HOMOLOGY DECOMPOSITIONS AND GROUPS INDUCING FUSION SYSTEMS

ASSAF LIBMAN AND NORA SEELIGER

Abstract. We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson’s construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.

1. The main results

The starting point of this paper is the results of Leary-Stancu in [9] and Robinson in [19]. For any saturated fusion system F on a finite p-group S they give recipes to construct groups π_{LS} (for Leary-Stancu) and π_{R} (for Robinson) which are generally infinite and which contain S as a Sylow p-subgroup, see §2.2, and realize F in the sense that F = F_{S}(π). Also, for primes q = 3, 5 mod (8), Aschbacher and Chermak constructed in [1] an infinite group π_{AC} which contains a Sylow subgroup of Spin_{7}(q^{n}) as its Sylow p-subgroup and which realizes Solomon’s fusion system F_{Sol}(q^{n}), see [10].

The primary goal of this paper is to give a uniform and conceptual treatment for all these constructions. Our approach is geometric rather than algebraic. The main observation in this paper is Theorem 1.1 below which we prove in the end of section 2. Also, it was observed by Aschbacher that if a saturated fusion system F has the form F_{S}(π) where π is possibly infinite, then a signaliser functor on π, see §2.2 for details, gives rise to an associated linking system L. A partial converse to this observation is point (2) of Theorem 1.1.

Theorem 1.1. Fix a p-local finite group (S, F, L) and let π be a group which contains S as a Sylow p-subgroup. Assume that F ⊆ F_{S}(π) and that there exists a map f: Bπ → |L|_{p}^{∧} whose restriction to BS ⊆ Bπ is homotopic to the natural map θ: BS → |L|_{p}^{∧}. Then

(1) F = F_{S}(π)

(2) There is a signaliser functor Θ on π which induces L, namely L is a quotient of the transporter system T_{S}^{c}(π).

(3) H^{∗}(|L|; 𝔽_{p}) is a retract of H^{∗}(Bπ; 𝔽_{p}) in the category of 𝔽_{p}-algebras. It is equal to the image of H^{∗}(π; 𝔽_{p}) ⟶^{res} H^{∗}(S; 𝔽_{p}).

The point of Propositions 4.1, 4.5 and 4.9 below is that the groups π_{LS}, π_{R} and π_{AC} satisfy the hypotheses of this theorem, hence recovering the results of [9, 19] and [1]; See Remarks 1.3–1.6 below. The conceptual reason that Theorem 1.1 applies to these groups, which is the main message of this paper, is that all these groups are built from different homology decompositions which express |L| as the homotopy colimit of a diagram of classifying spaces of finite groups. By taking homotopy colimits of carefully chosen subdiagrams we obtain classifying spaces Bπ together with maps Bπ → |L|_{p}^{∧} where π induces the fusion system F. The groups π_{LS} are built in §4.2 from the subgroup decomposition [4, §2]. The groups π_{R} are built in §4.1 from the normalizer decomposition with respect to the collection of the F-centric subgroups [12]. The groups π_{AC} are related to the normalizer decomposition with respect to the collection of the elementary abelian subgroup of F_{Sol}(q^{n}) as we explain in §4.3.

Date: June 30, 2010.

An interesting conclusion of Theorem 1.1(2) is the following result whose proof is given in §§4.1, 4.2. It settles a question which was open for some time.

Corollary 1.2. Fix a saturated fusion system F on S and let π be either Robinson’s group π_{R} or Leary-Stancu’s group π_{LS} so that F = F_{S}(π). Then the following are equivalent.

(1) F has an associated linking system L.

(2) There exists a signaliser functor on π which induces L.

Remark 1.3. We explain in Remark 4.2 below, see also Proposition 2.2, that the structure of a p-local finite group is not essential to deduce point (1) of Theorem 1.1 for arbitrary saturated fusion systems. Thus, even though Theorem 1.1 is not quite stated in this way, our results are as strong as those of [9] and [19] which apply to arbitrary saturated fusion systems.

Remark 1.4. Our approach shows that Robinson’s original construction isn’t quite the natural one. The group π_{R} that we construct in §4.1 is a factor group of Robinson’s original construction [19]. See Remark 4.3 below. Our costruction of π_{R} is inspired by the results of the first author and Antonio Viruel in [13].

Remark 1.5. The construction of π_{AC} in [1] requires q ≡ 3 or 5 mod (8). This is particularly important for the construction of the signaliser functors. The group π_{AC} that we construct in §4.3 is a variation of Aschbacher-Chermak’s original construction. Our group which is motivated by the geometric approach, requires no restriction on q mod (8), yet it admits a signaliser functor, see Remark 4.10. Thus, our construction improves Aschbacher-Chermak’s albeit at the price of not being as explicit as their’s.

Remark 1.6. Finally, point (3) of Theorem 1.1 applies to π_{LS}, π_{R} and π_{AC} and as far as we know, this cohomological information is new.

From the cohomological point of view, the groups π_{R} and π_{AC} are very close to their “target” namely the ring H^{∗}(F) of the stable elements, see [4, Theorem B], in a sense made precise in the next result which is proven in §§4.1, 4.3.

Proposition 1.7. Let π be either the group π_{R} associated to a saturated fusion system F (see §4.1) or the group π_{AC} if F = F_{Sol}(q^{n}) and p = 2 (see §4.3). Then

(1) Bπ is p-good in the sense of [6, I.§5].

(2) In the kernel of H^{∗}(Bπ; 𝔽_{p}) ⟶^{Bi^{∗}} H^{∗}(BS; 𝔽_{p}) the product of any two elements is zero.

2. Saturated fusion systems and p-local finite groups

In this section we will very briefly recall the notion of p-local finite groups and of transporter systems. We will take the approach of Broto Levi and Oliver in [4] and Oliver-Ventura [16, §§2,3]. Also see [5]. Fusion systems were first defined by Puig, e.g. [17].

2.1. Fusion systems. A fusion system on a finite p-group S is a small category F whose object set is the set of subgroups of S. Morphism sets F(P, Q) are group monomorphisms which include the set Hom_{S}(P, Q) of conjugations by elements of S. In addition, every morphism P → Q in F is the composition of an isomorphism P → P′ in F, which is necessarily a group isomorphism, followed by an inclusion P′ ≤ Q. From the definitions, every endomorphism in F is an automorphism.

Subgroups P, P′ ≤ S are called F-conjugate if they are isomorphic as objects of F. A subgroup P ≤ S is called fully centralized if P has the largest S-centralizer among all the subgroups in its F-conjugacy class. Similarly, P is fully normalized if its S-normalizer has the largest possible order among all the groups in its Fconjugacy class. We say that F is saturated if the following two conditions hold. First, every fully normalized subgroup P ≤ S is fully centralized and in this case Aut_{S}(P) = N_{S}(P)/C_{S}(P) is a Sylow p-subgroup of Aut_{F}(P). Second, if φ ∈ F(P, S) is such that φ(P) is fully centralized then φ extends to a morphism φ̃ ∈ F(N_{φ}, S) where N_{φ} ≤ N_{S}(P) consists of the elements g such that φ ∘ c_{g} ∘ φ^{−1} ∈ Aut_{S}(φ(P)) where c_{g} is the inner automorphism x ↦ gxg^{−1} of S.

A subgroup P ≤ S is called F-centric if P and all its F-conjugates contain their S-centralizer. We let F^{c} denote the full subcategory of F on the F-centric objects. A subgroup P ≤ S is called F-radical if Inn(P) is the maximal normal p-subgroup of Aut_{F}(P). Alperin’s fusion theorem [4, Theorem A.10] says that any saturated fusion system F is generated by the F-automorphism groups of its F-centric Fradical subgroups. Namely, every morphism in F is obtained by composition of restrictions of F-automorphisms of these groups.

An important source of saturated fusion systems are finite groups. If S is a Sylow p-subgroup of a finite group G, one forms the category F_{S}(G) whose objects are the subgroups of S and whose morphisms are the group monomorphisms c_{g}: P → Q of the form x ↦ gxg^{−1} for some g ∈ G.

2.2. Transporter systems, linking systems and signaliser functors. A finite p-subgroup S of a group π, possibly infinite, is a Sylow p-subgroup if any finite psubgroup Q ≤ π is conjugate to a subgroup of S.

Define F_{S}(π) where S ≤ π is a Sylow p-subgroup as we did above for finite groups; Up to isomorphism the choice of S is immaterial.

The objects of the transporter system T_{S}(π) are the subgroups of S; its morphism sets are

T_{S}(π)(P, Q) = N_{π}(P, Q) = {g ∈ π: gP g^{−1} ≤ Q}.

There is an obvious surjective functor T_{S}(π) → F_{S}(π) which is the identity on objects and is the quotient by the action of C_{π}(P) on morphism sets. We denote by T_{S}^{c}(π) the full subcategory whose objects are the F-centric subgroups of S for some fusion system F on S which will be understood from the context.

An associated centric linking system L to a saturated fusion system F on S is a category whose objects are the F-centric subgroups. It is equipped with a surjective functor π: L → F^{c} and an injective functor δ: T_{S}^{c}(S) → L, both induce the identity on object sets. In addition the following three conditions must be satisfied. First, the image of Z(P) under δ: N_{S}(P) → Aut_{L}(P) acts freely on L(P, Q), and F(P, Q) is isomorphic to the orbit set via π. Second, for any g ∈ N_{S}(P, Q) we have π(δ(g)) = c_{g}. Third, for any f ∈ L(P, Q) and any g ∈ P ≤ N_{S}(P) = Aut_{T_{S}^{c}(S)}(P) we have δ(π(f)(g)) ∘ f = f ∘ δ(g) in L.

A p-local finite group is a triple (S, F, L) of a saturated fusion system on S together with an associated linking system. It is not known if any saturated fusion system admits a centric linking system, and if so, if it is unique.

Now suppose that F = F_{S}(π) for a group π which contains S as a Sylow psubgroup. A signaliser functor on π is an assignment P ↦ Θ(P) for every Fcentric P ≤ S such that Θ(P) is a complement of Z(P) in C_{π}(P) and such that if gP g^{−1} ≤ Q then Θ(Q) ≤ gΘ(P)g^{−1}. Indeed, Θ is a contravariant functor from T_{S}^{c}(π) to the category of groups. A signaliser functor Θ, if it exists, gives rise to a centric linking system L = L_{Θ} where L(P, Q) = N_{π}(P, Q)/Θ(P). See [1] and also [13, Section 3] for more details.

2.3. Homotopy theoretic constructions. Fix a finite p-group S and a map f: BS → X. By [4, Definition 7.1] f gives rise to a fusion system F_{S}(f) on S whose objects are the subgroups of S and a group monomorphism φ: P → Q belongs to F_{S}(f) if and only if BP ⟶^{Bφ} BS ⟶^{f} X is homotopic to BP ⟶^{Bi} BS ⟶^{f} X. It follows directly from the definitions that any map g: X → Y induces an inclusion of fusion systems (not saturated in general)

F_{S}(f) ⊆ F_{S}(g ∘ f).

The category L_{S}(f), see [4, Definition 7.2], has the same object set as F = F_{S}(f). The morphisms P → Q in L_{S}(f) are pairs (φ, [H]) where φ ∈ F(P, Q) and [H] is the homotopy class of a path H in map(BP, X) from BP ⟶^{Bφ} BS ⟶^{f} X to BP ⟶^{Bi} BS ⟶^{f} X. There is a surjective functor π: L_{S}(f) → F_{S}(f) which “forgets” [H].

A map g: X → Y gives rise, by composition, to a functor

g_{∗}: L_{S}(f) → L_{S}(g ∘ f)

over the projection functors to F_{S}(f) ⊆ F_{S}(g ∘ f), i.e. π ∘ g_{∗} = incl_{F_{S}(f)}^{F_{S}(gf)} ∘ π.

Define L_{S}^{c}(f) as the full subcategory of L_{S}(f) on the F_{S}(f)-centric objects. If g: X → Y is a map then F_{S}(f) ⊆ F_{S}(g ∘ f) so every F_{S}(g ∘ f)-centric subgroup of S is also F_{S}(f)-centric and we obtain a functor

g_{∗}: L_{S}^{c}(f) → L_{S}^{c}(g ∘ f)

defined on the F_{S}(g ∘ f)-centric objects of L_{S}^{c}(f).

2.4. Classifying spaces. The classifying space of a p-local finite group is |L|_{p}^{∧}. The inclusion δ: S → Aut_{L}(S) thought of as a full subcategory of L gives rise to a map

(1)  θ: BS → |L|_{p}^{∧}

which will play a fundamental role in this paper. One of the key features of the map θ is [4, Proposition 7.3]

F = F_{S}(θ)  and  L = L_{S}^{c}(θ).

For any group G let BG denote the category with one object o_{G} and G as its automorphism set. A homomorphism φ: G′ → G induces a functor Bφ: BG′ → BG.

It is well known that if H ≤ G then the functor BC_{G}(H) × BH ⟶^{(x,y)↦xy} BG gives rise, after realization and adjunction, to a homotopy equivalence BC_{G}(H) ⟶^{≃} map(BH, BH)_{Bi} where the space on the right is the path component of the inclu-

sion BH ⟶^{Bi} BG.

Proposition 2.1. Fix an inclusion S ≤ π of a Sylow p-subgroup and consider Bi: BS → Bπ. Then F_{S}(Bi) = F_{S}(π) and L_{S}(Bi) = T_{S}(π).

Proof. The first isomorphism is clear because [BP, Bπ] = Hom(P, π)/ Inn(π). Define a functor Φ: T_{S}(π) → L_{S}(Bi) as follows. It is the identity on objects and for a morphism g ∈ N_{π}(P, Q) consider c_{g}: P → Q and the natural transformation g: Bc_{g} → Bi_{P}^{π} between the functors c_{g}, i_{P}^{S} : BP → Bπ given by o_{π} ⟶^{g} o_{π}. This natural transformation gives rise to a homotopy I × BP → Bπ from Bc_{g} and Bi_{P}^{π}, whence to a path H in map(BP, Bπ) from Bc_{g} to Bi_{P}^{π}. Define Φ(g) = (c_{g}, [H]).

The functoriality of Φ is easy to check. It is bijective because Φ maps the coset C_{π}(P)g ≤ N_{π}(P, Q) bijectively onto π_{1}^{#}(map(BP, Bπ); Bc_{g}, Bi_{P}^{π}) ≅ π_{1}BC_{π}(P) via BC_{π}(P) × BP → Bπ. □

2.5. The subgroup decomposition. Recall that the orbit category O = O(F) of a fusion system F has the same object set as F and O(P, Q) = F(P, Q)/ Inn(Q). See [4, Definition 2.1]. Let O(F^{c}) denote the full subcategory on the F-centric objects.

Fix a p-local finite group (S, F, L). By [4, Proposition 2.2] there is a functor called the subgroup decomposition

β: O(F^{c}) → Top

which as a functor into hoTop - the homotopy category of spaces - is isomorphic to the functor P ↦ BP. Moreover, there is a homotopy equivalence

hocolim_{O(F^{c})} β ⟶^{≃} |L|

whose restriction to β(S) ≃ BS is homotopic to the map BS → |L| induced by the inclusion S ≤ Aut_{L}(S). Hence θ_{p}^{∧} is homotopic to the map in display (1).

2.6. The normalizer decomposition. Let R be a set of F-centric subgroups of S which is closed under F-conjugation. The subdivision category S̄(R) is the following poset; see [12] for more details where it is denoted s̄d(R). The objects of S̄(R) are the F-conjugacy classes [P_{•}] of chains of inclusions P_{0} ⪇ ⋯ ⪇ P_{k} of subgroups in R called “k-simplices” (k ≥ 0). We say that P_{•} is F-conjugate to P′_{•} = P′_{0} ⪇ ⋯ ⪇ P′_{k} if there is an isomorphism P_{k} → P′_{k} in F which carries P_{i} isomorphically onto P′_{i} for all i. There is a unique morphism [P_{•}] → [Q_{•}] in S̄(R) if Q_{•} is F-conjugate to a sub-sequence (a “subsimplex”) of P_{•}.

Let Aut_{F}(P_{•}) be the subgroup of Aut_{F}(P_{k}) which leaves P_{0}, …, P_{k} invariant. Let Aut_{L}(P_{•}) be its preimage in Aut_{L}(P_{k}). Suppose that Q_{•} = Q_{0} ⪇ ⋯ ⪇ Q_{m} is a sub-simplex of P_{•} where Q_{m} = P_{i} for some i. Consider e ∈ N_{π}(P_{i}, P_{k}) and set ê = δ(e) ∈ L(P_{i}, P_{k}). For every φ ∈ Aut_{L}(P_{•}) we have π(φ) ∈ Aut_{F}(P_{•}) and therefore π(φ)|_{P_{i}} ∈ Aut_{F}(Q_{•}). It follows from [4, Lemma 1.10(a)] and the fact that ê is both an epimorphism and a monomorphism in L, see [2, Corollary 3.10] that there exists a unique φ′ ∈ Aut_{L}(Q_{•}) such that φ ∘ ê = ê ∘ φ′. This sets up a canonical group monomorphism Aut_{L}(P_{•}) ↪ Aut_{L}(Q_{•}). See [12] for more details. We obtain a functor [P_{•}] ↦ BAut_{L}(P_{•}) into the homotopy category of spaces (here we use the fact that the self-map that an inner automorphism of a group induces on its classifying space is homotopic to the identity).

The normalizer decomposition with respect to the collection R of F-centric subgroups, see [12, Theorem A], is a functor

δ: S̄(R) → Top

which as a functor into hoTop is isomorphic to the functor [P_{•}] ↦ BAut_{L}(P_{•}). In addition there is a map

hocolim_{S̄(R)} δ ⟶^{g} |L|

whose restriction to δ([P_{•}]) is induced, upon taking nerves, from the inclusion of categories BAut_{L}(P_{•}) ⊆ L. Hence, the restriction of hocolim_{S̄(R)} δ ≃ |L| → |L|_{p}^{∧} to BS ≤ BAut_{L}(S) is homotopic to θ in display (1). If R contains all the F-centric F-radical subgroups of S then g is a homotopy equivalence, [12, Theorem A].

Proof of Theorem 1.1. (1) Let Bi: BS → Bπ denote the inclusion. By Proposition 2.1 the map f induces an inclusion

F_{S}(π) = F_{S}(Bi) ⊆ F_{S}(f ∘ Bi) = F_{S}(θ) = F.

By hypothesis F ⊆ F_{S}(π) and therefore equality holds.

(2) We deduce from Proposition 2.1 and from §2.3 that L_{S}^{c}(Bi) = T_{S}^{c}(π), that L_{S}^{c}(f ∘ Bi) = L because f ∘ Bi ≃ θ, and that f induces a functor ρ: T_{S}^{c}(π) → L such that

[figure: T_{S}^{c}(π); π; ρ; L; π; F^{c}; F^{c}]

commutes. We will now prove, and this is the key observation, that ρ is surjective.

Set T = T_{S}^{c}(π). Since Z(P) ≤ Aut_{L}(P) acts on L(P, Q) with orbit set F(P, Q) and since the functors π in the square above are surjective, it suffices to prove that Z(P) is in the image of ρ_{P} : Aut_{T} (P) → Aut_{L}(P) for any F-centric P ≤ S. To do this consider the inclusion Bi_{P}^{S} : BP → BS. The composition

map(BP, BS)_{Bi_{P}^{S}} ⟶^{Bi_{∗}} map(BP, Bπ)_{Bi_{P}^{π}} ⟶^{f_{∗}} map(BP, |L|_{p}^{∧})_{f∘Bi_{P}^{π}}

is a homotopy equivalence because P is F-centric so the first space is homotopy equivalent to BC_{S}(P) = BZ(P) and because the third space is homotopy equivalent to BZ(P) by [4, Theorem 4.4(c)] since Bi ∘ f ≃ θ. We see from the definition of L_{S}(−), see §2.3, that ρ_{P} carries Z(P) ≤ N_{π}(P) isomorphically onto Z(P) ≤ Aut_{L}(P).

We will now define the signaliser functor Θ. For every F-centric P ≤ S set

Θ(P) =^{def} Ker(Aut_{T} (P) → Aut_{L}(P)).

Now, Aut_{T} (P) = N_{π}(P) so we obtain a commutative diagram with exact rows

[figure: 1; C_{π}(P); N_{π}(P); ρ_{P}; π; Aut_{F}(P); 1; 1; Z(P); Aut_{L}(P); π; Aut_{F}(P); 1]

which implies that Θ(P) ≤ C_{π}(P) and that it is a complement of Z(P) because we have seen that ρ_{P} (Z(P) = Z(P). For g ∈ π such that gP g^{−1} ≤ S we obtain a commutative diagram

\xymatrix{1 \ar[r] & Θ(P) \ar[r] \ar[d] & N_{π}(P) \ar[r] \ar[d]^{x↦gxg^{−1}}_{≅} & Aut_{L}(P) \ar[r] \ar[d]^{φ↦ρ(g)φρ(g)^{−1}}_{≅} & 1 \\ 1 \ar[r] & Θ(gP g^{−1}) \ar[r] & N_{π}(gP g^{−1}) \ar[r] & Aut_{L}(gP g^{−1}) \ar[r] & 1}

which shows that g Θ(P) g^{−1} = Θ(gP g^{−1}).

Now suppose that P ≤ Q and set N = N_{π}(P) ∩ N_{π}(Q). Set ê = δ(e) ∈ Aut_{L}(P, Q) where e ∈ N_{S}(P, Q) is the identity element. Let Aut_{L}(Q, ↓_{P}) be the subgroup of Aut_{L}(Q) of the automorphisms φ which restrict to an automorphism of P in the sense that there exists φ′ ∈ Aut_{L}(P) such that φ ∘ ê = ê∘ φ′. Note that φ′ is unique because ê is a monomorphism in L by [2, Corollary 3.10]. Similarly, let Aut_{L}(P, ↑^{Q}) denote the subgroup of Aut_{L}(P) of the automorphisms φ′ which extend to an automorphism of Q in the sense that there exists φ ∈ Aut_{L}(Q) such that φ ∘ ê = ê ∘ φ′. Note that φ is unique because ê is an epimorphism in L, [2, Corollary 3.10]. Thus φ ↦ φ′ sets up an isomorphism Aut_{L}(Q, ↓_{P}) → Aut_{L}(P, ↑^{Q}). Note that if φ = ρ_{Q}(g) for some g ∈ N_{π}(Q) then g ∈ N and φ′ = ρ_{P} (g).

Clearly, the preimage under ρ of Aut_{L}(Q, ↓_{P}) in Aut_{T} (Q) = N_{π}(Q) is N as is the preimage of Aut_{L}(P, ↑^{Q}) in Aut_{T} (P) = N_{π}(P). Also, C_{π}(Q) ≤ N. We obtain

[figure: 1; Θ(Q); N; ρ_{Q}; Aut_{L}(Q, ↓_{P}); ≅; φ↦φ′; 1; 1; Θ(P) ∩ N; N; ρ_{P}; Aut_{L}(P, ↑^{Q}); 1]

We deduce that Θ(Q) = Θ(P) ∩ N ≤ Θ(P), hence Θ is a signaliser functor.

(3) Fix some φ ∈ F(P, Q). There is some g ∈ π such that φ is the restriction of c_{g} ∈ Inn(π) to P because F = F_{S}(π). Since Bc_{g} is homotopic to the identity on Bπ, then following square commutes up to homotopy

[figure: BP; Bi_{P}^{S}; Bφ; BS; Bi; Bπ; Bc_{g}≃id; BQ; Bi_{P}^{S}; BS; Bi; Bπ.]

Applying H^{∗}(−; 𝔽_{p}) to this diagram for every φ ∈ F^{c} shows that that the image of

H^{∗}(Bπ; 𝔽_{p}) ⟶^{Bi^{∗}} H^{∗}(BS; 𝔽_{p}) is contained in the ring of stable elements H^{∗}(F^{c}; 𝔽_{p}), namely the subring of all the classes α ∈ H^{∗}(S; 𝔽_{p}) such that φ^{∗}(α|_{Q}) = α|_{P} for all φ ∈ F^{c}(P, Q).

Now we consider BS ⟶^{Bi} Bπ ⟶^{f} |L|_{p}^{∧} which is homotopic to θ. By [4, Theorem 5.8] θ^{∗} is an isomorphism of H^{∗}(|L|_{p}^{∧}; 𝔽_{p}) onto H^{∗}(F^{c}) ≤ H^{∗}(BS; 𝔽_{p}). But θ^{∗} = (Bi)^{∗} ∘ f^{∗} so Bi^{∗} is left inverse of f^{∗}, both are 𝔽_{p}-algebra homomorphisms. □

Ragnarsson defined in [18] a classifying spectrum 𝔹F for any saturated fusion system F. It is equipped with a structure map σ_{F}: 𝔹S → 𝔹F where 𝔹S = Σ^{∞}BS. In the presence of a linking system, 𝔹F ≃ Σ^{∞}|L|_{p}^{∧} and σ_{F} = Σ^{∞}θ, see display (1). He showed that the assignment of F ↦ 𝔹F is a functor from the category of fusions systems to the homotopy category of spectra. In addition he showed that F can be recovered from (𝔹F, σ_{F}) in the sense that the morphisms in F are exactly the group monomorphisms φ: P → Q such that 𝔹P ⟶^{𝔹φ} 𝔹S ⟶^{σ_{F}} 𝔹F is homotopic to 𝔹P ⟶^{𝔹i} 𝔹S ⟶^{σ_{F}} 𝔹F where 𝔹P = Σ^{∞}BP. Compare this with the construction in §2.3.

Here is a variant of Theorem 1.1(1). Note that while linking systems are not known to exists, the classifying spectra of saturated fusion systems always exist.

Proposition 2.2. Suppose F is a saturated fusion system on S and that S is a Sylow p-subgroup of some group π. Assume that there exists a map f: Bπ → Ω^{∞}𝔹F whose restriction to BS is homotopic to the right adjoint σ_{F}^{#} : BS → Ω^{∞}𝔹F of σ_{F}: 𝔹S → 𝔹F. Then F_{S}(π) ⊆ F.

Proof. By Proposition 2.1 and the constructions in §2.3, it only remains to prove that F = F_{S}(σ_{F}^{#}). This is straightforward from the definition of F_{S}(σ_{F}^{#}) and the adjunction [𝔹P, 𝔹F] ≅ [Σ^{∞}BP, 𝔹F] ≅ [BP, Ω^{∞}𝔹F]. □

3. Graphs and homotopy colimits

3.1. Categories of dimension 1. A small category C is called 1-dimensional if its nerve is 1-dimensional. Thus, in any pair of composable morphisms in C at least one of them is an identity morphism. Equivalently, there is a partition Obj(C) = C_{1} ∐ C_{2} such that if φ: x → y is a non-identity morphism in C then x ∈ C_{1} and y ∈ C_{2}.

Since there are no non-trivial compositions in C, it is clear that if D is any category then any assignment F: Obj(C) → Obj(D) together with an assignment of a morphism F (c_{1}) → F (c_{2}) to any non-identity morphism c_{1} → c_{2} in C, gives rise to a functor F: C → D.

Proposition 3.1. Let C be a 1-dimensional category and let F, F′: C → Top be functors whose values are CW-complexes. Assume that there is an isomorphism q: F ⟶^{≃} F′ in the homotopy category of spaces hoTop. Then there is a homotopy equivalence q̄: hocolim F ⟶^{≃} hocolim F′ with the property that the squares

\xymatrix{F (c) \ar[r] \ar[d]_{q_{c}}^{≃} & hocolim F \ar[d]^{q̄}_{≃} \\ F′(c) \ar[r] & hocolim F′}

commute up to homotopy for any c ∈ C.

Proof. Let C_{1} ∐ C_{2} be a partition of Obj(C) as described above. The isomorphism q: F → F′ in hoTop gives rise to the following maps which we will fix once and for all. Homotopy equivalences q_{c}: F (c) ⟶^{≃} F′(c) for every object c ∈ C and, for every non-identity morphism φ: c_{1} → c_{2} in C, a homotopy h_{φ}: F (c_{1}) × I → F′(c_{2}) from q_{c_{2}} ∘ F (φ) to F′(φ) ∘ q_{c_{1}}.

Define functors H, H_{0}, H_{1} : C → Top by describing their effect on objects of C_{1} and C_{2} and on non-identity morphisms as follows

\begin{matrix}H_{0} : c_{1} ↦ F (c_{1}), & H_{0} : c_{2} ↦ F′(c_{2}), & H_{0}(c_{1} ⟶^{φ} c_{2}) = q_{c_{2}} ∘ F (φ). \\ H_{1} : c_{1} ↦ F (c_{1}), & H_{1} : c_{2} ↦ F′(c_{2}), & H_{1}(c_{1} ⟶^{φ} c_{2}) = F′(φ) ∘ q_{c_{1}} \\ H: c_{1} ↦ F (c_{1}) × I, & H: c_{2} ↦ F′(c_{2}), & H(c_{1} ⟶^{φ} c_{2}) = h_{φ}. \end{matrix}

Now we will define natural transformations, all of which are homotopy equivalences:

F ⟶_{≃}^{(id,q_{2})} H_{0} ⟶_{≃}^{(i_{0},id)} H ⟵_{≃}^{(i_{1},id)} H_{1} ⟶_{≃}^{(q_{1},id)} F′

On objects of C_{1} and C_{2} the effect of these natural transformations is described by the rows of the diagram below and the columns show the effect of F, H_{0}, H, H_{1} and F′ on a non-identity morphism φ. The commutativity of all the squares shows that these are indeed natural transformations.

\xymatrix{F (c_{1}) \ar[r]^{id} \ar[d]_{F (φ)} & F (c_{1}) \ar[r]^{i_{0}} \ar[d]^{q_{c_{2}} ∘F (φ)} & F (c_{1}) × I \ar[d]^{h_{φ}} & F (c_{1}) \ar[l]_{i_{1}} \ar[r]^{q_{c_{1}}} \ar[d]^{F′(φ)∘q_{c_{1}}} & F′(c_{1}) \ar[d]^{F′(φ)} \\ F (c_{2}) \ar[r]^{≃}_{q_{c_{2}}} & F′(c_{2}) \ar[r]_{id} & F′(c_{2}) & F′(c_{2}) \ar[l]^{id} \ar[r]_{id} & F′(c_{2})}

We now obtain homotopy equivalences

hocolim_{C}F ⟶^{≃} hocolim_{C}H_{0} ⟶^{≃} hocolim_{C}H ⟵^{≃} hocolim_{C}H_{1} ⟶^{≃} hocolim_{C}F′.

It is also clear from the commutative ladder above that the second diagram in the statement of this proposition is homotopy commutative for all c ∈ C. □

3.2. Graphs and their associated categories. A special class of 1-dimensional categories is associated to graphs. Recall from Serre [20] that a graph Y consists of the following data. A set V of vertices also denoted vert(Y) and a set E of

(directed) edges denoted edge(Y). There is a function E ⟶^{y↦(o(y),t(y))} V ×V whose components are the origin and terminus of an edge y. It is also equipped with an involution E ⟶^{y↦ȳ} E which has no fixed points and “reverses the direction” of an edge, namely o(ȳ) = t(y) and t(ȳ) = o(y). The orbits of this involution [y] are called “geometric edges”. The set of the geometric edges of Y is denoted [edge(Y)].

A walk in Y from a vertex u to a vertex v is a sequence y_{1}, …, y_{n} of edges such that o(y_{1}) = u and t(y_{n}) = v and o(y_{i+1}) = t(y_{i}) for all i. A loop is a walk from a vertex to itself. A path from u to v is a walk which contains no loops.

An orientation of Y is a choice A ⊆ E of one representative from each geometric edge. Given an orientation A we will write |y| for either the edge y or ȳ which belongs to A.

Any graph Y gives rise to a small category C_{Y} whose object set is the disjoint union of vert(Y) and [edge(Y)]. Apart from identity morphisms there is a unique morphism [y] ⟶^{y} t(y) for any y ∈ edge(Y).

3.3. Graphs of groups. Let Y be a graph. Recall from [20, Def. 8 in §4.4] that a graph of groups (G, Y) is an assignment of groups G_{v} to every v ∈ vert(Y) and G_{y} to every y ∈ edge(Y) such that G_{y} = Gȳ and every edge is associated with a group monomorphism G_{y} → G_{t(y)} which is denoted a ↦ a^{y} (here a ∈ G_{y} and a^{y} is its image in G_{t(y)}, [20, pp. 41–42]).

We obtain an associated functor G: C_{Y} → {Groups} defined by

G(v) = G_{v},  G([y]) = G_{y},  G([y] ⟶^{y} t(y)) = G_{y} ⟶^{a↦a^{y}} G_{t(y)}.

Clearly, G carries morphisms in C_{Y} to group monomorphisms. Conversely, any functor G: C_{Y} → {Groups} with this property defines a graph or groups (G, Y).

3.4. The fundamental group. Fix a graph of groups (G, Y). Let F (edge(Y)) denote the free group whose generating set is edge(Y) and set

F̃(G, Y) = F (edge(Y)) ∗ ( ∗_{v∈vert Y} G_{v}).

Define F (G, Y) as the quotient group

F (G, Y) = F̃(G, Y)/⟨ȳ = y^{−1}, ya^{y}y^{−1} = aȳ⟩

for all y ∈ edge(Y) and all a ∈ G_{y} = Gȳ. Let T be a maximal tree in Y. The fundamental group of (G, Y) is defined as the factor group, see [20, p. 42]

π_{1}(G, Y, T) = F (G, Y)/⟨y = 1: y ∈ T ⟩.

The image of y in this group is denoted g_{y}. It is shown in [20, p. 44] that different choices of T yield isomorphic groups. By [20, Cor. 1, p. 45], the inclusions G_{v} ≤ F (G, Y), for any v ∈ vert(Y), induce injections

G_{v} ↪ π_{1}(G, Y, T).

It also follows from the definitions that for any y ∈ edge Y and for any a ∈ G_{y}

g_{y} · a^{y} · g_{y}^{−1} = aȳ.

Thus g_{y} conjugates the image of G_{y} ⟶^{a↦a^{y}} G_{t(y)} onto the image of G_{y} ⟶^{a↦aȳ} G_{o(y)}.

3.5. Homotopy colimits and graphs of groups. By applying the classifyingspace functor B(−) to the functor G: C_{Y} → {Groups} associated to (G, Y) we obtain a functor BG: C_{Y} → Top. The following result was observed by Dror-Farjoun in [7, §4.1].

Proposition 3.2. Consider G: C_{Y} → Top associated with (G, Y). Then

hocolim_{C_{Y}} BG ≃ Bπ

where π = π_{1}(G, Y, T) for some choice of a maximal tree T and the maps BG_{v} → hocolim_{C_{Y}} BG ≃ Bπ are homotopic to B(incl_{G_{v}}^{π}).

Proposition 3.3. Let (G, Y) be a graph of groups and suppose that

(i) The groups G_{v} and G_{y} contain Sylow p-subgroups P_{v} and P_{y} for every v ∈ vert Y and every y ∈ edge Y.

(ii) There exists a vertex v_{0} such that for any other vertex u of Y there exists a path (directed, without loops) y_{1}, …, y_{n} from v_{0} to u such that for any i the map G_{y_{i}} ⟶^{a↦a^{y_{i}}} G_{t(y_{i})} carries P_{y_{i}} onto a Sylow p-subgroup of G_{t(y_{i})}.

Set π = π_{1}(G, Y, T) and set S = P_{v_{0}}. Then hocolim_{C_{Y}} BG ≃ Bπ and (1) S is a Sylow p-subgroup of π.

(2) If the groups G_{v} are finite and if Y contains no loops then (a) Bπ is a p-good space ([6, I.§5]), and

(b) In the kernel K of H^{∗}(Bπ; 𝔽_{p}) ⟶^{Bi^{∗}} H^{∗}(BS; 𝔽_{p}) the product of any two elements is zero.

Proof. (1). By [20, I.§5.3] the group π acts without inversions on a tree X̃ such that the isotropy groups of the vertices of X̃ are conjugate to the subgroup G_{v} of π where v ∈ vert Y. If P is a finite p-subgroup of π then by [20, Prop. 19, §I.4.3] it fixes a vertex of X̃ and it is therefore conjugate in π to a subgroup of G_{v} for some vertex v. We therefore assume that P ≤ G_{v}.

By hypothesis there is a a path y_{1}, …, y_{n} from v_{0} to v with the properties listed in (ii). We will use induction on n to prove that P is conjugate to a subgroup of S. If n = 0 then P ≤ G_{v_{0}} and the result holds because S is a Sylow p-subgroup of G_{v_{0}}. Suppose that n > 0 and note that by hypothesis the image of G_{y_{n}} in G_{t(y_{n})} = G_{v}, which we denote by G_{y_{n}}^{y_{n}}, contains a Sylow p-subgroup. Thus, by conjugating P inside G_{v} we may assume that P ≤ G_{y_{n}}^{y_{n}}. As we have seen in §3.4 the element g_{y_{n}} ∈ π conjugates G_{y_{n}}^{y_{n}} onto Gȳ_{n} ; y_{n} ≤ G_{o(y_{n})} = G_{t(y_{n−1})}. We deduce that P is conjugate in π to a subgroup of G_{t(y_{n−1})} and we can now apply the induction hypothesis to the path y_{1}, …, y_{n−1} and deduce that P is conjugate in π to a subgroup of S.

(2) Since Y has no loops, it is a tree. Orient the tree Y in the only way so that there is a path from v_{0} to any vertex of Y which only uses edges from the orientation class A. We now claim that by possibly replacing (G, Y) with an isomorphic graph, the groups P_{v} and P_{y} can be chosen such that (P, Y) is a sub-graph of (G, Y),

namely G_{y} ⟶^{a↦a^{y}} G_{t(y)} carries P_{y} into P_{t(y)}. In fact, by hypothesis (ii) P_{y} ≅ P_{t(y)} if y ∈ A.

To prove this, consider the poset T of the subtrees Y′ of Y with root v_{0} for which is it possible to replace (G, Y) with an isomorphic tree of groups and to find Sylow p-subgroup P_{v} ≤ G_{v} and P_{y} ≤ G_{y} such that (P, Y′) is a subtree of (G, Y′). First, T is not empty because clearly {v_{0}} ∈ T . Next, consider a maximal element Y′ ∈ T and assume that Y′ ≠ Y. Thus, we may assume, possibly by replacing (G, Y) with an isomorphic tree, that (P, Y′) is a subtree of (G, Y′). Choose an edge y′ ∈ A such that v′ = o(y′) ∈ Y′ and v = t(y′) ∉ Y′.

Let Gȳ′ ; ȳ′ denote the image of G_{y′} in G_{o(y′)} and let P ȳ′ ; ȳ′ be the image of P_{y′}. Then

gP ȳ′ ; ȳ′g^{−1} ≤ P_{o(y′)} for some g ∈ G_{o(y′)} by the Sylow condition. Let c_{g} denote the

inner automorphism x ↦ gxg^{−1}. By replacing G_{y′} with gGȳ′ ; ȳ′g^{−1} we obtain an

isomorph (G̃, Y) of (G, Y). Now replace P_{y′} with gP ȳ′ ; ȳ′g^{−1} and replace P_{t(y′)} with the image of P_{y′} in G_{t(y′)} = G_{v} which is a Sylow p-subgroup by hypothesis (ii). Note that o(y′) ≠ t(y′) so we have left the trees (P, Y′) and (G, Y′) unchanged. By construction (P, Y′ ∪ {v}) is a subtree of (G̃, Y′ ∪ {v}) which contradicts the maximality of Y′. Thus, Y ∈ T which is what we wanted to prove.

(2)a. Since Y is a tree, π is the amalgamated product of finite groups which is generated by the groups G_{v}. Also notice that the images of P_{v} in π are subgroups of S because P_{y} → P_{t}(y) are isomorphisms for y ∈ A (consider all that rooted subtrees Y′ ⊆ Y for which P_{v} ≤ S and argue as above). Let K be the normal subgroup of π generated by the elements of order prime to p. Clearly K contains O^{p}(G_{v}) and therefore together with the groups P_{v} it generates π. Hence π = KS, so π/K is a finite p-group.

Now BK is p-good by [6, Proposition VII.3.2] since K is p-perfect [6, VII.3.2] because its abelianisation must be a torsion group with no elements of order p. The Bousfield-Kan spectral sequence E_{ij}^{2} = \underrightarrow{lim}_{i}H_{j}(BG; 𝔽_{p}) ⇒ H_{i+j}(Bπ; 𝔽_{p}) where G is the diagram of groups associated to (G, Y), clearly consists of finitely generated 𝔽_{p}- modules and therefore H_{∗}(Bπ; 𝔽_{p}) is finite in every dimension. The action of π/K on H^{∗}(BK; 𝔽_{p}) must be nilpotent by [6, II.5.2] so by the strong convergence of the Eilenberg-Moore spectral sequence for the fibration BK → Bπ → Bπ/K proven by Dwyer in [8], we deduce that H_{∗}(BK; 𝔽_{p}) is finite in every dimension. By [6, Chapter II, 5.1 and 5.2] we obtain a nilpotent fibration BK_{p}^{∧} → Bπ_{p}^{∧} → Bπ/K in which both the base and the fibre are p-complete, whence so is the total space. This shows that Bπ is p-good.

(2)b. Let G, P: C_{Y} → Groups be the associated functors for (G, Y) and (P, Y). Observe that hocolim_{C_{Y}} BP = BS because P_{y} → P_{t(y)} are isomorphisms for all y ∈ A (consider the poset T of rooted subtrees Y′ ⊆ Y for which hocolim BP|_{C_{Y ′}} ≃ BS and argue as above). We obtain a natural transformation of functors BP → BG which gives rise to a morphism of Bousfield-Kan spectral sequences E_{2}^{∗,∗}(G) → E_{2}^{∗,∗}(P) where

E_{2}^{i,j}(P) = \underleftarrow{lim}_{C_{Y}} ^{i}H^{j}(BP; 𝔽_{p}) ⇒ H^{i+j}(BS; 𝔽_{p}),
E_{2}^{i,j}(G) = \underleftarrow{lim}_{C_{Y}} ^{i}H^{j}(BG; 𝔽_{p}) ⇒ H^{i+j}(Bπ; 𝔽_{p})).

Since P_{v} ≤ G_{v} and P_{y} ≤ G_{y} are Sylow p-subgroups and G_{v} are finite groups, the maps H^{∗}(G_{v}; 𝔽_{p}) → H^{∗}(P_{v}; 𝔽_{p}) and H^{∗}(G_{y}; 𝔽_{p}) → H^{∗}(P_{y}; 𝔽_{p}) are injective. Since \underleftarrow{lim}^{0} is left exact, E_{2}^{0,∗}(G) → E_{2}^{0,∗}(P) is injective. Also E_{2}^{i,∗}(P) = E_{2}^{i,∗}(G) = 0 for all i ≥ 2 because C_{Y} is 1-dimensional. Therefore E_{∞}^{∗∗} = E_{2}^{∗∗}. We obtain the following diagram with exact rows from which it is clear that K = Ker(Bi^{∗}) is contained in E_{∞}^{1,∗}(G).

\xymatrix{0 \ar[r] & E_{2}^{1,∗−1}(G) \ar[r] \ar[d] & H^{∗}(Bπ) \ar[r] \ar[d]_{Bi^{∗}} & E_{2}^{0,∗}(G) \ar@{^{(}->}[d] \ar[r] & 0 \\ 0 \ar[r] & E_{2}^{0,∗−1}(P) \ar[r] & H^{∗}(BS) \ar[r] & E_{2}^{0,∗}(P) \ar[r] & 0}

It follows that xy = 0 for any x, y ∈ K because xy ∈ E_{∞}^{2,∗}(G) = 0. □

4. Groups inducing fusion systems

4.1. Robinson’s construction - the normalizer decomposition. Fix a p-local finite group (S, F, L) and a collection R of F-centric subgroups which contains all the F-centric F-radical subgroups (e.g. all the F-centric subgroups of S). Consider the normalizer decomposition δ: S̄(R) → Top, see §2.3, and recall that in the homotopy category of spaces, δ is isomorphic to the functor [R_{•}] ↦ BAut_{L}(R_{•}).

Fix fully normalized representatives R_{0}, R_{1}, …, R_{n} for the F-conjugacy classes of R and choose for convenience R_{0} = S. Let D be the subposet of S̄(R) generated by the objects [R_{0}], [R_{1}], …, [R_{n}] and [R_{1} < S], …, [R_{n} < S]. It is easy to check that D = C_{Y} where Y is the tree of height 1 with root [S] and edges [R_{i} < S] to the leaves [R_{i}] (i = 1, … n). We obtain a tree of groups (G, Y), see §2.6

(2)  [figure: Aut_{L}(S); Aut_{L}(R_{1}<S); Aut_{L}(R_{n}<S); Aut_{L}(R_{1}); Aut_{L}(R_{i}); Aut_{L}(R_{n})]

This tree of groups satisfies the conditions of Proposition 3.3 because the groups R_{i} are fully normalized so N_{S}(R_{i}) is a Sylow p-subgroup of both Aut_{L}(R_{i}) and Aut_{L}(R_{i} < S) = N_{Aut_{L}(S)}(R_{i}). Also note that S is a Sylow p-subgroup of Aut_{L}(S). Therefore hocolim BG ≃ Bπ_{R} for a group π_{R} which is the amalgamated product of the tree (2) and which contains S as a Sylow p-subgroup.

The crux, now, is that BG is the functor [R_{•}] ↦ BAut_{L}(R_{•}) and therefore it is isomorphic to δ|_{D} in hoTop. Proposition 3.1 now yields a composite map f

Bπ_{R} ≃ hocolim_{D} BG ⟶^{≃} hocolim_{D} δ|_{D} ⟶^{incl} hocolim_{S̄(R)} δ ⟶^{≃} |L| → |L|_{p}^{∧}

whose restriction to BS is homotopic to θ in display (1). We now claim that F ⊆ F_{S}(π_{R}).

Proof of claim. First, F_{S}(π_{R}) contains all the groups Aut_{F}(R_{i}) because π_{R} contains Aut_{L}(R_{i}) for all i = 0, …, n. Therefore F_{S}(π_{R}) contains the fusion system F′ on S generated by Aut_{F}(R_{i}). Clearly F′ ⊆ F and we next claim that, in fact F′ = F. To do this, we will prove by induction on |S : P | that if φ ∈ F(P, S) then φ ∈ F′(P, S). If |S : P | = 1 then P = S = R_{0} and therefore φ ∈ Aut_{F}(R_{0}) ⊆ F′.

Assume that |S : P | > 1. If P is not F-centric or not F-radical then by Alperin’s fusion theorem [4, Theorem A.10] it is a composition of restriction of automorphisms of F-centric F-radical subgroups whose order must be strictly bigger than the order of P and therefore, by the induction hypothesis, all these automorphisms must belong to F′. Hence φ ∈ F.

If P is F-centric F-radical then it is F-conjugate to R_{i} for some i. We now claim that P is F′-conjugate to R_{i}. Indeed, fix some F-isomorphism α: R_{i} → P. By Alperin’s fusion theorem α is the composition of restrictions of F-automorphisms β_{1}, …, β_{k} of some F-centric subgroups Q_{1}, …, Q_{k}, thus P = β_{k}(… (β_{1}(R_{i})) …). If |Q_{j}| = |P | for some j then β_{j} does not move β_{j−1}(… β_{1}(R_{i}) …) and by omitting it we obtain another F-isomorphism R_{i} → P. Thus, by possibly replacing α with another F-isomorphism, we may assume that |Q_{1}|, …, |Q_{k}| > |P |, whence by induction hypothesis β_{1}, …, β_{k} ∈ F′ and therefore α ∈ F′(R_{i}, P).

Finally, consider φ ∈ F(P, S) where P is F-centric F-radical. Note that P and φ(P) are F-conjugate to R_{i} for some i. Choose F′-isomorphisms α′ ∈ F′(R_{i}, P) and β′ ∈ F′(φ(P), R_{i}). Then β′ ∘ φ ∘ α′ ∈ Aut_{F}(R_{i}) ⊆ F′ and therefore φ ∈ F′. This completes the induction step. □

We summarize this in the following proposition.

Proposition 4.1. Fix a p-local finite group (S, F, L). Then the group π_{R} constructed above as the amalgamated product of (2) contains S as a Sylow p subgroup, it is equipped with a map f: Bπ_{R} → |L|_{p}^{∧} whose restriction to BS is homotopic to θ and furthermore, F ⊆ F_{S}(π_{R}).

We can now apply Theorem 1.1(1) to π_{R} and deduce that F = F_{S}(π_{R}). Corollary 1.2 holds for π_{R} because the implication (2)⇒(1) is a triviality and (1)⇒(2) follows from point (2) of Theorem 1.1.

Proposition 1.7 for π_{R} follows from Proposition 3.3(2) because the graph of groups (2) has no loops.

Remark 4.2. The groups which appear in the amalgam (2) can be constructed directly from F even without the existence of an associated linking system. Thus, the group π_{R} can be constructed for any saturated fusion system.

Indeed, since the groups R_{i} are F-centric the normalizer fusion systems F_{i} := N_{F}(R_{i}), see [4, §6], are constrained in the sense of [2, Proposition C]. Therefore F_{i} admit unique linking systems L_{i} and there are canonically defined finite groups L_{i} which contain N_{S}(R_{i}) as a Sylow p-subgroup such that F_{i} = F_{N_{S}(R_{i})}(L_{i}). In fact L_{i} = Aut_{L_{i}}(R_{i}). Similarly F_{i,0} := N_{F}(R_{i} < S) = N_{F_{0}}(R_{i}) are constrained fusion sub-systems of F_{0} = N_{F}(S) and therefore they have unique linking systems L_{i,0} and groups L_{i,0} such that F_{i,0} = F_{N_{S}(R_{i})}(L_{i,0}) and L_{i,0} = Aut_{L_{i,0}}(R_{i}).

By construction, every morphism P → Q in F_{0} extends to an automorphism of S and the arguments in §2.6 using the fact that ê ∈ L(R_{i}, S) is both a monomorphism and and epimorphism, show that there is a canonical isomorphism Aut_{L_{0}}(R_{i}) = N_{L_{0}}(R_{i}). Since F_{i,0} = N_{F_{0}}(R_{i}) the uniqueness of the linking system for F_{i,0} and [4, Definition 6.1] yield

L_{i,0} = Aut_{L_{i,0}}(R_{i}) = Aut_{N_{L_{0}}} (R_{i}) = Aut_{L_{0}}(R_{i}) = N_{L_{0}}(R_{i}) ≤ L_{0}.

Observe that F_{i,0} is a subsystem of F_{i} with the same Sylow N_{S}(R_{i}) which consists of the morphisms P → Q which extend to an F-automorphism of S. Let K denote the collection of all the subgroups of N_{S}(R_{i}) which contain R_{i}. Since both F_{i} and F_{i,0} contain R_{i} as a normal centric subgroup, K contains all the F_{i}-centric-radical subgroups and all the F_{i,0}-centric-radical subgroups. Define a sub-category L′_{i} ⊆ L_{i} with object set K and such that L′_{i}(P, Q) is the preimage of F_{i,0}(P, Q) under the projection L_{i}(P, Q) → F_{i}(P, Q). It is clear that L′_{i} is a “partial” linking system for F_{i,0} in the sense that it is only defined on the subgroups in K, but satisfies the axioms for a linking system. Our argument now follows [3, Step 7 of Theorem 4.6].

There is a functor Z: O(F_{i,0})^{op} → Ab defined by P ↦ Z(P). By [4, Proposition 3.1], elements of \underleftarrow{lim}^{3}Z|_{K} classify, up to isomorphism, the partial linking systems associated to F_{i,0} defined on K. Since K is closed to over-groups and contains all the F_{i,0}-centric-radicals, then [4, Proposition 3.2] and well known properties of the Λ-functors imply that

\underleftarrow{lim}_{O(F_{i,0}^{c})}^{3}Z ⟶^{≅} \underleftarrow{lim}_{O(F_{i,0}^{K})}^{3}Z|_{K}

is an isomorphism. By [2, Proposition 4.2] the left hand side vanishes. This shows that the partial linking system L′_{i} must coincide with L_{i,0}^{K} and in particular

L_{i,0} = Aut_{L_{i,0}}(R_{i}) = Aut_{L′_{i}}(R_{i}) ≤ Aut_{L_{i}}(R_{i}) = L_{i}.

We obtain the following tree of groups (G, Y) which is defined purely by F without any reference to an associated linking system

(3)  [figure: L_{0}; L_{1,0}; L_{n,0}; L_{1}; L_{i}; L_{n}.]

and define π_{R} as its amalgamated product. By Proposition 3.3 it contains S ≤ L_{0} as a Sylow p-subgroup because N_{S}(R_{i}) are Sylow p-subgroups of L_{i} and L_{i,0}. In the presence of a linking system for L, this tree of groups coincides with (2).

We can now prove that F = F_{S}(π_{R}) using our geometric methods. The assignment [R_{i}] ↦ F_{i} and [R_{i} < S] ↦ F_{i,0} define a functor ν from C_{Y} to the category of fusion sub-systems of F. By applying Ragnarsson’s functor 𝔹 we obtain a functor 𝔹ν: C_{Y} → Spectra together with a map 𝔹ν → 𝔹F. Since F_{i} and F_{i,0} are constrained, 𝔹ν ≃ Σ^{∞}BG_{p}^{∧} for the functor G associated to the tree (3). By adjunction we obtain BG → Ω^{∞}𝔹F in hoTop and by Propositions 3.1 and 3.3 we obtain f: Bπ_{R} → Ω^{∞}𝔹F whose restriction to BS is homotopic to σ_{F}^{#}. Proposition 2.2 implies that F_{S}(π_{R}) ⊆ F and the argument we used above exploiting Alperin’s fusion theorem shows that F ⊆ F_{S}(π_{R}).

Remark 4.3. Robinson’s originally constructed in [19] a group Γ as the amalgamated product (3) but with the groups L_{i,0} replaced with their Sylow p-subgroup N_{S}(R_{i}). The group π_{R} that we construct is therefore a quotient of Γ. Note however that if π_{R} admits a signaliser functor, then F has an associated linking system and we can apply Theorem 1.1 to BΓ → Bπ_{R} → |L|_{p}^{∧} in order to obtain a signaliser functor on Γ.

Remark 4.4. In [13, Theorem 5.5] it is shown that a p-local finite group with a “compressible F-centric” collection R admits a group Γ which contains S as a Sylow p-subgroup with the property that F = F_{S}(Γ) and that L is induced by some signaliser functor on Γ. The group Γ is simply the group π_{R}. Our techniques are inspired by this paper.

4.2. Leary-Stancu’s construction - the subgroup decomposition. The following argument is given explicitly in [21]. Fix a p-local finite group (S, F, L) and let Φ = {φ_{1}, …, φ_{n}} be a set of morphisms φ_{i}: P_{i} → S in F^{c} which, together with Inn(S), generate F (recall Alperin’s fusion theorem). For example, Φ is the set of all the morphisms in F^{c}.

Consider the oriented graph Y with one vertex v_{0} and oriented edges y_{1}, …, y_{n}. Define a grapg of group (G_{,}Y) where G_{v_{0}} = S, G_{y_{i}} = Gȳ_{i} = P_{i} and y_{i} ↦ P_{i} ⟶^{φ_{i}} S and ȳ_{i} ↦ P_{i} ⟶^{incl} S. Thus, the functor G: C_{Y} → Groups is depicted by

\xymatrix{P_{1} \ar[dr]^{φ_{1}} \ar[dr]_{incl} & ⋯ & P_{n} \ar[dl]^{incl} \ar[dl]_{φ_{n}} \\  & S}

Define a functor J: C_{Y} → O(F^{c}) by

J(v_{0}) = S,  J([y_{i}]) = P_{i},  J(ȳ_{i}) = P_{i} ⟶^{incl} S,  J(y_{i}) = P_{i} ⟶^{φ_{i}} S.

Define π_{LS} := π_{1}(G, Y, T) where the maximal tree T must be {v_{0}}.

Observe that J^{∗}β ≅ BG in hoTop where β is the subgroup decomposition, see §2.5. Now we obtain from Proposition 3.3, whose conditions are satisfied trivially, and Proposition 3.1 and the properties of β described in §2.6, a sequence of maps

Bπ_{LS} ≃ hocolim_{C_{Y}} BG ⟶^{≃} hocolim_{C_{Y}} J^{∗}β ⟶^{J_{∗}} hocolim_{O(F^{c})} β ⟶^{≃} |L| → |L|_{p}^{∧}

whose restriction to BS is homotopic to θ. We also observe that F ⊆ F_{S}(π_{LS}) because S ≤ π_{LS} and by construction of π_{1}(G, Y, T), see §3.4, every φ_{i}: P → S in Φ is obtained by conjugation by the element g_{y_{i}}. To summarize, we have shown:

Proposition 4.5. Fix a p-local finite group (S, F, L). Then the group π_{LS} contains S as a Sylow p-subgroup, it is equipped with a map f: Bπ_{LS} → |L|_{p}^{∧} whose restriction to BS is homotopic to θ and F ⊆ F_{S}(π_{LS}).

Thus, we can apply Theorem 1.1(1) to recover the results of [9], namely F = F_{S}(π_{LS}). Also Corollary 1.2 for π_{LS} follows since the implication (2)⇒(1) is a triviality and (1)⇒(2) follows from point (2) of Theorem 1.1.

Remark 4.6. The construction of π_{LS} appears in [9]. It is clear that it is defined purely in terms of F. The same ideas of Remark 4.2 can be used to obtain F = F_{S}(π_{LS}) for any saturated fusion system without the assumption that a linking system exists.

4.3. Aschbacher-Chermak’s construction - the normalizer decomposition. Fix an odd prime q and some n ≥ 1. We now recall from [10], see also [11], the construction of Solomon’s fusion system F_{Sol}(q^{n}) at the prime 2. This fusion system is obtained from the fusion system of Spin_{7}(q^{n}) as follows. Throughout we write q^{n} for the finite field 𝔽_{q^{n}} and q^{∞} for its algebraic closure. There is a group homomorphism ω: SL_{2}(q^{∞})^{3} → Spin_{7}(q^{∞}) whose kernel is the central subgroup of order 2 generated by (−I, −I, −I). We will denote ω(X_{1}, X_{2}, X_{3}) = [[X_{1}, X_{2}, X_{3}]].

Let H(q^{∞}) be the image of ω. The centre of Spin_{7}(q^{∞}) is a group of order 2 generated by z = [[I, I, −I]]. The set {[[±I, ±I, ±I]]} forms an elementary abelian group U of rank 2. By [10, Lemma 2.3] there is an element τ ∈ N_{Spin_{7}(q)}(U) such that c_{τ} ([[X_{1}, X_{2}, X_{3}]]) = [[X_{2}, X_{1}, X_{3}]] where c_{τ} denotes conjugation g ↦ τgτ^{−1}. Note that τ normalizes H(q^{∞}).

Fix a copy of the quaternion group Q_{8} in SL_{2}(q) and choose generators A, B of order 4. Set (cf. [11, §1])

C(q^{∞}) = {X ∈ C_{SL_{2}(q^{∞})}(A): X^{2^{k}} = I for some k} ≅ ℤ/2^{∞}
Q(q^{∞}) = ⟨C(q^{∞}), B⟩
S_{0}(q^{∞}) = ω(Q(q^{∞})^{3}) ≤ H(q^{∞})
S(q^{∞}) = S_{0}(q^{∞}) · ⟨τ⟩.

We will write Θ(q^{n}) = Θ(q^{∞})∩Spin_{7}(q^{n}) where Θ denotes one of C(−), Q(−), S_{0}(−) or S(−) etc. By [10, Lemma 2.7], S(q^{n}) is a Sylow 2-subgroup of Spin_{7}(q^{n}) hence also of H(q^{n}) · ⟨τ⟩. Also S_{0}(q^{n}) is a Sylow 2-subgroup of H(q^{n}).

There are automorphisms γ̂ and δ̂_{u} of S_{0}(q^{∞}) defined in [11, Definition 1.6] by

γ̂([[X_{1}, X_{2}, X_{3}]]) = [[X_{3}, X_{1}, X_{2}]],  δ̂_{u}([[X_{1}, X_{2}, A′B^{j}]]) = [[X_{1}, X_{2}, (A′)^{u}B^{j}]]

where A′ ∈ C(q^{∞}) and X_{1}, X_{2}, X_{3} ∈ Q(q^{∞}) and u is a carefully chosen 2-adic unit. Set

Γ_{n} = ⟨Inn(S_{0}(q^{n})), c_{τ} , δ̂_{u} ∘ γ̂ ∘ δ̂_{u}^{−1}⟩ ≤ Aut(S_{0}(q^{n})).

The fusion system F_{Sol}(q^{n}) on S = S(q^{n}) is the fusion system generated by F_{S}(Spin_{7}(q^{n})) and by F_{S_{0}(q^{n})}(Γ_{n}).

Set S = S(q^{n}) and S_{0} = S_{0}(q^{n}). It easily follows from [10, Proposition 2.5] and the definitions that C_{S}(U) = S_{0}. Also, |S: S_{0}| = 2 because |S(q^{∞}): S_{0}(q^{∞})| = 2 and therefore S_{0} is fully normalized. It is also F-centric by the next proposition because by construction of F, the F-conjugates of S_{0} are Spin_{7}(q^{n})-conjugates.

Proposition 4.7. C_{Spin_{7}(q^{n})}(S_{0}(q^{n})) ⊆ S_{0}(q^{n}).

Proof. Since U ⊆ S_{0} we deduce from [10, Proposition 2.5] that C_{Spin_{7}(q^{n})}(S_{0}) ⊆ C_{Spin_{7}(q^{n})}(U) = H(q^{n}) which contains S_{0} as a Sylow 2-subgroup and therefore C_{Spin_{7}(q^{n})}(S_{0}) = Z(S_{0})×C′ where C′ has odd order. We need to show that C′ = 1. Choose x ∈ C′ of order r. Since Ker ω is central of order 2 then ω^{−1}(x) consists of an element x̃_{1} of order r and an element x̃_{2} of order 2r. Now (Q_{8})^{3} ⊆ ω^{−1}(S_{0}) must leave {x̃_{1}, x̃_{2}} invariant because S_{0} centralizes x and since the orders of x̃_{1}, x̃_{2} are different (Q_{8})^{3} centralizes these elements. By Lemma 4.8 below x̃_{1} ∈ Q_{8} and therefore x̃_{1} = 1 which implies that x = 1. □

Lemma 4.8. Fix an odd prime q. Then

(1) The centralizer of any element X of order 4 in SL_{2}(q^{n}) is a cyclic group.

(2) Any quaternion group Q_{8} in SL_{2}(q^{∞}) contains its centralizer.

Proof. (1) The minimal polynomial of X is x^{2} + 1 which splits in 𝔽_{q^{n+1}} which contains ζ = \sqrt{−1}. Thus, X is conjugate in GL_{2}(q^{n+1}) to the diagonal matrix diag(ζ, −ζ) whose centralizer is the subgroup of diagonal matrices and therefore its SL_{2}(q^{n+1}) centralizer is the cyclic group 𝔽_{q^{n+1}}^{×}. Conjugating back we see that C_{SL_{2}(q^{n+1})}(X) is cyclic hence so is C_{SL_{2}(q^{n})}(X).

(2) Fix any n such that Q_{8} ≤ SL_{2}(q^{n}) and consider X in its centralizer. Now Q_{8} is generated by elements A, B of order 4, so X belongs to the cyclic groups C_{SL_{2}(q^{n})}(A) and C_{SL_{2}(q^{n})}(B). If ⟨A⟩ ⊆ ⟨X⟩ and ⟨B⟩ ⊆ ⟨X⟩ then Q_{8} ≤ ⟨X⟩ which is absurd. So either X ∈ ⟨A⟩ or X ∈ ⟨B⟩. □

Set F = F_{Sol}(q^{n}) and L = L_{Sol}(q^{n}) and Z is the subgroup generated by the element z (this is the centre of Spin_{7}(q^{n})). Set Aut_{L}(S_{0}; Z) = C_{Aut_{L}(S_{0})}(Z). By [10, Propositions 2.11, 3.3] the linking system L_{Sol}(q^{n}) contains L_{S}^{c}(Spin_{7}(q^{n})) as a subcategory. In fact, L_{S}^{c}(Spin_{7}(q^{n})) = C_{L}(Z) is the centralizer linking system [4, Definition 2.4] and by Proposition 4.7 and [4, remarks before Definiton 1.6]

Aut_{L}(S_{0}; Z) = Aut_{C_{L}(Z)}(S_{0}) = N_{Spin_{7}(q^{n})}(S_{0})/C′_{Spin_{7}(q^{n})}(S_{0})

= N_{Spin_{7}(q^{n})}(S_{0}) ≤ Spin_{7}(q^{n}).

We obtain inclusions of subcategories of L

\xymatrix{L_{S}(Spin_{7}(q^{n})) & BAut_{L}(S_{0}; Z) \ar@{^{(}->}[r] \ar@{^{(}->}[l] & BAut_{L}(S_{0})}

It gives rise to the diagram of spaces below, all of whose squares commute, except the one on the top left which only commutes up to homotopy.

[figure: B Spin_{7}(q^{n}); mod 2 equivalence; BN_{Spin_{7}(q^{n})}(S_{0}); BAut_{L}(S_{0}); |L_{S}(Spin_{7}(q^{n}))|_{2}^{∧}; BAut_{L}(S_{0}; Z)_{2}^{∧}; BAut_{L}(S_{0})_{2}^{∧}; |L|_{2}^{∧}; |L|_{2}^{∧}; |L|_{2}^{∧}.]

We observe that N_{Spin_{7}(q^{n})}(S_{0}) contains the Sylow subgroup S because |S: S_{0}| = 2. By Proposition 3.3, the homotopy colimit of the first row is homotopy equivalent to Bπ_{AC} where π_{AC} is the amalgam of the tree

(4)  Spin_{7}(q^{n}) ^{N_{Spin_{7}(q^{n})}(S_{0})} Aut_{L}(S_{0})

which contains S as a Sylow 2-subgroup. By Proposition 3.1 there is a map f: Bπ_{AC} → |L|_{2}^{∧} whose restriction to BS is homotopic to θ. Moreover, Spin_{7}(q^{n}) and Aut_{L}(S_{0}) are subgroups of π_{AC} and also Aut_{F}(S_{0}) = Aut_{Aut_{L}(S_{0})}(S_{0}) contains c_{τ} and δ̂_{u} ∘ γ̂ ∘ δ̂_{u}^{−1} because these homomorphisms clearly normalize S_{0}(q^{n}). We therefore deduce that F ⊆ F_{S}(π). Thus we have proven:

Proposition 4.9. The group π_{AC} satisfies the hypotheses of Theorem 1.1. In particular F = F_{S}(π_{AC}) and π_{AC} has a signaliser functor which induces L. It also satisfies the condition of Proposition 1.7 because (4) has no loops.

Remark 4.10. The group Γ that Aschbacher and Chermak constructed in [1] is the amalgamated product

Spin_{7}(q^{n}) ^{N_{Spin_{7}(q^{n})}(U)}  K

where |K: N_{Spin_{7}(q^{n})}(U)| = 3. They need the hypothesis that q ≅ 3 or 5 mod 8, however, which we don’t.

Indeed, the groups Γ and π_{AC} are different. We will show below that U is the unique normal subgroup of S_{0} = S_{0}(q^{n}) which is isomorphic to the Klein group ℤ/2 ⊕ ℤ/2 and therefore N_{Spin_{7}(q^{n})}(S_{0}) ⊆ N_{Spin_{7}(q^{n})}(U). The inclusion is proper because H(q^{n}) = C_{Spin_{7}(q^{n})}(U) does not normalize S_{0}(q^{n}). In addition, Aut_{Spin_{7}(q^{n})}(U) = C_{2} and Aut_{F_{Sol}(q^{n})}(U) = Σ_{3} and Aut_{F_{Sol}(q^{n})}(S_{0}) contains δ̂_{u}γ̂δ̂_{u}^{−1} from which we deduce that |Aut_{L}(S_{0}): N_{Spin_{7}(q^{n})}(S_{0})| = 3. Thus the groups we use in the amalgam (4) are smaller than those used by Aschbacher and Chermak. It is interesting to see how our approach interprets the signaliser functors geometrically in Theorem 1.1 rather than algebraically.

Claim. U is the unique normal subgroup of S_{0}(q^{n}) isomorphic to the Klein group.

Proof. Consider a Klein group E ◁ S_{0} and set Ẽ = ω^{−1}(E). Note that Ẽ is normalized by (Q_{8})^{3} ≤ Q(q^{∞})^{3}. Fix an element X̃ = (X_{1}, X_{2}, X_{3}) ∈ Ẽ and note that X_{i}^{2} = ±I because all the elements of E have order 2 and Ker ω = ±(I, I, I). If X_{i} has order 4 then X_{1}^{2} = X_{2}^{2} = X_{3}^{2} = −I so X_{1}, X_{2}, X_{3} have order 4 and Lemma 4.8 implies that Ẽ contains at least 2^{3} different (Q_{8})^{3} conjugates of X̃. This is impossible, so the order of X̃ is at at most 2.

If E ≠ U then we may assume that X_{1} ≠ ±I hence X_{1} = W B^{±1} for some W ∈ C(q^{∞}). Now it follows that X̃ · BX̃B^{−1} = (−W^{2}, I, I) ∈ Ẽ must also have order 2. This can only happen if W ∈ ⟨A⟩ ≤ C(q^{∞}) which implies that X_{1} ∈ Q_{8} whose only element of order 2 is −I and this is absurd. □

Remark 4.11. The construction which we have presented here is motivated by the normalizer decomposition with respect to the collection of the elementary abelian subgroups E, see [12]. If D is the subposet of S̄(E) of the form [Z] ← [Z < U] → [U] then the restriction of the normalizer decomposition to D has the form

|C_{L}(Z)| ⟵ |C_{L}(U)|_{hAut_{F} (Z<U)} ⟶ |C_{L}(U)|_{hAut_{F} (U)}.

Up to 2-completion, the space on the left is B Spin_{7}(q^{n}). A careful examination of the constructions in [12, Theorem B] shows that these maps are induces by inclusion of categories which contain BAut_{C_{L}(Z)}(S_{0}) and BAut_{L}(S_{0}) as very natural full subcategories to look at. We omit the details.

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

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

L.A.G.A-Laboratoire de Mathematiques Institut Galilee 99 Avenue J.B. Clement 93430 Villetaneuse France

   E-mail address: seeliger@math.univ-paris13.fr