EXTENSIONS OF LINKING SYSTEMS WITH p-GROUP KERNEL

BOB OLIVER AND JOANA VENTURA

Abstract. We study extensions of p-local finite groups where the kernel is a pgroup. In particular, we construct examples of saturated fusion systems F which do not come from finite groups, but which have normal p-subgroups A ◁ F such that F/A is the fusion system of a finite group. One of the tools used to do this is the concept of a “transporter system”, which is modelled on the transporter category of a finite group, and is more general than a linking system.

Let G be a finite group, with Sylow p-subgroup S ∈ Syl_{p}(G). The fusion system of G (at p) is the category F_{S}(G) whose objects are the subgroups of G, and where Mor_{F_{S}(G)}(P, Q) is the set of monomorphisms from P to Q induced by conjugation by elements of G. The transporter system of G at p is the category T_{S}(G) with the same objects as F_{S}(G), and with morphism sets Mor_{T_{S}(G)}(P, Q) = N_{G}(P, Q): the set of elements of G which conjugate P into Q. A subgroup P ≤ S is called p-centric in G if C_{G}(P) = Z(P) × C′_{G}(P) for a (unique) subgroup C′_{G}(P) of order prime to p; and the centric linking system of G is the category L_{S}^{c}(G) whose objects are the subgroups of S which are p-centric in G, and where Mor_{L_{S}^{c}(G)}(P, Q) = N_{G}(P, Q)/C′_{G}(P). All of these definitions are repeated in more detail at the beginning of Section 1. In several papers, such as [BLO1] and [O2], the fusion and linking systems of G are shown to play a central role in describing homotopy theoretic properties of the p-completed classifying space BG_{p}^{∧}.

Abstract fusion and linking systems have also been defined and studied, and are shown in [BLO2] to have many of the same properties as the fusion and linking systems of finite groups. A p-local finite group is defined to be a triple (S, F, L), where S is a finite p-group, F is a saturated fusion system over S (Definitions 1.2 and 1.3), and L is a centric linking system associated to F (Definition 1.6). Normal and central p-subgroups of fusion systems and linking systems are also defined (Definition 1.4).

Certain types of extensions of p-local finite groups, and in particular central extensions, were studied in [BCGLO2]. One hope was that extensions could provide a new way to construct exotic examples. But in the case of central extensions, this was shown to be impossible. By [BCGLO2, Theorem 6.13 and Corollary 6.14], if A is a central subgroup in (S, F, L), and (S/A, F/A, L/A) is induced by a group G, then (S, F, L) is induced by a group G̃ such that A ≤ Z(G̃) and G̃/A ≅ G.

In this paper, we look at the more general situation of extensions with p-group kernel. Equivalently, given a p-local finite group (S, F, L) and a finite p-group A, we want to find p-local finite groups (S̃, F̃, L̃) such that A ◁ F and (S, F, L) ≅ (S̃/A, F̃/A, L̃/A). One problem when doing this is that the fusion system F/A contains too little information: F cannot be described as an extension of F/A by A in any sense.

2000 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D20.

Key words and phrases. Classifying space, p-completion, finite groups, fusion.

B. Oliver is partially supported by UMR 7539 of the CNRS.

J. Ventura is partially supported by FCT/POCTI/FEDER and grant PDCT/MAT/58497/2004.

Both authors were partially supported by the Mittag-Leffler Institute in Sweden.

Another problem is that in general, when L̃ is a linking system and A ◁ L̃, then L̃ can be thought of as an extension of L̃/A by A, but L̃/A need not be a linking system. As explained in Section 2, it can contain much more information than a linking system does. Conversely, if we take a linking system L and try naively to extend it, then the resulting category will in general have too few objects to be a linking system. So we were forced to look at a larger class of categories to extend.

A transporter system is a category whose objects are subgroups of a given p-group S, associated to a given fusion system, which satisfies axioms motivated by the two main examples: the transporter category of a finite group, and categories of the form L/A when L is a linking system and A ◁ L. A transporter system T for which Ob(T ) is the set of all subgroups of S is always the transporter category of the finite group Aut_{T} (1) (Proposition 3.12), so we are interested mainly in the cases where not all subgroups of S are objects. The precise definition is given at the start of Section 3.

Nerves of transporter systems have many of the topological properties which are already known for linking systems. For example, if T is a transporter system associated to the fusion system F, and Ob(T ) includes all F-centric subgroups, then T induces a centric linking system L associated to F and |T |_{p}^{∧} ≃ |L|_{p}^{∧} (Proposition 4.6). As another example, if T^{r} ⊆ T is the full subcategory whose objects are the “T -radical subgroups” (Definition 3.9), then |T^{r}| ≃ |T | (Proposition 4.7).

Extensions of transporter systems are defined and studied in Section 5. If T is a transporter system, and τ: T̃ → T is a functor which satisfies certain category theoretic properties (Definition 5.1), then T̃ is also a transporter system and T̃/A ≅ T for a certain normal p-subgroup A ◁ T̃. Moreover, in this situation, |T̃| → |T | is a fibration with fiber BA. Once this has been established, then conditions are described (Theorem 5.11) which imply that T̃ is in fact a centric linking system, or at least a full subcategory of a centric linking system which includes all subgroups which are centric and radical.

Finally, in Section 6, we look at extensions 1 → A → T̃ → T → 1 of this type, when T is a full subcategory of the transporter category of a finite group G. We first show that if the induced action of π_{1}(|T |) on A factors through G, then T̃ will be a full subcategory of the transporter category of some group G̃ such that G̃/A ≅ G. Afterwards, we give examples (Example 6.2) of such extensions where the action does not factor through G, and where T̃ and its associated fusion system F̃ are exotic in in the sense that F̃ is not the fusion system of any finite group.

We would like to thank the University of Aberdeen, the Universitat Autònoma de Barcelona, and especially the Bernoulli Center in Lausanne and the Mittag-Leffler Institut near Stockholm for their hospitality, allowing the two authors to meet together and work on this project. We would also like to thank Albert Ruiz for his very timely discovery of some examples of exotic fusion systems which helped lead to our Example 6.2.

1. Background: fusion and linking systems

We first fix some notation. For any group G, and any x ∈ G, c_{x} denotes conjugation by x (c_{x}(g) = xgx^{−1}). For H, K ≤ G, we write

N_{G}(H, K) = {x ∈ G | xHx^{−1} ≤ K}

to denote the transporter set, and

Hom_{G}(H, K) = {c_{x} ∈ Hom(H, K) | x ∈ N_{G}(H, K)} ≅ N_{G}(H, K)/C_{G}(H).

We also set Aut_{G}(H) = Hom_{G}(H, H) ≅ N_{G}(H)/C_{G}(H).

Definition 1.1. Fix a finite group G and a Sylow subgroup S ∈ Syl_{p}(G).

(a) F_{S}(G) and T_{S}(G) denote the categories where Ob(F_{S}(G)) = Ob(T_{S}(G)) is the set of all subgroups of S, and where

Mor_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) ≅ N_{G}(P, Q)/C_{G}(P)

and

Mor_{T_{S}(G)}(P, Q) = N_{G}(P, Q).

Let ρ: T_{S}(G) ⟶ F_{S}(G) be the functor which is the identity on objects, and which sends x ∈ N_{G}(P, Q) to c_{x} ∈ Hom_{G}(P, Q).

(b) A p-subgroup P ≤ G is p-centric in G if Z(P) ∈ Syl_{p}(C_{G}(P)); equivalently, if C_{G}(P) = Z(P) × C′_{G}(P) for some (unique) subgroup C′_{G}(P) of order prime to p. Define L_{S}^{c}(G) to be the category whose objects are the subgroups of S which are p-centric in G, and where

Mor_{L_{S}^{c}(G)}(P, Q) = N_{G}(P, Q)/C′_{G}(P).

We call F_{S}(G) the fusion system (or fusion category) of G, T_{S}(G) the transporter system, and L_{S}^{c}(G) the centric linking system. In this paper, we will be looking at abstract versions of all three of these systems, starting with fusion systems.

Definition 1.2 ([Pg], [BLO2, Definition 1.1]). A fusion system over a finite p-group S is a category F, where Ob(F) is the set of all subgroups of S, and which satisfies the following two properties for all P, Q ≤ S:

• Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q); and

• each φ ∈ Hom_{F}(P, Q) is the composite of an isomorphism in F followed by an inclusion.

Fusion systems as defined above are too general for our purposes, and additional axioms are needed for them to be very useful. When F is a fusion system over a finite p-subgroup S, then two subgroups P, Q ≤ S are said to be F-conjugate if they are isomorphic as objects of the category F. A subgroup P ≤ S is called fully centralized in F if |C_{S}(P)| ≥ |C_{S}(P′)| for all P′ ≤ S which is F-conjugate to P. Similarly, a subgroup P ≤ S is called fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for all P′ ≤ S which is F-conjugate to P.

Definition 1.3 ([Pg], [BLO2, Definition 1.2]). A fusion system F over a finite p-group S is saturated if the following two conditions hold:

(I) For all P ≤ S which is fully normalized in F, P is fully centralized in F and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(II) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φP is fully centralized, and if we

set

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(φP)},

then there is φ ∈ Hom_{F}(N_{φ}, S) such that φ|_{P} = φ.

If G is a finite group and S ∈ Syl_{p}(G), then F_{S}(G) is a saturated fusion system. Axioms (I) and (II) follow mostly as consequences of the Sylow theorems (cf. [BLO2, Proposition 1.3]).

We next specify certain collections of subgroups relative to a given fusion system.

Definition 1.4. Let F be a fusion system over a finite p-subgroup S.

• A subgroup P ≤ S is F-centric if C_{S}(P′) = Z(P′) for all P′ ≤ S which is Fconjugate to P. We let F^{c} ⊆ F denote the full subcategory with objects the F-centric subgroups of S.

• A subgroup P ≤ S is F-radical if Out_{F}(P) is p-reduced; i.e., if O_{p}(Out_{F}(P)) = 1.

• A subgroup A ≤ S is normal in F (denoted A ◁ F) if for all P, Q ≤ S and all f ∈ Hom_{F}(P, Q), f extends to a morphism f ∈ Hom_{F}(P A, QA) such that f(A) = A.

• A subgroup A ≤ S is central in F if for all P, Q ≤ S and all f ∈ Hom_{F}(P, Q), f extends to a morphism f ∈ Hom_{F}(P A, QA) such that f|_{A} = Id_{A}.

If F = F_{S}(G) for some finite group G, then P ≤ S is F-centric if and only if P is p-centric in G (i.e., Z(P) ∈ Syl_{p}(C_{G}(P))), and P is F-radical if and only if N_{G}(P)/(P ·C_{G}(P)) is p-reduced. However, P being F-radical is not the same as being a radical p-subgroup.

In fact, it turns out that saturated fusion systems defined only on the centric subgroups are equivalent to saturated fusion systems defined on all subgroups. In other words, when constructing saturated fusion systems over a finite p-group S, we really need only define it on the centric subgroups of S, and check that it satisfies axioms (I) and (II) for those subgroups. The next theorem describes how a category constructed in this way can then be extended in a unique way to a saturated fusion system over S.

For any fusion system F over S, and any set H of subgroups of S which is closed under F-conjugacy, we say that F is H-saturated if conditions (I) and (II) in Definition 1.3 are satisfied for all P ∈ H. We say that F is H-generated if each morphism in F is a composite of restrictions of morphisms between subgroups in H.

Theorem 1.5. Fix a p-group S and a fusion system F over S.

(a) Assume F is saturated, and let H be the set of F-centric F-radical subgroups of S. Then F is H-generated. More precisely, for each P, P′ ≤ S and each φ ∈ Iso_{F}(P, P′), there are subgroups P = P_{0}, P_{1}, …, P_{k} = P′, subgroups Q_{i} ≥ ⟨P_{i−1}, P_{i}⟩ (i = 1, …, k) which are F-centric, F-radical, and fully normalized in F, and automorphisms φ_{i} ∈ Aut_{F}(Q_{i}), such that φ_{i}(P_{i−1}) = P_{i} for all i and φ = (φ_{k}|_{P_{k−1}}) ∘ ⋯ ∘ (φ_{1}|_{P_{0}}).

(b) Let F be a fusion system over a finite p-group S. Let H be a set of subgroups of S closed under F-conjugacy such that F is H-saturated and H-generated. Assume also that each F-centric subgroup of S not in H is F-conjugate to some subgroup P ≤ S such that Out_{S}(P) ∩ O_{p}(Out_{F}(P)) ≠ 1. Then F is saturated.

Proof. Part (a) is Alperin’s fusion theorem for saturated fusion systems, in the form shown in [BLO2, Theorem A.10]. Part (b) is proven in [BCGLO1, Theorem 2.2]. □

We now turn to linking systems associated to abstract fusion systems.

Definition 1.6 ([BLO2, Definition 1.7]). Let F be a fusion system over the p-group S. A centric linking system associated to F is a category L whose objects are the F-centric subgroups of S, together with a functor π: L ⟶ F^{c}, and distinguished monomorphisms P ⟶^{δ_{P}} Aut_{L}(P) for each F-centric subgroup P ≤ S, which satisfy the following conditions.

(A) π is the identity on objects. For each pair of objects P, Q ∈ L, Z(P) acts freely on Mor_{L}(P, Q) by composition (upon identifying Z(P) with δ_{P} (Z(P)) ≤ Aut_{L}(P)), and π induces a bijection

Mor_{L}(P, Q)/Z(P) ⟶^{≅} Hom_{F}(P, Q).

(B) For each F-centric subgroup P ≤ S and each x ∈ P, π(δ_{P} (x)) = c_{x} ∈ Aut_{F}(P).

(C) For each f ∈ Mor_{L}(P, Q) and each x ∈ P, the following square commutes in L:

\xymatrix{P \ar[r]^{f} \ar[d]^{δ_{P} (x)} & Q \ar[d]^{δ_{Q}(π(f)(x))} \\ P \ar[r]^{f} & Q .}

A p-local finite group is defined to be a triple (S, F, L), where S is a finite p-group, F is a saturated fusion system over S, and L is a centric linking system associated to F. The classifying space of the triple (S, F, L) is the p-completed nerve |L|_{p}^{∧}.

For any finite group G with Sylow p-subgroup S, the category L_{S}^{c}(G) (Definition 1.1) is easily seen to satisfy conditions (A), (B), and (C) above, and hence is a centric linking system associated to F_{S}(G). Thus (S, F_{S}(G), L_{S}^{c}(G)) is a p-local finite group, with classifying space |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧} (see [BLO1, Proposition 1.1]).

The following lifting lemma for linking systems is used frequently.

Lemma 1.7. Let (S, F, L) be a p-local finite group. Fix F-centric subgroups P, Q, R in S, and let P ⟶^{φ} Q ⟶^{ψ} R be any sequence of morphisms in F. Then the following hold.

(a) Let ψ̃ and ψ̃φ be arbitrary liftings to L of ψ and ψφ, respectively. Then there is a unique morphism φ̃ ∈ Mor_{L}(P, Q) such that

ψ̃ ∘ φ̃ = ψ̃φ;  (1)

and furthermore π_{P,Q}(φ̃) = φ.

(b) Choose liftings ψ̃φ ∈ Mor_{L}(P, R) and φ̃ ∈ Mor_{L}(P, Q) of ψφ and φ, respectively. Then there is a unique morphism ψ̃ ∈ Mor_{L}(Q, R) such that ψ̃ ∘ φ̃ = ψ̃φ, and such that π(ψ̃) = ψ ∘ c_{q} for some q ∈ φ(Z(P)).

Proof. (a) See [BLO2, Lemma 1.10] or Lemma A.7(a).

(b) Let α ∈ Mor_{L}(Q, R) be a lifting of ψ, i.e., π(α) = ψ. Then by axiom (A) of a linking system, there is z ∈ Z(P) such that ψ̃φ = α ∘ φ̃ ∘ δ_{P} (z), and by axiom (C) we have φ̃ ∘ δ_{P} (z) = δ_{Q}(φ(z)) ∘ φ̃. Set ψ̃ = α ∘ δ_{Q}(φ(z)), and note that π(ψ̃) = ψ ∘ c_{φ(z)}.

If ψ̃′ is another morphism satisfying the same conditions, then by assumption, π(ψ̃′) = π(ψ̃) ∘ c_{x} for some x ∈ φ(Z(P)). Then by axiom (A), ψ̃′ = ψ̃ ∘ δ_{Q}(φ(y)) for some y ∈ Z(P) such that φ(y) ∈ x·Z(Q). Also,

ψ̃ ∘ φ̃ = ψ̃φ = ψ̃′ ∘ φ̃ = ψ̃ ∘ δ_{Q}(φ(y)) ∘ φ̃ = ψ̃ ∘ φ̃ ∘ δ_{P} (y),

where the last equality follows from (C). Since the action of Z(P) on Mor_{L}(P, R) is free, it follows that y = 1, and hence that ψ̃′ = ψ̃. □

The following is an easy corollary to Lemma 1.7(a).

Corollary 1.8. Let F be a fusion system (not necessarily saturated) over a p-group S, and let L be a centric linking system associated to F. For each F-centric subgroup P ≤ S, choose an “inclusion” morphism ι_{P} ∈ Mor_{L}(P, S) such that π(ι_{P}) = incl_{P,S}. Then there are unique injections δ_{P,Q}: N_{G}(P, Q) ⟶ Mor_{L}(P, Q), for each pair of subgroups P, Q ∈ Ob(L), with the property that ι_{Q} ∘ δ_{P,Q}(g) = δ_{S}(g) ∘ ι_{P} for all g ∈ N_{G}(P, Q). Also, δ_{P} is the restriction to P ≤ N_{S}(P) of δ_{P,P} for each P, and the δ_{P,Q} define a functor from the transporter category of S (restricted to the objects of L) to L.

Proof. See [BLO2, Proposition 1.11]. □

We finish the section by noting the following standard result in group theory which will be needed later.

Lemma 1.9. (a) If Q ◁ P are finite p-groups and α ∈ Aut(P) is such that α|_{Q} = Id_{Q} and α/Q = Id_{P/Q}, then α has p-power order.

(b) If S is a p-subgroup of the finite group G, and H ◁ G, then S ∈ Syl_{p}(G) if and only if S ∩ H ∈ Syl_{p}(H) and SH/H ∈ Syl_{p}(G/H).

Proof. Point (a) is shown in [Go, Corollary 5.3.3]. Point (b) follows since S/(S ∩ H) ≅ SH/H and hence

[G : S] = [H : S ∩ H] · [G/H : SH/H].  □

2. Quotients of linking systems

In this section, we show that whenever (S, F, L) is a p-local finite group and A ◁ F (Definition 1.4), then we can define a quotient p-local finite group (S/A, F/A, (L/A)_{∗}^{c}) as a quotient of (S, F, L) by A. We also show that L/A, defined as the quotient of the free action of A on L, is not in general a linking system; and this will motivate the concept of a transporter system defined in the next section.

We first consider quotients of fusion systems. Recall that for any fusion system F over S, a subgroup A ≤ S is weakly closed in F if A is the only subgroup in its Fconjugacy class. Clearly, any normal subgroup in F is weakly closed, and any weakly closed subgroup is normal in S. When A is weakly closed in F, then we define F/A to be the fusion system over S/A where

Hom_{F/A}(P/A, Q/A) = {f/A | f ∈ Hom_{F}(P, Q)}.

Lemma 2.1. Fix a saturated fusion system F over a p-group S. Then for any subgroup A ◁ S which is weakly F-closed in S, F/A is a saturated fusion system over S/A.

Proof. This is shown, for example, in [O2, Lemma 2.6]. But since it plays a central role in this paper, we repeat the proof here.

Proof of (II): Fix φ ∈ Hom_{F/A}(P/A, S/A) such that Im(φ) is fully centralized in F/A. Set P′/A = φ(P/A), and

N_{φ} = {g ∈ N_{S/A}(P/A) | φc_{g}φ^{−1} ∈ Aut_{S/A}(P′/A)}.

Choose Q ≤ S which is F-conjugate to P and P′ and fully normalized in F. By [BLO2, Proposition A.2(b)], there is

χ ∈ Hom_{F}(N_{S}(P′), N_{S}(Q))

such that χ(P′) = Q. Let χ ∈ Hom_{F/A}(N_{S/A}(P′/A), N_{S/A}(Q/A)) be the induced homomorphism. Thus χ(C_{S/A}(P′/A)) = C_{S/A}(Q/A) since P′/A is fully centralized, and this restricts to an isomorphism C_{S}(P′)/A ≅ C_{S}(Q)/A. We have now proven

Q fully normalized in F ⟹ Q/A fully centralized in F/A.  (1)

Set ψ = χ ∘ φ ∈ Iso_{F/A}(P/A, Q/A) for short, and choose a lifting ψ̃_{0} ∈ Iso_{F}(P, Q) of ψ. Consider the subgroups

Ñ_{ψ}/A = N_{ψ} = {g ∈ N_{S/A}(P/A) | ψc_{g}ψ^{−1} ∈ Aut_{S/A}(Q/A)}

and

K_{Q} = Ker[Aut_{F}(Q) ⟶ Aut_{F/A}(Q/A)].

Thus ψ̃_{0}AutÑ_{ψ}(P)ψ̃_{0}^{−1} ≤ Aut_{S}(Q)·K_{Q}. Since Q is fully normalized in F, K_{Q} ◁ Aut_{S}(Q)·K_{Q} and Aut_{S}(Q) ∈ Syl_{p}(Aut_{S}(Q)·K_{Q}), and hence all Sylow p-subgroups of Aut_{S}(Q)·K_{Q} are conjugate by elements of K_{Q}. In particular, there is ω ∈ K_{Q} such that

(ωψ̃_{0})AutÑ_{ψ}(P)(ωψ̃_{0})^{−1} ≤ Aut_{S}(Q).  (2)

Set ψ̃ = ωψ̃_{0}. This is also a lifting of ψ since ω ∈ K_{Q} (ω/A = Id_{Q/A}); and by (2),

Ñ_{ψ} ≤ Nψ̃ =^{def} {g ∈ N_{S}(P) | ψ̃c_{g}ψ̃^{−1} ∈ Aut_{S}(Q)}.

Since Q is fully centralized in F (by axiom (I) for F), axiom (II) for F now implies that ψ̃ extends to a morphism ψ̂ ∈ Hom_{F}(Ñ_{ψ}, N_{S}(Q)), and hence that ψ = χ ∘ φ extends to

ψ ∈ Hom_{F/A}(N_{ψ}, N_{S/A}(Q/A)).

We claim that ψ(N_{φ}) ≤ Im(χ). To see this, fix g ∈ N_{φ}, and let h ∈ N_{S/A}(P′/A) be such that φc_{g}φ^{−1} = c_{h}. Then

c_{ψ(g)} = ψc_{g}ψ^{−1} = χc_{h}χ^{−1} = c_{χ(h)} ∈ Aut(Q/A),

so ψ(g) ∈ χ(h)·C_{S/A}(Q/A). We have already seen that C_{S/A}(Q/A) = χ(C_{S/A}(P′/A)), and this finishes the proof that ψ(g) ∈ Im(χ). Thus there is

φ ∈ Hom_{F/A}(N_{φ}, N_{S/A}(P′/A))

such that χ ∘ φ = ψ, and φ|_{P/A} = φ. This finishes the proof of condition (II) for F/A.

Proof of (I): Assume P/A is fully normalized in F/A. Since N_{S/A}(P′/A) = N_{S}(P′)/A for all P′ in the same F-conjugacy class, P is also fully normalized in F. Then P/A is fully centralized in F/A by (1). Also, Aut_{S/A}(P/A) ∈ Syl_{p}(Aut_{F/A}(P/A)) since Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) (by condition (I) again for F), and since a surjection of finite groups sends Sylow subgroups onto Sylow subgroups. □

Whenever (S, F, L) is a p-local finite group and A ◁ S is normal in F, we let L/A be the category whose objects are the subgroups P/A ≤ S/A such that P is F-centric, and where

Mor_{L/A}(P/A, Q/A) = Mor_{L}(P, Q)/A.

Here, g ∈ A acts on Mor_{L}(P, Q) via composition with δ_{P} (g) ∈ Aut_{L}(P). Let (L/A)^{c} be the subcategory of L/A whose objects are the F/A-centric subgroups of S/A. We will define (L/A)_{∗}^{c} to be a certain quotient category of (L/A)^{c}.

Lemma 2.2. Fix a p-local finite group (S, F, L), and assume A ◁ S is normal in F. Let F/A be the induced fusion system over S/A, and let (L/A)^{c} be defined as above. Then for each F/A-centric subgroup P/A ≤ S/A, P is F-centric, and there is a unique subgroup E_{0}(P) ≤ Aut_{L/A}(P/A) of order prime to p such that

C_{Aut_{L/A}(P/A)}(P/A) = E_{0}(P) × Z(P/A).

Let (L/A)_{∗}^{c} be the category whose objects are the F/A-centric subgroups of S/A, and where

Mor_{(L/A)_{∗}^{c}}(P/A, Q/A) = Mor_{L/A}(P/A, Q/A)/E_{0}(P).

Then (L/A)_{∗}^{c} is a well defined category, and is a centric linking system associated to F/A.

Proof. Fix an F/A-centric subgroup P/A ≤ S/A. For all P′/A which is F/A-conjugate to P/A, P′ is F-conjugate to P, and C_{S}(P′)/A ≤ C_{S/A}(P′/A) ≤ P′/A. Thus C_{S}(P′) ≤ P′ for all such P′, and this shows that P is F-centric. In particular, P ∈ Ob(L).

Set

K′_{P} = Ker[Aut_{L/A}(P/A) ⟶ Aut_{F/A}(P/A)] ,  (3)

and let δ̂_{P/A} denote the homomorphism

δ̂_{P/A} = δ_{P,P} /A: N_{S/A}(P/A)_{=N_{S}(P)/A} ⟶ Aut_{L/A}(P/A)_{=Aut_{L}(P)/A} .

If P is fully normalized in F, then

Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) ⟹ δ_{P,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{L}(P))

⟹ δ̂_{P/A}(N_{S/A}(P/A)) ∈ Syl_{p}(Aut_{L/A}(P/A)) .

Hence by Lemma 1.9(b), K′_{P} ∩δ̂_{P/A}(N_{S/A}(P/A)) = δ̂_{P/A}(Z(P/A)) is a Sylow p-subgroup of K′_{P}. Since |Z(P/A)| and |K′_{P} | are both invariant under F-conjugacy, this implies that δ̂_{P/A}(Z(P/A)) ∈ Syl_{p}(K′_{P}) whether or not P is fully normalized.

For each f ∈ Aut_{L}(P) and each g ∈ P,

δ_{P} (π(f)(g)) = f ∘ δ_{P} (g) ∘ f^{−1}

by axiom (C) for L. Upon passing to the quotient group Aut_{L/A}(P/A), this shows that each element of K′_{P} centralizes the subgroup δ̂_{P/A}(P/A). In particular, the Sylow p-subgroup δ̂_{P/A}(Z(P/A)) is central in K′_{P}, and hence

K′_{P} = E_{0}(P) × Z(P/A)  (4)

for a unique subgroup E_{0}(P) ≤ Aut_{L/A}(P/A).

Now define (L/A)_{∗}^{c} to be the category whose objects are the F/A-centric subgroups of S/A, and where

Mor_{(L/A)_{∗}^{c}}(P/A, Q/A) = Mor_{L/A}(P/A, Q/A)/E_{0}(P).

To see that composition is well defined, we must show, for each f ∈ Mor_{L/A}(P/A, Q/A) between F/A-centric subgroups, that

E_{0}(Q) ∘ f ⊆ f ∘ E_{0}(P).

For each α ∈ E_{0}(Q), there is by Lemma 1.7(a) (applied to L) a unique morphism β ∈ Aut_{L/A}(P/A) such that α ∘ f = f ∘ β, and β ∈ E_{0}(P) since the induced square in F/A must commute.

This shows that (L/A)_{∗}^{c} is a well defined category. The distinguished monomorphisms

δ_{P/A}: P/A ⟶ Aut_{L/A}(P/A) = Aut_{L}(P)/A

are induced by the distinguished monomorphisms δ_{P} for L. Conditions (B) and (C) for (L/A)^{c} follow directly from the corresponding conditions for L. It remains to prove condition (A), and this follows by (3) and (4). □

Note that by Proposition A.10, whenever (S, F, L) is a p-local finite group with normal p-subgroup A ◁ F, then the sequence

BA ⟶ |L| ⟶ |L/A|

is a homotopy fibration sequence.

It is easy to construct examples of linking systems L with normal subgroup A for which (L/A)^{c} is not a centric linking system. As a rather trivial example, set p = 2, fix a finite group H of even order, and set G = A_{4}×H. Let A ◁ G be the normal subgroup of order 4 in A_{4}, and fix Sylow subgroups S_{0} ∈ Syl_{2}(H) and S = A × S_{0} ∈ Syl_{2}(G). Let F = F_{S}(G), L = L_{S}^{c}(G), and L_{0} = L_{S_{0}}^{c}(H). Then A is normal in F. For any P ≤ S which is F-centric and contains A, P = A × P_{0} for some P_{0} ≤ S_{0}, Aut_{L}(P) = A_{4} × Aut_{L_{0}}(P_{0}), and hence Aut_{L/A}(P/A) ≅ C_{3} × Aut_{L_{0}}(P_{0}). Thus (L/A)^{c} is not a linking system.

Note also, in the above example, that the kernel of the map from Aut_{F}(P) to Aut_{F/A}(P/A) is not a p-group for any P containing A. This helps to motivate the following general criterion for L/A to be a linking system.

Proposition 2.3. Let (S, F, L) be a finite p-group, and assume A ◁ F. Then (L/A)^{c} is a linking system associated to F/A if and only if Ker[Aut_{F}(P) ⟶ Aut_{F/A}(P/A)] is a p-group for all P ≤ S such that P ≥ A and P/A is F/A-centric.

Proof. Let P be the set of all subgroups P ≤ S such that P ≥ A and P/A is F/Acentric. Consider the following subgroups for all P ∈ P:

K_{P} = Ker[Aut_{F}(P) ⟶ Aut_{F/A}(P/A)]
K′_{P} = Ker[Aut_{L/A}(P/A) ⟶ Aut_{F/A}(P/A)] .

By Lemma 2.2, K′_{P} = E_{0}(P) × Z(P/A) for some subgroup E_{0}(P) of order prime to p, and (L/A)^{c} is a linking system associated to F/A if and only if K′_{P} is a p-group for all P ∈ P.

Consider the diagram

\xymatrix{Aut_{L}(P) \ar@{->>}[r]^{/A} \ar@{->>}[d]^{/Z(P)} & Aut_{L/A}(P/A) \ar@{->>}[d]^{/K′_{P}} \\ Aut_{F}(P) \ar@{->>}[r]^{/K_{P}} & Aut_{F/A}(P/A) .}

Here, in all cases, “/H” means dividing out by the subgroup H. Since A and Z(P) are both p-groups, this shows that K_{P} is a p-group if and only if K′_{P} is a p-group, and thus (L/A)^{c} is a linking system if and only if K_{P} is a p-group for all P ∈ P. □

The following proposition describes one more very simple way to construct such examples.

Proposition 2.4. Fix a finite group G and Sylow subgroup S ∈ Syl_{p}(G). Assume there is a normal p-subgroup A ◁ G which is centric in G; i.e., C_{G}(A) = Z(A). Let L_{S}^{≥A}(G) ⊆ L_{S}^{c}(G) be the full subcategory with objects those P ≤ S containing A. Then A is normal in F_{S}(G), and L_{S}^{≥A}(G)/A ≅ T_{S/A}(G/A).

Proof. Since A is centric in G, so is every subgroup which contains A. Thus C_{G}(P) = Z(P) for every P ∈ Ob(L_{S}^{≥A}(G)), and so L_{S}^{≥A}(G) is a full subcategory of the transporter category T_{S}(G). It follows that L_{S}^{≥A}(G)/A ≅ T_{S}^{≥A}(G) ≅ T_{S/A}(G/A). □

One easily finds examples of groups whose (centric) transporter category is not a linking category. For example, there are subgroups P ≤ A_{7} such that P ≅ C_{2}^{2}, and C_{A_{7}}(P) ≅ C_{2}^{2} × C_{3}. Thus P is 2-centric in A_{7}, but for S ∈ Syl_{2}(A_{7}), L_{S}^{c}(A_{7}) is not a full subcategory of T_{S}(A_{7}). So if we set G = C_{2}^{7} ⋊ A_{7}, where A_{7} acts on A = C_{2}^{7} by permuting a basis, then (L_{S}^{c}(G)/A)^{c} is not a linking system for G/A ≅ A_{7}.

3. Transporter systems

In the last section, we saw that for a centric linking system L with normal subgroup A, the quotient category (L/A)^{c} need not be a linking system associated to any fusion system. This motivates us to define what we call “transporter systems”: categories with extra structures satisfying some properties similar to those of the transporter categories associated to finite groups, but without necessarily having such a group.

Recall (Definition 1.1) that for any finite group G and any S ∈ Syl_{p}(G), T_{S}(G) denotes the category with objects the subgroups of G, and with morphism sets the transporter sets N_{G}(P, Q). For any set H of subgroups of S, we let T_{H}(G) be the full subcategory of T_{S}(G) with object set H.

Definition 3.1. Let F be a fusion system over a p-group S. A transporter system associated to a fusion system F is a nonempty finite category T , together with a pair of functors

T_{Ob(T )}(S) ⟶^{ε} T ⟶^{ρ} F ,

satisfying the following conditions:

(A1) Ob(T ) ⊆ Ob(F), and Ob(T ) is closed under F-conjugacy and overgroups. Also, ε is the identity on objects and ρ is the inclusion on objects.

(A2) For each P, Q ∈ Ob(T ), the kernel

E(P) =^{def} Ker[ρ_{P,P} : Aut_{T} (P) ⟶ Aut_{F}(P)]

acts freely on Mor_{T} (P, Q) by right composition, and ρ_{P,Q} is the orbit map for this action. Also, E(Q) acts freely on Mor_{T} (P, Q) by left composition.

(B) For each P, Q ∈ Ob(T ), ε_{P,Q}: N_{S}(P, Q) ⟶ Mor_{T} (P, Q) is injective, and the composite ρ_{P,Q} ∘ ε_{P,Q} sends g ∈ N_{S}(P, Q) to c_{g} ∈ Hom_{F}(P, Q).

(C) For all φ ∈ Mor_{T} (P, Q) and all g ∈ P, the diagram

\xymatrix{P \ar[r]^{φ} \ar[d]_{ε_{P,P} (g)} & Q \ar[d]^{ε_{Q,Q}(ρ(φ)(g))} \\ P \ar[r]^{φ} & Q}

commutes in T .

(I) ε_{S,S}(S) ∈ Syl_{p}(Aut_{T} (S)).

(II) Let φ ∈ Iso_{T} (P, Q), P ◁ P ≤ S, and Q ◁ Q ≤ S be such that φ∘ε_{P,P} (P)∘φ^{−1} ≤ ε_{Q,Q}(Q). Then there is some φ ∈ Mor_{T} (P , Q) such that φ∘ε_{P,P} (1) = ε_{Q,Q}(1)∘φ.

We will often write “(S, F, T ) is a transporter system” to mean that T is a transporter system associated to the fusion system F over the p-group S. The above axioms are clearly labelled to show their connection with axioms (A), (B), and (C) of a linking system, and axioms (I) and (II) of a saturated fusion system. Note, however, that the concepts of fully normalized and fully centralized subgroups do not appear in the axioms of a transporter system — which does help simplify some of our proofs that if one category is a transporter system then another one is too.

For any transporter system (S, F, T ), axiom (A2) implies that the functor ρ: T → F is “source regular” in the sense of Definition A.5. So by Lemma A.6, a morphism in T is an isomorphism if and only if its image in F is an isomorphism. In particular, T is an EI category (all endomorphisms are automorphisms) since F is one.

We will show in Proposition 3.5 that transporter categories of finite groups, linking systems associated to saturated fusion systems, and (more generally) categories of the form L/A when L is a linking system and A a normal subgroup, are all examples of transporter systems. These examples provided our main motivation for the above definition, and the axioms for a transporter system are clearly related to those for fusion and linking systems.

We will prove soon, as Proposition 3.4(a), the following stronger form of axiom (I), which is more closely analogous to the axiom (I) in Definition 1.3: (I′) If P is fully normalized in F, then ε_{P,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)). The weaker axiom (I) we use here is motivated by the alternative set of axioms for a saturated fusion system due to Radu Stancu [St].

For all P ≤ Q ≤ S objects in T , we set ι_{P,Q} = ε_{P,Q}(1), and think of these as the inclusion morphisms in T . By condition (B), ρ sends inclusions in T to inclusions in F. Whenever P_{0} ≤ P ≤ S and Q_{0} ≤ Q ≤ S are in Ob(T ), and

\xymatrix{P_{0} \ar[r]^{φ_{0}} \ar[d]_{ι_{P_{0},P}} & Q_{0} \ar[d]^{ι_{Q_{0},Q}} \\ P \ar[r]^{φ} & Q ,}

is a commutative square in T , we say that φ_{0} is a restriction of φ (and sometimes write φ|_{P_{0},Q_{0}} = φ_{0}), and also that φ is an extension of φ_{0}. Thus axiom (II) gives conditions under which a morphism can be extended.

This terminology suggests that the restriction φ|_{P_{0},Q_{0}} should always exist and be unique, whenever P_{0} and Q_{0} are objects in T and ρ(φ)(P_{0}) ≤ Q_{0}. This is shown in the following lifting lemma, which is the analog for transporter systems of Lemma 1.7(a). We will see later (Lemma 3.8) that extensions are also unique when they exist.

Lemma 3.2. Let (S, F, T ) be a transporter system, and let ρ : T ⟶ F be the projection functor.

(a) Fix morphisms φ ∈ Hom_{F}(P, Q) and ψ ∈ Hom_{F}(Q, R), where P, Q, R ∈ Ob(T ). Then for any pair of liftings ψ̃ ∈ ρ_{Q,R}^{−1}(ψ) and ψ̃φ ∈ ρ_{P,R}^{−1}(ψφ), there is a unique lifting φ̃ ∈ ρ_{P,Q}^{−1}(φ) such that ψ̃ ∘ φ̃ = ψ̃φ.

(b) All morphisms in T are monomorphisms in the categorical sense. In other words, for all P, Q, R ∈ Ob(T ) and all φ_{1}, φ_{2} ∈ Mor_{T} (P, Q) and ψ ∈ Mor_{T} (Q, R), ψ ∘ φ_{1} = ψ ∘ φ_{2} implies φ_{1} = φ_{2}.

(c) For every morphism φ ∈ Mor_{T} (P, Q), and every P_{0}, Q_{0} ∈ Ob(T ) such that P_{0} ≤ P, Q_{0} ≤ Q, and ρ(φ)(P_{0}) ≤ Q_{0}, there is a unique morphism φ_{0} ∈ Mor_{T} (P_{0}, Q_{0}) such that φ ∘ι_{P_{0},P} = ι_{Q_{0},Q} ∘φ_{0}. In particular, every morphism in T is the composite of an isomorphism followed by an inclusion.

Proof. Since morphisms in F are all group monomorphisms, they are also monomorphisms in the categorical sense. Hence points (a) and (b) are special cases of Lemma A.7(a,b).

In the situation of (c), by definition of a fusion system, ρ(φ)|_{P_{0},Q_{0}} ∈ Hom_{F}(P_{0}, Q_{0}). So the result follows from (a), with P, Q, R replaced by P_{0}, Q_{0}, Q, ψ̃φ by φ ∘ ι_{P_{0},P}, etc. The last statement is the special case where P_{0} = P and Q_{0} = ρ(φ)(P). □

The following technical lemma can be thought of as a converse to axiom (II). It shows that the condition for extending a morphism, which is sufficent by axiom (II), is also necessary.

Lemma 3.3. Fix a transporter system (S, F, T ), and objects P ◁ P and Q ≤ Q of T . If φ ∈ Mor_{T} (P , Q) is an extension of φ ∈ Iso_{T} (P, Q), then

\xymatrix{P \ar[r]^{φ} \ar[d]_{ε_{P,P} (x)} & Q \ar[d]^{ε_{Q,Q}(ρ(φ)(x))} \\ P \ar[r]^{φ} & Q}  (1)

commutes in T for all x ∈ P.

Proof. By axiom (C), the following square commutes in T :

\xymatrix{P \ar[r]^{φ} \ar[d]_{ε_{P ,P} (x)} & Q \ar[d]^{ε_{Q,Q}(ρ(φ)(x))} \\ P \ar[r]^{φ} & Q}  (2)

Each morphism in (1) is the restriction of the corresponding morphism in (2). So square (1) also commutes by the uniqueness of restriction morphisms (Lemma 3.2(c)). □

We next prove that axiom (I) can be replaced by the stronger axiom (I′) stated above.

Proposition 3.4. The following hold for any transporter system (S, F, T ) and any subgroup P ≤ S.

(a) P is fully normalized in F if and only if ε_{P,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)).

(b) P is fully centralized in F if and only if ε_{P,P} (C_{S}(P)) ∈ Syl_{p}(E(P)).

Proof. The “if” part in both (a) and (b) is clear, since |Aut_{T} (P)| and |E(P)| depend only on the F-conjugacy class of P.

(a) Suppose otherwise: that T does not satisfy axiom (I′). Let P be a maximal counterexample. Thus P is fully normalized, and ε_{P,P} (N_{S}(P)) is not a p-Sylow subgroup of Aut_{T} (P). Since axiom (I′) holds for the group S by assumption, P ≨ S, and so P ≨ N_{S}(P). Choose Q ∈ Syl_{p}(Aut_{T} (P)) such that ε_{P,P} (N_{S}(P)) ≤ Q. We know that

ε_{P,P} (N_{S}(P)) ≠ Q, so ε_{P,P} (N_{S}(P)) ≨ Q_{0} =^{def} N_{Q}(ε_{P,P} (N_{S}(P))). We thus have strict inclusions of p-subgroups of Aut_{T} (P):

ε_{P,P} (P) ≨ ε_{P,P} (N_{S}(P)) ◁ Q_{0}  (3)

Pick any morphism φ ∈ Q_{0} not in ε_{P,P} (N_{S}(P)). Then φ^{−1} ∘ε_{P,P} (x)∘φ ∈ ε_{P,P} (N_{S}(P)) for all x ∈ N_{S}(P) since ε_{P,P} (N_{S}(P)) ◁ Q_{0}. So by axiom (II), there is an extension φ ∈ Aut_{T} (N_{S}(P)) of φ; i.e., a morphism such that the following square commutes:

\xymatrix{N_{S}(P) \ar[r]^{φ} & N_{S}(P) \\ P \ar[u]^{ι_{P,N_{S} (P)}} \ar[r]_{φ} & P . \ar[u]_{ι_{P,N_{S} (P)}}}

Set |φ| = p^{k}m where p∤m. Choose r such that r ≡ 0 (mod m) and r ≡ 1 (mod p^{k}). Then φ^{r} has order p^{k} and (since |φ||p^{k}) is again an extension of φ^{r} = φ. We can thus assume that φ has p-power order.

Choose P fully normalized and F-conjugate to N_{S}(P). Let γ ∈ Iso_{T} (N_{S}(P), P) be any lifting of an isomorphism γ′ ∈ Iso_{F}(N_{S}(P), P). Since |P | = |N_{S}(P)| > |P | and P is a maximal counterexample, we have ε_{P ,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)). Hence γ ∘φ ∘γ^{−1}

is conjugate to ε_{P ,P} (x) for some x ∈ N_{S}(P), because φ has p-power order (and so does γ ∘ φ ∘ γ^{−1}) and all p-Sylow subgroups are conjugate. By replacing γ by an appropriate element of Aut_{T} (P) ∘ γ, we can arrange that γ ∘ φ ∘ γ^{−1} = ε_{P ,P} (x).

Now, φ ∈ Aut_{T} (P) restricts to φ ∈ Aut_{T} (P). Set R = ρ(γ)(P) ≤ P and γ_{0} = γ|_{P,R} ∈ Iso_{T} (P, R) (using Lemma 3.2(c)). Then γ_{0}φγ_{0}^{−1} ∈ Aut_{T} (R) is a restriction of ε_{P ,P} (x). Hence ρ(ε_{P ,P} (x)) = c_{x} restricts to an automorphism of R (in F), which means that x ∈ N_{S}(R) and γ_{0}φγ_{0}^{−1} = ε_{R,R}(x). Also, |N_{S}(R)| ≤ |N_{S}(P)| since P is fully normalized and R is F-conjugate to P, so ρ(γ)(N_{S}(P)) = N_{S}(R), and x = ρ(γ)(y) for some y ∈ N_{S}(P). We thus have the following two commutative squares of isomorphisms in T :

\xymatrix{P \ar[r]^{γ_{0}} \ar[d]_{φ} & R \ar[d]^{ε_{R,R}(x)} & P \ar[r]^{γ_{0}} \ar[d]_{ε_{P,P} (y)} & R \ar[d]^{ε_{R,R}(x)} \\ P \ar[r]^{γ_{0}} & R & P \ar[r]^{γ_{0}} & R ,}

where the second commutes by Lemma 3.3. Upon comparing the two squares, we finally get φ = ε_{P,P} (y), which contradicts our assumption that φ ∉ ε_{P,P} (N_{S}(P)).

(b) Again fix P ≤ S, and let P′ be any subgroup which is F-conjugate to P and fully normalized in F. Then ε_{P′,P′}(N_{S}(P′)) ∈ Syl_{p}(Aut_{T} (P′)) by (a), and hence ε_{P′,P′}(C_{S}(P′)) = ε_{P′,P′}(N_{S}(P′)) ∩ E(P′) is a Sylow p-subgroup of E(P′) by Lemma 1.9(b). Also, E(P) ≅ E(P′), so P is fully centralized if and only if |C_{S}(P)| = |C_{S}(P′)|; equivalently, ε_{P,P} (C_{S}(P)) ∈ Syl_{p}(E(P)). □

We next check that the examples which motivated Definition 3.1 really are transporter systems.

Proposition 3.5. (a) For any p-local finite group (S, F, L), L is a transporter system associated to F. More generally, if A ◁ F is a normal subgroup, then L/A is a transporter system associated to F/A.

(b) For any finite group G and any S ∈ Syl_{p}(G), T_{S}(G) is a transporter system associated to F_{S}(G).

(c) Let (S, F, T ) be a transporter system, and let T_{0} ⊆ T be any nonempty full subcategory such that Ob(T_{0}) is closed under F-conjugacy and overgroups. Then T_{0} is also a transporter system associated to F.

Proof. Point (c) follows immediately from Definition 3.1.

We next check point (b). For finite G and S ∈ Syl_{p}(G), define

T (S) ⟶^{ε} T_{S}(G) ⟶^{ρ} F_{S}(G)

to be the inclusion, and the functor g ↦ c_{g}, respectively. The axioms of Definition 3.1 are easily checked.

It remains to prove (a). Let (S, F, L) be a p-local finite group. We prove here only that L is a transporter system associated to F. The last statement, that L/A is a transporter system for any A ◁ F, will then follow as a special case of Proposition 3.11, to be shown later.

Fix morphisms ι_{P} ∈ Mor_{L}(P, S), for all P ∈ Ob(L), such that π(ι_{P}) = incl_{P,S} ∈ Hom_{F}(P, S), and such that ι_{S} = Id_{S}. By Corollary 1.8, there is a unique functor

ε: T_{Ob(L)}(S) ⟶ L

such that ε_{P,S}(1) = ι_{P} and (ε_{P,P})|_{P} = δ_{P} for all P. Furthermore, ε_{P,Q} is an injection of N_{S}(P, Q) into Mor_{L}(P, Q) for all P, Q ∈ Ob(L) (again by Corollary 1.8). For each P ≤ Q ≤ S such that P, Q ∈ Ob(L), we set ι_{P,Q} = ε_{P,Q}(1) ∈ Mor_{L}(P, Q). We think of these as the inclusion morphisms, and define restriction and extension in L with respect to them.

We are now ready to check that the axioms of a transporter system hold for L. Axioms (A1) and (C) follow immediately from axioms (A) and (C) for a linking

system.

Axiom (A2) : By axiom (A) for a linking system, for any P, Q ∈ Ob(L), E(P) = δ_{P} (Z(P)) acts freely on Mor_{L}(P, Q), and ρ_{P,Q} is the orbit map of that action. It remains to show that E(Q) = δ_{Q}(Z(Q)) acts freely on Mor_{L}(P, Q). Assume f ∈ Mor_{L}(P, Q) and x ∈ Z(Q) are such that δ_{Q}(x) ∘ f = f. Then x centralizes π(f)(P), so x = π(f)(y) for some y ∈ Z(P) since P is F-centric, f = δ_{Q}(x) ∘ f = f ∘ δ_{P} (y) by axiom (C) for a linking system, and so y = 1 by axiom (A) of a linking system. Thus x = 1, and the action is free.

Axiom (B) : For any P, Q ∈ Ob(L) and any x ∈ N_{S}(P, Q),

ι_{Q} ∘ ε_{P,Q}(x) = ε_{Q,S}(1) ∘ ε_{P,Q}(x) = ε_{S,S}(x) ∘ ε_{P,S}(1) = δ_{S}(x) ∘ ι_{P} .

Since π(ι_{P}) and π(ι_{Q}) are inclusions in F, this shows that π(ε_{P,Q}(x)) is the restriction to Hom_{F}(P, Q) of π(δ_{S}(x)), and hence is conjugation by x by axiom (B) for L as a linking system. Thus π(ε_{P,Q}(x)) = c_{x}, and this proves axiom (B) for L as a transporter system.

Axiom (I′) : Assume P is fully normalized in F. Then Aut_{S}(P) ≅ N_{S}(P)/Z(P) is a Sylow p-subgroup of Aut_{F}(P) ≅ Aut_{L}(P)/δ_{P} (Z(P)) by axiom (I) for the saturated fusion system F (and axiom (A) for L as a linking system), and so ε_{P,P} (N_{S}(P)) is also a Sylow p-subgroup of Aut_{L}(P).

Axiom (II) : Let f ∈ Iso_{L}(P, Q), and P ◁ P ≤ S and Q ◁ Q ≤ S, be such that f ∘ ε_{P,P} (P) ∘ f^{−1} ≤ ε_{Q,Q}(Q). If Q is fully centralized in F, then axiom (II) for the saturated fusion system F implies that π(f) extends to a homomorphism ψ ∈ Hom_{F}(P , S), whose image must be contained in Q since π(f) conjugates Aut_{P} (P) into Aut_{Q}(Q). Hence by Lemma 1.7(b), there is f ∈ Mor_{L}(P , Q) (not necessarily a lifting of ψ) which extends f.

Now assume Q is not fully centralized. Choose R which is F-conjugate to P and Q and fully normalized in F. Then ε_{R,R}(N_{S}(R)) is a Sylow p-subgroup of Aut_{L}(R) (by axiom (I′)), and hence contains every p-subgroup of Aut_{L}(R) up to conjugacy. For any isomorphism φ ∈ Iso_{L}(Q, R), φ ∘ ε_{Q,Q}(N_{S}(Q)) ∘ φ^{−1} is a p-subgroup of Aut_{L}(R), so there is χ ∈ Aut_{L}(R) such that φ′ =^{def} χ ∘ φ conjugates ε_{Q,Q}(N_{S}(Q)) into ε_{R,R}(N_{S}(R)). Since R is fully centralized, the result of the last paragraph implies that there are morphisms f′ ∈ Mor_{L}(P , N_{S}(R)) and φ ∈ Mor_{L}(Q, N_{S}(R)) such that f′|_{P,R} = φ′ ∘ f and φ|_{Q,R} = φ′. By axiom (C) for a linking system,

f′ ∘ δ_{P} (P) ∘ f′^{−1} = δ_{N_{S}(R)}(π(f′)(P)) and φ ∘ δ_{Q}(Q) ∘ φ^{−1} = δ_{N_{S}(R)}(π(φ)(Q)).

After restriction, this shows that in Aut_{T} (R),

ε_{R,R}(π(f′)(P)) = (φ′ ∘ f) ∘ ε_{P,P} (P) ∘ (φ′ ∘ f)^{−1}

≤ φ′ ∘ ε_{Q,Q}(Q) ∘ φ′^{−1} = ε_{R,R}(π(φ)(Q)),

where the inequality holds since f ∘ ε_{P,P} (P) ∘ f^{−1} ≤ ε_{Q,Q}(Q) by assumption. Thus π(f′)(P) ≤ π(φ)(Q). By definition of a fusion system, there is λ ∈ Hom_{F}(P , Q) such that π(φ) = π(f′) ∘ λ, and so Lemma 1.7(a) now implies that there is f ∈ Mor_{L}(P , Q) such that φ = f′ ∘ f. Upon restricting these morphisms to P, this implies that φ′ ∘ f = φ′ ∘ f|_{P,Q}; and hence by Lemma 1.7(a) that f|_{P,Q} = f. □

More generally, one can also show that any quasicentric linking system in the sense of [BCGLO1, §3]) is a transporter system.

Later, in Proposition 3.12, we prove a partial converse to Proposition 3.5(b), by showing that any transporter system which “has enough objects” in a certain sense to be made precise is a full subcategory of the transporter category of a finite group.

Transporter systems were defined associated to arbitrary fusion systems. But in fact, the conditions on the definition are sufficiently restrictive that if a transporter system T is associated to a fusion system F, then F is saturated — at least with respect to the objects of T .

Proposition 3.6. Let F be a fusion system over a p-group S (not necessarily saturated), and let T be a transporter system associated to F. Then F is Ob(T )-saturated. If F is also Ob(T )-generated, and if Ob(T ) ⊇ Ob(F^{c}), then F is saturated. More generally, F is saturated if it is Ob(T )-generated, and every F-centric subgroup P ≤ S not in Ob(T ) is F-conjugate to some P′ such that Out_{S}(P′) ∩ O_{p}(Out_{F}(P′)) ≠ 1.

Proof. Assume P ∈ Ob(T ). If P is fully normalized in F, then by Proposition 3.4(a), ε_{P,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)). Hence by Lemma 1.9(b), applied with G = Aut_{T} (P) and H = E(P), ε_{P,P} (C_{S}(P)) ∈ Syl_{p}(E(P)) (hence P is fully centralized by Proposition 3.4(b)) and Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)). Thus F satisfies axiom (I) for the subgroup P.

Now fix f ∈ Iso_{F}(P, Q) such that Q is fully centralized in F, and let φ ∈ Iso_{T} (P, Q) be any isomorphism such that ρ(φ) = f. Set

N_{f} = {x ∈ N_{S}(P) | f ∘ c_{x} ∘ f^{−1} ∈ Aut_{S}(Q)}.

Then φ(ε_{P,P} (N_{f}))φ^{−1} is a p-subgroup of E(Q)·ε_{Q,Q}(N_{S}(Q)). Since Q is fully centralized, ε_{Q,Q}(C_{S}(Q)) ∈ Syl_{p}(E(Q)) (Proposition 3.4(b) again), and so ε_{Q,Q}(N_{S}(Q)) is a Sylow p-subgroup of E(Q)·ε_{Q,Q}(N_{S}(Q)). Thus there is χ ∈ E(Q) such that (χ ∘ φ)(ε_{P,P} (N_{f}))(χ ∘ φ)^{−1} is contained in ε_{Q,Q}(N_{S}(Q)). So by axiom (II) for T , χ ∘ φ extends to a T -morphism defined on N_{f}, ρ(χ ∘ φ) = ρ(φ) = f, and hence f extends to an F-morphism defined on N_{f}. Thus F satisfies axiom (II) for the subgroup P.

We have now shown that F is Ob(T )-saturated. The last two statements (F is saturated under additional hypotheses) follow from Theorem 1.5(b). □

For any transporter system (S, F, T ), we let T^{c} ⊆ T denote the full subcategory whose objects are those P ∈ Ob(T ) which are F-centric. We want to show that T determines a unique linking system L associated to F with object set Ob(T^{c}). When doing this, we use the term “linking system” associated to F in a slightly more general way than previously: to refer to any category L which satisfies all of the conditions in Definition 1.6, except that Ob(L) need not contain all F-centric subgroups.

Proposition 3.7. Let (S, F, T ) be a transporter system. Let E(P), for P ∈ Ob(T ), be as in Definition 3.1. Then for every F-centric subgroup P ≤ S, E(P) = E_{0}(P)×Z(P), where E_{0}(P) is the subgroup generated by all elements in E(P) of order prime to p. We can thus define a centric linking system L associated to F by setting Ob(L) = Ob(T^{c}), and by setting

Mor_{L}(P, Q) = Mor_{T} (P, Q)/E_{0}(P)

for all P, Q ∈ Ob(L).

Proof. By axiom (C), for all P ∈ Ob(T ), E(P) commutes with ε_{P,P} (P) in Aut_{T} (P). Hence if P is F-centric, then ε_{P,P} (Z(P)) is central in E(P), and is a Sylow p-subgroup by Proposition 3.4. This implies that E(P) splits as a product Z(P) × E_{0}(P), where E_{0}(P) consists of all elements in E(P) of order prime to p.

It is now straightforward to check that L, when defined as above, is a quotient category of T^{c} (i.e., composition is well defined). Also, axioms (A), (B), and (C) for a transporter system imply that L satisfies the corresponding axioms for a linking system, and thus is a linking system associated to F. □

We have already shown that every morphism in a transporter system is a monomorphism in the categorical sense. We now show that every morphism is also an epimorphism.

Lemma 3.8. Let (S, F, T ) be a transporter system. Fix subgroups P, Q, R ∈ T , together with morphisms ψ ∈ Mor_{T} (P, Q) and φ_{1}, φ_{2} ∈ Mor_{T} (Q, R) such that φ_{1} ∘ ψ = φ_{2} ∘ ψ. Then φ_{1} = φ_{2}. In other words, all morphisms in T are epimorphisms in the categorical sense.

Proof. Since ψ is the composite of an isomorphism followed by an inclusion (Lemma 3.2(c)), it suffices to prove this when P ≤ Q and ψ = ι_{P,Q} is the inclusion. Also, it suffices to do this when P ◁ Q: otherwise it can be shown in several steps using a chain of subgroups linking P to Q, each normal in the following one.

Set P′ = ρ(φ_{1})(P). By Lemma 3.2(c), φ_{1} ∘ ι_{P,Q} = φ_{2} ∘ ι_{P,Q} has a unique restriction β = φ_{1}|_{P,P′} = φ_{2}|_{P,P′} ∈ Iso_{T} (P, P′). Fix x ∈ Q, set y_{i} = ρ(φ_{i})(x), and consider the following two squares:

\xymatrix{Q \ar[r]^{φ_{i}} \ar[d]_{ε_{Q,Q}(x)} & R \ar[d]^{ε_{R,R}(y_{i})} & P \ar[r]^{β} \ar[d]_{ε_{P,P} (x)} & P′ \ar[d]^{ε_{P′,P′}(y_{i})} \\ Q \ar[r]^{φ_{i}} & R & P \ar[r]^{β} & P′}

The first square commutes by axiom (C), and the second square is defined to be a restriction of the first. Note that ε_{P,P} (x) is the restriction of ε_{Q,Q}(x) since ε is a functor (and since ι_{A,B} = ε_{A,B}(1) for all A ≤ B ≤ S). Hence the second square commutes by the uniqueness of restriction morphisms (Lemma 3.2(c)). Thus ε_{P′,P′}(y_{1}) = ε_{P′,P′}(y_{2}) = β ∘ ε_{P,P} (x) ∘ β^{−1}. Since ε_{P′,P′} is injective, this shows that y_{1} = y_{2}. Since this holds for all x ∈ Q, ρ(φ_{1}) = ρ(φ_{2}).

By axiom (A2), we now get φ_{2} = φ_{1} ∘ α for some α ∈ E(Q). Hence

φ_{1} ∘ α ∘ ι_{P,Q} = φ_{2} ∘ ι_{P,Q} = φ_{1} ∘ ι_{P,Q},

so α∘ι_{P,Q} = ι_{P,Q} by Lemma 3.2 again, and α = Id_{Q} since E(Q) acts freely on Mor_{T} (P, Q) (axiom (A2)). It follows that φ_{1} = φ_{2}. □

By analogy with the definition of radical and normal subgroups in a fusion system, we define:

Definition 3.9. Let (S, F, T ) be a transporter system.

• A subgroup Q ∈ Ob(T ) is called T -radical if ε_{Q,Q}(Q) = O_{p}(Aut_{T} (Q)).

• An arbitary subgroup Q ≤ S (not necessarily an object in T ) is called normal in T , denoted Q ◁ T , if for every morphism φ ∈ Mor_{T} (P, P′) in T , there is a morphism φ ∈ Mor_{T} (P Q, P′Q) such that ι_{P′,P′Q} ∘ φ = φ ∘ ι_{P,P Q} and ρ(φ)(Q) = Q.

If T is a full subcategory of T_{S}(G) for some finite group G and S ∈ Syl_{p}(G), then for Q ∈ Ob(T ), Q is T -radical if and only if Q is a radical p-subgroup of G in the usual sense. If (S, F, L) is a p-local finite group and we regard L as a transporter system, then P ∈ Ob(L) is L-radical if and only if it is F-centric and F-radical. More generally, if (S, F, T ) is an abstract transporter system and P ≤ S is F-centric and F-radical, then it is not hard to see that P is also T -radical, but not conversely. For example, assume G = H ⋊ S where H has order prime to p, S ∈ Syl_{p}(G), and C_{G}(H) ≤ H. Set F = F_{S}(G) and T = T_{S}(G). Then every subgroup of S is T -radical, but no proper subgroup of S is F-radical (nor F-centric if S is abelian).

The following proposition is the version for transporter systems of Alperin’s fusion theorem.

Proposition 3.10. Let (S, F, T ) be a transporter system. For each P, P′ ∈ Ob(T ) and each φ ∈ Iso_{T} (P, P′), there are subgroups

P = P_{0}, P_{1}, …, P_{k} = P′  and  Q_{i} ≥ ⟨P_{i−1}, P_{i}⟩ (i = 1, …, k)

where each Q_{i} is T -radical and fully normalized in F, and also automorphisms α_{i} ∈ Aut_{T} (Q_{i}) and isomorphisms φ_{i} ∈ Iso_{T} (P_{i−1}, P_{i}), such that φ_{i} = α_{i}|_{P_{i−1},P_{i}} for each i, and φ = φ_{k} ∘ ⋯ ∘ φ_{1}.

Proof. Fix φ, and assume inductively that the result holds for all isomorphisms between larger subgroups of S. If P = P′ = S, there is nothing to prove, so we assume that P, P′ ≨ S. Choose a fully normalized subgroup Q in the F-conjugacy class of P and P′. Since ε_{Q,Q}(N_{S}(Q)) is a Sylow p-subgroup in Aut_{T} (Q) by axiom (I′), there are isomorphisms ψ ∈ Iso_{T} (P, Q) and ψ′ ∈ Iso_{T} (P′, Q) such that ψ ∘ ε_{P,P} (N_{S}(P)) ∘ ψ^{−1} ≤ ε_{Q,Q}(N_{S}(Q)) and similarly for ψ′. So by axiom (II), ψ extends to a morphism from N_{S}(P) to N_{S}(Q), and similarly for ψ′, and so the proposition holds for ψ and ψ′ both. Thus to prove the proposition for φ, it suffices to prove it for ψ′ ∘ φ ∘ ψ^{−1} ∈ Aut_{T} (Q). In other words, we are reduced to the case where P = P′ is fully normalized.

If P is T -radical, then we are done. Otherwise, let R ≩ P be such that ε_{P,P} (R) = O_{p}(Aut_{T} (P)). By axiom (II) again, any φ ∈ Aut_{T} (P) extends to an automorphism of R, and again we are done by the induction hypothesis. □

If (S, F, T ) is a transporter system and A ◁ T , then we define the quotient category T /A by letting Ob(T /A) be the set of all P/A for A ≤ P ∈ Ob(T ), and setting

Mor_{T /A}(P/A, Q/A) = Mor_{T} (P, Q)/ε_{P,P} (A) = ε_{Q,Q}(A)\Mor_{T} (P, Q).

The equivalence between these two formulas for Mor_{T /A}(P/A, Q/A) follows from axiom (C). We next show that T /A is itself a transporter system.

Proposition 3.11. If (S, F, T ) is a p-local finite group, and A is a normal subgroup in T , then T /A is a transporter system associated to the fusion system F/A.

Proof. We will denote by [f] the morphism in Mor_{T /A}(P/A, Q/A) represented by f in Mor_{T} (P, Q), and by f/A the morphism in Mor_{F/A}(P/A, Q/A) induced by f in Mor_{F}(P, Q). So all morphisms in T /A have the form [f] for some f ∈ Mor(T ), and all morphisms in F/A have the form f/A for some f ∈ Mor(F).

Let ρ : T /A ⟶ F/A be the functor induced by ρ : T ⟶ F, i.e., ρ([f]) = ρ(f)/A. Let ε : T_{Ob(T /A)}(S/A) ⟶ T /A be the functor which is the identity on objects, and where

ε_{P/A,Q/A}: N_{S/A}(P/A, Q/A)_{=N_{S}(P,Q)/A} ⟶ Mor_{T /A}(P/A, Q/A)_{=Mor_{T} (P,Q)/A}

is defined by setting ε_{P/A,Q/A}(gA) = [ε_{P,Q}(g)], for all g ∈ N_{S}(P, Q). Axiom (A1) holds by definition of ρ. Axiom (A2): E(P/A) acts freely on Mor_{T /A}(P/A, Q/A). Assume φ ∈ Mor_{T} (P, Q) and χ ∈ Aut_{T} (P) are such that [φ]∘[χ] = [φ] and [χ] ∈ E(P/A). Then φ∘χ = φ∘ε_{P,P} (a) for some a ∈ A, χ = ε_{P,P} (a) by Lemma 3.2(b), and hence [χ] is the identity in E(P/A).

ρ_{P,Q} is the orbit map for the E(P)-action on Mor_{T /A}(P/A, Q/A). Fix P and Q, and φ, ψ ∈ Mor_{T} (P, Q). If [ψ] = [φ]∘[χ] for some [χ] ∈ E(P/A), then clearly ρ([ψ]) = ρ([φ]). Conversely, assume that ρ([φ]) = ρ([ψ]); we must show that they are in the same E(P/A)-orbit. We have

ρ(φ)(P)/A = Im(ρ([φ])) = Im(ρ([ψ])) = ρ(ψ)(P)/A,

and so ρ(φ)(P) = ρ(ψ)(P). Since every morphism in F factors as an F-isomorphism followed by an inclusion, this implies that there is α ∈ Aut_{F}(P) such that ρ(ψ) = ρ(φ) ∘ α. So by Lemma 3.2(a), there is χ ∈ Aut_{T} (P) such that ψ = φ ∘ χ and ρ(χ) = α. Then ρ([χ]) = α/A = Id_{P/A}, and hence [χ] ∈ E(P/A).

E(Q) acts freely on Mor_{T} (P, Q). Assume φ ∈ Mor_{T} (P, Q) and χ ∈ Aut_{T} (Q) are such that [χ] ∘ [φ] = [φ] and [χ] ∈ E(Q/A). Then for some a ∈ A,

χ ∘ φ = φ ∘ ε_{P,P} (a) = ε_{Q,Q}(ρ(φ)(a)) ∘ φ

where the second equality holds by axiom (C) for a transporter system; ρ(φ)(a) ∈ A since A is normal in F; χ = ε_{Q,Q}(ρ(φ)(a)) by Lemma 3.8; and hence [χ] is the identity in E(Q/A). Axiom (B) for T /A follows directly from axiom (B) for T . Axiom (C) for T /A is a consequence of axiom (C) applied to T . More precisely, for each f ∈ Mor_{T} (P, Q) and each g ∈ P,

[f] ∘ ε_{P/A,P/A}(gA) = [f] ∘ [ε_{P,P} (g)] = [f ∘ ε_{P,P} (g)] = [ε_{Q,Q}(ρ(f)(g)) ∘ f]
= ε_{Q/A,Q/A}(ρ(f)(g)A) ∘ [f] = ε_{Q/A,Q/A}(ρ([f])(gA)) ∘ [f] .

Axiom (I): By axiom (I) for T , ε_{S,S}(S) is a Sylow p-subgroup of Aut_{T} (S). So upon diving out by A, we get that ε_{S/A,S/A}(S/A) is a Sylow p-subgoup of Aut_{T /A}(S/A).

Axiom (II): Fix [f] ∈ Iso_{T /A}(P/A, Q/A), and let P/A ◁ P /A ≤ S/A and Q/A ◁ Q/A ≤ S/A be such that

[f] ∘ ε_{P/A,P/A}(P /A) ∘ [f]^{−1} ≤ ε_{Q/A,Q/A}(Q/A).

Then for any lifting of [f] to f ∈ Iso_{T} (P, Q),

f ∘ ε_{P,P} (P) ∘ f^{−1} ≤ ε_{Q,Q}(Q).

So f extends to some f ∈ Mor_{T} (P , Q) by axiom (II) applied to T , and thus [f] extends to [f] in Mor_{T /A}(P /A, Q/A). □

The next proposition shows that a transporter system over S which contains all subgroups of S as objects is the transporter category of a finite group.

Proposition 3.12. Let T be a transporter system over a p-group S for which Ob(T ) is the set of all subgroups of S. Set G = Aut_{T} (1), and identify S as a subgroup of G via ε_{1,1}. Then S ∈ Syl_{p}(G), and there is an isomorphism of 4-tuples

(T , F, ρ, ε) ≅ (T_{S}(G), F_{S}(G), (g ↦ c_{g}), incl).

Proof. Note first that S ∈ Syl_{p}(G) by axiom (I′) (Proposition 3.4), applied with P = 1. By (II) (applied with P = Q = 1), for each P, Q ≤ S and each g ∈ N_{G}(P, Q), there is some ε̂_{P,Q}(g) ∈ Mor_{T} (P, Q) which extends g ∈ Aut_{T} (1), and this extension is unique by Lemma 3.8. If R is another subgroup and h ∈ N_{G}(Q, R), then ε̂_{P,R}(hg) = ε̂_{Q,R}(h) ∘ ε̂_{P,Q}(g) by the uniqueness of the construction.

We have thus defined a functor ε̂ from T_{S}(G) to T which is the identity on objects. Consider the following diagram:

\xymatrix{T_{S}(S) \ar[r]^{incl} \ar[d]^{Id} & T_{S}(G) \ar[r]^{(g↦c_{g})} \ar[d]^{ε̂} & F_{S}(G) \ar[d]^{Id} \\ T_{S}(S) \ar[r]^{ε} & T \ar[r]^{ρ} & F .}

By Lemma 3.3 (applied with P = Q = 1), for g ∈ N_{G}(P, Q), ρ(ε̂_{P,Q}(g)) is conjugation by g. This proves that F_{S}(G) ⊆ F, and that the right hand square in the diagram commutes. For g ∈ N_{S}(P, Q), ε_{P,Q}(g) is an extension of ε_{1,1}(g) (since ε is a functor). So ε is the restriction of ε̂ to T_{S}(S), and thus the left hand square commutes.

It remains to show that ε̂ is an isomorphism of categories; it then follows from the surjectivity of ρ that F_{S}(G) = F. For all P, Q ≤ S and φ ∈ Mor_{T} (P, Q), there is a uniquely defined restriction φ|_{1,1} ∈ Aut_{T} (1) = G by Lemma 3.2(c). Set g = φ|_{1,1} for short. By Lemma 3.3 again, gxg^{−1} = ρ(φ)(x) ∈ Q for all x ∈ P (recall that we identify S as a subgroup of G via ε_{1,1}), and thus g ∈ N_{G}(P, Q). It follows that φ = ε̂_{P,Q}(g); g is unique by the uniqueness of the restriction; and thus ε̂_{P,Q} is a bijection. □

More generally, if (S, F, T ) is a transporter system, and there is some Q ∈ Ob(T ) such that Q ◁ T , then one can show that T is isomorphic to a full subcategory of T_{S}(Aut_{T} (Q)).

We next describe another way to construct new transporter systems as quotients of other transporter systems; a construction which will be useful in Section 6. As in Definition 3.1, for a transporter system (S, F, T ), we let E(P) be the kernel

E(P) = Ker[Aut_{T} (P) ⟶ Aut_{F}(P)]

for all P ∈ Ob(T ). We regard E as a functor E: T^{op} ⟶ Gps: for each φ ∈ Mor_{T} (P, Q) and each x ∈ E(Q), E(φ)(x) ∈ E(P) is the unique morphism such that

φ ∘ E(φ)(x) = x ∘ φ ∈ Mor_{T} (P, Q).

Proposition 3.13. Fix a transporter system (S, F, T ). Let E_{0} : T^{op} ⟶ Gps be a subfunctor of E which satisfies the following two conditions for each P, Q ∈ Ob(T ) and each φ ∈ Mor_{T} (P, Q):

(a) E_{0}(P) has order prime to p, and

(b) E_{0}(Q) = E(φ)^{−1}(E_{0}(P)).

Let T /E_{0} be the quotient category which has the same objects as T , and where

Mor_{T /E_{0}}(P, Q) = Mor_{T} (P, Q)/E_{0}(P).

Then T /E_{0} is a transporter system associated to F.

Proof. The assumption that E_{0} is a subfunctor implies that T /E_{0} is a category (i.e., that composition is well defined). The structure functors

T_{Ob(T )}(S) ⟶^{ε′} T /E_{0} ⟶^{ρ′} F

are defined in the obvious way. Most of the axioms in Definition 3.1 are easily carried over from T to T /E_{0}. The injectivity of ε′, and axiom (I) (ε′_{S,S}(S) ∈ Syl_{p}(Aut_{T /E_{0}}(S))) follow from (a). The freeness of the action of E(Q)/E_{0}(Q) on Mor_{T} (P, Q)/E_{0}(P) (part of axiom (A2)) follows from (b).

The only axiom which requires a little more explanation is (II). Fix φ′ = [φ] in Iso_{T /E_{0}}(P, Q) (where φ ∈ Iso_{T} (P, Q)). Assume P and Q are such that P ◁ P, Q ◁ Q, and

φ′ ∘ ε′_{P,P} (P) ∘ φ′^{−1} ≤ ε′_{Q,Q}(Q) ≤ Aut_{T /E_{0}}(Q).

Equivalently, in T , this means that

φ ∘ ε_{P,P} (P) ∘ φ^{−1} ≤ ε_{Q,Q}(Q)·E_{0}(Q).

Since |E_{0}(Q)| is prime to p, ε_{Q,Q}(Q) is a Sylow p-subgroup of ε_{Q,Q}(Q)·E_{0}(Q). Hence there is α ∈ E_{0}(Q) such that (αφ) ∘ ε_{P,P} (P) ∘ (αφ)^{−1} ≤ ε_{Q,Q}(Q). Thus αφ can be extended to some φ ∈ Mor_{T} (P , Q) by axiom (II) for T , and so φ′ = [φ] = [αφ] extends to the morphism φ′ = [φ] ∈ Mor_{T /E_{0}}(P , Q). □

The following is an example of how Proposition 3.13 will be applied in Section 6. Fix a finite group G, a normal subgroup H ◁ G, and a Sylow subgroup S ∈ Syl_{p}(G). Let H be the set of all subgroups P ≤ S such that (Z(P) ∩ H) ∈ Syl_{p}(C_{H}(P)). For each P ∈ H, C_{H}(P) = (Z(P) ∩ H) × C′_{H}(P), where C′_{H}(P) is the set (a subgroup) of all elements of C_{H}(P) of order prime to p. Then C′_{H} is a subfunctor of C_{G} which satisfies conditions (a) and (b) in Proposition 3.13, and hence T_{H}(G)/C′_{H} is a transporter system associated to F_{S}(G). When H = G, then this is of course just the centric linking system L_{S}^{c}(G).

The next proposition describes the opposite of this construction: it describes general conditions for an extension of a transporter system to again be a transporter system associated to the same fusion system. However, its main purpose is to show how to construct any centric transporter system associated to a given F — any transporter system whose objects are the F-centric subgroups — as an extension of a linking system associated to F.

We refer to Definition A.5 for the definition of a source regular extension of a category. For any source regular extension T′ ⟶^{ν} T of T , we let K̃: T′^{op} ⟶ Gps denote the “kernel functor”, defined on objects by setting K̃(P) = Ker[Aut_{T′}(P) ⟶ Aut_{T} (P)], and on morphisms by sending φ ∈ Mor_{T′}(P, Q) to the unique homomorphism K̃(φ) ∈ Hom(K̃(Q), K̃(P)) such that α ∘ φ = φ ∘ K̃(φ)(α) for all α ∈ K̃(Q). (Note that K̃ as defined here is a functor from T′ to groups, as opposed to the functor K^{∗} from T to groups up to conjugacy defined in Lemma A.7(c).)

Proposition 3.14. Fix a transporter system (S, F, T ) and a source regular extension T′ ⟶^{ν} T of T . Let K̃: T′^{op} ⟶ Gps be the kernel functor. Then T′ can be given a structure of a transporter system associated to F if the following conditions hold for all P ∈ Ob(T′) = Ob(T ):

(a) K̃(P) is finite of order prime to p.

(b) K̃(φ) is a monomorphism for all φ ∈ Mor(T ).

(c) For all g ∈ P, K̃(ε_{P,P} (g)) = Id_{K̃(P)}.

(d) If P ≤ P ≤ N_{S}(P) and x ∈ K̃(P) are such that K̃(δ_{P,P} (g))(x) = x for all g ∈ P, then x ∈ Im(K̃(ι_{P,P})).

Proof. Let T_{Ob(T )}(S) ⟶^{ε} T ⟶^{ρ} F be the structure functors for T , and set ρ′ = ρ∘ν. By (c) (applied with P = S), there is a unique homomorphism ε′_{S,S}: S ⟶ Aut_{T′}(S) such that ν_{S,S} ∘ ε′_{S,S} = ε_{S,S}. For each P ≤ S, choose some ε′_{P,S}(1) ∈ Mor_{T′}(P, S) such that ν_{P,S} ∘ ε′_{P,S}(1) = ε_{P,S}(1). By Lemma A.7(a), for each P, Q ∈ Ob(T′) and each g ∈ N_{S}(P, Q), there is a unique morphism ε′_{P,Q}(g) ∈ Mor_{T′}(P, Q) such that

ε′_{Q,S}(1) ∘ ε′_{P,Q}(g) = ε′_{S,S}(g) ∘ ε′_{P,S}(1),

and this defines a functor ε′ such that ν ∘ ε′ = ε.

Axiom (A1) for T′ follows immediately, (A2) by (b) and since ν is source regular, (B) since ρ′ ∘ε′ = ρ ∘ε, and (I) from (a) and the corresponding axiom for the transporter system T ).

We next check axiom (C). Fix φ ∈ Mor_{T′}(P, Q) and g ∈ P, and set h = ρ′(φ)(g) for short. By Lemma A.7(a), there is a unique automorphism γ ∈ Aut_{T′}(P) such that ν_{P,P} (γ) = ε_{P,P} (g), and such that the following square commutes in T′:

\xymatrix{P \ar[r]^{φ} \ar[d]_{γ} & Q \ar[d]^{ε′_{Q,Q}(h)} \\ P \ar[r]^{φ} & Q .}

Set |h| = p^{k}. By juxtaposing copies of the above square (and by the uniqueness in Lemma A.7(a)), γ^{p^{k}} = Id. By (c), ν_{P,P}^{−1} (ε_{P,P} (P)) is the product of K̃(P) with a pgroup, hence ε_{P,P} (g) has a unique lifting to Aut_{T′}(P) of p-power order, and this proves that γ = ε′_{P,P} (g).

It remains to prove axiom (II) for T′. Fix P ◁ P ≤ S, Q ◁ Q ≤ S, and φ ∈ Iso_{T′}(P, Q) such that φε′_{P,P} (P)φ^{−1} ≤ ε′_{Q,Q}(Q). Then ν(φ) extends to a morphism on P by axiom (II) for T , and hence there is ψ ∈ Mor_{T′}(P , Q) such that ν(ψ|_{P,Q}) = ν(φ). Let α ∈ K̃(P) be the unique element such that ψ|_{P,Q} = φ ∘ α; then α normalizes ε′_{P,P} (P). Since α ∈ K̃(P) which is normal in Aut_{T′}(P), this shows that

[α, ε′_{P,P} (P)] ∈ K̃(P) ∩ ε′_{P,P} (P) = 1,

where the intersection is trivial since one of the groups has order prime to p and the other is a p-group. We can now apply condition (d) to show that α extends to α ∈ Aut_{T′}(P), and hence ψ ∘ α^{−1} is an extension of φ to P. □

4. Homotopy properties of transporter systems

We collect here some results about the fundamental group, the homotopy type, and the cohomology of nerves of transporter systems. Several results from the appendix will be needed when proving these.

By Proposition A.3(a), for any transporter system (S, F, T ), π_{1}(|T |) is the free group on the morphisms in T , modulo relations given by composition, and by setting inclusion morphisms ι_{P,Q} equal to the identity. More precisely, let

θ: Mor(T ) ⟶ π_{1}(|T |)

be the map which sends φ ∈ Mor_{T} (P, Q) to the loop (based at the vertex S) formed by the edges ι_{Q,S}·φ·ι_{P,S}^{−1} (composed from right to left). This clearly sends composites to products and sends inclusions to the identity, and Proposition A.3 says that θ is universal among all maps defined on Mor(T ) with these properties.

If φ_{0}, φ ∈ Mor(T ) are such that φ_{0} is a restriction of φ, then they differ by composition with inclusion morphisms, and hence θ(φ_{0}) = θ(φ). Thus by Proposition 3.10 (Alperin’s fusion theorem for transporter systems), π_{1}(|T |) is generated by the subgroups θ(Aut_{T} (P)) for fully normalized T -radical subgroups P ∈ Ob(T ).

The following proposition describes one way to construct transporter subsystems, as “kernels” of homomorphisms defined on the fundamental group.

Proposition 4.1. Let (S, F, T ) be a transporter system. Fix a finite group Γ, and a homomorphism Φ: π_{1}(|T |) ⟶ Γ. We also identify Φ with Φ ∘ θ; i.e., as a function defined on Mor(T ) which sends composites to products and inclusion morphisms to the identity. For each subgroup H ≤ Γ, set S_{H} = (Φ ∘ ε_{S,S})^{−1}(H), a subgroup of S, and assume that S_{1} ∈ Ob(T ). Let T_{H} ⊆ T be the subcategory defined by setting

Ob(T_{H}) = {P ∈ Ob(T ) | P ≤ S_{H}} and Mor_{T_{H}} (P, Q) = {φ ∈ Mor_{T} (P, Q) | Φ(φ) ∈ H}.

Let F_{H} ⊆ F be the fusion system over S_{H} generated by ρ(T_{H}), and let

T_{Ob(T_{H})}(S_{H}) ⟶^{ε_{H,H}} T_{H} ⟶^{ρ_{H}} F_{H}

be the restrictions of ε and ρ. Then the following hold for each H ≤ Γ.

(a) Φ(Mor(T )) = Φ(Aut_{T} (S_{1})).

(b) T_{H} is a transporter system associated to the fusion system F_{H} if and only if ε_{S_{1},S_{1}}(S_{H}) ∈ Syl_{p}(Aut_{T_{H}}(S_{1})). In particular, T_{H} is a transporter system whenever H ◁ Γ.

(c) Assume that ε_{S_{1},S_{1}}(S_{H}) ∈ Syl_{p}(Aut_{T_{H}}(S_{1})); and also that for all P ≤ S fully centralized in F, C_{S_{1}}(P) ≤ P implies P ∈ Ob(T ) (Φ is an admissible homomorphism in the sense of Definition 5.10). Then F_{H} is a saturated fusion system.

(d) Assume P ∈ Ob(T ) implies P ∩ S_{1} ∈ Ob(T ). Then |T_{H}| has the homotopy type of the covering space of |T | with fundamental group Φ^{−1}(H).

Proof. In general, for any P ≤ S, we write P_{1} = P ∩ S_{1} for short. We first claim, for all P, Q ≤ S such that P_{1}, Q_{1} ∈ Ob(T ), that there is a well defined restriction map

r = r_{P,Q}: Mor_{T} (P, Q) ⟶ Mor_{T} (P_{1}, Q_{1}).

In other words, for all ψ ∈ Mor_{T} (P, Q),

ι_{Q_{1},Q} ∘ r(ψ) = ψ ∘ ι_{P_{1},P}  for all ψ ∈ Mor_{T} (P, Q).  (1)

By Lemma 3.2(c), this means proving that ρ(ψ)(P_{1}) ≤ Q_{1}. For all g ∈ P_{1},

Φ(ε_{S,S}(ρ(ψ)(g))) = Φ(ε_{Q,Q}(ρ(ψ)(g))) = Φ(ψ)·Φ(ε_{P,P} (g))·Φ(ψ)^{−1} = 1

by axiom (C) for T , so ρ(ψ)(g) ∈ S_{1}. Thus ρ(ψ)(P_{1}) ≤ Q_{1}, and these restriction maps are all defined. Note also that (1) implies Φ(r(ψ)) = Φ(ψ) for all ψ.

(a) By Proposition 3.10 (Alperin’s fusion theorem for transporter systems), Φ(Mor(T )) is contained in the subgroup generated by all Φ(Aut_{T} (P)) for fully normalized P ∈ Ob(T ). So it suffices to show that Φ(Aut_{T} (P)) ≤ Φ(Aut_{T} (S_{1})) for all P ∈ Ob(T ) fully normalized in F. Fix such a P, and fix α ∈ Aut_{T} (P). We can assume inductively that Φ(Aut_{T} (Q)) ≤ Φ(Aut_{T} (S_{1})) for all Q ≤ S such that Q ≩ P. If P ≥ S_{1}, then r_{P,P} (α) ∈ Aut_{T} (S_{1}), and so Φ(α) = Φ(r(α)) ∈ Φ(Aut_{T} (S_{1})).

We are thus reduced to the case where P ≱ S_{1}. Then P ≨ P S_{1}, so P ≨ N_{P S_{1}}(P) = P ·N_{S_{1}}(P). Since P is fully normalized, ε_{P,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)) by axiom (I′). Set

Aut_{1}(P) = Ker[Aut_{T} (P) ⟶^{Φ} Γ],

so that

ε_{P,P} (N_{S_{1}}(P)) = Aut_{1}(P) ∩ ε_{P,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{1}(P)).

Since all Sylow p-subgroups of Aut_{1}(P) are conjugate,

αε_{P,P} (N_{S_{1}}(P))α^{−1} = βε_{P,P} (N_{S_{1}}(P))β^{−1}

for some β ∈ Aut_{1}(P), and thus β^{−1} ∘ α normalizes ε_{P,P} (N_{S_{1}}(P)). By axiom (II), β^{−1} ∘ α extends to an automorphism α ∈ Aut_{T} (N_{P S_{1}}(P)); i.e., α|_{P,P} = β^{−1} ∘ α. Then Φ(α) = Φ(β^{−1} ∘ α) = Φ(α). Since Φ(α) ∈ Φ(Aut_{T} (S_{1})) by the induction hypothesis, this finishes the proof of (a).

(b) The axioms (A1), (A2), (B), (C), and (II) of a transporter system for T_{H} follow easily from the corresponding axioms for T , without any restrictions on H. Hence T_{H} is a transporter system if and only if axiom (I) holds.

Set A = Aut_{T} (S_{1}) and A_{H} = Aut_{T_{H}}(S_{1}) = {α ∈ A | Φ(α) ∈ H} for short. Consider the restriction map

r_{S_{H},S_{H}}: Aut_{T} (S_{H}) ⟶ A = Aut_{T} (S_{1})

as defined above; an injective map by Lemma 3.8. By axiom (II) (and Lemma 3.3), Im(r_{S_{H},S_{H}}) is the normalizer in A of ε_{S_{1},S_{1}}(S_{H}). Thus

ε_{S_{H},S_{H}}(S_{H}) ∈ Syl_{p}(Aut_{T_{H}}(S_{H})) ⟺ ε_{S_{1},S_{1}}(S_{H}) ∈ Syl_{p}(N_{A_{H}}(ε_{S_{1},S_{1}}(S_{H}))).

Since a proper subgroup of a p-group is properly contained in its normalizer, this shows that axiom (I) holds if and only if ε_{S_{1},S_{1}}(S_{H}) ∈ Syl_{p}(A_{H}).

Now, S_{1} is fully normalized in F, since it is the only subgroup in its F-conjugacy class. Hence ε_{S_{1},S_{1}}(S) ∈ Syl_{p}(A) by axiom (I′). So if H ◁ Γ, then ε_{S_{1},S_{1}}(S_{H}) is a Sylow subgroup of A_{H} by Lemma 1.9(b).

(c) Now assume ε_{S_{1},S_{1}}(S_{H}) ∈ Syl_{p}(Aut_{T_{H}}(S_{1})). Hence T_{H} is a transporter system by (b). Assume also that for all P ≤ S fully centralized in F, C_{S_{1}}(P) ≤ P implies P ∈ Ob(T ). In particular, if P ≤ S_{H} is F_{H}-centric, then P ≥ C_{S_{H}}(P) ≥ C_{S_{1}}(P), and thus P ∈ Ob(T_{H}). So F_{H} is saturated by Proposition 3.6.

(d) We must prove that |T_{H}| is homotopy equivalent to a covering space of |T |. Let T′_{H} ⊆ T be the subcategory with Ob(T′_{H}) = Ob(T ), and where Mor_{T′_{H}}(P, Q) is the set of all φ ∈ Mor_{T} (P, Q) such that Φ(φ) ∈ H. By Proposition A.4, |T′_{H}| is homotopy equivalent to the covering space of |T | with fundamental group Φ^{−1}(H). Condition (1) in Proposition A.4 follows by point (a) here.

It remains to prove that |T′_{H}| ≃ |T_{H}|. For all P, Q ∈ Ob(T′_{H}), all φ ∈ Mor_{T′_{H}}(P, Q), and all g ∈ P, φ ∘ ε_{P,P} (g) ∘ φ^{−1} = ε_{Q,Q}(ρ(φ)(g)) by axiom (C), and thus Φ(φ) ∈ H conjugates Φ(ε_{P,P} (g)) to Φ(ε_{Q,Q}(ρ(φ)(g))). In particular, g ∈ S_{H} if and only if ρ(φ)(g) ∈ S_{H}, and thus ρ(φ)(P ∩ S_{H}) ≤ Q ∩ S_{H}. So by Lemma 3.2(a), there is a unique restriction map

r_{P,Q}^{H}: Mor_{T′_{H}}(P, Q) ⟶ Mor_{T_{H}}(P ∩ S_{H}, Q ∩ S_{H}).

Define a retraction functor r^{H}: T′_{H} ⟶ T_{H} by sending an object P to P ∩ S_{H}, and sending ψ ∈ Mor_{T} (P, Q) to r_{P,Q}^{H}(ψ). The inclusion morphisms define a natural transformation of functors from incl ∘ r^{H} to Id_{T′_{H}}, and thus |T′_{H}| ≃ |T_{H}|. □

We next look at the cohomology of the nerve of a transporter system T . By Proposition A.3(b), for any abelian group A with action of π_{1}(|T |), H^{∗}(|T |; A) ≅ \underleftarrow{lim}_{T}^{∗}(Φ)

for a certain functor Φ: T^{op} ⟶ Ab which sends all objects to A. So we must study the higher limits of functors on T . A more general discussion of higher limits is given in the appendix.

When (S, F, T ) is a transporter system, the orbit category O(T ) of T is defined to be the category whose objects are the objects of T , and where

Mor_{O(T )}(P, Q) = ε_{Q,Q}(Q)\Mor_{T} (P, Q).

When T = T_{S}(G) is the category of a finite group G with Sylow subgroup S, then O(T ) is the usual orbit category of G:

Mor_{O(T )}(P, Q) = Q\N_{G}(P, Q) ≅ map_{G}(G/P, G/Q).

The relation between higher limits of p-local functors on T and on O(T ) (i.e., functors which take values in ℤ_{(p)}-modules) is described in the following lemma.

Lemma 4.2. Fix a transporter system (S, F, T ), and let Φ: T^{op} ⟶ ℤ_{(p)}-mod be any functor. Then there is a spectral sequence

E_{2}^{ij} = \underleftarrow{lim}^{i}_{O(T )}(H^{j}(−; Φ(−))) ⟹ \underleftarrow{lim}_{T}^{i+j}(Φ).  (2)

Proof. This is a special case of Proposition A.11. □

The reason for working with higher limits over O(T ) instead of looking directly at higher limits over T is that certain techniques developed when studying higher limits over other orbit categories also apply to this situation. We refer to the appendix for the definition of graded groups Λ^{∗}(G; M), when G is any finite group and M is a ℤ_{(p)}-module (and the prime p is understood). Their relation to limits over O(T ) is described by the following lemma.

Lemma 4.3. Fix a transporter system (S, F, T ), and let

Φ: O(T )^{op} ⟶ ℤ_{(p)}-mod

be any functor which vanishes except on the F-conjugacy class of one subgroup Q ∈ Ob(T ). Then

\underleftarrow{lim}^{∗}_{O(T )}(Φ) ≅ Λ^{∗}(Aut_{O(T )}(Q); Φ(Q)).

Proof. Since the result is independent of the choice of Q in its F-conjugacy class, we can assume that Q is fully normalized. Set Γ = Aut_{O(T )}(Q), and set Σ = ε_{Q,Q}(N_{S}(Q))/Q ∈ Syl_{p}(Γ).

Let O_{p}(Γ) be the p-subgroup orbit category: the category whose objects are the psubgroups of Γ, and with morphism sets Mor_{O_{p}(Γ)}(P, Q) = Q\N_{Γ}(P, Q). By definition, for any ℤ_{(p)}[Γ]-module M,

Λ^{∗}(Γ; M) = \underleftarrow{lim}^{∗}_{O_{p}(Γ)}(F_{M}) where F_{M}(P) = \begin{cases}M & if P = 1 \\ 0 & if P ≠ 1. \end{cases}

(See the discussion before Proposition A.2.) Let O_{Σ}(Γ) ⊆ O_{p}(Γ) be the full subcategory with objects the subgroups of Σ; this is clearly a subcategory equivalent to O_{p}(Γ).

Define a functor

α: O_{Σ}(Γ) ⟶ O(T )

as follows. For each subgroup P/Q ≤ Σ, set α(P/Q) = P. For each P/Q, P′/Q ≤ Σ and each γ ∈ Aut_{T} (Q) (i.e., Qγ ∈ Γ) such that P′γ ∈ Mor_{O(Γ)}(P/Q, P′/Q), γ extends to a unique morphism φ ∈ Mor_{T} (P, P′) by axiom (II), and we set

α(Qγ) = P′φ ∈ Mor_{O(T )}(P, P′).

We want to apply Proposition A.2 ([BLO3, Proposition 4.3]) to this functor α. Point (a) (Γ ≅ End_{O(T )}(Q)) holds by construction. Point (c) follows from Lemmas 3.8 and A.8(b): all morphisms in O(T ) are epimorphisms in the categorical sense since all morphisms in T are epimorphisms. It remains to check the other two hypotheses.

(b) For each P ∈ Ob(O(T )) such that P and Q are not T -isomorphic, all isotropy subgroups of the Γ-action on Mor_{O(T )}(Q, P) (via right composition) are nontrivial p-subgroups. For f ∈ Mor_{T} (Q, P), the isotropy subgroup of [f] ∈ Mor_{O(T )}(Q, P) is the subgroup of all classes [α] for α ∈ Aut_{T} (Q) such that f ∘ α = ε_{P,P} (x) ∘ f for some x ∈ P. By Lemma 3.2, for any given x ∈ P, there is at most one α which satisfies this equation, and there is such α if and only if

ρ(f)(Q) = ρ(ε_{P,P} (x) ∘ f)(Q) = x·ρ(f)(Q)·x^{−1}.

Thus the stabilizer subgroup of [f] is isomorphic to N_{P} (ρ(f)(Q))/Q, which is a nontrivial p-group since ρ(f)(Q) ≨ P.

(d) For any P/Q ≤ Σ, any R ∈ Ob(O(T )), and any φ ∈ Mor_{O(T )}(Q, R) which is P/Q-invariant, there is some φ ∈ Mor_{O(T )}(P, R) such that φ = φ ∘ ι_{Q,P}. This follows from axiom (II).

Thus, by Proposition A.2, the natural map

\underleftarrow{lim}^{∗}_{O(T )}(Φ) ⟶_{≅}^{α^{∗}} \underleftarrow{lim}^{∗}_{O_{Σ}(Γ)}(Φ ∘ α) = Λ^{∗}(Γ; Φ(Q))

is an isomorphism. □

Lemma 4.3 can now be combined with results in [JMO] about the functors Λ^{∗}(−; −), to obtain the following corollary.

Corollary 4.4. Fix a transporter system (S, F, T ).

(a) Assume

Φ: O(T )^{op} ⟶ ℤ_{(p)}-mod

has the property that for all P ∈ Ob(T ) such that Φ(P) ≠ 0, there is an element of order p in Aut_{O(T )}(P) which acts trivially on Φ(P). Then

\underleftarrow{lim}^{∗}_{O(T )}(Φ) = 0.

(b) Assume

Ψ: T^{op} ⟶ ℤ_{(p)}-mod

has the property that for every fully centralized subgroup P ∈ Ob(T ) such that Ψ(P) ≠ 0, there is g ∈ C_{S}(P)∖P such that ε_{P,P} (g) acts trivially on Ψ(P). Then

\underleftarrow{lim}_{T}^{∗}(Ψ) = 0.

Proof. By [JMO, Proposition 5.5], for any finite group Γ and any ℤ_{(p)}[Γ]-module M such that some element g ∈ Γ of order p acts via the identity on M, Λ^{∗}(Γ; M) = 0. In particular, by the hypothesis in (a), Λ^{∗}(Aut_{O(T )}(P); Φ(P)) = 0 for each P ∈ Ob(T ).

Point (a) now follows from Lemma 4.3, together with an appropriate filtration of Φ and the long exact sequences in higher limits for extensions of functors on O(T ).

Point (b) follows from (a), together with Lemma 4.2. □

We can now look at consequences for the homology of |T |.

Proposition 4.5. Fix a finite group G, a Sylow subgroup S ∈ Syl_{p}(G), and a full subcategory T ⊆ T_{S}(G) whose set of objects is closed under G-conjugacy and overgroups. Let A be an ℤ_{(p)}[G]-module, regarded also as a functor from T to ℤ_{(p)}-mod which sends all objects to A and which sends a morphism g ∈ N_{G}(P, Q) to the action of g on A. Let S_{1} ≤ S be the subgroup of elements which act on A via the identity. Assume, for all P ≤ S such that C_{S_{1}}(P) ≤ P, that P ∈ Ob(T ) (the action of T on A is admissible in the sense of Definition 5.10). Then

H^{∗}(|T |; A) ≅ \underleftarrow{lim}_{T}^{∗}(A) ≅ H^{∗}(G; A);

where the first group means cohomology with coefficients twisted via the natural homomorphism π_{1}(|T |) ⟶ G.

Proof. The first isomorphism follows from Proposition A.3(b). To prove the second, let Φ_{0} ⊆ Φ: T_{S}(G) ⟶ ℤ_{(p)}-mod be the functors

Φ(P) = A for all P ≤ S and Φ_{0}(P) = \begin{cases}A & if P ∉ Ob(T ) \\ 0 & if P ∈ Ob(T ). \end{cases}

In both cases, g ∈ N_{G}(P, Q) acts on A via the ℤ_{(p)}[G]-module structure.

Let T_{{1}}(G) ⊆ T_{S}(G) be the full subcategory with the trivial subgroup as the only object. We claim that

\underleftarrow{lim}^{∗}_{T_{S}(G)}(Φ) ≅ \underleftarrow{lim}^{∗}_{T_{\{1\}}(G)}(Φ) ≅ H^{∗}(G; A) and  \underleftarrow{lim}^{∗}_{T_{S}(G)}(Φ_{0}) = 0.

The first isomorphism holds because T_{S}(G) contains T_{{1}}(G) as a deformation retract (every morphism restricts to a unique automorphism of 1), and the second because functors from T_{{1}}(G) to abelian groups are the same as ℤ[G]-modules. The last isomorphism follows from Lemma 4.4(b), since whenever P ∉ Ob(T ), C_{S_{1}}(P) ≰ P, and for any g ∈ C_{S_{1}}(P)∖P, ε_{P,P} (g) acts trivially on A.

It now follows that

\underleftarrow{lim}_{T}^{∗}(A) ≅ \underleftarrow{lim}^{∗}_{T_{S}(G)}(Φ/Φ_{0}) ≅ \underleftarrow{lim}^{∗}_{T_{S}(G)}(Φ) ≅ H^{∗}(G; A).

Here, the first isomorphism holds since C^{∗}(T ; A) ≅ C^{∗}(T_{S}(G); Φ/Φ_{0}) (the chain complexes defined in Lemma A.1). □

As a second application of the results in this section, we show that the nerve of a transporter system (S, F, T ) has the same p-completed homotopy type as the nerve of the associated linking system, in the sense of Proposition 3.7.

Proposition 4.6. Fix a transporter system (S, F, T ) such that Ob(T ) contains all subgroups P ≤ S which are F-centric and F-radical. Let T^{c} be the full subcategory with objects the F-centric subgroups which are in Ob(T ), and let L be the linking system associated to T^{c}. Then the inclusion T^{c} ↪ T and the projection T^{c} −↠ L induce homotopy equivalences

|T |_{p}^{∧} ≃ |T^{c}|_{p}^{∧} ≃ |L|_{p}^{∧}.

Proof. By [BLO1, Lemma 1.3], if τ: C̃ ⟶ C is any source regular functor (see Definition A.5) such that Ker(τ_{c̃,c̃}) is finite of order prime to p for each c̃ ∈ Ob(C̃), then |τ| induces an isomorphism from H_{∗}(|C̃|; 𝔽_{p}) to H_{∗}(|C|; 𝔽_{p}), and hence a homotopy equivalence |C̃| ≃ |C|. This is the situation for the projection of T^{c} onto L (by axiom (A2) for T ), and hence |T^{c}| ≃ |L|.

Let Φ: T^{op} ⟶ Ab be the constant functor which sends all objects to 𝔽_{p} and all morphisms to the identity. Let Φ_{0} ⊆ Φ be the subfunctor

Φ_{0}(P) = \begin{cases}Φ(P) & if P ∉ Ob(T^{c}) \\ 0 & if P ∈ Ob(T^{c}). \end{cases}

Then

H^{∗}(|T |; 𝔽_{p}) ≅ \underleftarrow{lim}_{T}^{∗}(Φ) and H^{∗}(|T |, |T^{c}|; 𝔽_{p}) ≅ \underleftarrow{lim}_{T}^{∗}(Φ_{0}).

For each P ∈ Ob(T )∖ Ob(T^{c}) which is fully centralized, and each g ∈ C_{S}(P)∖P, ε_{P,P} (g) acts trivially on Φ(P). So \underleftarrow{lim}^{∗}(Φ_{0}) = 0 by Lemma 4.4(b). It follows that the inclusion of |T^{c}| into |T | is a mod p homology equivalence, and hence induces a homotopy equivalence of p-completions. □

Since a space is p-good if its p-completion is p-good [BK, Proposition I.5.2], and the p-completion of a linking system is p-good by [BLO2, Proposition 1.12], Proposition 4.6 implies that |T | is p-good whenever Ob(T ) contains all F-centric subgroups. In fact, the same argument as that used in the proof of [BLO2, Proposition 1.12] can be used in this situation (together with Proposition 3.10) to show directly that for any transporter system T , |T | is p-good.

We finish the section by describing one situation where an inclusion of transporter systems actually induces a homotopy equivalence between their uncompleted nerves.

Let (S, F, T ) be a transporter system. Recall that a subgroup P ∈ Ob(T ) is T - radical if O_{p}(Aut_{T} (P)) = ε_{P,P} (P). We let T^{r} ⊆ T denote the full subcategory whose objects are the T -radical subgroups of S.

The following proposition generalizes [BCGLO1, Theorem 3.5].

Proposition 4.7. Let (S, F, T ) be a transporter system. Then |T^{r}| ≃ |T |. More generally, |T′| ≃ |T | for any full subcategory T′ ⊆ T which contains T^{r}.

Proposition 4.7 is an immediate consequence of the following lemma, whose proof is modelled on that of [BCGLO1, Proposition 3.11].

Lemma 4.8. Let (S, F, T ) be a transporter system. Let T_{0} ⊆ T be any full subcategory such that Ob(T_{0}) is closed under F-conjugacy. Let P ≤ S be maximal among those subgroups in Ob(T )∖ Ob(T_{0}), and let T_{1} ⊆ T be the full subcategory whose objects are the objects in T_{0} together with all subgroups F-conjugate to P. Assume furthermore that P is not T -radical. Then the inclusion of nerves |T_{0}| ⊆ |T_{1}| is a homotopy equivalence.

Proof. Throughout this proof, “extensions” and “restrictions” of morphisms are taken as usual with respect to the inclusions ι_{P,Q} for P ≤ Q ≤ S. Also, for φ ∈ Mor_{T} (Q, Q′) and R ≤ Q, we write φ(R) = ρ(φ)(R) ≤ Q′.

We must show that the inclusion functor I: T_{0} → T_{1} induces a homotopy equivalence |T_{0}| ≃ |T_{1}|. By Quillen’s Theorem A (see [Q]), it will be enough to prove that the undercategory P′↓I has contractible nerve (i.e., |P′↓I| ≃ ∗) for each P′ in T_{1}. This is clear when P′ is not isomorphic to P (since P′↓I has initial object (P′, Id) in that case), so it suffices to consider the case P′ = P. Since P was arbitrarily chosen in its isomorphism class, we can also assume that P is fully normalized.

By assumption, P is not T -radical, and thus O_{p}(Aut_{T} (P)) ≩ ε_{P,P} (P). Define

P̂ = ε_{P,P}^{−1} (O_{p}(Aut_{T} (P))) .

Since P is fully normalized, ε_{P,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)) contains O_{p}(Aut_{T} (P)). Hence ε_{P,P} (P̂) = O_{p}(Aut_{T} (P)), and so P̂ ≩ P.

Recall that the objects in P ↓I are the pairs (Q, φ) for Q ∈ Ob(T_{0}) and φ ∈ Mor_{T} (P, Q); and that Mor_{P ↓I}((Q, φ), (R, ψ)) is the set of all χ ∈ Mor_{T} (Q, R) such that χ ∘ φ = ψ. Consider the full subcategories

C_{2} ⊆ C_{1} ⊆ C_{0} ⊆ P ↓I,

defined by setting

Ob(C_{0}) = {(Q, φ) | φ(P) ◁ Q}
Ob(C_{1}) = {(Q, φ) | φ(P) = P ◁ Q}
Ob(C_{2}) = {(Q, φ) | φ(P) = P ◁ Q, P̂ ≤ Q} .

We will prove that |P ↓I| ≃ |C_{0}| ≃ |C_{1}| ≃ |C_{2}| ≃ ∗.

Let r_{1} : P ↓I ⟶ C_{0} be the retraction which sends (Q, φ) to (N_{Q}(φ(P)), φ′), where φ′ ∈ Mor_{T} (P, N_{Q}(φ(P))) is the restriction of φ, and which sends a morphism to its restriction. All of these restrictions are well defined by Lemma 3.2(a). There is a natural transformation of functors from incl ∘ r_{1} to the identity on P ↓I which sends an object (Q, φ) to the inclusion ι_{N_{Q}(φ(P)),Q}, and thus |P ↓I| ≃ |C_{0}|.

Now let (Q, φ) be an object in C_{0}, and set P′ = φ(P) ◁ Q and φ′ = φ|_{P,P′} ∈ Iso_{T} (P, P′). Since P is fully normalized, ε_{P,P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)), and hence the p-subgroup φ′^{−1}ε_{P′,P′}(Q)φ′ is conjugate in Aut_{T} (P) to a subgroup of ε_{P,P} (N_{S}(P)). Let α ∈ Aut_{T} (P) be such that (αφ′^{−1})ε_{P′,P′}(Q)(αφ′^{−1})^{−1} is contained in ε_{P,P} (N_{S}(P)). By axiom (II) for T , αφ′^{−1} extends to some morphism ψ ∈ Mor_{T} (Q, N_{S}(P)) such that ψ(P′) = P. Then (ψ(Q), ψ ∘ φ) is isomorphic to (Q, φ); and (ψ(Q), ψ ∘ φ) is in C_{1} since ψφ(P) = P. Thus every object in C_{0} is isomorphic to an object in C_{1}, and this proves that |C_{0}| ≃ |C_{1}|.

Now define a retraction r_{2} : C_{1} ⟶ C_{2} as follows. On objects, we set r_{2}(Q, φ) = (QP̂, ι_{Q,QP̂} ∘φ). Fix χ ∈ Mor_{C_{1}}((Q, φ), (R, ψ)); then χ(P) = P since φ(P) = P = ψ(P),

so the restriction χ_{0} =^{def} χ|_{P,P} ∈ Aut_{T} (P) is defined. Since χ_{0} normalizes ε_{P,P} (P̂) = O_{p}(Aut_{T} (P)), and since χ_{0} conjugates ε_{P,P} (Q) to ε_{P,P} (R) by Lemma 3.3, there is an extension r_{2}(χ) ∈ Mor_{T} (QP̂, RP̂) of χ_{0} by axiom (II), which is unique (and also an extension of χ) by Lemma 3.8. Thus r_{2}(χ) is a morphism in C_{2} from r_{2}(Q, φ) to r_{2}(R, ψ), and the uniqueness of this extension implies that r_{2} is a functor. There is a natural transformation of functors from incl ∘ r_{2} to Id_{C_{1}} which sends an object (Q, φ) to the morphism ι_{Q,QP̂}, and hence |C_{1}| ≃ |C_{2}|.

It remains to show that |C_{2}| is contractible. For any object (Q, φ) in C_{2}, axiom (II) can again be used to construct an extension of φ to φ ∈ Mor_{T} (P̂, Q), thus a morphism in C_{2} from (P̂, ι_{P,P̂}) to (Q, φ), and this morphism is unique by Lemma 3.8. Hence (P̂, ι_{P,P̂}) is an initial object in C_{2}, and |C_{2}| ≃ ∗. □

5. Extensions of transporter systems

We are now ready to study extensions of transporter systems and the resulting extensions of fusion systems. We first give a precise definition of what we mean by an extension of a transporter system. Afterwards, we show that any such extension is itself a transporter system, give conditions for when it is a linking system, and look at the obstruction theory for such extensions.

Definition 5.1. Let (S, F, T ) be a transporter system. An extension T consists of a category T̃ and a functor τ : T̃ ⟶ T , such that for all P , Q ∈ Ob(T̃),

(a) τ is a bijection on objects;

(b) K_{P} =^{def} Ker[τ_{P ,P} : AutT̃(P) ⟶ Aut_{T} (τ(P))] is a finite p-group;

(c) K_{P} acts freely on MorT̃(P , Q) by right composition and τ is the orbit map of this action; and

(d) K_{Q} acts freely on MorT̃(P , Q) by left composition, and τ is the orbit map of this action.

Conditions (a), (c), and (d) above are equivalent to saying that τ: T̃ ⟶ T is a source and target regular extension, in the sense of Definition A.5.

We want to show that every extension of a transporter system in this sense is again a transporter system; moreover, that an extension of T is equivalent to a transporter system T̃ with normal subgroup A ◁ T̃ such that T̃/A ≅ T . In order to show that T̃ is a transporter system, we need first to associate a finite p-group S̃, and a fusion system F̃ over S̃, to the category T̃.

Define S̃ to be the pull-back of τ_{S̄,S̄} and ε_{S,S}:

\xymatrix{S̃ \ar[r]^{ε̃S̃} \ar[d]_{q} & AutT̃(S̄) \ar[d]^{τ_{S̄,S̄}} \\ S \ar[r]^{ε_{S,S}} & Aut_{T} (S) .}  (1)

Thus S̃ = {(g, f) ∈ S × AutT̃(S̄) | ε_{S,S}(g) = τ(f)}. Let q : S̃ ⟶ S and ε̃S̃ : S̃ ⟶ AutT̃(S̄) be the structure maps of the pull-back, so q is surjective and ε̃S̃ is injective. Set A = Ker(q), and set P̃ = q^{−1}(P) for each P ≤ S. So A and P̃ are subgroups of S̃, and each P̃ contains A. In fact, for each P ∈ Ob(T ), the group P̃ is an extension of the form

1 ⟶ A ⟶^{i} P̃ ⟶^{q|P̃} P ⟶ 1

(including the case of S), where i is the inclusion map. Since the functor τ is a bijection on objects and we defined a group P̃ ≤ S̃ for each P ∈ Ob(T ), we will consider that the objects in T̃ are subgroups of S̃ (all containing A as a normal subgroup) and will denote by P̃ the object in T̃ such that τ(P̃) = P.

The pull-back square (1) also shows that ε̃S̃ restricts to an isomorphism from A to Ker(τS̃). In particular, A is a p-group by condition (b) in Definition 5.1, and so S̃ is also a p-group.

The following lifting lemma is the analogy for extensions of transporter systems to Lemmas 1.7 and Lemma 3.2.

Lemma 5.2. Let T be a transporter system and let T̃ be an extension of T by the p-group A. Fix morphisms P ⟶^{φ} Q ⟶^{ψ} R in T , and let ψ̃φ ∈ MorT̃(P̃, R̃) be any lifting of ψ ∘ φ.

(a) For any lifting φ̃ ∈ MorT̃(P̃, Q̃) of φ, there is a unique morphism ψ̃ ∈ MorT̃(Q̃, R̃) such that τ(ψ̃) = ψ and ψ̃ ∘ φ̃ = ψ̃φ.

(b) For any lifting ψ̃ ∈ MorT̃(Q̃, R̃) of ψ, there is a unique morphism φ̃ ∈ MorT̃(P̃, Q̃) such that τ(φ̃) = φ and ψ̃ ∘ φ̃ = ψ̃φ.

Proof. These are special cases of Lemmas A.7(a) and A.8(a). □

Next, we define the functors

T_{Ob(T̃)}(S̃) \underrightarrow{ε̃} T̃ \underrightarrow{ρ̃} Gps

which will give T̃ the structure of a transporter system. Notice that we already have a group monomorphism ε̃S̃ : S̃ ⟶ AutT̃(S̃), and we will be using it in the proofs of Lemmas 5.3 and 5.4.

For each P̃, choose a lifting ι̃P̃ ∈ MorT̃(P̃, S̃) of ι_{P} ∈ Mor_{T} (P, S). These are chosen arbitrarily, except that we require ι̃S̃ = IdS̃ in T̃. (Recall that ι_{P} = ε_{P,P} (1).)

Lemma 5.3. For each P̃, Q̃ ∈ Ob(T̃) and each x̃ ∈ NS̃(P̃, Q̃), there is a unique morphism ε̃_{P̃,Q̃}(x̃) in T̃ that makes the following diagram commute:

\xymatrix{P̃ \ar[r]^{ι̃P̃} \ar[d]_{ε̃_{P̃,Q̃}(x̃)} & S̃ \ar[d]^{ε̃S̃(x̃)} \\ Q̃ \ar[r]^{ι̃Q̃} & _{S̃ .}}  (2)

Furthermore, ε̃_{P̃,P̃}(A) = Ker(τ_{P̃,P̃}), and thus Ker(τ_{P̃,P̃}) ≅ A.

Proof. The first statement follows immediately from lifting Lemma 5.2(b), applied with ψ̃ = ι̃Q̃, ψ̃φ = ε̃S̃(x̃) ∘ ι̃P̃, and φ = ε_{P,P} (q(x̃)).

By Lemma 5.2, there is a unique bijection χ: Ker(τ_{P̃,P̃}) ⟶^{≅} Ker(τ_{S̃,S̃}) such that the square

\xymatrix{P̃ \ar[r]^{ι̃P̃} \ar[d]_{α} & S̃ \ar[d]^{χ(α)} \\ P̃ \ar[r]^{ι̃Q̃} & _{S̃ .}}

commutes for all α ∈ Ker(τ_{P̃,P̃}). Since χ(ε̃_{P̃,P̃}(a)) = ε̃S̃(a) for a ∈ A by (2), and ε̃S̃(A) = Ker(τ_{S̃,S̃}) by definition of ε̃S̃, this shows that ε̃_{P̃,P̃}(A) = Ker(τ_{P̃,P̃}). □

We thus get a well-defined map from Mor(T_{Ob(T̃)}(S̃)) to Mor(T̃) which sends x̃ to ε̃_{P̃,Q̃}(x̃). By putting together two squares of the form (2) we see that this defines a functor

ε̃: T_{Ob(T̃)}(S̃) ⟶ T̃

which is the identity on objects. Notice that ε̃_{P̃,Q̃} is defined so that the square

\xymatrix{NS̃(P̃, Q̃) \ar[r]^{ε̃_{P̃,Q̃}} \ar[d]^{q} & MorT̃(P̃, Q̃) \ar[d]^{τ_{P̃,Q̃}} \\ N_{S}(P, Q) \ar[r]^{ε_{P,Q}} & Mor_{T} (P, Q)}  (3)

commutes in T̃, i.e., ε̃_{P̃,Q̃} is a lifting of ε_{P,Q}.

Lemma 5.4. For all P̃, Q̃ ∈ Ob(T̃), the map ε̃_{P̃,Q̃} : NS̃(P̃, Q̃) ⟶ MorT̃(P̃, Q̃) is injective.

Proof. Let x̃, ỹ ∈ NS̃(P̃, Q̃) be such that ε̃_{P̃,Q̃}(x̃) = ε̃_{P̃,Q̃}(ỹ). Then

ε̃S̃(x̃) ∘ ι̃P̃ = ι̃Q̃ ∘ ε̃_{P̃,Q̃}(x̃) = ι̃Q̃ ∘ ε̃_{P̃,Q̃}(ỹ) = ε̃S̃(ỹ) ∘ ι̃P̃

by (2). Also, by (3), q(x̃) = q(ỹ), so ε̃S̃(x̃) and ε̃S̃(ỹ) are liftings of the same morphism ε_{S,S}(q(x̃)). Thus ε̃S̃(x̃) = ε̃S̃(ỹ) by Lemma 5.2(a). But ε̃S̃ is injective by construction (see (1)), so x̃ = ỹ. □

Lemma 5.5. Given a morphism f ∈ MorT̃(P̃, Q̃), there is a unique group homorphism φ ∈ Hom(P̃, Q̃) such that the diagram

\xymatrix{P̃ \ar[r]^{f} \ar[d]_{ε̃_{P̃,P̃}(x)} & Q̃ \ar[d]^{ε̃_{Q̃,Q̃}(φ(x))} \\ P̃ \ar[r]^{f} & Q̃}  (4)

commutes for all x ∈ P̃.

Proof. Set x̄ = q(x). By axiom (C) of the transporter system T ,

τ(f) ∘ ε_{P,P} (x̄) = ε_{Q,Q}(ρ(τ(f))(x̄)) ∘ τ(f) ,

for all x̄ ∈ P. So the result follows from the lifting Lemma 5.2(a). □

Now we can define the functor ρ̃ : T̃ ⟶ Gps to be the identity on objects, and to send a morphism f ∈ MorT̃(P̃, Q̃) to the unique group homomorphism defined in Lemma 5.5. Again, by putting together commutative squares of the form (4), one sees that ρ̃ is indeed a functor.

Let F̃ be the fusion system generated by the image of the functor ρ̃. Thus when P̃, Q̃ ∈ Ob(T̃), MorF̃(P̃, Q̃) is the image under ρ̃ of MorT̃(P̃, Q̃). For arbitrary subgroups R, R′ ≤ S̃, MorT̃(R, R′) ⊆ Inj(R, R′) is the set of all monomorphisms which are composites of restrictions of F̃-morphisms between objects in T̃.

Proposition 5.6. The category T̃, together with the maps ε̃_{P̃,Q̃} and the functor ρ̃ defined in Lemmas 5.3 and 5.5, define a transporter system associated to F̃. In particular, F̃ is an Ob(T̃)-saturated fusion system. Furthermore, A ◁ T̃, and (S̃/A, F̃/A, T̃/S) ≅ (S, F, T ).

Proof. The last statement is clear, and the saturation of F̃ follows from Proposition 3.6, once we know that T̃ is a transporter system. So we need only check that T̃ satisfies the axioms of Definition 3.1. Axiom (A1) follows from the conditions in Definition 5.1 for the extension T̃, and axiom (C) holds by construction of ρ̃.

For each P̃, Q̃ ≤ S̃, the injectivity of ε̃_{P̃,Q̃} was shown in Lemma 5.4. The composite ρ̃_{P̃,Q̃} ∘ ε̃_{P̃,Q̃} sends g ∈ NS̃(P̃, Q̃) to c_{g} ∈ HomF̃(P̃, Q̃) by the construction of ρ̃ using Lemma 5.5 (applied with f = ε_{P̃,Q̃}(g)). This proves axiom (B).

We next check axiom (A2). For each P̃ ∈ Ob(T̃), set

Ẽ(P̃) = Ker[AutT̃(P̃) ⟶ AutF̃(P̃)].

Then τ_{P̃,P̃}(Ẽ(P̃)) ≤ E(P). So the free actions of E(P) and E(Q) on Mor_{T} (P, Q) lift to free actions of Ẽ(P̃) and Ẽ(Q̃) on MorT̃(P̃, Q̃). If f, f′ are two morphisms such that ρ̃(f) = ρ̃(f′), then ρ(τ(f)) = ρ(τ(f′)), so τ(f′) = τ(f) ∘ α for some α ∈ E(P) by axiom (A2) for T . By the lifting Lemma 5.2(b), α lifts to a unique α̃ ∈ AutT̃(P̃) such that f′ = f ∘ α̃, and α̃ ∈ Ẽ(P̃) since ρ̃(f′) = ρ̃(f).

Axiom (I) for T̃ (ε̃_{S̃,S̃}(S̃) ∈ Syl_{p}(AutT̃(S̃))) follows from axiom (I) for T .

Axiom (II) is a consequence of the same axiom applied to T . Given a morphism φ ∈ MorT̃(P̃, Q̃), and subgroups P̃ ≤ P̃′ ≤ NS̃(P̃) and Q̃ ≤ Q̃′ ≤ NS̃(Q̃) such that φ ∘ ε̃_{P̃,P̃}(P̃′) ∘φ^{−1} ≤ ε̃_{Q̃,Q̃}(Q̃′), apply the functor τ : T̃ ⟶ T to the last inclusion. Then axiom (II) for T implies that there is some f ∈ MorT̃(P̃′, Q̃′) such that τ(f)|_{P,Q} = τ(φ). Hence by Lemma 5.3 (Ker(τ_{P̃,P̃}) = ε̃_{P̃,P̃}(A)), there is a ∈ A such that f|_{P̃,Q̃} = φ ∘ ε̃_{P̃,P̃}(a). So φ =^{def} f ∘ ε̃_{P̃′,P̃′}(a^{−1}) is an extension of φ. □

We now look at the action of T on A induced by an extension of transporter systems.

Lemma 5.7. Fix a transporter system (S, F, T ), and an extension

1 ⟶ A ⟶ T̃ ⟶^{τ} T ⟶ 1.

Then this defines an action Φ : Mor(T ) ⟶ Out(A) of Mor(T ) on A as follows. For each f ∈ Mor_{T} (P, Q), choose any f̃ ∈ MorT̃(P̃, Q̃) such that τ(f̃) = f, and set

Φ̃(f̃) = ρ̃(f̃)|_{A} ∈ Aut(A).

Then the class Φ(f) = [Φ̃(f̃)] ∈ Out(A) is well defined, independently of the choice of lifting f̃, and satisfies the following conditions.

(a) Φ(f) ∘ Φ(f′) = Φ(f ∘ f′) for any composable pair of morphisms f, f′ in T .

(b) For any lifting f̃ ∈ Mor(T̃) of f, the following square commutes for all a ∈ A:

\xymatrix{P̃ \ar[r]^{f̃} \ar[d]_{ε̃_{P̃,P̃}(a)} & Q̃ \ar[d]^{ε̃_{Q̃,Q̃}(Φ̃(f̃)(a))} \\ P̃ \ar[r]^{f̃} & _{Q̃ .}}  (5)

Proof. If f̃′ is another lifting of f, then there is some a ∈ A such that f̃′ = f̃∘ ε̃_{P̃,P̃}(a). Then ρ̃(f̃′) = ρ̃(f̃) ∘ c_{a}, and so [ρ̃(f̃′)|_{A}] = [ρ̃(f̃)|_{A}] in Out(A). This proves that Φ(f) is well defined in Out(A), independently of the choice of lifting.

Point (a) is immediate from the definition, and (b) follows from axiom (C) for T̃. □

Extensions of transporter systems in the sense of Definition 5.1 give one class of examples of extensions of categories which are both source regular and target regular, in the sense of Definition A.5. By that definition, an extension of the transporter system T is exactly the same as a source regular extension of T by a functor F: T^{op} ⟶ Gps_{∗} such that F sends all objects in T to finite p-groups and sends all morphisms to isomorphisms.

The general obstruction theory for constructing extensions of categories which are source and target regular and have abelian kernels is described in Proposition A.9 (with some remarks on the nonabelian case). The following proposition deals with the special case of extensions of transporter systems. Since any extension of a transporter system by a finite p-group can be factored as the composite of a sequence of extensions by abelian p-groups, the assumption that the kernel be abelian is not a major restriction.

Proposition 5.8. Fix a transporter system (S, F, T ). Then extensions of T by a given finite abelian p-group A are in one-to-one correspondence with actions Φ of π_{1}(|T |) on A, together with elements of H^{2}(|T |; A) (with coefficients twisted by Φ).

Proof. By Proposition A.3(a), an action of π_{1}(|T |) on A correspond to maps from Mor(T ) to Aut(A) which send composites to products and inclusions to the identity. Hence the action of π_{1}(|T |) on A is described in Lemma A.7(c); and is the same as that defined by Lemma 5.7 by point (b) of that lemma. By Proposition A.9, the set of extensions which realize a given action of π_{1}(|T |) on A are in one-to-one correspondence with \underleftarrow{lim}^{2}(K), where K(P) ≅ A for all P, and this group is isomorphic to H^{2}(|T |; A) by Lemma A.3(b). □

We now examine conditions for an extension of T to be a linking system, and also for its associated fusion system to be exotic. We first give an example of the complications which can arise if one doesn’t impose extra conditions. Set p = 2, and let L be the centric linking system of A_{6}. Since π_{1}(|L|) ≅ Σ_{4} ∗_{D_{8}} Σ_{4}, it surjects onto A_{6} ≅ P SL_{2}(9)

and onto P SL_{2}(7); in fact, onto P SL_{2}(q) for any q ≡ ±7 (mod 16) which is a prime or the square of a prime. Let |L| act on A = C_{2}^{7} via the faithful action of A_{6} on C_{2}^{4} and that of P SL_{2}(7) ≅ GL_{3}(2) on C_{2}^{3}. The resulting extension of L by A is an exotic transporter system; but not very interesting, since it has too few objects to generate a saturated fusion system.

We now want to define the concept of an “admissible” action or extension of a transporter system, in order to avoid such examples. We first need one more lemma to motivate the precise conditions in the definition.

When τ: T̃ ⟶ T is an extension of T by A, we let

Φ_{P,Q} : Mor_{T} (P, Q) ⟶ Out(A) and Φ_{P} : E(P) ⟶ Out(A)

be the restrictions of the action Φ to the corresponding sets of morphisms. Recall that E(P) = Ker(ρ_{P,P}) ≤ Aut_{T} (P), so Φ_{P} is a group homomorphism. We also write Ẽ(P̃) = Ker(ρ̃_{P̃,P̃}).

Lemma 5.9. Fix a transporter system (S, F, T ), and an extension

1 ⟶ A ⟶ T̃ ⟶^{τ} T ⟶ 1.

Let F̃ ⟶^{τ_{F}} F and S̃ ⟶^{q} S be the extensions of fusion systems and p-groups associated to T̃. Set

S_{1} = q(CS̃(A)) = Ker[S ⟶^{ε_{S,S}} Aut_{T} (S) ⟶^{Φ_{S,S}} Out(A)].

Then the following hold for any subgroup P ∈ Ob(T ), where P̃ = q^{−1}(P).

(a) If P is fully normalized in F, then Ẽ(P̃) ≤ ε̃_{P̃,P̃}(CS̃(P̃)) if and only if Ker(Φ_{P}) is a p-group.

(b) If P̃ ≤ S̃ is any subgroup which is F̃-centric and F̃-radical, or more generally any F̃-centric subgroup such that OutS̃(P̃) ∩ O_{p}(OutF̃(P̃)) = 1, then P̃ ≥ A and C_{S_{1}}(P) ≤ P.

(c) If C_{S_{1}}(P) ≤ P, then P̃ is F̃-centric.

Proof. (a) Consider the following commutative diagram:

\xymatrix{1 \ar[r] & Ẽ(P̃) \ar[r] \ar[d] & AutT̃(P̃) \ar[r]^{ρ̃} \ar@{->>}[d]^{τ_{P̃,P̃}} & AutF̃(P̃) \ar[d]^{ψ} \\ 1 \ar[r] & Ker(Φ_{P}) \ar[r] & Aut_{T} (P) \ar[r]^{(Φ,ρ)} & Out(A) × Aut_{F}(P)}

where ψ(f) = (f|_{A}, f/A) for f ∈ AutF̃(P̃). By Lemma 1.9(a), Ker(ψ) is a p-group. Hence for each x ∈ Ker(Φ_{P}), x^{p^{k}} ∈ τ_{P̃,P̃}(Ẽ(P̃)) for some k; and this implies that τ_{P̃,P̃}(Ẽ(P̃)) has p-power index in Ker(Φ_{P}). Since Ker(τ_{P̃,P̃}) ≅ A is a p-group, it now follows that Ẽ(P̃) is a p-group if and only if Ker(Φ_{P}) is a p-group. In particular, if Ẽ(P̃) ≤ ε_{P̃,P̃}(CS̃(P̃)), then Ker(Φ_{P}) is a p-group.

Now assume conversely that Ker(Φ_{P}) is a p-group. We have just shown that Ẽ(P̃) is a p-group. Since P is fully normalized, P̃ is also fully normalized, and so axiom (I′) for T̃ implies that ε̃_{P̃,P̃}(NS̃(P̃)) is a Sylow p-subgroup of AutT̃(P̃). Since a normal p-subgroup of a finite group is contained in all of its Sylow p-subgroups, this shows that

Ẽ(P̃) ≤ ε̃_{P̃,P̃}(NS̃(P̃)).

Furthermore, by axiom (C) for T̃, every element of Ẽ(P̃) commutes with every element of ε̃_{P̃,P̃}(P̃), and thus Ẽ(P̃) ≤ ε̃_{P̃,P̃}(CS̃(P̃)).

(b) Assume P̃ ≤ S̃ is F̃-centric and OutS̃(P̃) ∩ O_{p}(OutF̃(P̃)) = 1, and consider the following diagram with exact rows:

\xymatrix{1 \ar[r] & CS̃(P̃) \ar[r] \ar[d]^{κ} & NS̃(P̃) \ar[r]^{conj} \ar@{->>}[d]^{q} & AutF̃(P̃) \ar[d]^{ψ} \\ 1 \ar[r] & C_{S_{1}}(P) \ar[r] & N_{S}(P) \ar[r]^{conj} & Out(A) × Aut_{F}(P) .}

Then

{g ∈ NS̃(P̃) | c_{g} ∈ Ker(ψ)} ≤ CS̃(P̃) ≤ P̃ :  (6)

the first inclusion since Ker(ψ) ≤ O_{p}(AutF̃(P̃)), and the second since P̃ is F̃-centric. In particular, A ≤ P̃. Also, for all g ∈ C_{S_{1}}(P), there is g̃ ∈ NS̃(P̃) such that q(g̃) = g, g̃ ∈ P̃ by (6), and so g ∈ P.

(c) If C_{S_{1}}(P) ≤ P, then q(CS̃(P̃)) ≤ P, and so P̃ is F̃-centric. □

Conditions (b) and (c) above help to motivate the following definition.

Definition 5.10. Fix a transporter system (S, F, T ).

(a) A homomorphism Φ: π_{1}(|T |) ⟶ Γ is called admissible if, upon setting S_{1} = Ker(Φ ∘ ε_{S,S}), P ≤ S fully centralized in F and C_{S_{1}}(P) ≤ P imply P ∈ Ob(T ).

(b) An action Φ: π_{1}(|T |) ⟶ Out(A) of T on a p-group A is called admissible if the homomorphisms Φ is admissible.

(c) An extension 1 → A → T̃ → T → 1 is called admissible if the action of T on A defined in Lemma 5.7 is admissible.

Recall, for any fusion system F over S and any A ≤ S, the centralizer fusion system C_{F}(A) is the fusion system over C_{S}(A) where for each P, Q, Hom_{C_{F} (A)}(P, Q) is the set of those f ∈ Hom_{F}(P, Q) which extend to f ∈ Hom_{F}(P A, QA) such that f|_{A} = Id_{A} (see [BLO2, §2]). As a consequence of Lemma 5.9, when applied to admissible extensions, we now get:

Theorem 5.11. Fix a transporter system (S, F, T ) and an admissible extension 1 → A → T̃ → T → 1, and let S̃ and F̃ be the p-group and fusion system associated to T̃. Let Φ: π_{1}(|T |) ⟶ Out(A) be the action defined in Proposition 5.7, and let (S_{1}, F_{1}, T_{1}) be the “kernel” transporter system of Φ in the sense of Proposition 4.1. Then the following hold.

(a) Ob(T̃) contains all subgroups of S̃ which are F̃-centric and F̃-radical. Moreover, every F̃-centric subgroup P̃ ≤ S̃ not in Ob(T̃) is F̃-conjugate to some P̃′ such that OutS̃(P̃′) ∩ O_{p}(OutF̃(P̃′) ≠ 1.

(b) F̃ is a saturated fusion system.

(c) Let L̃ be the full subcategory of T̃ whose objects are those F̃-centric subgroups of S̃ in Ob(T̃). Then L̃ is a linking system if and only if Ker[Φ_{P} : E(P) ⟶ Out(A)] is a p-group for all P ∈ Ob(T ) such that P̃ ∈ Ob(L̃).

(d) If F̃ is the fusion system of a finite group G̃, then F is the fusion system of NG̃(A)/A, and F_{1} is the fusion system of CG̃(A)/A. Thus if F_{1} is exotic (not the fusion system of any finite group), then so is F̃.

Proof. Point (a) follows immediately from Lemma 5.9(b), and point (b) then follows from Proposition 3.6.

By definition, L̃ is a linking system associated to F̃ if and only if Ẽ(P̃) ≤ ε̃_{P̃,P̃}(Z(P̃)) for all P̃ ∈ Ob(L̃), and Z(P̃) = CS̃(P̃) for all such P̃. Also, if this condition holds for any subgroup in the F̃-conjugacy class of P̃, it also holds for P̃. So (c) follows from Lemma 5.9(a).

It remains to prove (d). By construction, A ◁ F̃, F̃/A ≅ F, and CF̃(A)/A ≅ F_{1}. If F̃ ≅ FS̃(G̃) for some finite group G̃, then since A ◁ F̃, F̃ is also the fusion system of NG̃(A). Thus F is the fusion system of NG̃(A)/A. Also, by definition, a morphism in F̃ lies in CF̃(A) if and only if it is conjugation by an element in CG̃(A). Hence CF̃(A) is the fusion system of CG̃(A), and so F_{1} is the fusion system of CG̃(A)/A. □

We finish the section with some tools which are useful when computing the obstruction groups which appear in Proposition 5.8.

Lemma 5.12. Fix a transporter system (S, F, T ), a finite abelian p-group A, and an admissible action Φ: T ⟶ Aut(A). Set S_{1} = Ker(Φ ∘ ε_{S,S}) ◁ S, and assume that P ∈ Ob(T ) implies P ∩ S_{1} ∈ Ob(T ). Let T_{1} ⊆ T be the transporter subsystem of Proposition 4.1: Ob(T_{1}) is the set of subgroups of S_{1} which are in Ob(T ), and Mor(T_{1}) is the set of morphisms ψ in T between objects of T_{1} such that Φ(ψ) = 1. Then there is a spectral sequence

E_{2}^{ij} = H^{i}(Γ; H^{j}(|T_{1}|; A)) ⟹ H^{i+j}(|T |; A) ,

where Γ = Im(Φ) ≤ Aut(A). Here, H^{∗}(|T_{1}|; A) is ordinary cohomology with untwisted coefficients. The action of Γ on H^{j}(|T_{1}|; A) is induced by the action of Γ on A, together with the homotopy action of Γ ≅ Aut_{T} (S_{1})/Aut_{T_{1}}(S_{1}) on |T_{1}| via restrictions.

Proof. This is the spectral sequence induced by the fibration sequence

X̃ ⟶ EΓ ×_{Γ} X̃ ⟶ BΓ,

where X̃ denotes the covering space of |T | with fundamental group Ker(Φ). By Proposition 4.1(d), X̃ ≃ |T_{1}|. The action of Γ on X̃ induces the homotopy action on |T_{1}| described above via the construction in the proof of Propositions 4.1(d) and A.4. □

In the above situation, if one wants to determine the action of Γ on H^{∗}(|T_{1}|; A) in concrete situations where the actions on |T_{1}| and on A are both nontrivial, then more precision is needed when formulating the above statement. But this formulation suffices for our purposes here.

The following is an immediate corollary to Lemma 5.12.

Corollary 5.13. Let (S, F, T ), Φ: T ⟶ Aut(A), S_{1} ◁ S, and T_{1} ⊆ T be as in Lemma 5.12, and set Γ = Im(Φ) ≤ Aut(A). Assume, furthermore, that H^{i}(|T_{1}|; 𝔽_{p}) = 0 for i = 1, 2. Then H^{i}(|T |; A) ≅ H^{i}(Γ; A) for i = 1, 2. Thus there is a bijective correspondence between extensions of T by A and group extensions of Γ by A (with the given actions of T and Γ on A). □

6. Examples

We first look at conditions which imply that an admissible extension of the transporter category of a finite group (or one of its full subcategories) is again the transporter category of a finite group. Afterwards, we give some examples of “exotic” extension, where the target is the transporter category of a finite group but the source is not.

Recall that for any transporter system (S, F, T ),

θ: Mor(T ) ⟶ π_{1}(|T |)

is the map which sends φ ∈ Mor_{T} (P, Q) to the loop ι_{Q,S}·φ·ι_{P,S}^{−1} (composed from right to left). When T = T_{S}(G) for some finite group G and some S ∈ Syl_{p}(G), there is also a map Mor(T ) ⟶ G induced by the inclusions of the transporter sets Mor_{T} (P, Q) = N_{G}(P, Q) into G. This map clearly sends composites to products and inclusions 1 ∈ N_{G}(P, S) to the identity. Hence by the universality property of θ (Proposition A.3(a)), there is a unique homomorphism

η_{G}: π_{1}(|T_{S}(G)|) ⟶ G

such that η_{G}(θ(x)) = x for all P, Q ≤ S and all x ∈ N_{G}(P, Q). To keep the notation simple, we also write η_{G} for the analogous homomorphism defined on π_{1}(|T |) when T is a full subcategory of T_{S}(G) which contains S.

Proposition 6.1. Fix a finite group G, a Sylow subgroup S ∈ Syl_{p}(G), and a set H of subgroups of S which is closed under G-conjugacy and overgroups. Let

1 ⟶ A ⟶ T̃ ⟶^{τ} T_{H}(G) ⟶ 1

be an admissible extension where A is a finite p-group. Assume furthermore that the homomorphism

Φ: π_{1}(|T_{H}(G)|) ⟶ Out(A)

of Lemma 5.7 factors through G: that Φ = Φ′ ∘ η_{G} for some Φ′ ∈ Hom(G, Out(A)). Then there is an extension of groups

1 ⟶ A ⟶ G̃ ⟶^{χ} G ⟶ 1

such that T̃ is a full subcategory of T_{χ^{−1}(S)}(G̃).

Proof. Assume first that A is abelian. In this case, by Proposition 5.8, the extensions of T = T_{H}(G) which realize the given action Φ are in bijective correspondence with elements of H^{2}(|T |; A). By Proposition 4.5, η_{G} induces an isomorphism

η_{G}^{∗}: H^{∗}(G; A) ⟶^{≅} H^{∗}(|T |; A).

Furthermore, by the construction of the cohomology invariants, η_{G}^{∗} sends the element in H^{2}(G; A) which describes a group extension to the element of H^{2}(|T |; A) ≅ \underleftarrow{lim}_{T}^{2}(A)

which describes the extension of categories. Thus each extension of categories is induced by an extension of groups, which is what we wanted to prove.

Now assume A is nonabelian, and set T = T̃/Z(A). We can assume, by induction on |A|, that the proposition holds for the extension

1 ⟶ A/Z(A) ⟶ T ⟶^{τ} T_{H}(G) ⟶ 1 .

Thus there is an extension G of G by A/Z(A), a Sylow subgroup S ∈ Syl_{p}(G), and a set H of subgroups of S, such that T ≅ T_{H}(G).

Consider the following diagram:

\xymatrix{π_{1}(|T_{H}(G)|) \ar[r]^{proj} \ar[d]_{η_{G}} & π_{1}(|T_{H}(G)|) \ar[r]^{Φ} \ar[d]_{η_{G}} & Out(A) \ar[r]^{Res} & Aut(Z(A)) \\ G \ar[r]^{proj} & G . \ar[ur]^{Φ′}} (1)

The triangle commutes by assumption, and the square by the naturality of η_{G}. The action of π_{1}(|T |) on Z(A) defined by the extension

1 ⟶ Z(A) ⟶ T̃ ⟶^{τ} T_{H}(G) ⟶ 1  (2)

is the composite of the upper row in (1) (see the definition of the action in Lemma 5.7), and it thus factors through G by the commutativity of (1). So by the abelian case of the proposition, applied to (2), there is an extension G̃ of G by Z(A) such that T̃ is a fully subcategory of TS̃(G̃). Also, G̃ is an extension of G by A, since T_{H}(G) ≅ T̃/A. □

Thus, in order to find an exotic extension, we must find an admissible action of some T_{H}(G) on a p-group A which does not factor through G. The following example was suggested to us by Albert Ruiz; it is just the simplest of a large family of such examples which he has constructed in [Rz]. The example is stated in terms of the linking system of G, but it also holds if L_{S}^{c}(G) is replaced by the transporter system T_{S}^{c}(G) with the same objects.

Example 6.2. Fix a prime p ≥ 5. Let q be another prime, let e be the order of q in 𝔽_{p}^{×} (thus e|(p − 1)), and assume e > 2. Set G = GL_{ep}(q), fix S ∈ Syl_{p}(G), and set F = F_{S}(G) and L = L_{S}^{c}(G). Let A be cyclic of order p. Then there is a homomorphism

Φ: π_{1}(|L|) = π_{1}(|L_{S}^{c}(G)|) ⟶ Aut(A),

where |Im(Φ)| = e, which does not factor through G. Also, there is an admissible extension

1 ⟶ A ⟶ L̃ ⟶ L_{S}^{c}(G) ⟶ 1,

where L̃ is a linking system associated to a saturated fusion system F̃ which is not the fusion system of any finite group.

Proof. Set V = (𝔽_{q^{e}})^{p} (with the canonical basis). Fix an 𝔽_{q}-basis for 𝔽_{q^{e}}, which also yields an 𝔽_{q}-basis for V. This allows us to identify GL_{p}(q^{e}) = Aut_{𝔽_{q^{e}}}(V) as a subgroup of GL_{pe}(q) = Aut_{𝔽_{q}}(V).

Let Γ ≤ 𝔽_{p}^{×} be the subgroup of order e generated by q. We identify this with Gal(𝔽_{q^{e}}/𝔽_{q}) (field automorphisms) by identifying q ∈ 𝔽_{p}^{×} with the automorphism x ↦ x^{q}. Set H = (𝔽_{q^{e}})^{×} ⋊ Γ, regarded as a subgroup of GL_{e}(q) = Aut_{𝔽_{q}}(𝔽_{q^{e}}) (vector space automorphisms). This defines an embedding of H ≀ Σ_{p} = H^{p} ⋊ Σ_{p} into GL_{ep}(q).

Now,

|G| = ∏_{i=0}^{ep−1}(q^{ep} − q^{i}) = q^{ep(ep−1)/2}· ∏_{i=1}^{ep}(q^{i} − 1),

where p|(q^{i} − 1) only when e|i, and (q^{ei} − 1)/(q^{e} − 1) ≡ i (mod p). We leave it as an exercise to show that (q^{ep} − 1)/(q^{e} − 1) is divisible by p but not by p^{2}. Thus if p^{ℓ} is the largest power of p dividing q^{e} − 1, then p^{pℓ+1} is the largest power of p dividing |G| and |H ≀ Σ_{p}|; and this proves that H ≀ Σ_{p} contains a Sylow p-subgroup of G.

Fix Δ ∈ Syl_{p}((𝔽_{q^{e}})^{×}), a cyclic group of order p^{ℓ}, and set S = Δ ≀ C_{p} ∈ Syl_{p}(H ≀ Σ_{p}) ⊆ Syl_{p}(G). Set T = Δ^{p} ≤ S: the set of elements of S which are diagonal in GL_{p}(q^{e}).

We now want to define a homomorphism Ψ′ from π_{1}(|F|) onto Γ, for which the kernel fusion system (in the sense of Proposition 4.1) is exotic. The existence of such a homomorphism is shown in [Rz], but we give a more explicit construction here.

Let V ≅ (𝔽_{q})^{ep} be the vector space upon which G acts. Consider the subset

X = {g ∈ S | |g| = p, Ker(g − Id_{V}) = 0};

i.e., the set of elements of S of order p whose action on V does not have 1 as eigenvalue. Every element of S∖T acts on V with characteristic polynomial (X^{p} − 1)^{e}, which has 1 as root, and thus X ⊆ T. So if we fix z ∈ (𝔽_{q^{e}})^{×} of order p, and again identify GL_{p}(q^{e}) ≤ GL_{ep}(q) (and (𝔽_{q^{e}})^{p} ≅ (𝔽_{q})^{pe}), then

X = {diag(z^{i_{1}}, …, z^{i_{p}}) | i_{1}, …, i_{p} ∈ 𝔽_{p}^{×}}.

By definition of X, any morphism in F = F_{S}(G) sends elements of X to elements of X.

Now, Aut_{F}(T) = Aut_{G}(Δ^{p}) ≅ Γ ≀ Σ_{p}: the subgroup generated by the field automorphisms Γ = Aut(𝔽_{q^{e}}) ≤ Aut(Δ), and by all permutations of the p factors. Define Ψ: Aut_{F}(T) ⟶ Γ to be the composite

Ψ: Aut_{F}(T) ≅ Γ^{p} ⋊ Σ_{p} −↠ Γ × Σ_{p} −↠^{pr_{1}} Γ.

Thus for all α ∈ Aut_{F}(T), if α(diag(z, z, …, z)) = diag(z^{i_{1}}, …, z^{i_{p}}), then Ψ(α) = i_{1} ⋯ i_{p} ∈ Γ ≤ 𝔽_{p}^{×}. So Ψ(α) depends only on α|_{Z(S)∩X}.

For each pair of subgroups P, Q ≤ S which are p-centric in G, define

Φ′_{P,Q}: Hom_{F}(P, Q) ⟶ Γ

as follows. For φ ∈ Hom_{F}(P, Q), φ|_{⟨P ∩X⟩} extends to some φ ∈ Aut_{F}(T) by axiom (II) (and since T is the unique abelian subgroup of index p in S), and we set Φ′_{P,Q}(φ) = Ψ(φ). Since Ψ(φ) depends only on φ|_{Z(S)∩X} and P ≥ Z(S), this is independent of the choice of extension φ. If ψ ∈ Hom_{F}(Q, R) is another morphism in F^{c}, and ψ ∈ Aut_{F}(T) is an extension of ψ|_{⟨Q∩X⟩}, then ψ ∘ φ is an extension of (ψ ∘ φ)|_{⟨P ∩X⟩} (since φ(P ∩ X) ⊆ Q ∩ X) and thus Φ′_{P,R}(ψ ∘ φ) = Φ′_{Q,R}(ψ)·Φ′_{P,Q}(φ).

We have thus defined a map Φ′ from Mor(F^{c}) to Γ which sends composites to products and sends inclusions to the identity. So by Proposition A.3(a), Φ′ can be identified with a homomorphism from π_{1}(|F^{c}|) to Γ. Set

Φ = Φ′ ∘ π_{1}(|π|): π_{1}(|L|) ⟶ Γ.

Let (S̃, F̃, L̃) be the semidirect product extension of (S, F, L) by A induced by Φ; i.e., the extension which corresponds to the zero element in H^{2}(|T |; A). In the notation of Theorem 5.11, S_{1} = S, and so the extension is admissible. Hence F̃ is a saturated fusion system by Theorem 5.11(b). Also, since L = L_{S}^{c}(G) is a linking system, E(P) is a p-group for all P ∈ Ob(L), and thus L̃ is a linking system by Theorem 5.11(c).

Let L_{1} ⊆ L and F_{1} ⊆ F be the “kernel” linking and fusion systems:

Mor(L_{1}) = {φ ∈ Mor(L) | Φ(φ) = 1} and Mor(F_{1}) = {φ ∈ Mor(F) | Φ′(φ) = 1}.

Thus by Proposition 4.1, F_{1} is a saturated fusion system over S and L_{1} is a linking system associated to F_{1}. By construction, Aut_{F_{1}}(T) ≅ Γ^{p−1} ⋊ Σ_{p}. So by [BLO2, Example 9.4] (or more directly by [BLO2, Proposition 9.5]), F_{1} is not the fusion system of any finite group. (This is where we need to assume that e > 2, and hence that p ≥ 5.) Hence by Theorem 5.11(d), F̃ is not the fusion system of any finite group. □

We next make some cohomology computations, to illustrate the use of Lemma 5.12 and Corollary 5.13, and to show that the extension in Example 6.2 is the only one (up to isomorphism) with that action on the kernel.

Lemma 6.3. Let G = GL_{ep}(q), S ∈ Syl_{p}(G), A ≅ C_{p}, and π_{1}(|L_{S}^{c}(G)|) ⟶^{Φ} Aut(A) be as in Example 6.2, set F = F_{S}(G) and L = L_{S}^{c}(G), and let F_{1} ⊆ F and L_{1} ⊆ L be the kernel subsystems. Then for i = 1, 2, H^{i}(|L|; A) ≅ H^{i}(|L_{1}|; 𝔽_{p}) = 0.

Proof. Set H = (𝔽_{q^{e}}^{×} ⋊ C_{e}) ≀ Σ_{p}, regarded as a subgroup of G which contains S via the embedding (𝔽_{q^{e}}^{×} ⋊ C_{e}) ≤ GL_{e}(q). Let Ψ denote the composite

Ψ: H = (𝔽_{q^{e}}^{×} ⋊ C_{e}) ≀ Σ_{p} −↠ C_{e} ≀ Σ_{p} −↠ C_{e} × Σ_{p} −↠^{pr_{1}} C_{e} ≤ 𝔽_{p}^{×},

where the second homomorphism takes the product of the p coordinates in C_{e}, and set H_{1} = Ker(Ψ). Set

H_{1} = H_{1} ∩ (𝔽_{q^{e}}^{×} ⋊ C_{e})^{p} ≅ (𝔽_{q^{e}}^{×})^{p} ⋊ (C_{e})^{p−1}.

Then

H^{i}(H_{1}; 𝔽_{p}) = H^{i}((𝔽_{q^{e}}^{×})^{p}; 𝔽_{p})^{(C_{e})^{p−1}}

vanishes for i = 1, 2, and similarly for

H^{i}(Σ_{p}; 𝔽_{p}) ⟶^{mono} H^{i}(C_{p} ⋊ C_{p−1}; 𝔽_{p}).

So by the spectral sequence for the semidirect product, H^{i}(H_{1}; 𝔽_{p}) = 0 for i = 1, 2. By [BLO2, Theorem B], the inclusion B(S) ⊆ L_{1} induces a monomorphism

H^{∗}(|L_{1}|; 𝔽_{p}) ⟶ H^{∗}(S; 𝔽_{p}),

and this factors through H^{∗}(H_{1}; 𝔽_{p}). Hence H^{i}(|L_{1}|; 𝔽_{p}) = 0 for i = 1, 2. So by Corollary 5.13, if π_{1}(|L|) acts on A via Φ, then H^{2}(|L|; A) ≅ H^{2}(Γ; A) = 0 (since Γ has order prime to p). □

By Lemma 6.3 and Proposition 5.8, it now follows that the only extension of L by A (with the given action) is the “semidirect product” constructed in Example 6.2.

Example 6.2 is interesting as an example of an exotic fusion system with normal psubgroup for which the quotient is the fusion system of a group. But it is “too simple” in some sense, and leaves a lot of questions as to what other sorts of extensions can occur. This motivates the following list of problems.

In all cases, in the following list, 1 → A → T̃ → T → 1 is an extension of transporter systems, associated to fusion systems F̃ and F over p-groups S̃ and S (and where A is abelian). We always let Φ denote the action of π_{1}(|T |) on A. The phrase “interesting extension” is intentionally left imprecise, but at least means an extension of transporter systems 1 → A → T̃ → T → 1 which is not induced by an extension of finite groups.

(1) Find interesting extensions for which Im(Φ) has order a multiple of p.

(2) Find interesting extensions where subgroups which are centric and radical in F̃ are not all sent to subgroups which are quasicentric in F.

(3) Find an extension where F and F_{1} (the kernel subsystem) are both fusion systems of groups, but where F̃ is exotic. Essentially, this means finding F_{1} ◁ F which are fusion systems of groups, but not fusion systems of a pair of groups G_{1} and G such that G_{1} ◁ G with appropriate quotient.

We refer to [BCGLO1, Definition 3.1] for the definition of an F-quasicentric subgroup. When F = F_{S}(G) for a finite group G, then a subgroup P ≤ S is F-quasicentric if and only if C_{G}(P) has a normal subgroup of order prime to p and of p-power index. Question (2) is motivated by [BCGLO1, Theorem B], which says that for any p-local finite group (S, F, L), there is a linking system L^{q} ⊇ L for which Ob(L^{q}) is the set of all F-quasicentric subgroups of S, and that for any full subcategory L_{0} ⊆ L^{q} which contains all F-centric F-radical subgroups of S, |L_{0}|, |L|, and |L^{q}| all have the same homotopy type.

In the remaining part of this section, we describe some constructions which are simple modifications of the one in Example 6.2, and which show that problems (1) and (2) have solutions if one keeps the requirements for “interesting extensions” to a minimum. What this really shows is that we want more restrictive conditions on what makes them “interesting”.

Let p, q, and e be as in Example 6.2: p is a prime with p ≥ 5, q is a prime power with p ∤ q, and e ≥ 3 is the order of q in 𝔽_{p}^{×}. Set H = GL_{pe}(q^{p}), and let G = H ⋊ C_{p} where C_{p} acts via the field automorphism of order p on 𝔽_{q^{p}}. Fix S_{0} ∈ Syl_{p}(H) and S = S_{0} ⋊ C_{p} ∈ Syl_{p}(G), and set L_{0} = L_{S_{0}}^{c}(H). Set

H = {P ≤ S | Z(P) ∩ H ∈ Syl_{p}(C_{H}(P))}.

For each P ∈ H, C_{H}(P) = (Z(P) ∩ H) × C′_{H}(P) where C′_{H}(P) is the subgroup of elements of order prime to p in C_{H}(P), and we set

T = T_{H}(G)/C′_{H}.

This is a transporter system by Proposition 3.13. Also, L_{0} is the kernel linking system of the homomorphism from π_{1}(|T |) to C_{p} induced by the surjection of π_{1}(|T_{H}(G)|) onto G/H. Hence the homomorphism of Example 6.2 extends to a homomorphism

Φ̂: π_{1}(|T |) ⟶ Γ × C_{p} ≤ Aut(A),  (A ≅ C_{p^{2}})

and the homomorphism is admissible by construction of T . The kernel linking system of Φ̂ is the same as the kernel system for π_{1}(|L_{0}|) ⟶^{Φ_{0}} Γ, which was shown to be exotic in Example 6.2. So by Theorem 5.11, for any extension

1 ⟶ A ⟶ L̃ ⟶ T ⟶ 1

which realizes this action of T on A, L̃ is a linking system associated to an exotic fusion system. Using Lemmas 5.12 and 6.3, we see that H^{2}(|L|; A) ≅ H^{2}(C_{e} × C_{p}; A) = 0, so there is a unique extension of this type.

A similar, but slightly more complicated example can be constructed by setting H = GL_{pe}(q)^{p}, G = GL_{pe}(q) ≀ Σ_{p} = H ⋊ Σ_{p}, S ∈ Syl_{p}(G), and T = T_{H}(G)/C′_{H} for an appropriate class H of subgroups of S. One can then define Φ from π_{1}(|T |) to Γ ≀ Σ_{p}, where Γ again is cyclic of order e, and look at the induced extensions of A = (C_{p})^{p} by T .

As yet another example, let p, q, and e be as in Example 6.2, set G_{1} = GL_{pe}(q), and set G_{2} = SL_{3}(p). Fix S_{i} ∈ Syl_{p}(G_{i}), and set G = G_{1} × G_{2} and S = S_{1} × S_{2} ∈ Syl_{p}(G). Set

H = {P ≤ S | Z(P) ∩ G_{1} ∈ Syl_{p}(C_{G_{1}}(P))}.

For each P ∈ H, C_{G_{1}}(P) = (Z(P) ∩ G_{1}) × C′_{G_{1}}(P), where C′_{G_{1}}(P) consists of all elements in C_{G_{1}}(P) of order prime to p. Set T = T_{H}(G)/C′_{G_{1}}, a transporter system by Proposition 3.13. Define

Φ: π_{1}(|T |) ⟶ π_{1}(|L_{S_{1}}^{c}(G_{1})|) × π_{1}(|T_{S_{2}}(G_{2})|) ⟶ Aut(C_{p}) × Aut(C_{p}^{3}),

where the first homomorphism is induced by the two projections, and the second is the product of the homomorphism of Example 6.2 with the canonical action of G_{2} = SL_{3}(p) on C_{p}^{3}. Set A = C_{p} × C_{p}^{3}, let

1 ⟶ A ⟶ L̃ ⟶ T ⟶ 1

be any extension which realizes Φ, and let S̃ and F̃ be the associated p-group and fusion system. By Theorem 5.11 again, F̃ is a saturated fusion system over S̃, but F̃ is not the fusion system of any finite group, since if it were the fusion system of some G̃, then CG̃(A)/A would have as fusion system the subsystem of index e in F_{S_{1}}(G_{1}). Also, for each P ∈ Ob(T ),

Ker[Φ: C_{G}(P)/C′_{G_{1}}(P) ⟶ Aut(A)] ≤ C_{G_{1}}(P)/C′_{G_{1}}(P)

is a p-group, and so the restriction of L̃ to F̃-centric subgroups is a linking system by Theorem 5.11(c).

If we now take P = S_{1} × 1 ∈ H, and let P̃ be such that P̃/A = P, then P̃ is centric and radical in F̃. However, P is not quasicentric in F_{S}(G), since its centralizer fusion system involves the fusion system of G_{2}.

Appendix A. General background on categories and higher limits

We begin with a description of the “bar resolution” for computing higher derived functors of inverse limits of diagrams of abelian groups. For any small category C, C-mod denotes the category of functors C^{op} ⟶ Ab.

Lemma A.1. Let C be any small category, and let F : C^{op} ⟶ Ab be any functor. Let C^{∗}(C; F) be the normalized chain complex for F:

C^{n}(C; F) = ∏_{c_{0}→···→c_{n}} F (c_{0}),

where the product is taken over all composable sequences of nonidentity morphisms. For ξ ∈ C^{n}(C; F), define

d(ξ)(c_{0} ⟶^{φ} c_{1} → ⋯ → c_{n+1}) = F (φ)(ξ(c_{1} → ⋯ → c_{n+1}))+

∑_{i=1}^{n+1}(−1)^{i}ξ(c_{0} → ⋯ ĉ_{i} ⋯ → c_{n+1}) ,

where we set ξ(c → ⋯ → c′) = 0 for any sequence containing an identity morphism.

Then \underleftarrow{lim}_{C}^{∗}(F) ≅ H^{∗}(C^{∗}(C; F), d).

Proof. The proof is the same as those in [GZ, Appendix II, Proposition 3.3] or [O1, Lemma 2] (where the result is shown for the unnormalized chain complex). We sketch the argument here. Let 𝔓_{−1} be the constant functor \underline{ℤ}. For each n ≥ 0, define 𝔓_{n}: C^{op} ⟶ Ab by letting 𝔓_{n}(c) be the free abelian group with basis the set of all sequences c ⟶^{φ} c_{0} ⟶^{φ_{1}} ⋯ ⟶^{φ_{n}} c_{n}, modulo the subgroup generated by those sequences for which some φ_{i} for i ≥ 1 is an identity morphism. Also, for ψ ∈ Mor(C), 𝔓_{n}(ψ) is defined by composing the first morphism with ψ. For all n ≥ 0 and all F in C-mod, Mor_{C-mod} (𝔓_{n}, F) ≅ C^{n}(C; F). Thus 𝔓_{n} is projective in C-mod for n ≥ 0. Boundary morphisms ∂: 𝔓_{n} ⟶ 𝔓_{n−1} are defined by setting

∂([c → c_{0} → ⋯ → c_{n}]) = ∑_{i=0}^{n}(−1)^{i}[c → c_{0} → ⋯ ĉ_{i} ⋯ → c_{n}].

This is seen to be exact (hence a projective resolution of \underline{ℤ}) via the splitting homomor-

phisms 𝔓_{n}(c) ⟶ 𝔓_{n+1}(c) which send [c → c_{0} → ⋯ → c_{n}] to [c ⟶^{Id} c → c_{0} → ⋯ →

c_{n}]. Since \underleftarrow{lim}_{C}^{0}(F) ≅ Mor_{C-mod} (\underline{ℤ}, F),

\underleftarrow{lim}_{C}^{∗}(F) ≅ Ext_{C-mod}^{∗} (\underline{ℤ}, F) ≅ H^{∗}(Mor_{C-mod} (𝔓_{∗}, F), ∂^{∗}) ≅ H^{∗}(C^{∗}(C; F), d). □

For any finite group G, we let O_{p}(G) be the p-subgroup orbit category of G: the category whose objects are the p-subgroups of G, and where

Mor_{O_{p}(G)}(P, Q) = Q\N_{G}(P, Q) ≅ map_{G}(G/P, G/Q).

Here, N_{G}(P, Q) is the transporter set:

N_{G}(P, Q) = {g ∈ G | gP g^{−1} ≤ Q}.

If S ∈ Syl_{p}(G), then O_{S}(G) ⊆ O_{p}(G) denotes the full subcategory whose objects are the subgroups of S. These two categories are clearly equivalent (every object in O_{p}(G) is isomorphic to an object in O_{S}(G)). Note that O_{S}(G) is not the same as the orbit category of the fusion system F_{S}(G).

For any ℤ[G]-module M, we define

Λ^{∗}(G; M) = \underleftarrow{lim}^{∗}_{O_{p}(G)}(F_{M}),

where F_{M}: O_{p}(G)^{op} ⟶ Ab is the functor F_{M}(P) = 0 if P ≠ 1 and F_{M}(1) = M. The role these groups play in computing higher limits of certain functors is illustrated by the following proposition, which is a special case of [BLO3, Proposition 4.3].

Proposition A.2. Fix a category C, a finite group G, a Sylow subgroup S ∈ Syl_{p}(G), and a functor

α: O_{S}(G) ⟶ C.

Set c_{0} = α(1). For each object d in C, we regard the set Mor_{C}(c_{0}, d) as a G-set via α and composition. Assume that the following conditions hold:

(a) α sends G = Aut_{O_{S}(G)}(1) bijectively to End_{C}(c_{0}).

(b) For each d ∈ Ob(C) such that d ≅ c̸_{0}, all isotropy subgroups of the G-action on Mor_{C}(c_{0}, d) are nontrivial p-subgroups.

(c) For each ξ ∈ Mor(O_{p}(G)), α(ξ) is an epimorphism in the categorical sense: φ ∘ α(ξ) = ψ ∘ α(ξ) implies φ = ψ.

(d) For each P ≤ S, each d ∈ Ob(C), and each φ ∈ Mor_{C}(c_{0}, d) which is fixed by the P -action, there is some φ ∈ Mor_{C}(α(P), d) such that φ = φ ∘ α(incl_{1}^{P}).

Let

Φ : C^{op} ⟶ Ab

be any functor which vanishes except on the isomorphism class of c_{0}. Then the natural map

\underleftarrow{lim}_{C}^{∗}(Φ) ⟶_{≅}^{α^{∗}} \underleftarrow{lim}^{∗}_{O_{S}(G)}(Φ ∘ α) ≅ Λ^{∗}(G; Φ(c_{0}))

is an isomorphism. □

We now look at fundamental groups of nerves of categories. The next proposition describes the relationship between an action of π_{1}(|C|) and an action of Mor(C).

For any group Γ, B(Γ) denotes the category with one object and with automorphism group Γ, and we identify |B(G)| = BG. This notation will be used in several of the proofs in this section.

Proposition A.3. Fix a small category C, and an object c_{0} in C. Assume, for each c ∈ Ob(C), that Mor_{C}(c, c_{0}) ≠ ∅, and fix some distinguished “inclusion” morphism ι_{c} ∈ Mor_{C}(c, c_{0}) (where ι_{c_{0}} = Id_{c_{0}}). For each morphism ψ ∈ Mor_{C}(c, d), let θ(ψ) ∈ π_{1}(|C|, c_{0}) be the element represented by the loop ι_{d}·ψ·ι_{c}^{−1} (where paths are composed from right to left). Then the following hold:

(a) For any group Γ, and any function F: Mor(C) ⟶ Γ such that F (ι_{c}) = 1 for all c and F (ψ ∘φ) = F (ψ)F (φ) when ψ ∘φ is defined, there is a unique homomorphsm F: π_{1}(|C|, c_{0}) ⟶ Γ such that F (θ(ψ)) = F (ψ) for all ψ ∈ Mor(C). In other words, π_{1}(|C|, c_{0}) is the free group on the morphisms in C modulo the relations given by composition and the ι_{c}.

(b) Let F: C ⟶ Ab be any functor which sends all morphisms in C to isomorphisms of groups. Set A = F (c_{0}), and let F′ be the functor which sends each object to A, and where F′(ψ) = F (ι_{d}) ∘ F (ψ) ∘ F (ι_{c})^{−1} for ψ ∈ Mor_{C}(c, d). Then F′ is naturally isomorphic to F. Also,

\underleftarrow{lim}_{C}^{∗}(F) ≅ H^{∗}(|C|; A),

where the second group is cohomology twisted via the linear action of π_{1}(|C|) on A induced by F′.

Proof. (a) By elementary homotopy theory, any element of π_{1}(|C|, c_{0}) is represented by a loop which follows the edges of |C|, and hence is a product of loops of the form θ(φ) for φ ∈ Mor(C). Thus π_{1}(|C|) = ⟨Im(θ)⟩. This also proves the uniqueness of the homomorphism F, and it remains to prove its existence.

Let F̃: C ⟶ B(Γ) be the functor which sends all objects of C to the unique object of B(Γ), and which sends morphisms via F. Set

F = π_{1}(|F̃|): π_{1}(|C|, c_{0}) ⟶ Γ .

For each ψ ∈ Mor(C),

F (θ(ψ)) = F (ι_{d}·ψ·ι_{c}^{−1}) = F (ι_{d})·F (ψ)·F (ι_{c})^{−1} = F (ψ)

since F (ι_{c}) = F (ι_{d}) = 1 by assumption.

(b) The functor F′ is naturally isomorphic to F via the natural isomorphism which sends each c ∈ Ob(C) to ι_{c}. The map from Mor(C) to Aut(A) defined by F′ factors through π_{1}(|C|) by (a). The chain complex C^{∗}(C; F′) of Lemma A.1 is precisely the same as the chain complex C^{∗}(|C|; A) for cohomology with twisted coefficients, and thus

\underleftarrow{lim}_{C}^{∗}(F) ≅ \underleftarrow{lim}_{C}^{∗}(F′) ≅ H^{∗}(|C|; A).  □

The following proposition describes very general conditions for the nerve of a subcategory of C to have the homotopy type of a covering space of |C|.

Proposition A.4. Fix a small category C, a group Γ, and a surjective homomorphism F: π_{1}(|C|) ⟶ Γ. Set F = F ∘ θ: Mor(C) ⟶ Γ, where θ is as in Proposition A.3. For each H ≤ Γ, let C_{H} ⊆ C be the subcategory with the same objects, where for all ψ ∈ Mor(C), ψ ∈ Mor(C_{H}) if and only if F (ψ) ∈ H. Assume that

(1) for each c ∈ Ob(C) and each g ∈ Γ, there is an object d ∈ Ob(C) and an isomorphism ψ ∈ Iso_{C}(c, d) such that F (ψ) = g.

Then for each H ≤ Γ, |C_{H}| is homotopy equivalent to the covering space of |C| whose fundamental group is F^{−1}(H).

Proof. For any set X with left Γ-action, let E_{Γ}(X) be the category with object set X, and where

Mor_{E_{Γ}(X)}(x, y) = {g ∈ Γ | gx = y}.

Let C̃_{H} be the pullback category in the following square:

\xymatrix{C̃_{H} \ar[r] \ar[d] & E_{Γ}(Γ/H) \ar[d]^{λ} \\ C \ar[r]^{F̂} & B(Γ) .}

Here, F̂ and λ send all objects to the unique object of B(Γ); F̂ sends morphisms via F, and λ sends g ∈ Mor_{E_{Γ}(Γ/H)}(aH, bH) (if gaH = bH) to g ∈ Mor(B(Γ)). Thus Ob(C̃_{H}) = Ob(C) × (Γ/H), and Mor(C̃_{H}) is the set of pairs of morphisms in C and E_{Γ}(Γ/H) which get sent to the same morphism in B(Γ). Then |C̃_{H}| is a covering space of |C| by construction; and is the covering space with fundamental group F^{−1}(H) since π_{1}(|E_{Γ}(Γ/H)|) = H.

The subcategory C_{H} can be identified with the full subcategory of C̃_{H} whose objects are the pairs (c, 1H) for c ∈ Ob(C). Under this identification, each object in C̃_{H} is isomorphic to an object in C_{H} by (1), and thus |C_{H}| ≃ |C̃_{H}|. □

We now discuss extensions of categories. We want to look at the following very general situation, which occurs in several different contexts throughout the paper.

Definition A.5. Let τ: C̃ ⟶ C be a functor between categories which is a bijection on objects and surjective on morphism sets. For each c̃ ∈ Ob(C̃), set

K(c̃) =^{def} Ker[τ_{c̃,c̃}: AutC̃(c̃) ⟶ Aut_{C}(τ(c̃))].

(a) We say that τ is source regular (or a source regular extension of C by K) if for all c̃, d̃ ∈ Ob(C̃), K(c̃) acts freely on MorC̃(c̃, d̃) and τ_{c̃,d̃} is the orbit map for this action.

(b) We say that τ is target regular (or a target regular extension of C by K) if for all c̃, d̃ ∈ Ob(C̃), K(d̃) acts freely on MorC̃(c̃, d̃) and τ_{c̃,d̃} is the orbit map for this action.

The next three lemmas list some of the elementary properties of these classes of functors. For simplicity in notation throughout the rest of this section, when working with functors C̃ ⟶ C which are source or target regular, we identify each c ∈ Ob(C̃) with τ(c) ∈ Ob(C).

Lemma A.6. If τ: C̃ ⟶ C is either source or target regular, then a morphism in C̃ is an isomorphism if and only if its image in C is an isomorphism. In particular, if all endomorphisms in C are automorphisms, then the same is true of C̃.

Proof. Clearly, τ sends isomorphisms to isomorphisms, and we need only prove the converse. Assume ψ̃ ∈ MorC̃(c, d) is such that τ_{c,d}(ψ̃) is an isomorphism, and choose φ̃ ∈ MorC̃(d, c) such that τ(φ̃) = τ(ψ̃)^{−1}. Since τ_{c,c} is the orbit map for a free action of K(c), and τ_{c,c}(φ̃∘ψ̃) = τ_{c,c}(Id_{c}), φ̃∘ψ̃ ∈ K(c), and hence is an automorphism. Similarly, ψ̃ ∘ φ̃ ∈ K(d) ≤ AutC̃(d), and thus ψ̃ and φ̃ are both isomorphisms. □

Let Gps be the category of groups with homomorphisms. Let Gps_{∗} be the category with the same objects, and where

Mor_{Gps_{∗}}(G, H) = Rep(G, H) =^{def} Inn(H)\Hom(G, H).

Lemma A.7. The following hold for any source regular functor τ: C̃ ⟶ C.

(a) For any sequence c_{0} ⟶^{φ} c_{1} ⟶^{ψ} c_{2} of morphisms in C, and any pair of morphisms ψ̃ ∈ MorC̃(c_{1}, c_{2}) and ψ̃φ ∈ MorC̃(c_{0}, c_{2}) such that τ(ψ̃) = ψ and τ(ψ̃φ) = ψφ, there is a unique morphism φ̃ ∈ MorC̃(c_{0}, c_{1}) such that τ(φ̃) = φ and ψ̃ ∘ φ̃ = ψ̃φ.

(b) For each ψ̃ ∈ Mor(C̃), ψ̃ is a monomorphism in C̃ in the categorical sense (i.e., ψ̃ ∘ φ̃_{1} = ψ̃ ∘ φ̃_{2} implies φ̃_{1} = φ̃_{2}) if and only if τ(ψ̃) is a monomorphism in C.

(c) There is a functor K^{∗}: C^{op} ⟶ Gps_{∗}, defined on objects by setting

K^{∗}(c) = Ker[AutC̃(c) ⟶^{τ_{c,c}} Aut_{C}(c)]

for all c ∈ Ob(C̃). On morphisms, for f ∈ MorC̃(c, d), K^{∗}(τ(f)) = [ψ], where ψ ∈ Hom(K^{∗}(d), K^{∗}(c)) is the unique homomorphism such that the following square commutes in C̃ for all α ∈ K^{∗}(d):

\xymatrix{c \ar[r]^{f} \ar[d]_{ψ(α)} & d \ar[d]^{α} \\ c \ar[r]^{f} & d.}  (1)

If τ is also target regular, then K^{∗} sends all morphisms in C to isomorphisms.

Proof. (a) Let φ̂ ∈ Mor_{T} (c_{0}, c_{1}) be any lifting of φ. Then there is an element ξ ∈ K(c_{0}) such that ψ̃φ = ψ̃ ∘ φ̂ ∘ ξ, and we set φ̃ = φ̂ ∘ ξ. If φ̃′ is any other lifting which satisfies the same condition, then φ̃′ = φ̃ ∘ η for some η ∈ K(c_{0}), and η = 1 since ψ̃ ∘ φ̃ = ψ̃ ∘ φ̃′ and K(c_{0}) acts freely on Mor_{T} (c_{0}, c_{2}).

(b) Fix ψ̃ ∈ MorC̃(c, d), and set ψ = τ_{c,d}(ψ̃). Assume first that ψ̃ is a monomorphism, and let φ_{1}, φ_{2} ∈ Mor_{C}(c_{0}, c) be such that ψ ∘ φ_{1} = ψ ∘ φ_{2}. By (a), applied to any given lifting of ψ ∘ φ_{1}, there are morphisms φ̃_{1}, φ̃_{2} ∈ MorC̃(c_{0}, c) such that ψ̃ ∘ φ̃_{1} = ψ̃ ∘ φ̃_{2} and τ_{c_{0},c}(φ̃_{i}) = φ_{i}. Then φ̃_{1} = φ̃_{2} since ψ̃ is a monomorphism, so φ_{1} = φ_{2}, and ψ is a monomorphism.

Conversely, assume ψ is a monomorphism, and let φ̃_{1}, φ̃_{2} ∈ MorC̃(c_{0}, c) be such that ψ̃ ∘ φ̃_{1} = ψ̃ ∘ φ̃_{2}. Then τ_{c_{0},c}(φ̃_{1}) = τ_{c_{0},c}(φ̃_{2}) since ψ is a monomorphism, and hence φ̃_{1} = φ̃_{2} by the uniqueness of the lifting in (a). So ψ̃ is a monomorphism.

(c) By (a), for each f ∈ MorC̃(c, d) and each α ∈ K^{∗}(d), there is a unique ψ(α) ∈ K^{∗}(c) which makes square (1) commute. By juxtaposing such squares, one sees that the resulting function ψ: K^{∗}(d) ⟶ K^{∗}(c) is a homomorphism which depends only on f, and that this defines a functor K̃ from C̃ to Gps. For each c ∈ Ob(C), K̃ sends each morphism in K^{∗}(c) to an inner automorphism in K^{∗}(c) (by (1) again), and hence K̃ factors through a functor K^{∗} from C to Gps_{∗}.

If τ is also target regular, then each ψ(α) determines a unique α in the situation of (1), and so K^{∗} sends each morphism to an isomorphism. □

The following lemma is proven in exactly the same way as Lemma A.7.

Lemma A.8. The following hold for any target regular functor τ: C̃ ⟶ C.

(a) For any sequence c_{0} ⟶^{φ} c_{1} ⟶^{ψ} c_{2} of morphisms in C, and any pair of morphisms φ̃ ∈ MorC̃(c_{0}, c_{1}) and ψ̃φ ∈ MorC̃(c_{0}, c_{2}) such that τ(φ̃) = φ and τ(ψ̃φ) = ψφ, there is a unique morphism ψ̃ ∈ MorC̃(c_{1}, c_{2}) such that τ(ψ̃) = ψ and ψ̃ ∘ φ̃ = ψ̃φ.

(b) For each ψ̃ ∈ Mor(C̃), ψ̃ is an epimorphism in C̃ in the categorical sense (i.e., φ̃_{1} ∘ ψ̃ = φ̃_{2} ∘ ψ̃ implies φ̃_{1} = φ̃_{2}) if and only if τ(ψ̃) is an epimorphism in C.

(c) There is a functor K_{∗}: C ⟶ Gps_{∗}, defined on objects by setting

K_{∗}(c) = Ker[AutC̃(c) ⟶^{τ_{c,c}} Aut_{C}(c)]

for all c ∈ Ob(C̃). On morphisms, for f ∈ MorC̃(c, d), K_{∗}(τ(f)) = [ψ], where ψ ∈ Hom(K_{∗}(c), K_{∗}(d)) is the unique homomorphism such that the following square commutes in C̃ for all α ∈ K_{∗}(c):

\xymatrix{c \ar[r]^{f} \ar[d]_{α} & d \ar[d]^{ψ(α)} \\ c \ar[r]^{f} & d.}

The obstruction theory for describing source or target regular extensions of C by a given functor K from C to Gps_{∗} has been studied by Hoff [Hf]. For simplicity here, we handle only the case where K takes values in abelian groups. The nonabelian case is handled in [Hf] by using higher limits of functors with values in nonabelian groups (when they exist). Alternatively, if one assumes that K restricts to a functor ZK (where ZK(c) = Z(K(c))), then there is an obstruction theory which involves the groups \underleftarrow{lim}^{i}(Z(K)) for i = 2, 3.

Now fix K: C^{op} ⟶ Ab, and let C̃ ⟶^{τ} C be a source regular extension of C by K. Choose a section

s: Mor(C) ⟶ Mor(C̃).

For each pair of composable morphisms c_{0} ⟶^{φ} c_{1} ⟶^{ψ} c_{2} in C, let ω(φ, ψ) ∈ K(c_{0}) be the unique element such that

s(ψ) ∘ s(φ) = s(ψφ) ∘ ω(φ, ψ) ∈ MorC̃(c_{0}, c_{2}).

This defines an element ω ∈ C^{2}(C; K) (see Lemma A.1).

We next check that dω = 0. Fix a triple c_{0} ⟶^{φ} c_{1} ⟶^{ψ} c_{2} ⟶^{χ} c_{3} of composable morphisms. Associativity in C̃ gives the relation

s(χ) ∘ (s(ψ) ∘ s(φ)) = s(χψφ) ∘ ω(ψφ, χ) ∘ ω(φ, ψ)
= (s(χ) ∘ s(ψ)) ∘ s(φ) = s(χψφ) ∘ ω(φ, χψ) ∘ K(φ)(ω(ψ, χ)).

This gives the relation

ω(ψφ, χ) ∘ ω(φ, ψ) = ω(φ, χψ) ∘ K(φ)(ω(ψ, χ)) ∈ K(c_{0}).

This proves that dω = 0, and hence that ω ∈ Z^{2}(C; K).

It remains to show that the class [ω] ∈ \underleftarrow{lim}^{2}(K) is independent of the choice of section s. Let s′ be another section, and let β ∈ C^{1}(C; K) be such that s′(φ) = s(φ) ∘ β(φ) for all φ ∈ Mor_{C}(c_{0}, c_{1}). Let ω′ ∈ C^{2}(C; K) be the cochain analogous to the cochain ω already defined. Then for φ and ψ as above,

s′(ψ) ∘ s′(φ) = s′(ψφ) ∘ ω′(φ, ψ) = s(ψφ) ∘ β(ψφ) ∘ ω′(φ, ψ)
= s(ψ) ∘ β(ψ) ∘ s(φ) ∘ β(φ) = s(ψ) ∘ s(φ) ∘ K(φ)(β(ψ)) ∘ β(φ) .

Thus, in the abelian group K(c_{0}),

ω(φ, ψ)^{−1} ∘ ω′(φ, ψ) = β(ψφ)^{−1} ∘ K(φ)(β(ψ)) ∘ β(φ) = dβ(φ, ψ);

and this proves that ω′ = ω·dβ.

We have now shown how to assign a unique element of \underleftarrow{lim}^{2}(K) to each extension of C by K. It is easy to show, using similar manipulations, that all elements of \underleftarrow{lim}^{2}(K) can be realized in this way, and that extensions corresponding to the same element of \underleftarrow{lim}^{2}(K) are isomorphic. This can all be summarized in the following proposition:

Proposition A.9. Fix a small category C, and a functor K: C^{op} ⟶ Ab. Then there is a bijective correspondence between isomorphism classes of source regular extensions of C by K and the group \underleftarrow{lim}_{C}^{2}(K). □

We next look at geometric realizations of source and target regular functors. The following, very general result can be used in many situations to prove that certain maps between linking or transporter systems are fibrations with fiber BA, for a certain p-group A.

Proposition A.10. Fix a group G, and a functor τ: C̃ ⟶ C between small categories which is both source regular and target regular. Assume |C| is connected, fix c_{0} ∈ Ob(C̃),

and set G = Ker(τ_{c_{0},c_{0}}). Then the map |C̃| ⟶^{|τ|} |C| of geometric realizations is a fiber bundle with fiber BG.

Proof. We prove this by constructing a topological group, together with a principal bundle over that group of which |τ| is an associated bundle. For each c ∈ Ob(C) = Ob(C̃), set K(c) = Ker(τ_{c,c}) ◁ AutC̃(c).

Let G be the category of self equivalences of B(G), where morphisms in G are natural isomorphisms of functors. Thus

Ob(G) = Aut(G) and Mor_{G}(α, β) = {g ∈ G | β = c_{g} ∘ α}.

Let P be the category of functors B(G) ⟶ C̃ which go isomorphically to one of the “fibers” of τ; more precisely:

Ob(P) = {(c, λ) | c ∈ Ob(C), λ ∈ Iso(G, K(c))}
Mor_{P}((c, λ), (d, μ)) = {φ ∈ MorC̃(c, d) | μ = c_{φ} ∘ λ} .

Consider the following square of categories and functors:

\xymatrix{P × B(G) \ar[r]^{^{pr^{2}}} \ar[d]_{ev} & P \ar[d]^{κ} \\ C̃ \ar[r]^{τ} & C ;}  (2)

where κ is the forgetful functor κ(c, λ) = c, and ev is the “evaluation functor”

ev(g, (c, λ) ⟶^{φ} (d, μ)) = (c ⟶_{=μ(g)∘φ}^{φ∘λ(g)} d).

For each n ≥ 0, let G_{n} denote the set of n-simplices in the nerve of G, and similarly for the other categories. Then G_{n} is a group with multiplication

(α_{0} ⟶^{g_{1}} α_{1} ⟶^{g_{2}} ⋯) · (β_{0} ⟶^{h_{1}} β_{1} ⟶^{h_{2}} ⋯) = (α_{0} ∘ β_{0} ⟶^{g_{1}·α_{0}(h_{1})} α_{1} ∘ β_{1} ⟶^{g_{2}·α_{1}(h_{2})} ⋯),

and the face and degeneracy maps are all homorphisms. So |G| is a topological group. Furthermore, G_{n} has a free right action on P_{n} defined by

((c_{0}, λ_{0}) ⟶^{φ} (c_{1}, λ_{1})) ∘ (α_{0} ⟶^{g} α_{1}) = ((c_{0}, λ_{0} ∘ α_{0}) ⟶^{φ∘λ_{0}(g)} (c_{1}, λ_{1} ∘ α_{1}));

and a left action on B(G)_{n} defined by

(α_{0} ⟶^{g} α_{1})·(∗ ⟶^{h} ∗) = (∗ ⟶^{g·α_{0}(h)} ∗);

actions which again commute with face and degeneracy maps. Together with the functors κ and ev, these actions induce bijections

P_{n}/G_{n} ≅ C_{n} and P_{n} ×_{G_{n}} B(G)_{n} ≅ C̃_{n}

for all n. Hence the geometric realizations |ev| and |κ| are principal |G|-fibrations (see, e.g., [May, §§18–20] or [GJ, Corollary V.2.7]). In particular, ev induces a homeomorphism |C̃| ≅ |P| ×_{|G|} BG, and hence |τ| is a fiber bundle with fiber BG associated to

the principal bundle |P| ⟶^{|κ|} |C|. □

We now finish by describing how the Grothedieck spectral sequence applies to describe higher limits over a target regular extension.

Proposition A.11. Fix small categories C̃ and C, and let τ: C̃ ⟶ C be a target regular functor. For each c ∈ Ob(C), set

K(c) = Ker[AutC̃(c) ⟶^{τ} Aut_{C}(c)].

Let Φ: C̃^{op} ⟶ Ab be any functor. Then there is a spectral sequence

E_{2}^{ij} = \underleftarrow{lim}_{C}^{i}(H^{j}(K(−); Φ(−))) ⟹ \underleftarrow{lim} ; C̃^{i+j}(Φ).  (3)

Proof. This is shown using the Grothendieck spectral sequence (cf. [Wb, §5.8]). Consider the following triangle of categories and functors:

[figure: C̃-mod; R_{τ}; \underleftarrow{lim}; C-mod; \underleftarrow{lim}; _{Ab.}]  (4)

Here, R_{τ}: C̃-mod ⟶ C-mod is right Kan extension by τ. Since R_{τ} is right adjoint to an exact functor (composition with τ^{op}), it sends injectives to injectives.

Let R_{τ}^{i} be the i-th derived functor of R_{τ}. Thus for all Φ in C̃-mod and all c ∈ Ob(C),

R_{τ}^{j}(Φ)(c) = \underleftarrow{lim}^{j}_{(τ↓c)^{op}}(Φ ∘ κ_{c}^{op}),

where κ_{c}: τ↓c ⟶ C̃ is the forgetful functor. The Grothendieck spectral sequence for the triangle (4) takes the form

E_{2}^{ij} = \underleftarrow{lim}_{C}^{i}(R_{τ}^{j}(Φ)) ⟹ \underleftarrow{lim} ; C̃^{i+j}(Φ).  (5)

Now, τ↓c is the overcategory with objects the pairs (d, φ) for d ∈ Ob(C̃) and φ ∈ Mor_{C}(τ(d), c), and where

Mor_{τ↓c}((d, φ), (d′, ψ)) = {f ∈ MorC̃(d, d′) | ψ ∘ τ(f) = φ}.

Consider the functors

B(K(c)) ⟶^{S} τ↓c ⟶^{T} B(K(c)),

where S sends the unique object of B(K(c)) to (c, Id) and sends α ∈ K(c) to α ∈ Aut_{τ↓c}(c, Id); and where T is defined as follows. Fix an arbitrary map of sets

s: Ob(τ↓c) ⟶ Mor(C̃)

such that for all (d, φ), s(d, φ) ∈ MorC̃(d, c) and τ_{d,c}(s(d, φ)) = φ. For each morphism f in τ↓c from (d, φ) to (d′, ψ), there is (since τ is target regular) a unique element T (f) ∈ K(c) such that

T (f) ∘ s(d, φ) = s(d′, ψ) ∘ f.

This defines T. Furthermore, T ∘S = Id_{B(K(c))}, while s defines a natural transformation of functors from S ∘ T to Id_{τ↓c}.

It now follows that for all c and j,

R_{τ}^{j}(Φ)(c) = \underleftarrow{lim}^{j}_{(τ↓c)^{op}}(Φ ∘ κ_{c}^{op}) ≅ \underleftarrow{lim}^{j}_{B(K(c))}(Φ ∘ κ_{c} ∘ S) ≅ H^{j}(K(c); Φ(c)).

The spectral sequence (3) now follows from this together with (5). □

References

[BK] P. Bousfield & D. Kan, Homotopy limits, completions, and localizations, Lecture notes in math. 304, Springer-Verlag (1972)

[BCGLO1] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. (to appear)

[BCGLO2] C. Broto, N. Castellana, J. Grodal, R. Levi, & B. Oliver, Extensions of p-local finite groups, Trans. Amer. Math. Soc. (to appear)

[BLO1] C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed classifying spaces of finite groups, Invent. math. 151 (2003), 611–664

[BLO2] C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, Journal Amer. Math. Soc. 16 (2003), 779–856

[BLO3] C. Broto, R. Levi, & B. Oliver, Discrete models for the p-local homotopy theory of compact Lie groups, in preparation

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[JMO] S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-maps of BG via Gactions, Annals of Math. 135 (1992), 184–270

[May] P. May, Simplicial objects in algebraic topology, Van Nostrand (1967)

[O1] B. Oliver, Higher limits via Steinberg representations, Comm. in Algebra 22 (1994), 1381-1393 [O2] B. Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Camb. Phil.

Soc. 137 (2004), 321–347 [Pg] Ll. Puig, Unpublished notes (ca. 1990)

[Q] D. Quillen, Algebraic K-theory I, Lecture notes in mathematics 341 (1973), 77–139

[Rz] A. Ruiz, Exotic subsystems of finite index in the fusion systems of general linear groups over finite fields, preprint

[St] R. Stancu, Equivalent definitions of fusion systems, preprint

[Wb] C. Weibel, An introduction to homological algebra, Cambridge Univ. Press (1994)

LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

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

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