 and 3.3 and Equation (3.4) it is enough to prove that for any P ≤ Q ≤ R we have

H^{n,m}(B)(ι_{Q}^{R}) ∘ H^{n,m}(B)(ι_{P}^{Q}) = H^{n,m}(B)(ι_{P}^{R})

and for any P ≤ Q →^{φ} φ(Q) we have

H^{n,m}(B)(ι_{P}^{Q}) ∘ H^{n,m}(A)(φ_{|P}) = H^{n,m}(A)(φ) ∘ H^{n,m}(B)(ι_{φ(P)}^{φ(Q)}).

We can check both conditions at the level of cochains: For the first condition, the definitions (3.5) of B^{n,m}(ι_{P}^{Q}), B^{n,m}(ι_{Q}^{R}) and B^{n,m}(ι_{P}^{R}) depend upon choices of representatives for the right cosets P \Q and \overline{P} \\overline{Q}, Q\R and \overline{Q}\\overline{R} and P \R and \overline{P} \\overline{R} respectively. Fix choices of representatives for the first four right cosets. Then the bijections P \Q × Q\R → P \R and \overline{P} \\overline{Q} × \overline{Q}\\overline{R} → \overline{P} \\overline{R} provide choices for the last two right cosets. With these choices we have

B^{n,m}(ι_{Q}^{R}) ∘ B^{n,m}(ι_{P}^{Q}) = B^{n,m}(ι_{P}^{R}).

For the second condition, the maps B^{n,m}(ι_{P}^{Q}) and B^{n,m}(ι_{φ(P)}^{φ(Q)}) depend on choices

of representatives for the right cosets P \Q and \overline{P} \\overline{Q}, and φ(P)\φ(Q) and \overline{φ(P)}\\overline{φ(Q)} respectively. Fix representatives in P \Q and \overline{P} \\overline{Q} and force the other choices via the bijections P \Q →^{φ} φ(P)\φ(Q) and \overline{P} \\overline{Q} →^{\overline{φ}} \overline{φ(P)}\\overline{φ(Q)}. Then we have

B^{n,m}(ι_{P}^{Q}) ∘ A^{n,m}(φ_{|P}) = A^{n,m}(A)(φ) ∘ B^{n,m}(ι_{φ(P)}^{φ(Q)}).
□

3.10. Proposition. For each p, q ≥ 0 the functor F → ℤ_{(p)} - mod with values

H^{p}(\overline{P}; H^{q}(P ∩ T; M)

and taking φ ∈ Hom_{F}(P, Q) to H^{p,q}(A)(φ) is a cohomological Mackey functor with covariant part taking φ ∈ Hom_{F}(P, Q) to H^{p,q}(B)(φ).

Proof. Property (1) from Definition 2.2 holds by Equation (3.4). Property (2) follows from property (1), the well known fact that conjugation induces the identity on cohomology, from Lemma 3.7 and from \overline{c_{p}} = c\overline{p} for p ∈ P ≤ S. Now we check property (3), also known as Mackey condition or double coset formula. So let Q, R ≤ P ≤ S. We will prove this condition at the level of cochains, i.e.:

A^{n,m}(ι_{Q}^{P}) ∘ B^{n,m}(ι_{R}^{P}) = ∑_{x∈Q\P/R} B^{n,m}(ι_{Q∩^{x}R}^{Q}) ∘ A^{n,m}(ι_{Q∩^{x}R}^{^{x}R}) ∘ A^{n,m}(c_{x^{−1}}_{|^{x}R}).

So let f ∈ A^{n,m}(R) = Hom_{R}(Bn ; \overline{R} ⊗ B_{R}^{m}, M), y ∈ Bn ; \overline{Q} and x ∈ B_{Q}^{m}. Then:

A^{n,m}(ι_{Q}^{P})(B^{n,m}(ι_{R}^{P})(f))(y⊗x) = B^{n,m}(ι_{R}^{P})(f)(\overline{ι_{Q}^{P}}(y)⊗ι_{Q}^{P}(x)) = B^{n,m}(ι_{R}^{P})(f)(y⊗x).

This equals

∑_{w∈P/R} f(\overline{τ}_{n}^{P,R}(\overline{w}^{−1}y) ⊗ τ_{m}^{P,R}(w^{−1}x)),

where w runs over a set of representatives of the left cosets P/R, \overline{τ}_{n}^{P,R} : Bn ; \overline{P} → Bn ; \overline{R} and τ_{m}^{P,R} : B_{P}^{m} → B_{R}^{m}. Now we let Q acts on the left on P/R and we group together the terms corresponding to a given Q-orbit in P/R:

∑_{p∈Q\P/R} ∑_{q∈Q/Q∩^{p}R} f(\overline{τ}_{n}^{P,R}(\overline{qp}^{−1}y) ⊗ τ_{m}^{P,R}((qp)^{−1}x)),

where now p runs over a set of representatives for the double cosets Q\P/R and q runs over a set of representatives of the left cosets Q/Q ∩^{n}R. This equals

∑_{p∈Q\P/R} ∑_{q∈Q/Q∩^{p}R} f(\overline{τ}_{n}^{P,R}(\overline{p}^{−1}\overline{q}^{−1}y) ⊗ τ_{m}^{P,R}(p^{−1}q^{−1}x)).

The right-hand side of the Mackey formula is

∑_{p∈Q\P/R} ∑_{q∈Q/Q∩^{p}R} f(\overline{p}^{−1}\overline{τ}_{n}^{Q,Q∩^{p}R}(\overline{q}^{−1}y)\overline{p} ⊗ p^{−1}τ_{q}^{Q,Q∩^{p}R}(q^{−1}x)p)

with \overline{τ}_{n}^{Q,Q∩^{p}R} : Bn ; \overline{Q} → Bn ; \overline{Q∩^{p}R}, τ_{m}^{Q,Q∩^{p}R} : B_{Q}^{m} → B_{Q∩^{p}R}^{m} and where we have realized

c_{p^{−1}}_{|^{p}R} at the level of cochains as

A^{n,m}(c_{p^{−1}}_{|^{p}R})(y ⊗ x) = (\overline{p}^{−1}y\overline{p} ⊗ p^{−1}xp).

The map τ_{m}^{P,R} depends on a choice of representatives for the right cosets R\P. Analogously, for any representative p ∈ Q\P/R, the map τ_{m}^{Q,Q∩^{p}R} is built out of a set of representatives of Q ∩^{p}R\Q. We want to choose representatives of R\P and of Q ∩^{p}R\Q for each double coset QpR such that

\xymatrix{Q \ar[r]^{ρ_{p}} \ar[d]_{q↦p^{−1}q} & Q ∩^{p}R \ar[d]^{q↦p^{−1}qp} \\ P \ar[r]^{ρ} & R}

commutes for each double coset QpR. For this is enough to choose arbitrary representatives q of Q ∩^{p}R\Q for each double coset QpR and build the representatives in R\P via the bijection

⨆_{p∈Q\P/R} Q ∩^{p}R\Q → R\P

that takes (Q∩^{p}R)q to Rp^{−1}q. The same argument for \overline{τ}_{n}^{P,R} and the maps \overline{τ}_{n}^{Q,Q∩^{p}R} finishes the proof of property (3).

To prove property (4) we go back to the level of cohomology. Let P ≤ Q ≤ S. By Lemmas 3.7 and 3.8 the composition H^{n,m}(B)(ι_{P}^{Q}) ∘ H^{n,m}(A)(ι_{P}^{Q}) is equal to

H^{n}(tr) ∘ H^{m}(tr′) ∘ H^{m}(ι_{P}^{Q}) ∘ H^{n}(\overline{ι_{P}^{Q}}).

Because cohomology over finite groups is a cohomological Mackey functor we know that H^{m}(tr′) ∘ H^{m}(ι_{P}^{Q}) is multiplication by |Q ∩ T |/|P ∩ T |. Moving out this factor we are left with H^{n}(tr) ∘ H^{n}(\overline{ι_{P}^{Q}}). As \overline{ι_{P}^{Q}} = ι\overline{Q} ; \overline{P} we obtain again by properties of cohomology for finite groups that this composition is multiplication by |\overline{Q}|/|\overline{P} |. So finally we obtain that H^{n,m}(B)(ι_{P}^{Q}) ∘ H^{p,q}(A)(ι_{P}^{Q}) is multiplication by

\frac{|Q ∩ T |}{|P ∩ T |} \frac{|\overline{Q}|}{|\overline{P} |} = \frac{|Q|}{|P |}.

□

3.11. Remark. In view of the proof of Lemma 3.8 we could have used

∑_{\overline{q}_{i}∈\overline{Q}/\overline{P}} ∑_{t_{j}∈(Q∩T)/(P ∩T)} f(\overline{τ}_{n}^{Q,P} (\overline{q_{i}}^{−1}y) ⊗ τ_{m}^{Q,P} (t_{j}^{−1}q_{i}^{−1}x)).

as an alternative definition in Equation 3.5.

4. Construction of the spectral sequence.

In this section we prove the main theorem of this paper:

4.1. Theorem. Let F be a fusion system over the p-group S, T a strongly F-closed subgroup of S and M a ℤ_{(p)}-module with trivial S-action. Then there is a first quadrant cohomological spectral sequence with second page

E_{2}^{n,m} = H^{n}(S/T; H^{m}(T; M))^{F}

and converging to H^{n+m}(F; M).

Proof. For each subgroup P ≤ S we have the short exact sequence

P ∩ T → P → \overline{P} = P/P ∩ T.

The construction of the Lyndon-Hochschild-Serre spectral sequence in MacLane’s book [11, XI.10.1] associates to this short exact sequence a double complex naturally isomorphic (Remark 3.1) to the double complex

A^{n,m}(P) = Hom_{P} (Bn ; \overline{P} ⊗ B_{P}^{m}, M)

defined in Section 3. This double complex we can filter either by columns or rows. If we filter by columns we obtain a spectral sequence {^{c}E_{k}^{∗,∗}(P), d_{k}}_{0≤k≤∞} whose second page is ^{c}E_{2}^{n,m}(P) = H^{n}(\overline{P}; H^{m}(P ∩ T; M)). If we filter by rows we obtain a spectral sequence {^{r}E_{r}^{∗,∗}(P), d_{k}}_{0≤k≤∞} whose second page collapses as^{r}E_{2}^{n,m}(P) = H^{m}(P; M) for n = 0 and ^{r}E_{2}^{n,m}(P) = 0 for n > 0.

For each morphism φ ∈ Hom_{F}(P, Q) we have morphisms of double complexes A^{n,m}(φ): A^{n,m}(Q) → A^{n,m}(P) and B^{n,m}(φ): A^{n,m}(P) → A^{n,m}(Q) defined in Section 3. These morphisms of double complexes induce morphism of spectral sequences consisting of a sequence of morphism of differential bigraded ℤ_{(p)}-modules

cE_{k}^{∗,∗}(A)(φ): ^{c}E_{k}^{∗,∗}(Q) → ^{c}E_{k}^{∗,∗}(P)
cE_{k}^{∗,∗}(B)(φ): ^{c}E_{k}^{∗,∗}(P) → ^{c}E_{k}^{∗,∗}(Q)

and

rE_{k}^{∗,∗}(A)(φ): ^{r}E_{k}^{∗,∗}(Q) → ^{r}E_{k}^{∗,∗}(P)
rE_{k}^{∗,∗}(B)(φ): ^{r}E_{k}^{∗,∗}(P) → ^{r}E_{k}^{∗,∗}(Q)

for 0 ≤ k ≤ ∞. We deal now with the filtration by columns spectral sequences. The second page ^{c}E_{2}^{∗,∗} is obtained by computing vertical cohomology followed by horizontal cohomology in in the double complex A^{∗,∗}. Hence we have

cE_{2}^{n,m}(P) = H^{n}(\overline{P}; H^{m}(P ∩ T; M)),
cE_{2}^{n,m}(A)(φ) = H^{n,m}(A)(φ) and ^{c}E_{2}^{n,m}(B)(φ) = H^{n,m}(B)(φ),

for P ≤ S and φ ∈ Hom_{F}(P, Q), where H^{n,m}(A) and H^{n,m}(B) are functors F → ℤ_{(p)} - mod by Remark 3.3 and Corollary 3.9 respectively. Hence, for each 2 ≤ k ≤ ∞, we have a contravariant functor

cE_{k}^{∗,∗}(A) : F → Differential bigraded ℤ_{(p)}-modules

and a covariant functor

cE_{k}^{∗,∗}(B) : F → Differential bigraded ℤ_{(p)}-modules.

On the one hand, we can take invariants for each 2 ≤ k ≤ ∞ to obtain a bunch of differential bigraded ℤ_{(p)}-modules

cE_{k}^{∗,∗}^{F} = {z ∈ ^{c}E_{k}^{∗,∗}(S)|^{c}E_{k}^{∗,∗}(A)(φ)(z) = ^{c}E_{k}^{∗,∗}(A)(ι_{P}^{S})(z) for P →^{φ} S}.

On the other hand, for k = 2, we have by Proposition 3.10 that (^{c}E_{2}^{∗,∗}(A),^{c}E_{2}^{∗,∗}(B)) is a cohomological Mackey functor. Because ^{c}E_{k+1}^{∗,∗} = H^{∗}(^{c}E_{k}^{∗,∗}, d_{k}) and because passing to cohomology preserves cohomological Mackey functors we deduce that cE_{k}^{∗,∗}(A) is a cohomological Mackey functor with covariant part ^{c}E_{k}^{∗,∗}(B) for 2 ≤ k < ∞. By Proposition 2.10 we obtain then that

(4.2)  ^{c}E_{k+1}^{∗,∗}^{F} = H^{∗}(^{c}E_{k}^{∗,∗}, d_{k})^{F} = H^{∗}(^{c}E_{k}^{∗,∗}^{F}, d_{k})

for 2 ≤ k < ∞. In fact, because the Lyndon-Hochschild-Serre spectral sequence is a first quadrant cohomological spectral sequence, for each subgroup P ≤ S and each n, m ≥ 0 we have that

cE_{k}^{n,m}(P) = ^{c}E_{k+1}^{n,m}(P) = … = ^{c}E_{∞}^{n,m}(P)

for k big enough. Because there are a finite number of subgroups of S we deduce that

cE_{k}^{n,m}^{F} = ^{c}E_{k+1}^{n,m}^{F} = … = ^{c}E_{∞}^{n,m}^{F}

for k big enough. Hence Equation (4.2) also holds for k = ∞ and we have obtained a spectral sequence

{^{c}E_{k}^{∗,∗}^{F}, d_{k}}_{2≤k≤∞}.

To study whether this spectral sequence converges recall that for P ≤ S the spectral sequence {^{c}E_{k}^{∗,∗}(P), d_{k}}_{0≤k≤∞} converges to H^{∗}(P; M). Hence we have short exact sequences

\xymatrix{0 \ar[r] & F^{n}H^{n+m}(P; M) \ar[r] & F^{n+1}H^{n+m}(P; M) \ar[r] & ^{c}E_{∞}^{n,m}(P) \ar[r] & 0}

where

0 ⊆ … ⊆ F^{n}H^{n+m}(P; M) ⊆ F^{n+1}H^{n+m}(P; M) ⊆ … ⊆ H^{n+m}(P; M)

is the filtration induced on H^{∗}(P; M) by the filtration by columns on the double complex A^{∗,∗}(P). This short exact sequence is natural with respect to morphisms of double complexes. Hence for each φ ∈ Hom_{F}(P, Q) we have morphisms of short exact sequences

\xymatrix{0 \ar[r] & F^{n}H^{n+m}(Q; M) \ar[d]^{F^{n}H^{n+m}(A)(φ)} \ar[r] & F^{n+1}H^{n+m}(Q; M) \ar[r] \ar[d]^{F^{n+1}H^{n+m}(A)(φ)} & ^{c}E_{∞}^{n,m}(Q) \ar[d]^{^{c}E_{∞}^{n,m}(A)(φ)} \ar[r] & 0 \\ 0 \ar[r] & F^{n}H^{n+m}(P; M) \ar[r] & F^{n+1}H^{n+m}(P; M) \ar[r] & ^{c}E_{∞}^{n,m}(P) \ar[r] & 0}

and

\xymatrix{0 \ar[r] & F^{n}H^{n+m}(P; M) \ar[d]^{F^{n}H^{n+m}(B)(φ)} \ar[r] & F^{n+1}H^{n+m}(P; M) \ar[r] \ar[d]^{F^{n+1}H^{n+m}(B)(φ)} & ^{c}E_{∞}^{n,m}(P) \ar[d]^{^{c}E_{∞}^{n,m}(B)(φ)} \ar[r] & 0 \\ 0 \ar[r] & F^{n}H^{n+m}(Q; M) \ar[r] & F^{n+1}H^{n+m}(Q; M) \ar[r] & ^{c}E_{∞}^{n,m}(Q) \ar[r] & 0.}

We want to show that the morphisms H^{n}(A)(φ): H^{n}(Q; M) → H^{n}(P; M) and H^{n}(B)(φ): H^{n}(P; M) → H^{n}(Q; M) induced by A and B on the targets of the spectral sequences are the usual maps in cohomology of groups. We consider the total complex of the double complex A^{∗,∗}(P) defined as usual by Tot^{s}(A) = ⨁_{n+m=s} A^{n,m}(P) and with total differential d^{h}+d^{v}. There is a chain map given by ζ: Hom_{P} (B_{P}^{∗} , M) → Tot^{∗}(A) sending f ∈ Hom_{P} (B_{P}^{m}, M) to ζ(f) ∈ A^{0,m} defined by

ζ(f)(\overline{p} ⊗ x) = f(x), \overline{p} ∈ \overline{P}, x ∈ B_{P}^{m}.

The map ζ induces an isomorphism between the cohomology of the total complex and H^{∗}(P; M) (cf. [11, p. 352]). Now, from the definitions of the maps A^{0,m}(φ): A^{0,m}(Q) → A^{0,m}(P) and B^{0,m}(ι_{P}^{Q}): A^{0,m}(P) → A^{0,m}(Q) it is easy to check that H^{n}(A)(φ) and H^{n}(B)(ι_{P}^{Q}) are the usual maps in cohomology of groups (see [4, (D), page 82]).

By properties of cohomology for finite groups (H^{n}(A), H^{n}(B)): F → ℤ_{(p)} - mod is a cohomological Mackey functor for each n ≥ 0. Hence so are the functors (F^{n}H^{n+m}(A), F^{n}H^{n+m}(B)): F → ℤ_{(p)} - mod induced in the filtration for n, m ≥ 0. By the arguments above also the pair (^{c}E_{∞}^{n,m}(A),^{c}E_{∞}^{n,m}(B)): F → ℤ_{(p)} - mod is a cohomological Mackey functor for n, m ≥ 0. Then by Lemma 2.9 we have a short exact sequence of ℤ_{(p)}-modules

\xymatrix{0 \ar[r] & (F^{n}H^{n+m})^{F} \ar[r] & (F^{n+1}H^{n+m})^{F} \ar[r] & ^{c}E_{∞}^{n,m}^{F} \ar[r] & 0.}

It is immediate that taking invariants and filtering commute and hence we have

\xymatrix{0 \ar[r] & F^{n}(H^{n+m}^{F}) \ar[r] & F^{n+1}(H^{n+m}^{F}) \ar[r] & ^{c}E_{∞}^{n,m}^{F} \ar[r] & 0,}

for the filtration of H^{n+m}^{F} = H^{n+m}(S)^{F} given by F^{n}(H^{n+m}^{F}) = F^{n}(H^{n+m}(S))∩ H^{n+m}(S)^{F}. This finishes the proof. □

As noted in the introduction each morphism φ : P → Q induces a morphism from the E_{2}-page of the Lyndon-Hochschild-Serre spectral sequence of the short exact sequence

Q ∩ T → Q → \overline{Q}

to that of

P ∩ T → P → \overline{P} .

This a morphism of differential graded algebras and hence the E_{2}-page of the spectral sequence in the statement of the theorem is a differential bigraded subalgebra of H^{p}(S/T; H^{q}(T; M)). In particular, the differential in the E_{2}-page of the spectral sequence of the theorem is the restriction to this subalgebra of the differential in the E_{2}-page of the Lyndon-Hochschild-Serre spectral sequence for T → S → S/T.

5. Comparison.

In this section we compare our spectral sequence and Lyndon-Hochschild-Serre spectral sequence. Let G be a finite group, K⊴G and S ∈ Syl_{p}(G). Then T = K∩S is a Sylow p-subgroup of K. Moreover, T is strongly F_{S}(G)-closed. Fix a ℤ_{(p)}- module M with trivial G-action. The Lyndon-Hochschild-Serre spectral sequence E_{∗,G} of the extension K → G → G/K is

H^{n}(G/K; H^{m}(K; M)) ⇒ H^{n+m}(G; M)

meanwhile the spectral sequence E_{∗} from Theorem 1.1 associated to T is

H^{n}(S/T; H^{m}(T; M))^{F_{S}(G)} ⇒ H^{n+m}(F; M).

Note that by the classical stable elements theorem [6, XII.10.1], attributed to Tate by Cartan and Eilenberg, H^{∗}(G; M) = H^{∗}(F; M) and both spectral sequences converge to the same target. Recall that, by construction, E_{∗} is a sub-spectral sequence of the Lyndon-Hochschild-Serre spectral sequence E_{∗,S} of T → S → S/T.

5.1. Theorem. The spectral sequences E_{∗,G} and E_{∗} are isomorphic.

Proof. Consider the category F_{G}(G) with objects the subgroups of G and morphisms given by Mor_{F_{G}(G)}(H, I) = Hom_{G}(H, I). Clearly F_{S}(G) is a full subcategory of F_{G}(G). For each subgroup H ≤ G we have a short exact sequence

H ∩ K → H → \overline{H} = H/H ∩ K.

If φ = c_{g}: H → I is a morphism in F_{G}(G) then, as K is normal in G, conjugation by g ∈ G takes H ∩K → H → \overline{H} to I ∩K → I → \overline{I}. Exactly the same construction of Section 3 gives a cohomological Mackey functor (A, B): F_{G}(G) → CCh^{2}(ℤ_{(p)}) with values H ↦ A^{n,m}(H) = Hom_{H}(Bn ; \overline{H} ⊗ B_{H}^{m}, M), where B_{H}^{∗} and B∗ ; \overline{H} are the bar

17

resolutions for H and \overline{H} respectively. Moreover, for H ≤ S, as T = K ∩ S, we have H ∩ K = H ∩ T and this functor over F_{G}(G) extends the one built in Section 3 over F_{S}(G).

The inclusion of the short exact sequence T → S → S/T into K → G → G/K gives a morphism {res_{r}}_{r≥2} of spectral sequences from E_{∗,G} into E_{∗,S}. The morphism of differential graded algebras res_{2} : E_{2,G} → E_{2,S} coincides with the morphism induced in cohomology by the functor A applied to the inclusion morphism S ≤ G of F_{G}(G), H^{∗,∗}(A)(ι_{S}^{G}). Applying the functor B to the same inclusion S ≤ G we get another morphism going in the opposite direction (transfer)

\xymatrix{H^{n}(G/K; H^{m}(K; M)) \ar[r]^{H^{n,m}(A)(ι_{S}^{G})} & H^{n}(S/T; H^{m}(T; M)). \ar[l]^{H^{n,m}(B)(ι_{S}^{G})}}

Recall that E_{2} ≤ E_{2,S} are exactly the F-stable elements H^{p}(S/T; H^{q}(T; M))^{F_{S}(G)}. Because conjugation by g ∈ G induces the identity on H^{p}(G/K; H^{q}(K; M)) it is straightforward that res_{2}(E_{2,G}) ≤ E_{2}. Hence {res_{r}}_{r≥2} is a morphism of spectral sequences E_{∗,G} → E_{∗}. If we prove that res_{2}(E_{2,G}) = E_{2} then res_{2} is an isomorphism and hence res_{r} is an isomorphism for each r ≥ 2 and we are done. To see that res_{2}(E_{2,G}) = E_{2} we proceed as usual when there is a Mackey functor available (cf. [4, Theorem III.10.3]). Let z ∈ H^{n}(S/T; H^{m}(T; M))^{F_{S}(G)} and consider w = H^{n,m}(B)(ι_{S}^{G})(z) ∈ H^{n}(G/K; H^{m}(K; M)). By the double coset formula 2.2(3) and the cohomological condition 2.2(4) and because z is F_{S}(G)-stable we obtain

H^{n,m}(A)(ι_{S}^{G})(w) = ∑_{x∈S\G/S} B(ι_{S∩^{x}S}^{S})A(ι_{S∩^{x}S}^{^{x}S})c_{x^{−1}}^{∗}(z)
= ∑_{x∈S\G/S} B(ι_{S∩^{x}S}^{S})A(ι_{S∩^{x}S}^{S})(z)
= ∑_{x∈S\G/S} |S : S ∩^{x}S|z = |G : S|z.

As q = |G : S| is a p′-number it follows that z = res(\frac{w}{q}). □

6. Tate’s Theorem.

In this section we prove Tate’s theorem following his original cohomological proof. Associated to every first quadrant spectral sequence there is a five terms exact

sequence. In the case of the Lyndon-Hochschild-Serre spectral sequence for K ⊴ G and the G-module M we obtain the inflation-restriction exact sequence:

(6.1) 0 → H^{1}(G/K; M^{K}) → H^{1}(G; M) → H^{1}(K, M)^{G/K} → H^{2}(G/K; M^{K}) → H^{2}(G; M),

where the second arrow from the right is the transgression. The five terms exact sequence for the spectral sequence of Theorem 1.1 for the fusion system F over the p-subgroup S with strongly F-closed subgroup T and ℤ_{(p)}-module M with trivial S-action is

(6.2) 0 → H^{1}(S/T; M)^{F} → H^{1}(F; M) → H^{1}(T; M)^{F} → H^{2}(S/T; M)^{F} → H^{2}(F; M),

where the arrow H^{1}(T; M)^{F} → H^{2}(S/T; M)^{F} is the transgression. Recall that H^{1}(F; M) = H^{1}(S; M)^{F} and H^{2}(F; M) = H^{2}(S; M)^{F}. The inclusion of the E_{2}- page of the spectral sequence of Theorem 1.1 into the E_{2}-page of the Lyndon-Hochschild-Serre spectral sequence for T → S → S/T induces a restriction map of five terms exact sequences:

\xymatrix{0 \ar[r] & H^{1}(S/T; M)^{F} \ar[r] \ar[d]^{f_{1}} & H^{1}(F; M) \ar[r] \ar[d]^{g_{1}} & (H^{1}(T; M)^{S/T})^{F} \ar[r] \ar[d]^{h_{1}} & H^{2}(S/T; M)^{F} \ar[r] \ar[d]^{f_{2}} & H^{2}(F; M) \ar[d]^{g_{2}} \\ 0 \ar[r] & H^{1}(S/T; M) \ar[r] & H^{1}(S; M) \ar[r] & H^{1}(T; M)^{S/T} \ar[r] & H^{2}(S/T; M) \ar[r] & H^{2}(S; M).}

Notice the following:

(1) The maps g_{1} and g_{2} are injections as H^{1}(S; M)^{F} and H^{2}(S; M)^{F} are subgroups of H^{1}(S; M) and H^{2}(S; M) respectively.

(2) H^{1}(S; 𝔽_{p}) = Hom_{Ab}(S, 𝔽_{p}) = Hom_{Ab}(S/S^{p}[S, S], 𝔽_{p}) and

H^{1}(F; 𝔽_{p}) = Hom_{Ab}(S/S^{p}[S, F], 𝔽_{p})

where [S, F] = ⟨[P, Aut_{F}(P)]|P ≤ S⟩ is the focal subgroup of S.

(3) H^{1}(T; 𝔽_{p})^{S/T} = Hom_{S/T} (T/T^{p}[T, T], 𝔽_{p}) = Hom_{Ab}(T/T^{p}[T, S], 𝔽_{p}) and

hence

(H^{1}(T; 𝔽_{p})^{S/T})^{F} = Hom_{Ab}(T/T^{p}[T, F], 𝔽_{p}).

(4) H^{1}(S/T; M)^{F} = H^{1}(S/T; M)^{F/T} and H^{2}(S/T; M)^{F} = H^{2}(S/T; M)^{F/T}

by Remark 3.2. Now we are ready to prove Tate’s theorem.

6.3. Theorem ([14]). Let F be a fusion system over the p-group S. If the restriction map H^{1}(F; 𝔽_{p}) → H^{1}(S; 𝔽_{p}) is an isomorphism then F = F_{S}(S).

Proof. The isomorphism in the statement is equivalent to S^{p}[S, S] = S^{p}[S, F] by (2) above. Throughout we will need to consider the hyperfocal subgroup of F

O_{F}^{p}(S) = ⟨[P, O^{p}(Aut_{F}(P)]|P ≤ S⟩.

Define a series of subgroups of S by S_{0} = S and S_{n+1} = S_{n}^{p}[S_{n}, S]. Define another series of subgroups of S by T_{0} = S and T_{n+1} = T_{n}^{p}[T_{n}, F]. The hypothesis reads now as S_{1} = T_{1}. Moreover, T_{1} is strongly F-closed and contains O_{F}^{p}(S) by [8, A.6]. We proof by induction that this is the case for any n ≥ 1, i.e., that T_{n} is strongly F-closed, contains O_{F}^{p}(S) and S_{n} = T_{n} for any n ≥ 1.

So assume the hypothesis holds for T_{n}. As T_{n} is strongly F-closed in S we have a restriction map as above for M = 𝔽_{p} coefficients:

\xymatrix{0 \ar[r] & H^{1}(S/T_{n})^{F} \ar[r] \ar[d]^{f_{1}} & H^{1}(F) \ar[r] \ar[d]^{g_{1}} & (H^{1}(T_{n})^{S/T_{n}})^{F} \ar[r] \ar[d]^{h_{1}} & H^{2}(S/T_{n})^{F} \ar[r] \ar[d]^{f_{2}} & H^{2}(F) \ar[d]^{g_{2}} \\ 0 \ar[r] & H^{1}(S/T_{n}) \ar[r] & H^{1}(S) \ar[r] & H^{1}(T_{n})^{S/T_{n}} \ar[r] & H^{2}(S/T_{n}) \ar[r] & H^{2}(S).}

Because O_{F}^{p}(S) ≤ T_{n} the quotient F/T_{n} is a p-group, i.e., F/T_{n} = F_{S/T_{n}}(S/T_{n}). Hence by point (4) before the proof we have that both f_{1} and f_{2} are isomorphisms. Also by hypothesis we have that T_{1} = S_{1} and hence g_{1} is an isomorphisms by (2) above. Hence by the five lemma and (1) we get that h_{1} is an isomorphism. Then by (3) we deduce that T_{n}^{p}[T_{n}, S] = T_{n}^{p}[T_{n}, F], i.e., that S_{n+1} = T_{n+1}. To prove that O_{F}^{p}(S) is contained in T_{n+1} consider the unique p-power index fusion subsystem F_{T_{n}} of F on T_{n} [3, 4.3]. Then by [8, A.14] we have that O_{F_{Tn}}^{p} (T_{n}) = O_{F}^{p}(S). By [8, A.6] we have that O_{F}^{p}(S) ≤ T_{n}^{p}[T_{n}, F_{T_{n}}] ≤ T_{n}^{p}[T_{n}, F] = T_{n+1}. Finally, as S_{n+1} is normal in S so is T_{n+1}. Then by [8, A.7(1)] T_{n+1} is strongly F-closed in F. This finishes the induction.

We have proven in particular that O_{F}^{p}(S) ≤ T_{n} = S_{n} for each n ≥ 1. Clearly S_{n} is the trivial group for n big enough as S is a finite p-group. Hence we deduce that O_{F}^{p}(S) = 1 and this implies that there are no p′-automorphisms in F, i.e., F = F_{S}(S). □

References

[1] Aschbacher M., Normal subsystems of fusion systems, PLMS (3) 97 (2008), pp. 239-271.

[2] C. Broto, R. Levi and B.Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779-856.

[3] C. Broto, N. Castellana, J. Grodal, R. Levi and B. Oliver Extensions of p-local finite groups, Trans. Amer. Math. Soc. 359 (2007), 3791-3858.

[4] K.S. Brown, Cohomology of groups, Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin, 1982.

[5] J. Cantarero, J. Scherer and A. Viruel, Nilpotent p-local finite groups, preprint 2011, arXiv:1107.5158v1 [math.AT]

[6] H. Cartan and S. Eilenberg, Homological Algebra, Princenton University Press 1956.

[7] D. Craven, Control of Fusion and Solubility in Fusion Systems, J. Algebra 323 (2010), no. 9, 2429–2448.

[8] A. Díaz, A. Glesser, R. Stancu and S. Park, Tate’s and Yoshida’s theorems on control of transfer for fusion systems, J. Lond. Math. Soc. (2) 84 (2011), no. 2, 475-494.

[9] A.Díaz, A. Ruiz, A. Viruel, All p-local finite groups of rank two for odd prime p, Trans. Amer. Math. Soc. 359 (2007), no. 4, 1725-1764.

[10] R.J. Flores, R.M. Foote, Strongly closed subgroups of finite groups, Adv. Math. 222 (2009), no. 2, 453-484.

[11] S. Mac Lane, Homology, Springer-Verlag 1963.

[12] S. Park, Realizing a fusion system by a single finite group, Archiv der Mathematik, Volume 94, Number 5, 405-410.

[13] R. Solomon, R. Stancu, Conjectures on finite and p-local groups, L’Enseignement Mathematique 54, (2008), 61-66.

[14] J. Tate, Nilpotent quotient groups, Topology 3, (1964) suppl. 1, 109-111.

[15] P. Webb, A Guide to Mackey Functors, Handbook of Algebra, vol. 2, editor M. Hazewinkel, Elsevier 2000, 805-836.

Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Apdo correos 59, 29080 Málaga, Spain.

E-mail address: adiaz@agt.cie.uma.es