AN ALGEBRAIC MODEL FOR FINITE LOOP SPACES

CARLES BROTO, RAN LEVI, AND BOB OLIVER

Contents

1. Background on fusion and linking systems over discrete p-toral groups 3 2. Normalizer fusion subsystems 6 3. The “bullet" construction 10 4. Centric and radical subgroups determine saturation 13 5. Extensions of p-local compact groups 19 6. The category Aut_{typ}(L) 29 7. The proof of Theorem A 36 Appendix A. Transporter systems over discrete p-toral groups 38 References 47

The theory of p-local compact groups, developed in [BLO3], is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the earlier theory of p-local finite groups.

Compact Lie groups, and the p-compact groups of Dwyer and Wilkerson [DW], are examples of loop spaces which are finite CW complexes, at least homologically. In this paper, we show that in fact, every homologically finite loop space X gives rise to a p-local compact group, and thus enjoys all of the important properties shared by these spaces (see [BLO3] for details).

The key idea is very simple. If X is a finite loop space, then its group of components is finite, and its identity component is a connected finite loop space. As such, after pcompletion in the sense of Bousfield and Kan [BK], the identity component is a p-compact group. This leads us to investigate a much more general question: whether a space B with a finite regular covering E → B, such that E is the classifying space of a p-local compact group, is itself the classifying space of a p-local compact group.

Before stating our main theorem, we briefly describe the objects of study. Let ℤ/p^{∞} denote the union of all ℤ/p^{n} under the obvious inclusions. A discrete p-toral group is a group S containing a normal subgroup of the form (ℤ/p^{∞})^{r} (r ≥ 0) with p-power index. A (saturated) fusion system over S is a category whose objects are the subgroups of S, and whose morphisms are monomorphisms of groups which are required to satisfy certain axioms. We refer the reader to Section 1 (especially Definition 1.4 and Corollary 1.8) for details.

Date: 10 December 2012.

2000 Mathematics Subject Classification. Primary 55R35. Secondary 20D20, 20E22.

Key words and phrases. Finite loop spaces, Classifying spaces, p-local compact groups, fusion.

C. Broto is partially supported by FEDER-MICINN grant MTM 2010-20692.

R. Levi is partially supported by EPSRC grant EP/J014524/1.

B. Oliver is partially supported by UMR 7539 of the CNRS, and by project ANR BLAN08-2_338236, HGRT.

Given a saturated fusion system F over S, a centric linking system associated to F is a category L whose objects are those subgroups of S which are F-centric (Definition 1.5), and whose morphism sets are, in an appropriate sense, extensions of the corresponding morphism sets between the same objects in F. This extra structure allows us to associate a “classifying space” to the fusion system in question. Thus a p-local compact group is a triple (S, F, L), where S is a discrete p-toral group, F is a saturated fusion system over S, and L is a centric linking system associated to F. The classifying space of a p-local compact group is the p-completion of the geometric realisation of its linking system.

Our main theorem is the following.

Theorem A. Assume that f: X ⟶ Y is a finite regular covering space with covering group G, where X is the classifying space of a p-local compact group (S_{∗}, F_{∗}, L_{∗}). Then Y _{p}^{∧} is the classifying space of a p-local compact group (S, F, L).

Let p be a prime. A space Y is p-good if its p-completion Y _{p}^{∧} is p-complete [BK]. A space Y is 𝔽_{p}-finite if H_{∗}(Y, 𝔽_{p}) is a finite dimensional 𝔽_{p}-vector space. In [DW], Dwyer and Wilkerson define a p-compact group to be an 𝔽_{p}-finite loop space X whose classifying space BX is p-complete. In [BLO3], we show that every p-compact group X gives rise to a p-local compact group (S, F, L) whose classifying space has the homotopy type of BX.

Theorem B. Let X be any path connected space. Then for each prime p such that ΩX is 𝔽_{p}-finite, the space X_{p}^{∧} has the homotopy type of the classifying space of a p-local compact group. In particular, this holds for each prime p if ΩX has the homotopy type of a finite complex.

Proof. Fix a prime p such that ΩX is 𝔽_{p}-finite. In particular, H_{0}(ΩX, 𝔽_{p}) is finite, so π_{1}(X) is a finite group, and hence X is p-good by [BK, § VII.5]. Set π = π_{1}(X), let X̃ be the universal cover of X, and consider the following fibration:

X̃ → X → Bπ.

Then ΩX̃ is a connected component of ΩX, and hence is also 𝔽_{p}-finite. Applying fibrewise p-completion, we obtain a fibration

X̃_{p}^{∧} → X → Bπ,

where X̃_{p}^{∧} is p-complete and ΩX̃_{p}^{∧} is 𝔽_{p}-finite. Thus X̃_{p}^{∧} is the classifying space of a pcompact group. By [BLO3, Theorem 10.7], X̃_{p}^{∧} is the classifying space of a p-local compact group. So by Theorem A, X_{p}^{∧} ≃ X_{p}^{∧} is the classifying space of a p-local compact group.

In particular, if X is a path connected space such that ΩX has the homotopy type of a finite CW complex, then this holds for X at all primes. □

The paper is organized as follows. We start in Section 1 with the basic definitions and general background on p-local compact groups. In Sections 2–4, certain constructions and results which are already known for p-local finite groups are generalized to the p-local compact case. In Section 5, we describe how to construct extensions of p-local compact groups algebraically, and in Section 6, we study a category Aut_{typ}(L) which allows us to translate those results to a topological setting. Finally, in Section 7, we prove Theorem A. We end the paper with an appendix where we collect necessary results on transporter systems over discrete p-toral groups.

Notation: When G and H are groups, H < G always means that H is a proper subgroup of G. Whenever F: C ⟶ D is a functor and c, c′ ∈ Ob(C), we write F_{c,c′} for the restriction of F to Mor_{C}(c, c′), and write F_{c} = F_{c,c}.

The authors wish to acknowledge support from the Universitat Autònoma de Barcelona, the Institute of Mathematics in Aberdeen and the University of Copenhagen for allowing the authors to meet and providing support at various times while this project was developing.

1. Background on fusion and linking systems over discrete p-toral groups

In this section, we collect the definitions and some basic facts on p-local compact groups which will be useful throughout the paper. We refer to [BLO3] for more details on many of the results described here.

Definition 1.1. A discrete p-torus is a group which is isomorphic to (ℤ/p^{∞})^{n} for some finite n. A discrete p-toral group is a group P with a normal subgroup P_{0} ⊴ P such that P_{0} is a discrete p-torus and P/P_{0} is a finite p-group. The subgroup P_{0} will be called the identity component of P, and P will be called connected if P = P_{0}. Set π_{0}(P) = P/P_{0}, and set rk(P) = n (the rank of P) if P_{0} ≅ (ℤ/p^{∞})^{n}. The order of P is the pair |P | =^{def} (rk(P), |π_{0}(P)|), regarded as an element of ℕ^{2} ordered lexicographically.

Let 𝔖𝔲𝔟(S) be the set of all subgroups of a (discrete p-toral) group S. For any group G and any H, K ≤ G, Hom_{G}(H, K) ⊆ Hom(H, K) denotes the set of homomorphisms induced by conjugation in G.

Definition 1.2. A fusion system F over a discrete p-toral group S is a category with Ob(F) = 𝔖𝔲𝔟(S), whose morphism sets Hom_{F}(P, Q) satisfy the following conditions:

(a) Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q) for all P, Q ∈ 𝔖𝔲𝔟(S).

(b) Every morphism in F factors as an isomorphism in F followed by an inclusion.

With motivation from group theory, we make the following definition.

Definition 1.3. Let F be a fusion system over a discrete p-toral group S. Two subgroups P, Q ∈ 𝔖𝔲𝔟(S) are F-conjugate if they are isomorphic as objects of the category F. Let P^{F} denote the set of all subgroups of S which are F-conjugate to P.

We are now ready to recall the definition of saturation of a fusion system.

Definition 1.4. Let F be a fusion system over a discrete p-toral group S.

• A subgroup P ≤ S is fully centralized in F if |C_{S}(P)| ≥ |C_{S}(Q)| for all Q ∈ P^{F}.

• A subgroup P ≤ S is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(Q)| for all Q ∈ P^{F}.

• F is a saturated fusion system if the following three conditions hold:

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

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

set

N_{φ} = {g ∈ N_{S}(P) |^{φ}c_{g} ∈ Aut_{S}(φ(P))},

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

(III) If P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ is an increasing sequence of subgroups of S, with P_{∞} = ⋃_{n=1}^{∞} P_{n}, and if φ ∈ Hom(P_{∞}, S) is any homomorphism such that φ|_{P_{n}} ∈ Hom_{F}(P_{n}, S) for all n, then φ ∈ Hom_{F}(P_{∞}, S).

We next define certain classes of subgroups which play an important role in generating the morphisms in a fusion system.

Definition 1.5. Let F be a fusion system over a discrete p-toral group S. A subgroup P ∈ 𝔖𝔲𝔟(S) is F-centric if C_{S}(Q) = Z(Q) for all Q ∈ P^{F}. A subgroup P ∈ 𝔖𝔲𝔟(S) is F-radical if Out_{F}(P) =^{def} Aut_{F}(P)/Inn(P) is p-reduced; i.e., contains no nontrivial normal p-subgroup. Let F^{c} denote the full subcategory of F whose objects are the F-centric subgroups of S.

In this paper it will be convenient to also use a different but equivalent definition of saturation, based on that due to Roberts and Shpectorov [RS] in the finite case. We recall their definitions.

Definition 1.6. Let F be a fusion system over a p-group S.

• A subgroup P ≤ S is fully automized in F if the index of Aut_{S}(P) in Aut_{F}(P) is finite and prime to p.

• A subgroup P ≤ S is receptive in F if it has the following property: for each Q ≤ S and each φ ∈ Iso_{F}(Q, P), if we set N_{φ} = {g ∈ N_{S}(Q) | ^{φ}c_{g} ∈ Aut_{S}(P)}, then there is φ̄ ∈ Hom_{F}(N_{φ}, S) such that φ̄|_{P} = φ.

In this terminology, axioms (I) and (II) in Definition 1.4 say that each fully normalized subgroup is fully centralized and fully automized, and each fully centralized subgroup is receptive.

By definition, if P is fully automized in F, then Aut_{F}(P) is an extension of a discrete p-torus by a finite group. Hence it does contain maximal discrete p-toral subgroups, unique up to conjugation, which we regard as its Sylow p-subgroups. As usual, we let Syl_{p}(Aut_{F}(P)) denote the set of its Sylow p-subgroups.

The next lemma describes the relation between these concepts and those already defined.

Lemma 1.7. The following hold for any fusion system F over a discrete p-toral group S.

(a) Every receptive subgroup of S is fully centralized.

(b) If P ≤ S is fully automized and receptive in F, then it is fully normalized.

(c) If P ≤ S is fully automized and receptive in F, and Q ∈ P^{F}, then there is a morphism φ ∈ Hom_{F}(N_{S}(Q), N_{S}(P)) such that φ(Q) = P.

Proof. The proofs are identical to those given in [RS] and [AKO, Lemma I.2.6] in the finite case. A generalization of these statements will be proven in Lemma 2.2 below. □

The following is an immediate consequence of Lemma 1.7.

Corollary 1.8. A fusion system F over a discrete p-toral group S is saturated if and only if

• each subgroup of S is F-conjugate to one which is fully automized and receptive in F; and

• axiom (III) holds for F: if P_{1} < P_{2} < ⋯ are subgroups of S, P = ⋃_{i=1}^{∞} P_{i}, and φ ∈ Hom(P, S) is such that φ|_{P_{i}} ∈ Hom_{F}(P_{i}, S) for each i, then φ ∈ Hom_{F}(P, S).

When G is a finite group and H is a set of subgroups of G, T_{H}(G) denotes the H-transporter category for G: Ob(T_{H}(G)) = H, and for H, K ∈ H,

Mor_{T_{H}(G)}(H, K) = {g ∈ G |^{g}H ≤ K} .

Definition 1.9. Let F be a fusion system over a discrete p-toral group S. Let H be a set of subgroups of S which is closed under F-conjugacy and overgroups, and includes all subgroups which are F-centric and F-radical. An H-linking system associated to F is a category L with object set H, together with a pair of functors

T_{H}(S) ⟶^{δ} L ⟶^{π} F ,

such that each object is isomorphic (in L) to one which is fully centralized in F, and such that the following conditions are satisfied:

(A) The functor δ is the identity on objects, and π is the inclusion on objects. For each P, Q ∈ H such that P is fully centralized in F, C_{S}(P) acts freely on Mor_{L}(P, Q) via δ_{P} and right composition, and π_{P,Q} induces a bijection

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

(B) For each P, Q ∈ H and each g ∈ N_{S}(P, Q), π_{P,Q} sends δ_{P,Q}(g) ∈ Mor_{L}(P, Q) to c_{g} ∈ Hom_{F}(P, Q).

(C) For all ψ ∈ Mor_{L}(P, Q) and all g ∈ P, ψ ∘ δ_{P} (g) = δ_{Q}(π(ψ)(g)) ∘ ψ.

A centric linking system is an Ob(F^{c})-linking system; i.e., a linking system whose objects are the F-centric subgroups of S.

When P ≤ Q, we set ι_{P,Q} = δ_{P,Q}(1). The morphisms ι_{P,Q} are regarded as the inclusions in L.

Definition 1.10. A linking triple is a triple of the form (S, F, L), where S is a discrete p-toral group, F is a saturated fusion system over S, and L is an H-linking system for some family H. A p-local compact group is a linking triple where L is a centric linking system.

Definition 1.11. Let F be a fusion system over a discrete p-toral group S, and let H ⊆ 𝔖𝔲𝔟(S) be a family of subgroups. Then H is closed in 𝔖𝔲𝔟(S) if for each increasing sequence P_{1} < P_{2} < P_{3} < ⋯ of subgroups in H, ⋃_{i=1}^{∞} P_{i} is also in H.

Let H ⊆ 𝔖𝔲𝔟(S) be a closed family. Then

(a) F is H-closed if for each sequence P_{1} < P_{2} < ⋯ in H with P = ⋃_{i=1}^{∞} P_{i}, and each homomorphism φ ∈ Hom(P, S) such that φ|_{P_{i}} ∈ Hom_{F}(P_{i}, S) for each i, φ ∈ Hom_{F}(P, S).

(b) F is H-generated if every morphism in F is a composite of restrictions of morphisms in F between subgroups in H.

(c) F is H-saturated if it is H-closed and if every subgroup of H is F-conjugate to a subgroup which is fully automized and receptive.

The following two results, both generalizations to discrete p-toral groups of well known properties of p-groups, will be useful.

Lemma 1.12 ([BLO3, Lemma 1.8]). If P < Q are discrete p-toral groups, then P < N_{Q}(P).

Lemma 1.13. Let Q ⊴ P be discrete p-toral groups, where |P/Q| < ∞. Then the group

{α ∈ Aut(P) | α|_{Q} = Id, [α, P] ≤ Q}

is discrete p-toral.

Proof. Let A denote this group of automorphisms. For each α ∈ A and each g ∈ P, α(g) = gχ(g) for some χ(g) ∈ Q since [α, P] ≤ Q, c_{g} = c_{α(g)} ∈ Aut(Q) since α|_{Q} = Id, and hence χ(g) ∈ Z(Q). Also, for a ∈ Q, α(ga) = gχ(g)a = (ga)χ(g) since α(a) = a and [a, χ(g)] = 1, so χ(ga) = χ(g). Thus α is determined by the map χ: P/Q ⟶ Z(Q). The resulting injection A ⟶ Map(P/Q, Z(Q)) is a homomorphism, and so A is discrete p-toral since Map(P/Q, Z(Q)) ≅ Z(Q)^{|P/Q|} is. □

2. Normalizer fusion subsystems

Let F be a fusion system over a discrete p-toral group S. For each Q ≤ S and each K ≤ Aut(Q), define

Aut_{F}^{K}(Q) = K ∩ Aut_{F}(Q)
Aut_{S}^{K}(Q) = K ∩ Aut_{S}(Q)
N_{S}^{K}(Q) = { x ∈ N_{S}(Q) | c_{x} ∈ K } (the K-normalizer of Q in S).

Definition 2.1. Let F be a fusion system over a discrete p-toral group S. Fix a subgroup Q ≤ S and a group of automorphisms K ≤ Aut(Q).

• Q is fully K-automized in F if Aut_{S}^{K}(Q) ∈ Syl_{p}(Aut_{F}^{K}(Q)).

• Q is fully K-normalized in F if for each φ ∈ Hom_{F}(Q, S),

|N_{S}^{K}(Q)| ≥ |N_{S}^{^{φ}K}(φ(Q))| ,

where ^{φ}K = {φαφ^{−1} | α ∈ K } ≤ Aut(φ(Q)).

• N_{F}^{K}(Q) ⊆ F is the fusion system over N_{S}^{K}(Q) where for P, R ≤ N_{S}^{K}(Q),

Hom_{N_{F}^{K}(Q)}(P, R) = {φ ∈ Hom_{F}(P, R) | ∃ φ ∈ Hom_{F}(P Q, RQ)

with φ|_{P} = φ, φ(Q) = Q, and φ|_{Q} ∈ K}.

As special cases of the above definition, set N_{F}(Q) = N_{F}^{Aut(Q)}(Q) and C_{F}(Q) = N_{F}^{{1}}(Q): the normalizer and centralizer fusion systems, respectively, of Q.

The next lemma is a generalization of results in [RS] to fusion systems over discrete p-toral groups.

Lemma 2.2. Let F be a fusion system over a discrete p-toral group S, and let P be an F-conjugacy class of subgroups of S. Assume either that F is saturated, or (more generally) that P contains a subgroup which is fully automized and receptive in F. Then the following hold for each P ∈ P and each K ≤ Aut(P).

(a) The subgroup P is fully centralized if and only if it is receptive.

(b) The subgroup P is fully K-normalized if and only if it is fully K-automized and receptive. In this case, for each Q ∈ P and each φ ∈ Iso_{F}(Q, P), there are χ ∈ Aut_{F}^{K}(P) and φ ∈ Hom_{F}(N_{S}^{K^{φ}}(Q)·Q, N_{S}^{K}(P)·P) such that φ|_{Q} = χ ∘ φ.

Proof. (a) For fusion systems over finite p-groups, this is shown in [RS, Propositions 3.7 & 4.6] and in [AKO, Lemma I.2.6(c)]. Those proofs carry over unchanged to the discrete p-toral case.

(b) Assume P is fully K-automized and receptive in F. Fix Q ∈ P and φ ∈ Iso_{F}(Q, P). Then^{φ}Aut_{S}^{K^{φ}}(Q) is a discrete p-toral subgroup of Aut_{F}^{K}(P), and since P is fully K-automized, there is χ ∈ Aut_{F}^{K}(P) such that ^{χφ}Aut_{S}^{K^{φ}}(Q) ≤ Aut_{S}^{K}(P). Note that K^{φ} = K^{χφ} since χ ∈ K.

Set ψ = χ ∘ φ. Then ^{ψ}Aut_{S}^{K^{ψ}}(Q) ≤ Aut_{S}^{K}(P), and since P is receptive, there is a morphism ψ ∈ Hom_{F}(N_{S}^{K^{ψ}}(Q)·Q, N_{S}^{K}(P)·P) which extends ψ. In particular, |N_{S}^{K^{φ}}(Q)| = |N_{S}^{K^{ψ}}(Q)| ≤ |N_{S}^{K}(P)|. Since Q and φ were arbitrary, this proves that P is fully K-normalized in F.

Now assume P is fully K-normalized. Fix R ∈ P and ψ ∈ Iso_{F}(P, R) such that R is fully automized and receptive in F. Fix T ∈ Syl_{p}(Aut_{F}^{^{ψ}K}(R)) such that T ≥ ^{ψ}Aut_{S}^{K}(P). Since R is fully automized, there is α ∈ Aut_{F}(R) such that ^{α}T ≤ Aut_{S}(R). Set φ = α ∘ ψ. Then ^{φ}Aut_{S}^{K}(P) ≤ ^{α}T ≤ Aut_{S}(R), and since R is receptive, φ extends to φ ∈ Hom_{F}(N_{S}^{K}(P)·P, N_{S}(R)), where Im(φ) ≤ N_{S}^{^{φ}K}(R).

Consider the following commutative diagram

\xymatrix{1 \ar[r] & C_{S}(P) \ar[r] \ar[d] & N_{S}^{K}(P) \ar[r] \ar[d]_{φ} & Aut_{S}^{K}(P) \ar[d]^{c_{φ}} \ar[r] & 1 \\ 1 \ar[r] & C_{S}(R) \ar[r] & N_{S}^{^{φ}K}(R) \ar[r] & Aut_{S}^{^{φ}K}(R) \ar[r] & 1}

where the rows are exact and all vertical maps are monomorphisms. Since P is fully K-normalized, φ(N_{S}^{K}(P)) = N_{S}^{^{φ}K}(R), and hence φ(C_{S}(P)) = C_{S}(R) and ^{φ}Aut_{S}^{K}(P) = Aut_{S}^{^{φ}K}(R). Thus P is receptive by (a) (and since R is receptive).

Now, T ∈ Syl_{p}(Aut_{F}^{^{ψ}K}(R)) by assumption, so ^{α}T ∈ Syl_{p}(Aut_{F}^{^{φ}K}(R)). Also, ^{α}T ≤ Aut_{S}(R), and hence ^{α}T = Aut_{S}^{^{φ}K}(R). Thus Aut_{S}^{K}(P) ∈ Syl_{p}(Aut_{F}^{K}(P)), so P is fully K-automized, and this finishes the proof of (b). □

The main result in this section is that normalizer fusion subsystems over discrete p-toral groups are saturated. The proof given here is modelled on that of [AKO, Theorem I.5.5].

Theorem 2.3. Fix a saturated fusion system F over a discrete p-toral group S. Assume Q ≤ S and K ≤ Aut(Q) are such that Q is fully K-normalized in F. Then N_{F}^{K}(Q) is a saturated fusion system over N_{S}^{K}(Q).

Proof. Set S_{+} = N_{S}^{K}(Q) and F_{+} = N_{F}^{K}(Q) for short. For each P ≤ S_{+}, set

K_{P} = {α ∈ Aut(P Q) | α(P) = P, α(Q) = Q, α|_{Q} ∈ K}.

We need to show the following statements.

(a) Each subgroup of S_{+} is F_{+}-conjugate to a subgroup P such that P Q is fully K_{P} - normalized in F.

(b) If P ≤ S_{+} and P Q is fully K_{P} -normalized in F, then P is fully automized in F_{+}.

(c) If P ≤ S_{+} and P Q is fully K_{P} -normalized in F, then P is receptive in F_{+}.

(d) Axiom (III) holds for N_{F}^{K}(Q).

The theorem will then follow immediately from Corollary 1.8.

The proofs of (a), (b), and (c) are identical to the corresponding proofs in [AKO, Theorem I.5.5] (which is stated for fusion systems over finite p-groups). It remains to prove (d).

Fix subgroups P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ ≤ S_{+}, and set P = ⋃_{i=1}^{∞} P_{i}. Assume φ ∈ Hom(P, S_{+}) is such that φ_{i} =^{def} φ|_{P_{i}} ∈ Hom_{F_{+}}(P_{i}, S_{+}) for each i.

For each i, set

X_{i} = {ψ ∈ Hom_{F}(P_{i}Q, S) | ψ|_{P_{i}} = φ_{i}, ψ(Q) = Q, ψ|_{Q} ∈ K} ≠ ∅ ,

and let X_{i} be the image of X_{i} in Rep_{F}(P_{i}Q, S) (equivalently, the set of C_{S}(P_{i})-conjugacy classes of morphisms in X_{i}). Since Rep_{F}(P_{i}Q, S) is finite by [BLO3, Lemma 2.5], so is X_{i}. There are natural restriction maps X_{i} ⟶ X_{i−1}, and since the sets are finite and nonempty, the inverse limit is nonempty. Fix an element ([ψ_{i}])_{i=1}^{∞} in the inverse limit. Thus ψ_{i} ∈ Hom_{F}(P_{i}Q, S), ψ_{i}|_{P_{i}} = φ_{i}, ψ_{i}|_{Q} ∈ K, and c_{g_{i−1}} ∘ ψ_{i}|_{P_{i−1}Q} = ψ_{i−1} for some g_{i−1} ∈ C_{S}(P_{i−1}).

Since S is artinian (cf. [BLO3, Proposition 1.2]), there is N such that C_{S}(P_{i}) = C_{S}(P) for each i ≥ N. For each i ≥ N, set ψ′_{i} = c_{g_{N}} ∘ c_{g_{N+1}} ∘ ⋯ ∘ c_{g_{i−1}} ∘ ψ_{i}. Then ψ′_{i}|_{P_{i−1}} = ψ′_{i−1}, and ψ′_{i}|_{P_{i}} = φ_{i}. Set ψ′ = ⋃_{i=1}^{∞} ψ′_{i}. Then ψ′ ∈ Hom_{F}(P Q, S) by axiom (III) for F, ψ′|_{P} = φ, ψ′|_{Q} = ψ_{N}|_{Q} ∈ K, and so φ ∈ Hom_{F_{+}}(P, S). □

The following is one easy application of Theorem 2.3.

Lemma 2.4. Let F be a saturated fusion system over a discrete p-toral group S. Assume Q ⊴ P ≤ S, where Q is F-centric. Let φ, φ′ ∈ Hom_{F}(P, S) be such that φ|_{Q} = φ′|_{Q}. Then there is x ∈ Z(Q) such that φ′ = φ ∘ c_{x}.

Proof. Since φ ∘ c_{g} = c_{φ(g)} ∘ φ for each g ∈ Q, it suffices to show that φ′ = c_{y} ∘ φ for some y ∈ Z(φ(Q)). Upon replacing P by φ′(P), Q by φ(Q) = φ′(Q), and φ by φ ∘ (φ′)^{−1}, we can assume that φ′ = incl_{P}^{S} and φ|_{Q} = Id_{Q}. We must show that φ = c_{x} for some x ∈ Z(Q).

Set K = Aut_{P} (Q). Since Q is F-centric, it is fully centralized. Since Aut_{F}^{K}(Q) = Aut_{S}^{K}(Q) = K, Q is fully K-automized, and hence fully K-normalized by Lemma 2.2(a,b). Hence by Theorem 2.3, the normalizer subsystem N_{F}^{K}(Q) over N_{S}^{K}(Q) = P ·C_{S}(Q) = P is saturated. Also, since φ|_{Q} = Id, Aut_{φ(P)}(Q) = Aut_{P} (Q) = K. Thus φ(P) ≤ N_{S}^{K}(Q), and φ ∈ Mor(N_{F}^{K}(Q)). Set F_{0} = N_{F}^{K}(Q) for short.

It thus suffices to prove that F_{0} = F_{P} (P). Assume otherwise: then by [BLO3, Theorem 3.6], there is R ≤ P such that O_{p}(Aut_{F_{0}}(R)) = 1, and in particular, such that Aut_{F_{0}}(R) is not a p-group. By definition of F_{0}, we can assume that R ≥ Q. Hence there is Id ≠ α ∈ Aut_{F_{0}}(R) which has finite order prime to p. Since K is a discrete p-toral group and α|_{Q} ∈ K, α|_{Q} = Id_{Q}. Hence for g ∈ R, g and α(g) have the same conjugation action on Q, and g^{−1}α(g) ∈ C_{R}(Q) ≤ Q. Thus α induces the identity on R/Q, so by [BLO3, Lemma 1.7(a)], each α-orbit in R has p-power order. This contradicts the assumption that α ≠ Id has order prime to p, so F_{0} = F_{P} (P), and this finishes the proof. □

We will need the following application of Lemma 2.4.

Lemma 2.5. Let F be a saturated fusion system over a discrete p-toral group S. Then for each P ≤ S, the set of S-conjugacy classes of fully normalized subgroups F-conjugate to P is finite of order prime to p.

Proof. Since the conclusion depends only on the F-conjugacy class of P, we can assume that P is fully normalized.

Let Rep_{F}(P, S)_{fc} ⊆ Rep_{F}(P, S) be the subset consisting of all classes of homomorphisms whose image is fully centralized. Recall that Rep_{F}(P, S) is finite by [BLO3, Lemma 2.5].

Step 1: Assume first that P is F-centric. We first prove that Rep_{F}(P, S)_{fc} = Rep_{F}(P, S) has order prime to p in this case.

Recall from [BLO3, § 3] that there is a functor (P ↦ P^{•}) from F to itself, with the properties that P^{•} ≥ P for each P ≤ S, (P^{•})^{•} = P^{•}, and the image F^{•} of the functor contains finitely many S-conjugacy classes. (We will be studying this functor in more detail in the next section.)

Let P = P_{0} < P_{1} < P_{2} < … < P_{m} = S be such that P_{i} = N_{S}(P_{i−1})^{•} for each i. The sequence is finite since there are only finitely many conjugacy classes in F^{•} (and since N_{S}(P_{i}) > P_{i} whenever P_{i} < S by Lemma 1.12). Fix 0 ≤ i < m, and consider the restriction map

Res: Rep_{F}(N_{S}(P_{i}), S) ⟶ Rep_{F}(P_{i}, S).

This is injective by Lemma 2.4, and Out_{S}(P_{i}) ≅ N_{S}(P_{i})/P_{i} ≠ 1 is a finite p-group by [BLO3, Proposition 1.5(c)]. Let Q/P_{i} be the stabilizer subgroup of [φ] ∈ Rep_{F}(P_{i}, S) under the action of N_{S}(P_{i})/P_{i}; then φ extends to an F-morphism on Q by axiom (II) in Definition 1.4. Thus Im(Res) is the fixed subset for the N_{S}(P_{i})/P_{i}-action, and hence |Rep_{F}(P_{i}, S)| ≡ |Rep_{F}(N_{S}(P_{i}), S)| (mod p).

By [BLO3, Proposition 3.3], each ψ ∈ Hom_{F}(N_{S}(P_{i}), S) extends to a unique morphism ψ^{•} ∈ Hom_{F}(N_{S}(P_{i})^{•}, S). Thus restriction defines a bijection from Hom_{F}(N_{S}(P_{i})^{•}, S) to Hom_{F}(N_{S}(P_{i}), S), and hence (after dividing out by the conjugation action of S) from Rep_{F}(N_{S}(P_{i})^{•}, S) to Rep_{F}(N_{S}(P_{i}), S). In other words, |Rep_{F}(P_{i}, S)| ≡ |Rep_{F}(P_{i+1}, S)| (mod p). Thus

|Rep_{F}(P, S)| = |Rep_{F}(P_{0}, S)| ≡ |Rep_{F}(P_{m}, S)| = |Out_{F}(S)|,

where |Out_{F}(S)| is prime to p by axiom (I) in Definition 1.4.

Step 2: Now assume P < S is not F-centric; we claim that |Rep_{F}(P, S)_{fc}| is prime to p. Set

Γ = {γ ∈ Aut_{F}(P C_{S}(P)) | γ|_{P} = Id_{P} } and T = Aut_{C_{S}(P)}(P C_{S}(P)) ⊴ Γ .

Then Γ ≅ Aut_{C_{F} (P)}(C_{S}(P)) and T ≅ Inn(C_{S}(P)) via restriction to C_{S}(P), so T ∈ Syl_{p}(Γ).

Now Γ/T ≤ Out_{F}(P C_{S}(P)) acts on Rep_{F}(P C_{S}(P), S) by right composition. If φ ∈ Hom_{F}(P C_{S}(P), S) and γ ∈ Γ are such that [φ] = [φ ∘ γ] in Rep_{F}(P C_{S}(P), S), then there is g ∈ S such that φ = c_{g} ∘ φ ∘ γ, so φ = φ ∘ γ^{p^{n}} for some n, hence |γ| is a power of p, and γ ∈ T. Thus the Γ/T -action on Rep_{F}(P C_{S}(P), S) is free. The restriction map from Rep_{F}(P C_{S}(P), S) to Rep_{F}(P, S)_{fc} is surjective by axiom (II), and the inverse image of any [ψ] ∈ Rep_{F}(P, S)_{fc} is one of the Γ/T -orbits. Thus |Rep_{F}(P, S)_{fc}| = |Rep_{F}(P C_{S}(P), S)|/|Γ/T |. Since |Rep_{F}(P C_{S}(P), S)| is prime to p by Step 1, |Rep_{F}(P, S)_{fc}| is also prime to p.

Step 3: We are now ready to prove the lemma. Let P_{fc} be the set of S-conjugacy classes of subgroups fully centralized in F and F-conjugate to P, and let P_{fn} ⊆ P_{fc} be the subset of classes of fully normalized subgroups. Let ρ: Rep_{F}(P, S)_{fc} ⟶ P_{fc} be the map which sends the class of φ to the class of φ(P).

Let N||Out_{F}(P)| be the largest divisor prime to p. For each Q ∈ P^{F} such that [Q] ∈ P_{fc}, |ρ^{−1}([Q])| = |Aut_{F}(Q)|/|Aut_{S}(Q)|. Thus p||ρ^{−1}(Q)| if Q is not fully normalized, and |ρ^{−1}(Q)| = N if Q is fully normalized. So |ρ^{−1}(P_{fn})| ≡ |Rep_{F}(P, S)_{fc}| (mod p), hence is prime to p by Steps 1 and 2; and thus |P_{fn}| = |ρ^{−1}(P_{fn})|/N is also prime to p. □

3. The “bullet" construction

The bullet construction plays an important role in the theory of p-local compact groups, and is studied at length in [BLO3, Sec. 3]. However, in that paper, we always assumed that the ambient fusion system is saturated. In this one, we will need some of these properties to prove saturation in certain situations, and hence must know them in a more general setting. We start by recalling the definition.

Definition 3.1. Fix a discrete p-toral group S, set T = S_{0}, and let W ≤ Aut(T) be a subgroup which contains Aut_{S}(T). Let m ≥ 0 be such that exp(S/T) = p^{m}.

• For each A ≤ T, set I(A) = C_{T} (C_{W} (A)) =^{def} {t ∈ T | w(t) = t, ∀w ∈ C_{W} (A)} ≥ A.

• For each P ≤ S, set P^{[m]} = ⟨g^{p^{m}} | g ∈ P ⟩, and set P^{•} = P ·I(P^{[m]})_{0}.

• Let P, Q ≤ S and φ ∈ Inj(P, Q) be such that φ|_{P^{[m]}} = w|_{P^{[m]}} for some w ∈ W. Then φ^{•}: P^{•} ⟶ Q^{•} denotes the unique map of sets, if it exists, such that φ^{•}(gh) = φ(g)w(h) for each g ∈ P and h ∈ I(P^{[m]})_{0}.

Note that the definition of P^{•} for P ≤ S depends only on S and on the choice of a subgroup W ≤ Aut(S_{0}). For any fusion system F over S, we associate to F the construction (P ↦ P^{•}) with W = Aut_{F}(S_{0}), and set F^{•} = { P^{•} | P ≤ S }.

We will show later that under certain additional conditions, the map φ^{•} is defined and a homomorphism for each morphism φ.

Lemma 3.2. Let F be a fusion system over a discrete p-toral group S. Then

(a) P ≤ Q ≤ S implies P^{•} ≤ Q^{•}, and

(b) P ≤ S implies (P^{•})^{•} = P^{•} and N_{S}(P) ≤ N_{S}(P^{•}).

The following also hold if Aut_{F}(S_{0}) is finite.

(c) The set F^{•} contains finitely many S-conjugacy classes.

(d) Assume P_{1} ≤ P_{2} ≤ ⋯ are subgroups of S, and P = ⋃_{i=1}^{∞} P_{i}. Then there is N > 0 such that P_{i}^{•} = P^{•} for each i > N.

Proof. By definition, the function P ↦ P^{•} depends only on Aut_{F}(S_{0}) (and on S). Hence (a), (b), and (c) hold by exactly the same arguments as those used to prove them in [BLO3, Lemma 3.2]. The only exception is the last statement in (b) (not shown in [BLO3]), which follows since for g ∈ N_{S}(P), g ∈ N_{S}(P^{[m]}), and hence g ∈ N_{S}(I(P^{[m]})). Note that the assumption |Aut_{F}(S_{0})| < ∞ is needed to prove (c).

In the situation of (d), we have P_{1}^{•} ≤ P_{2}^{•} ≤ ⋯ by (a), and this sequence contains only finitely many subgroups by (c). Thus there are N > 0 and Q ≤ S such that P_{i}^{•} = Q for each i > N. In particular, P = ⋃_{i=1}^{∞} P_{i} ≤ Q, so P^{•} ≤ Q^{•} = Q by (a) and (b). Since P_{i}^{•} ≤ P^{•} for each i by (a) again, this shows that P^{•} = Q = P_{i}^{•} for i > N. □

While we do not assume here that fusion systems are saturated, we will, in most cases, assume the following condition on a fusion system F:

for each P ≤ S_{0} and each φ ∈ Hom_{F}(P, S_{0}), φ = w|_{P} for some w ∈ Aut_{F}(S_{0}). (∗)

By [BLO3, Lemma 2.4(b)], (∗) always holds if F is saturated.

Lemma 3.3. Let F be a fusion system over a discrete p-toral group S such that F satisfies (∗). Then for each P ≤ S and each φ ∈ Hom_{F}(P^{•}, S), φ(P^{•}) = φ(P)^{•} ∈ F^{•}. In particular, F^{•} is invariant under F-conjugacy.

Proof. Let m be such that p^{m} = exp(S/S_{0}), and set W = Aut_{F}(S_{0}). Set Q = φ(P). We must show that φ(P^{•}) = Q^{•}.

Set R = P^{•} ∩ S_{0} ∩ φ^{−1}(S_{0}). Thus R is the largest subgroup of P^{•} such that R and φ(R) are both contained in S_{0}. By (∗), there is w ∈ Aut_{F}(S_{0}) such that φ|_{R} = w|_{R}. In particular, w(P^{[m]}) = Q^{[m]}, so wC_{W} (P^{[m]})w^{−1} = C_{W} (Q^{[m]}), and w sends I(P^{[m]}) = C_{T} (C_{W} (P^{[m]})) isomorphically to I(Q^{[m]}). Also, φ(I(P^{[m]})_{0}) ≤ S_{0} since it is connected, so I(P^{[m]})_{0} ≤ R and φ(I(P^{[m]})_{0}) = w(I(P^{[m]})_{0}) = I(Q^{[m]})_{0}. So φ(P^{•}) = Q^{•}. □

Throughout the rest of the section, we need to consider the following condition on a fusion system F over a discrete p-toral group S:

for each P ≤ S and each φ ∈ Hom_{F}(P, S_{0}),

there exists φ ∈ Hom_{F}(P ·C_{S}(P)_{0}, S_{0}) such that φ|_{P} = φ.

(∗∗)

As will be seen in the next lemma, this is a stronger version of (∗).

Lemma 3.4. Fix a fusion system F over a discrete p-toral group S which satisfies (∗∗). Then the following hold.

(a) The fusion system F satisfies condition (∗).

(b) For each P, R ≤ S_{0} such that P ≤ R ≤ P ·I(P)_{0}, each w ∈ W, and each ψ ∈ Hom_{F}(R, S) such that ψ|_{P} = w|_{P}, ψ(R) ≤ S_{0} and ψ = w|_{R}.

(c) For each P, Q ≤ S and each φ ∈ Hom_{F}(P, Q), φ^{•} is defined and φ^{•} ∈ Hom(P^{•}, Q^{•}).

(d) Assume P ≤ Q ≤ P^{•}, φ ∈ Hom_{F}(P, S), and φ ∈ Hom_{F}(Q, S) are such that φ = φ|_{P}. Then φ = φ^{•}|_{Q}.

Proof. Set W = Aut_{F}(S_{0}). Let m be such that p^{m} = exp(S/S_{0}).

(a) For each P ≤ S_{0} and φ ∈ Hom_{F}(P, S_{0}), φ extends to w ∈ Aut_{F}(S_{0}) by (∗∗).

(b) Let P ≤ S_{0}, P ≤ R ≤ P ·I(P)_{0}, w ∈ W, and ψ ∈ Hom_{F}(R, S) be as above. Set P_{∗} = ψ(P) = w(P) and R_{∗} = ψ(R). Then P_{∗} ≤ R_{∗} are abelian, so each element of Aut_{R_{∗}}(S_{0}) ≤ W acts via the identity on P_{∗}, and hence via the identity on I(P_{∗}). In other words, [I(P_{∗}), R_{∗}] = 1. By (∗∗), ψ^{−1} ∈ Iso_{F}(R_{∗}, R) extends to a morphism χ ∈ Hom_{F}(R_{∗}·I(P_{∗})_{0}, S_{0}).

Now, χ(P_{∗}) = P, and χ|_{P_{∗}} = ψ^{−1}|_{P_{∗}} = w^{−1}|_{P_{∗}}. Also, χ|_{P_{∗}·I(P_{∗})_{0}} = u|_{P_{∗}·I(P_{∗})_{0}} for some u ∈ W by (a), u|_{P_{∗}} = w^{−1}|_{P_{∗}}, and hence u|_{I(P_{∗})} = w^{−1}|_{I(P_{∗})} by definition of I(−). Thus χ|_{P_{∗}·I(P_{∗})_{0}} = w^{−1}|_{P_{∗}·I(P_{∗})_{0}}, and so χ(P_{∗}·I(P_{∗})_{0}) = P ·I(P)_{0} ≥ R. Since χ is injective and χ(R_{∗}) = R, this implies that R_{∗} ≤ P_{∗}·I(P_{∗})_{0} ≤ S_{0} and ψ = (χ|_{R_{∗}})^{−1} = w|_{R}.

(c) Fix a morphism φ ∈ Hom_{F}(P, Q), and let w ∈ W be such that φ|_{P^{[m]}} = w|_{P^{[m]}}. Set R = P^{[m]}·(P ∩I(P^{[m]})_{0}). Then P^{[m]} ≤ R ≤ P^{[m]}·I(P^{[m]})_{0}, and (b) implies φ|_{R} = w|_{R}. Hence φ^{•}: P^{•} ⟶ Q^{•} is well defined as a map of sets by the formula φ^{•}(gh) = φ(g)w(h) for all g ∈ P and h ∈ I(P^{[m]})_{0}.

To prove that φ^{•} is a homomorphism, it remains to show that w(ghg^{−1}) = φ(g)w(h)φ(g)^{−1} for all g ∈ P and h ∈ I(P^{[m]})_{0}. Set u = c_{φ(g)}^{−1} ∘ w ∘ c_{g} ∈ W; we must show that u|_{I(P^{[m]})_{0}} = w|_{I(P^{[m]})_{0}}. But u|_{P^{[m]}} = w|_{P^{[m]}} since φ|_{P^{[m]}} = w|_{P^{[m]}}, and so u|_{I(P^{[m]})} = w|_{I(P^{[m]})} by definition of I(−).

(d) Fix P, Q ≤ S such that P ≤ Q ≤ P^{•}. Assume φ ∈ Hom_{F}(P, S) and φ ∈ Hom_{F}(Q, S) are such that φ = φ|_{P}. Set R = P^{[m]} and Q′ = Q ∩ R·I(R)_{0}. By definition, φ^{•}|_{R·I(R)_{0}} = w|_{R·I(R)_{0}} for any w ∈ W such that φ|_{R} = w|_{R}. By (b), φ|_{Q′} = w|_{Q′} = φ^{•}|_{Q′}. Since

Q = Q ∩ P^{•} = Q ∩ (P ·I(R)_{0}) = P ·(Q ∩ R·I(R)_{0}) = P Q′

and φ|_{P} = φ, this proves that φ = φ^{•}|_{Q}. □

Lemma 3.4(c) allows us to extend each φ ∈ Hom_{F}(P, Q) to a homomorphism φ^{•} ∈ Hom(P^{•}, Q^{•}). In the next lemma, we add some more hypotheses, enough to ensure that φ^{•} ∈ Mor(F).

Lemma 3.5. Fix a fusion system F over a discrete p-toral group S. Let H be a family of subgroups of S invariant under F-conjugacy. Assume the following:

(i) Aut_{F}(S_{0}) is finite.

(ii) F satisfies (∗∗).

(iii) For all P ∈ H and P ≤ Q ≤ P^{•}, Q ∈ H.

(iv) F is H-generated and H-saturated.

Then the following hold.

(a) For each φ ∈ Mor(F), φ^{•} ∈ Mor(F).

(b) If, in addition, Ĥ ⊇ H is also invariant under F-conjugacy, and is such that P ∈ Ĥ implies P^{•} ∈ H, then F is Ĥ-saturated.

Proof. (a) Fix φ ∈ Hom_{F}(P, Q); we must show φ^{•} ∈ Hom_{F}(P^{•}, Q^{•}). It suffices to do this when φ is an isomorphism. Since F is H-generated, φ is a composite of isomorphisms which are restrictions of F-isomorphisms between subgroups in H. So it suffices to consider the case where φ extends to φ ∈ Iso_{F}(P , Q) for some P , Q ∈ H. If φ^{•} ∈ Mor(F), then φ^{•} ∈ Mor(F) since it is a restriction of φ^{•}. It thus suffices to prove (a) when P and Q are in H.

By axiom (III), which holds by hypothesis for all subgroups in H, there is a subgroup P′ ≤ P^{•} containing P which is maximal among subgroups of P^{•} satisfying φ′ =^{def} φ^{•}|_{P′} ∈ Hom_{F}(P′, Q^{•}). Assume P′ < P^{•}, and set Q′ = φ′(P′).

Since P, Q ∈ H, and since P ≤ P′ ≤ P^{•} and Q ≤ Q′ ≤ Q^{•}, P′, Q′ ∈ H by (iii). Since F is H-saturated, there are R′ ≤ S and ψ′ ∈ Iso_{F}(Q′, R′) such that R′ is receptive in F. Set R = ψ′(Q) and ψ = ψ′|_{Q} ∈ Iso_{F}(Q, R). By Lemma 3.4(d), ψ′ = ψ^{•}|_{Q′}, and ψ′φ′ = (ψφ)^{•}|_{P′}. We thus have isomorphisms of triples

(P^{•} ≥ P′ ≥ P) ⟶_{≅}^{(φ^{•},φ′,φ)} (Q^{•} ≥ Q′ ≥ Q) ⟶_{≅}^{(ψ^{•},ψ′,ψ)} (R^{•} ≥ R′ ≥ R) .

Set P′′ = N_{P^{•}}(P′) and Q′′ = N_{Q^{•}}(Q′). Since P^{•} > P′, P′′ > P′ and Q′′ > Q′ by Lemma 1.12. Also, N_{ψ′} ≥ Q′′ and N_{ψ′φ′} ≥ P′′ since there are (abstract) homomorphisms which extend ψ′ to Q′′ and ψ′φ′ to P′′. Hence ψ′ and ψ′φ′ extend to homomorphisms ψ′′ ∈ Hom_{F}(Q′′, S) and χ′′ ∈ Hom_{F}(P′′, S), and ψ′′ = ψ^{•}|_{Q′′} and χ′′ = (ψφ)^{•}|_{P′′} by Lemma 3.4(d) again. So ψ′′(Q′′) = χ′′(P′′) = N_{R^{•}}(R′), and φ^{•}|_{P′′} = (ψ′′)^{−1}χ′′ is a morphism in F, which contradicts the maximality of P′. We now conclude that P′ = P^{•}, and hence that φ^{•} ∈ Iso_{F}(P^{•}, Q^{•}).

(b) We are assuming F is H-saturated, and want to show it is Ĥ-saturated. Fix P in Ĥ∖H. Thus P^{•} ∈ H by assumption. Choose R which is F-conjugate to P^{•}, and receptive and fully automized. Fix φ ∈ Iso_{F}(P^{•}, R) and set Q = φ(P). By Lemma 3.4(d), φ = (φ|_{P})^{•}, and so Q^{•} = R.

Set H = {α ∈ Aut_{F}(Q^{•}) | α(Q) = Q}. Since Q^{•} = R is fully automized, Aut_{S}(Q^{•}) ∈ Syl_{p}(Aut_{F}(Q^{•})). Thus H acts by translation on Aut_{F}(Q^{•})/Aut_{S}(Q^{•}), a finite set of order prime to p, and there is β ∈ Aut_{F}(Q^{•}) such that the coset β^{−1}Aut_{S}(Q^{•}) is in an H-orbit of order prime to p. In other words, the stabilizer subgroup β^{−1}Aut_{S}(Q^{•})β ∩ H has index prime to p in H, and thus lies in Syl_{p}(H).

By (a) (and the definition of H), restriction to Q sends βHβ^{−1} surjectively onto Aut_{F}(β(Q)). Also, Aut_{S}(β(Q)) is the image of Aut_{S}(Q^{•}) ∩ βHβ^{−1} under this surjection, and hence is a Sylow p-subgroup in Aut_{F}(β(Q)). Thus β(Q) is fully automized. (Note that β(Q)^{•} = Q^{•}.)

For any R ∈ P^{F} = Q^{F} and any φ ∈ Iso_{F}(R, Q), N_{φ} ≤ N_{φ^{•}}: if φc_{g}φ^{−1} = c_{h} ∈ Aut_{S}(Q), then φ^{•}c_{g}φ^{•}^{−1} = c_{h} ∈ Aut_{S}(Q^{•}) by Lemma 3.2(d). Hence Q and β(Q) are both receptive in F since Q^{•} is. Thus the conjugacy class P^{F} contains a subgroup which is receptive and fully automized.

Assume P_{∞} = ⋃_{i=1}^{∞} P_{i}, where {P_{i}} is an increasing sequence of subgroups in Ĥ. Fix φ ∈ Hom_{F}(P_{∞}, S) such that φ|_{P_{i}} ∈ Hom_{F}(P_{i}, S) for each i. By Lemma 3.2(d), for i large enough, P_{i}^{•} = P_{∞}^{•}. By Lemma 3.4(d), φ is the restriction of (φ|_{P_{i}})^{•}, which is in F by (a). So φ ∈ Hom_{F}(P_{∞}, S). □

4. Centric and radical subgroups determine saturation

The main result in this section is Theorem 4.2, which gives sufficient conditions for a fusion system over a discrete p-toral group to be saturated.

We will frequently refer to the conditions (∗) and (∗∗) of Section 3, which we recall here:

∀ P ≤ S_{0}, ∀ φ ∈ Hom_{F}(P, S_{0}), φ = w|_{P} for some w ∈ Aut_{F}(S_{0}).  (∗)
∀ P ≤ S and φ ∈ Hom_{F}(P, S_{0}), ∃ φ ∈ Hom_{F}(P ·C_{S}(P)_{0}, S_{0}) with φ|_{P} = φ.  (∗∗)

By Lemma 3.4(a), condition (∗∗) implies (∗).

The following finiteness result will be needed.

Lemma 4.1. Let F be a fusion system over a discrete p-toral group S such that Aut_{F}(S_{0}) is finite and (∗) holds. Then for each P ≤ S, there are only finitely many S-conjugacy classes of subgroups of S which are F-conjugate to P.

Proof. By (∗) and since |Aut_{F}(S_{0})| < ∞, P_{0}^{F} is finite. By [BLO3, Lemma 1.4(a)], for each R ∈ P_{0}^{F}, there are only finitely many N_{S}(R)/R-conjugacy classes of finite subgroups of N_{S}(R)/R of any given order. (Compare [BLO3, Lemma 2.5].) □

The main result in this section is the following theorem. We refer to Definition 1.11 for the definitions of H-saturated and H-generated fusion systems.

Theorem 4.2. Fix a fusion system F over a discrete p-toral group S such that Aut_{F}(S_{0}) is finite and condition (∗∗) holds. Let H be a family of subgroups of S which satisfies the following:

(i) H is invariant under F-conjugacy.

(ii) H is closed in 𝔖𝔲𝔟(S), and F is H-generated and H-saturated.

(iii) For all P ∈ H and P ≤ Q ≤ P^{•}, Q ∈ H.

(iv) If P ∈ F^{•} is F-centric and P ∉ H, then there is Q ∈ P^{F} such that

O_{p}(Out_{F}(Q)) ∩ Out_{S}(Q) ≠ 1.

Then F is saturated.

Theorem 4.2 will be shown by following as closely as possible the proof of [BCGLO1, Theorem 2.2]. The main difference is that since a discrete p-toral group can have infinitely many subgroups, the induction arguments used in [BCGLO1] cannot be used here. But the beginning steps are mostly unchanged: they are based on the concept of proper P-pairs for an F-conjugacy class P of subgroups of S.

Definition 4.3. Let F be a fusion system over a discrete p-toral group S, and let P be an F-conjugacy class of subgroups of S.

• A proper P-pair is a pair of subgroups (Q, P) such that P < Q ≤ N_{S}(P) and P ∈ P.

• Two proper P-pairs (Q, P) and (Q′, P′) are F-conjugate if there is an isomorphism φ ∈ Iso_{F}(Q, Q′) such that φ(P) = P′. We let (Q, P)^{F} denote the set of proper P-pairs which are F-conjugate to (Q, P).

• A proper P-pair (Q, P) is fully normalized if |N_{N_{S}(P)}(Q)| ≥ |N_{N_{S}(P′)}(Q′)| for all (Q′, P′) ∈ (Q, P)^{F}.

Some basic properties of proper P-pairs are shown in the following lemma.

Lemma 4.4. Fix a fusion system F over a discrete p-toral group S. Assume H ⊆ 𝔖𝔲𝔟(S) is closed, and invariant under F-conjugacy. Assume also that F is H-generated and Hsaturated. Let P be an F-conjugacy class of subgroups of S, maximal among those not in H.

(a) If (Q, P) is a fully normalized proper P-pair, then Q is receptive in F and

Aut_{N_{S}(P)}(Q) ∈ Syl_{p}(Aut_{N_{F} (P)}(Q)).

(b) For each proper P-pair (Q, P), and each (Q′, P′) ∈ (Q, P)^{F} which is fully normalized, there is a morphism

φ ∈ Hom_{F}(N_{N_{S}(P)}(Q), N_{S}(P′))

such that φ(P) = P′ and φ(Q) = Q′.

(c) Assume that Aut_{F}(S_{0}) is finite, condition (∗∗) holds, and P ∈ H, P ≤ Q ≤ P^{•} imply Q ∈ H. Then for each P, P̂ ∈ P such that P̂ is fully normalized in F, there is a morphism φ ∈ Hom_{F}(N_{S}(P), N_{S}(P̂)) such that φ(P) = P̂.

Proof. (a) For each proper P-pair (Q, P), define

K_{P} = {φ ∈ Aut(Q) | φ(P) = P } ≤ Aut(Q) .

Notice that

Aut_{S}^{K_{P}} (Q) = Aut_{S}(Q) ∩ K_{P} = Aut_{N_{S}(P)}(Q)
Aut_{F}^{K_{P}} (Q) = Aut_{F}(Q) ∩ K_{P} = Aut_{N_{F} (P)}(Q)

(1)

N_{S}^{K_{P}} (Q) = N_{N_{S}(P)}(Q) .  (2)

Assume the pair (Q, P) is fully normalized. Fix Q′ ∈ Q^{F} and α ∈ Iso_{F}(Q, Q′), and set P′ = α(P). Then (Q′, P′) ∈ (Q, P)^{F}, and

αK_{P} = K_{P′} =^{def} {φ ∈ Aut(Q′) | φ(P′) = P′}.

Hence by (2),

|N_{S}^{K_{P}} (Q)| = |N_{N_{S}(P)}(Q)| ≥ |N_{N_{S}(P′)}(Q′)| = |N_{S}^{K_{P′}}(Q′)| = |N_{S}^{^{α}K_{P}} (Q′)|,

and so Q is fully K_{P} -normalized in F. By Lemma 2.2(b) (and since Q ∈ H and F is H-saturated), Q is receptive and fully K_{P} -automized in F. So by (1),

Aut_{N_{S}(P)}(Q) ∈ Syl_{p}(Aut_{N_{F} (P)}(Q)).

(b) Let (Q, P), and (Q′, P′) ∈ (Q, P)^{F}, be proper P-pairs such that (Q′, P′) is fully normalized. Let K_{P} ≤ Aut(Q) and K_{P′} ≤ Aut(Q′) be as in (a). Since (Q′, P′) is fully normalized, Q′ is receptive and fully K_{P′}-automized by (a).

Choose some ψ ∈ Iso_{F}(Q, Q′) such that ψ(P) = P′. By Lemma 2.2(b), there are χ ∈ Aut_{F}^{K_{P′}}(Q′) and φ ∈ Hom_{F}(N_{S}^{K_{P}} (Q), N_{S}^{K_{P′}}(Q′)) such that φ|_{Q} = χ ∘ ψ. Then φ(P) = χ(P′) = P′, φ(Q) = Q′, and N_{S}^{K_{P}} (Q) = N_{N_{S}(P)}(Q) and N_{S}^{K_{P′}}(Q′) = N_{N_{S}(P′)}(Q′) by (2).

(c) Fix subgroups P, P̂ ∈ P such that P̂ is fully normalized in F. We will construct a morphism in F from N_{S}(P) to N_{S}(P̂) which sends P to P̂. In particular, P̂ will be fully centralized in F, since its centralizer contains an injective image of the centralizer of any other subgroup in P.

Let T̂ be the set of all sequences

ξ =(P = P_{0}, Q_{0}, φ_{0}; P_{1}, Q_{1}, φ_{1}; …; P_{k−1}, Q_{k−1}, φ_{k−1}; P_{k} = P̂)  (3)

such that for each 0 ≤ i ≤ k − 1, (Q_{i}, P_{i}) is a proper P-pair, φ_{i} ∈ Hom_{F}(Q_{i}, N_{S}(P_{i+1})), and φ_{i}(P_{i}) = P_{i+1}. Let T ⊆ T̂ be the subset of those sequences in which each φ_{i} is maximal, in the sense that it cannot be extended in F to a subgroup of N_{S}(P_{i}) which properly contains Q_{i}. We give T̂ the partial ordering by inclusion (of sequences of the same length with the same P_{i}’s); then T is the set of maximal elements in T̂. Since F is H-saturated, axiom (III) ensures that each element of T̂ is contained in an element of T .

We first check that T ≠ ∅. Choose any φ ∈ Iso_{F}(P, P̂). Since F is H-generated, there are subgroups P_{i} < R_{i} ∈ H for 0 ≤ i ≤ k, and morphisms ϕ_{i} ∈ Hom_{F}(R_{i}, R_{i+1}) for i ≤ k − 1, such that P_{0} = P, P_{k} = P̂, ϕ_{i}(P_{i}) = P_{i+1}, and φ = ϕ_{k−1}|_{P_{k−1}} ∘ ⋯ ∘ ϕ_{1}|_{P_{1}} ∘ ϕ_{0}|_{P_{0}}. For each i, let P_{i+1} = ϕ_{i}(P_{i}). Then N_{R_{i}}(P_{i}) > P_{i} by Lemma 1.12 (and since R_{i} > P_{i}). Set Q_{i} = N_{R_{i}}(P_{i}) and φ_{i} = ϕ_{i}|Q_{i}. Then (Q_{i}, P_{i}) is a proper P-pair, φ_{i} ∈ Hom_{F}(Q_{i}, N_{S}(P_{i+1})), and (P_{0}, Q_{0}, φ_{0}; P_{1}, Q_{1}, φ_{1}; …; P_{k}) ∈ T̂. Thus T̂ ≠ ∅, and T ≠ ∅ since each element is contained in a maximal element.

Lemma 3.5(a) applies in this situation by the extra hypotheses which were assumed. Hence for any ξ ∈ T as in (3), φ_{i} extends to φ_{i}^{•} ∈ Hom_{F}(Q_{i}^{•}, N_{S}(P_{i+1})^{•}) for each i, and therefore to Q_{i}^{•} ∩ N_{S}(P_{i}). The maximality of φ_{i} implies that

Q_{i} = Q_{i}^{•} ∩ N_{S}(P_{i}).  (4)

The same argument applied to φ_{i}^{−1} shows that

φ_{i}(Q_{i}) = φ_{i}(Q_{i})^{•} ∩ N_{S}(P_{i+1}).  (5)

Let T_{r} ⊆ T be the subset of those ξ for which there is no 1 ≤ i ≤ k − 1 such that Q_{i} = N_{S}(P_{i}) = φ_{i−1}(Q_{i−1}). Let T ⟶^{R} T_{r} be the “reduction” map, which for every 1 ≤ i ≤ k − 1 such that Q_{i} = N_{S}(P_{i}) = φ_{i−1}(Q_{i−1}), removes from the sequence the triple (P_{i}, Q_{i}, φ_{i}), and replaces φ_{i−1} by the composite φ_{i} ∘ φ_{i−1}. Since T ≠ ∅, the existence of the reduction map R shows that T_{r} ≠ ∅.

For each ξ ∈ T , define

I(ξ) = {i = 0, …, k − 1 | Q_{i} < N_{S}(P_{i}) and φ_{i}(Q_{i}) < N_{S}(P_{i+1})}.

If ξ ∈ T and I(ξ) ≠ ∅, define

λ(ξ) = min_{i∈I(ξ)}{ |Q_{i}^{•}| } ≥ (0, p) .

Since Aut_{F}(S_{0}) is assumed to be finite, F^{•} has finitely many F-conjugacy classes by Lemma 3.2(c), so there are only a finite number of values that λ(ξ) can take.

Assume there is a sequence ξ ∈ T_{r} as in (3) such that I(ξ) = ∅. For each 0 ≤ i < k, if Q_{i} < N_{S}(P_{i}), then φ_{i}(Q_{i}) = N_{S}(P_{i+1}) since i ∉ I(ξ), and hence Q_{i+1} < N_{S}(P_{i+1}) if i + 1 ≤ k − 1 since ξ is reduced. Thus |N_{S}(P_{i})| > |Q_{i}| = |N_{S}(P_{i+1})| > ⋯ > |N_{S}(P_{k})| in this case, which contradicts the assumption that P_{k} = P̂ is fully normalized. Hence Q_{i} = N_{S}(P_{i}) for each i < k, and φ_{k−1} ∘ ⋯ ∘ φ_{0} lies in Hom_{F}(N_{S}(P), N_{S}(P̂)) and sends P to P̂.

It remains to show that there exists ξ ∈ T_{r} such that I(ξ) = ∅. To see this, fix ξ ∈ T_{r} as in (3) such that I(ξ) ≠ ∅. We will construct ξ̂ ∈ T_{r} such that either I(ξ̂) = ∅ or λ(ξ̂) > λ(ξ). Since λ(ξ) can only take finitely many values, this will prove our claim.

Fix i ∈ I(ξ), set R_{i+1} = φ_{i}(Q_{i}), and choose a fully normalized proper P-pair (U_{i}, V_{i}) which is F-conjugate to (Q_{i}, P_{i}) (hence also to (R_{i+1}, P_{i+1})). By (b), there are morphisms

ψ_{i} ∈ Hom_{F}(N_{N_{S}(P_{i})}(Q_{i}), N_{S}(V_{i})), and θ_{i} ∈ Hom_{F}(N_{N_{S}(P_{i+1})}(R_{i+1}), N_{S}(V_{i}))

such that ψ_{i}(P_{i}) = θ_{i}(P_{i+1}) = V_{i} and ψ_{i}(Q_{i}) = θ_{i}(R_{i+1}) = U_{i}. Upon replacing (P_{i}, Q_{i}, φ_{i}) by

(P_{i}, N_{N_{S}(P_{i})}(Q_{i}), ψ_{i}; V_{i}, θ_{i}(N_{N_{S}(P_{i+1})}(R_{i+1})), θ_{i}^{−1}) ,

we get a new sequence ξ_{1} ∈ T̂. Upon applying axiom (III) again, we get a sequence ξ_{2} ∈ T where (P_{i}, Q_{i}, φ_{i}) has been replaced by

(P_{i}, Q̃_{i}, ψ̃_{i}; V_{i}, R̃_{i+1}, θ̃_{i}^{−1})

for some maximal extensions ψ̃_{i} of ψ_{i} and θ̃_{i} of θ_{i}. Also, Q_{i} < N_{S}(P_{i}) and R_{i+1} < N_{S}(P_{i+1}) since i ∈ I(ξ), so

Q̃_{i} ≥ N_{N_{S}(P_{i})}(Q_{i}) > Q_{i} and R̃_{i+1} ≥ N_{N_{S}(P_{i+1})}(R_{i+1}) > R_{i+1} ,

and Q̃_{i}^{•} > Q_{i}^{•} and R̃_{i+1}^{•} > R_{i+1}^{•} by (4) and (5). In particular, |Q̃_{i}^{•}| > λ(ξ) and |R̃_{i+1}^{•}| > λ(ξ).

Upon repeating this procedure for all i ∈ I(ξ), we obtain a new element ξ′ ∈ T such that either I(ξ′) = ∅ or λ(ξ′) > λ(ξ). Set ξ̂ = R(ξ′) ∈ T_{r}. Then either I(ξ̂) = ∅ or λ(ξ̂) > λ(ξ). Since the function λ can only take a finite number of possible values, it follows by induction that there is ξ ∈ T_{r} such that I(ξ) = ∅. □

Very roughly, Lemma 4.4 allows us to reduce the proof of Theorem 4.2 to showing that the saturation properties hold for certain subgroups that are normal in the fusion system. This case is handled by the following lemma.

Lemma 4.5. Let F be a fusion system over a discrete p-toral group S such that Aut_{F}(S_{0}) is finite and condition (∗) holds. Fix a subgroup Q ⊴ S, set H = {P ≤ S | P > Q}, and assume

(i) Q ⊴ F;

(ii) F is H-generated and H-saturated; and

(iii) either Q is not F-centric, or Out_{S}(Q) ∩ O_{p}(Out_{F}(Q)) ≠ 1.

Then Q is fully automized and receptive in F.

Proof. When Γ is a group containing a normal discrete p-torus P of finite index, we let O_{p}(Γ) be the inverse image in Γ of the maximal normal p-subgroup O_{p}(Γ/P) under the obvious projection. Equivalently, this is the largest normal discrete p-toral subgroup of Γ. Define

Q̂ = {x ∈ S | c_{x} ∈ O_{p}(Aut_{F}(Q))} .

Then Q̂ ⊴ S by definition. We claim that Q̂ is strongly closed in F. Assume that x ∈ Q̂ is F-conjugate to y ∈ S. Since Q is normal in F, there exists ψ ∈ Hom_{F}(⟨x, Q⟩, ⟨y, Q⟩) which satisfies ψ(Q) = Q and ψ(x) = y. In particular, ψ ∘ c_{x} ∘ ψ^{−1} = c_{y}. It follows that y ∈ Q̂, since c_{x} ∈ O_{p}(Aut_{F}(Q)).

Note also that Q·C_{S}(Q) ⊴ Q̂, and Q̂/Q·C_{S}(Q) ≅ Out_{S}(Q) ∩ O_{p}(Out_{F}(Q)). If Q is Fcentric, then this last group is nontrivial by (iii), and if not, then Q·C_{S}(Q) > Q by definition. Thus Q̂ > Q in either case, and so Q̂ ∈ H.

Consider the following statement:

each φ ∈ Aut_{F}(Q) extends to some φ ∈ Aut_{F}(Q̂).  (6)

We first prove that (6) implies the lemma, and then prove (6).

Point (6) implies the lemma: Since Q is normal in F and Q̂ is strongly closed, each of them is the only subgroup in its F-conjugacy class. So Q and Q̂ are both fully centralized and fully normalized in F. Also, Q̂ is receptive and fully automized in F, since F is H-saturated and Q̂^{F} = {Q̂} ⊆ H.

By (6), the restriction map from Aut_{F}(Q̂) to Aut_{F}(Q) is surjective, and so Aut_{S}(Q) ∈ Syl_{p}(Aut_{F}(Q)) since Aut_{S}(Q̂) ∈ Syl_{p}(Aut_{F}(Q̂)). Thus Q is fully automized in F.

Next we prove that Q is receptive in F. Fix φ ∈ Aut_{F}(Q). As usual, let N_{φ} be the group of all g ∈ N_{S}(Q) such that φc_{g}φ^{−1} ∈ Aut_{S}(Q). By (6), φ extends to some ψ ∈ Aut_{F}(Q̂). Consider the groups of automorphisms

K = {χ ∈ Aut_{S}(Q̂) | χ|_{Q} = c_{x} some x ∈ N_{φ}}
K_{0} = {χ ∈ Aut_{F}(Q̂) | χ|_{Q} = Id_{Q}} ⊴ Aut_{F}(Q̂).

By definition, for all x ∈ N_{φ}, we have (ψc_{x}ψ^{−1})|_{Q} = χ|_{Q} for some χ ∈ Aut_{S}(Q̂). In other words, as subgroups of Aut(Q̂),

ψK ≤ Aut_{S}(Q̂)·K_{0}.

Now, Aut_{S}(Q̂) ∈ Syl_{p}(Aut_{S}(Q̂)K_{0}) since Q̂ is fully automized, so there are ω ∈ Aut_{S}(Q̂) and χ ∈ K_{0} such that ^{ωχ}(^{ψ}K) ≤ Aut_{S}(Q̂). Hence ^{χψ}K ≤ Aut_{S}(Q̂). Since Q̂ is receptive in F, χψ ∈ Aut_{F}(Q̂) extends to a morphism φ defined on N_{χψ} ≥ N_{S}^{K}(Q̂) ≥ N_{φ}, and φ|_{Q} = ψ|_{Q} = φ since χ|_{Q} = Id_{Q}.

Proof of (6): Since F is H-generated, each φ ∈ Aut_{F}(Q) is a composite of automorphisms of Q which extend to strictly larger subgroups. So it suffices to show (6) when φ itself extends to some P > Q.

Let X be the set of all subgroups P ∈ H such that φ = φ|_{Q} for some φ ∈ Hom_{F}(P, S). We are assuming that X ≠ ∅. We claim that

P ∈ X ⟹ N_{Q̂P} (P) ∈ X.  (7)

Assume this, and fix P_{1} ∈ X. If |Q̂P_{1}/P_{1}| < ∞, then by repeated application of (7) and Lemma 1.12, we get that Q̂ ∈ X. If not, then |Q̂/Q| = ∞, and Q̂_{0}Q/Q is a nontrivial discrete p-torus. Set P_{2} = N_{Q̂_{0}P_{1}}(P_{1}) and P_{3} = P_{2} ∩Q̂_{0}Q. Then P_{2} ∈ X by (7), P_{1} ≤ P_{2} ≤ Q̂_{0}P_{1}, and P_{2} > P_{1} if P_{2} ≱ Q̂_{0} by Lemma 1.12. So P_{3} > Q, and hence P_{3} ∈ X. Now set P_{4} = NQ̂(P_{3}), so P_{4} ∈ X by (7) again, P_{4} ≥ Q̂_{0}Q since Q̂_{0}Q/Q is abelian, hence [Q̂:P_{4}] < ∞, and Q̂ ∈ X by earlier remarks. Since Q̂ ∈ X, φ extends to some φ̂ ∈ Hom_{F}(Q̂, S), and φ̂(Q̂) = Q̂ since Q̂ is strongly closed in F. This shows that (7) implies (6).

It remains to prove (7). Fix P ∈ X ⊆ H, and let φ̃ ∈ Iso_{F}(P, P_{2}) be an extension of φ. Choose some P_{3} ∈ P^{F} = P_{2}^{F} which is fully automized and receptive in F. Notice that P_{3} > Q, since Q ⊴ F by (i). Let ψ ∈ Iso_{F}(P_{2}, P_{3}) be any isomorphism. Upon replacing φ̃ by ψ ∘ φ̃, we can arrange that P_{2} be fully automized and receptive.

Consider the groups of automorphisms

L = {χ ∈ Aut_{F}(P_{2}) | χ|_{Q} ∈ O_{p}(Aut_{F}(Q))}
L_{0} = {χ ∈ Aut_{F}(P_{2}) | χ|_{Q} = Id_{Q}} .

Both L and L_{0} are normal subgroups of Aut_{F}(P_{2}). Also, L/L_{0} is a discrete p-toral group, since there is a monomorphism L/L_{0} → O_{p}(Aut_{F}(P_{2})). Since P_{2} is fully automized, Aut_{S}(P_{2}) ∈ Syl_{p}(Aut_{F}(P_{2})), and hence Aut_{S}^{L}(P_{2}) ∈ Syl_{p}(L) and L = Aut_{S}^{L}(P_{2})L_{0}.

Thus P_{2} is fully L-automized in F. By Lemma 2.2(b), there are g ∈ N_{S}^{L}(P_{2}) = NQ̂(P_{2}), χ ∈ L_{0}, and ψ ∈ Hom_{F}(N_{S}^{L^{φ}}(P)P, N_{S}^{L}(P_{2})P_{2}), such that ψ|_{P} = (c_{g} ∘χ) ∘φ. Also, N_{S}^{L^{φ}}(P)P = N_{Q̂P} (P). Upon replacing ψ by c_{g}^{−1} ∘ ψ, we can assume that g = 1. Then ψ|_{Q} = φ|_{Q}, and thus N_{Q̂P} (P) ∈ X. □

The following lemma combines Lemmas 4.4 and 4.5. Lemma 4.6. Let F be a fusion system over a discrete p-toral group S such that Aut_{F}(S_{0}) is finite and condition (∗∗) holds. Fix a family H ⫋ 𝔖𝔲𝔟(S), and a subgroup P ≤ S which is maximal in 𝔖𝔲𝔟(S)∖H. Assume

(i) H is invariant under F-conjugacy; (ii) H is closed in 𝔖𝔲𝔟(S), and F is H-generated and H-saturated; (iii) P ∈ H and P ≤ Q ≤ P^{•} imply Q ∈ H; and (iv) either P is not F-centric, or Out_{S}(P) ∩ O_{p}(Out_{F}(P)) ≠ 1.

Then F is (H ∪ P^{F})-saturated.

Proof. By assumption, all overgroups of subgroups in P^{F} are in H. Since P^{F} contains only finitely many S-conjugacy classes by Lemma 4.1, there is a subgroup Q̂ ∈ P^{F} which is fully normalized in F. By Lemma 4.4(c), for each Q ∈ P^{F}, there is χ_{Q} ∈ Hom_{F}(N_{S}(Q), N_{S}(Q̂)) such that χ_{Q}(Q) = Q̂. Let N ⊆ H be the family of all subgroups of N_{S}(Q̂) which strictly contain Q̂. We claim that the normalizer system N_{F}(Q̂) is N -saturated.

For each Q ∈ N, the pair (Q, Q̂) is a proper P^{F}-pair. Let (Q′, P′) be a proper P^{F}-pair which is F-conjugate to (Q, Q̂) and fully normalized in F. Then Q̂ = χ_{P′}(P′) < Q′′ =^{def} χ_{P′}(Q′) ≤ N_{S}(Q̂), so Q′′ ∈ N, and (Q′′, Q̂) is a proper P^{F}-pair. Furthermore, since (Q′, P′) is fully normalized, and since χ_{P′} is a monomorphism, (Q′′, Q̂) is also fully normalized in F. If α ∈ Iso_{F}(Q, Q′) is such that α(Q̂) = P′, then (χ_{P′}|_{Q′}) ∘ α is a morphism in N_{F}(Q̂) which sends (Q, Q̂) onto (Q′′, Q̂). Hence by Lemma 4.4(a), applied with with P = P^{F}, the subgroup Q′′ is receptive in F (hence in N_{F}(Q̂)), and is fully automized in N_{F}(Q̂). This shows that every Q ∈ N is N_{F}(Q̂)-conjugate to some Q′′ ∈ N which is fully automized and receptive in the normalizer fusion system. Axiom (III) holds for N_{F}(Q̂) with respect to the family N, since N is closed under overgroups, N ⊆ H, and F is H-saturated. Thus N_{F}(Q̂) is N -saturated.

Conjugation by (χ_{P})|_{P} ∈ Iso_{F}(P, Q̂) sends Aut_{F}(P) isomorphically to Aut_{F}(Q̂) and Aut_{S}(P) into Aut_{S}(Q̂). So if P and Q̂ are not F-centric, then (iv) implies that Out_{S}(Q̂) ∩ O_{p}(Out_{F}(Q̂)) ≠ 1. Thus by Lemma 4.5, Q̂ is receptive and fully automized in N_{F}(Q̂). Hence Q̂ is fully automized in F. It is receptive in F since for each Q ∈ P^{F}, there is an F-morphism which sends Q onto Q̂ and N_{S}(Q) into N_{S}(Q̂).

Axiom (III) holds for H ∪ P^{F} since it holds for H and since no subgroup in P^{F} contains any subgroups in H. Thus F is (H ∪ P^{F})-saturated. □

We are now ready to prove Theorem 4.2. The inductive Lemma 4.6 would suffice to prove the theorem for fusion systems over finite p-groups. But for fusion systems over discrete p-toral groups, because our groups have infinite chains of subgroups, the results of Section 3 are needed to allow an induction proof.

Proof of Theorem 4.2.Let 𝔎 be the set of all closed families of subgroups K ⊆ 𝔖𝔲𝔟(S) such that

• K ⊇ H,

• K is invariant under F-conjugacy,

• P ∈ K and P ≤ Q ≤ P^{•} implies Q ∈ K, and

• F is K-saturated.

We must show that 𝔖𝔲𝔟(S) ∈ 𝔎.

Assume otherwise, and choose K_{0} ∈ 𝔎 for which K_{0} ∩ F^{•} contains the largest possible number of F-conjugacy classes. Set K_{1} = {P ≤ S | P^{•} ∈ K_{0}}. Then K_{1} ∈ 𝔎 by Lemma 3.5(b). If K_{1} ⫋ 𝔖𝔲𝔟(S), then F^{•} ⊈ K_{0}; let P be maximal among subgroups in ∈ F^{•}∖K_{0}. Then P is maximal among subgroups in 𝔖𝔲𝔟(S)∖K_{1}, K_{1} ∪ P^{F} ∈ 𝔎 by Lemma 4.6, and this contradicts the maximality assumption on K_{0}. □

5. Extensions of p-local compact groups

The main result in this section is a version of [O1, Theorem 9] which describes how to extend a p-local compact group by a finite group. But before proving this, we need to show some of the basic properties of linking systems over discrete p-toral groups. All of the results in this section are generalizations of results in [O1] or earlier papers about linking systems over finite p-groups.

We first look at automorphisms of fusion and linking systems. The definitions are the same as in the finite case (e.g., [AOV, Definition 1.13]).

Definition 5.1. (a) For any fusion system over a discrete p-toral group S, an automorphism α of S is fusion preserving if there is an automorphism α̂ of F which sends an object P to α(P) and sends a morphism φ to αφα^{−1} (after restricting α in the obvious way). Let Aut(S, F) be the group of fusion preserving automorphisms of S, and set Out(S, F) = Aut(S, F)/Aut_{F}(S).

(b) For any linking system L over a discrete p-toral group S, an automorphism of categories α: L ⟶^{≅} L is isotypical if for each P ∈ Ob(L), α(δ_{P} (P)) = δ_{α(P)}(α(P)). Let Aut_{typ}^{I}(L) be the group of isotypical automorphisms of L which send inclusions to inclusions.

(c) Each γ ∈ Aut_{L}(S) acts on the set Mor(L) by composing on the left or right with γ and its restrictions. More precisely, for any φ ∈ Mor_{L}(P, Q), set

γφ = γ|_{Q,π(γ)(Q)} ∘ φ ∈ Mor_{L}(P, π(γ)(Q))  (1)

and

φγ = φ ∘ γ|_{π(γ)^{−1}(P),P} ∈ Mor_{L}(π(γ)^{−1}(P), Q).  (2)

(d) For each γ ∈ Aut_{L}(S), let c_{γ} ∈ Aut_{typ}^{I}(L) be the automorphism which sends an object P to π(γ)(P), and a morphism φ to γφγ^{−1}. Set

Out_{typ}(L) = Aut_{typ}^{I}(L)/{c_{γ} | γ ∈ Aut_{L}(S)} .

The argument that Aut_{typ}^{I}(L) is a group in this situation is exactly the same as that used in [AOV, Lemma 1.14] when S is finite.

Proposition 5.2. Let (S, F, L) be a linking triple, with structure functors

T_{Ob(L)}(S) ⟶^{δ} L ⟶^{π} F .

Fix α ∈ Aut_{typ}^{I}(L). Let β ∈ Aut(S) be the restriction of α to S under the identification S ≅ δ_{S}(S); thus α(δ_{S}(g)) = δ_{S}(β(g)) for all g ∈ S. Then β ∈ Aut(S, F). Furthermore, π ∘ α = β̂ ∘ π, where β̂ ∈ Aut(F) is the automorphism which sends P to β(P) and φ to βφβ^{−1}.

Proof. Clearly, α(S) = S, and hence α sends δ_{S}(S) to itself. Thus β is well defined. For each P ∈ Ob(L) and g ∈ P, since α sends inclusions to inclusions, it sends

\xymatrix{P \ar[r]^{ι_{P,S}} \ar[dd]_{δ_{P} (g)} & S \ar[dd]^{δ_{S}(g)} &  & α(P) \ar[r]^{ι_{α(P),S}} \ar[dd]_{α(δ_{P} (g))} & S \ar[dd]^{δ_{S}(β(g))} \\  &  & to \\ P \ar[r]^{ι_{P,S}} & S &  & α(P) \ar[r]^{ι_{α(P),S}} & S.}

The first square commutes by axiom (C) in Definition 1.9, so the second also commutes. Since restrictions in L are uniquely defined by Proposition A.4(d), this shows that α(δ_{P} (g)) = δ_{S}(β(g))|_{α(P)} = δ_{α(P)}(β(g)). Hence

δ_{α(P)}(β(P)) = α(δ_{P} (P)) = δ_{α(P)}(α(P)) ,

where the second equality holds since α is isotypical. Thus α(P) = β(P) since δ_{α(P)} is injective.

Fix P, Q ∈ Ob(L) and ψ ∈ Mor_{L}(P, Q), and set φ = π(ψ) ∈ Hom_{F}(P, Q). For each g ∈ P, α sends

\xymatrix{P \ar[r]^{ψ} \ar[dd]_{δ_{P} (g)} & Q \ar[dd]^{δ_{Q}(φ(g))} &  & β(P) \ar[r]^{α(ψ)} \ar[dd]_{δ_{β(P)}(β(g))} & β(Q) \ar[dd]^{δ_{β(Q)}(β(φ(g)))} \\  &  & to \\ P \ar[r]^{ψ} & Q &  & β(P) \ar[r]^{α(ψ)} & β(Q) .}

The first square commutes by axiom (C), so the second also commutes. By (C) again, δ_{β(Q)}(β(φ(g))) can be replaced by δ_{β(Q)}(π(α(ψ))(β(g))), leaving the second square commutative. Since δ_{β(Q)} is a monomorphism, and since morphisms in L are epimorphisms by Proposition A.4(g), it follows that β(φ(g)) = π(α(ψ))(β(g)). Thus

π(α(ψ)) = βφβ^{−1} = βπ(ψ)β^{−1}.  (3)

In particular, βφβ^{−1} ∈ Hom_{F}(β(P), β(Q)) for each P, Q ∈ Ob(L) and φ ∈ Hom_{F}(P, Q). Since Ob(L) includes all subgroups which are F-centric and F-radical, all morphisms in F are composites of restrictions of morphisms between objects of L by Alperin’s fusion theorem in the version of [BLO3, Theorem 3.6]. So βφβ^{−1} ∈ Mor(F) for all φ ∈ Mor(F), and β ∈ Aut(S, F).

Thus there is a well defined functor β̂ from F to itself which sends each P ≤ S to β(P) and sends each φ ∈ Hom_{F}(P, Q) to βφβ^{−1}. This is an automorphism of the category F by the same argument applied to α^{−1}, and π ∘ α = β̂ ∘ π by (3). □

We are now ready to define the structures which will be needed to construct extensions of linking systems.

Definition 5.3. Fix a linking triple (S, F, L) and a finite group G.

(a) An extension pair for L and G is a pair (Γ̂, τ), where Γ̂ is an extension of Γ =^{def} Aut_{L}(S) ⊴ Γ̂ by G, and where τ: Γ̂ ⟶ Aut_{typ}^{I}(L) is a homomorphism which makes both triangles in the following diagram commute:

[figure: Γ; conj; incl; (†); Aut_{typ}^{I}(L); (α↦α_{S}); (‡); Γ̂; conj; τ; Aut(Γ) ,]

Fix an extension pair U = (Γ̂, τ) for L and G. Let ρ: Γ̂ ⟶ G be the surjection with kernel Γ.

(b) Let L_{U} = L_{(Γ̂,τ)} be the category with Ob(L_{U}) = Ob(L), and with

Mor(L_{U}) = Mor(L) ×_{Γ} Γ̂ = (Mor(L) × Γ̂)/∼,

where (φ, γ) ∼ (φ′, γ′) if and only if there is λ ∈ Γ such that φ′ = φλ and γ′ = λ^{−1}γ. Here, φλ is as defined in (2). When φ ∈ Mor_{L}(γ(P), Q), the equivalence class of (φ, γ) is denoted [[φ, γ]] ∈ Mor_{L_{U}} (P, Q). Composition in L_{U} is defined by

[[ψ, η]] ∘ [[φ, γ]] = [[ψ ∘ τ(η)(φ), ηγ]].

Here, τ(η)(φ) ∈ Mor_{L}(ηγ(P), η(Q)) when φ ∈ Mor_{L}(γ(P), Q) (where we write η(P) = τ(η)(P), etc.).

Thus when U = (Γ̂, τ) is an extension pair for L and G, [[φλ, γ]] = [[φ, λγ]] in Mor(L_{U}) for all φ ∈ Mor(L), λ ∈ Γ, and γ ∈ Γ̂.

To show composition in L_{U} is well defined, we note that for all ψ, φ ∈ Mor(L), μ, λ ∈ Γ, and η, γ ∈ Γ̂ with appropriate domain and range,

[[ψμ, η]] ∘ [[φλ, γ]] = [[ψμ ∘ τ(η)(φλ), ηγ]] = [[ψμ ∘ τ(η)(φ), (ηλη^{−1})ηγ]]

= [[ψμ ∘ τ(η)(φ)μ^{−1}, μηλγ]] = [[ψ ∘ τ(μη)(φ), μηλγ]] = [[ψ, μη]] ∘ [[φ, λγ]]

The second equality follows from the commutativity of triangle (‡) in Definition 5.3, and the fourth from that of (†).

We are now ready to state and prove the main result of this section. The following theorem is a generalization to p-local compact groups of [O1, Theorem 9] (which in turn was a generalization of [BCGLO2, Theorem 4.6]).

Theorem 5.4 ([O1, Theorem 9]). Fix a linking triple (S, F, L). Set H = Ob(L), and assume it is closed under overgroups. Let

T_{H}(S) ⟶^{δ} L ⟶^{π} F

be the structure functors for L. Set Γ = Aut_{L}(S), and regard S as a subgroup of Γ via the inclusion δ: T_{H}(S) ⟶ L. Fix a finite group G and an extension pair U = (Γ, τ) for L and G, and choose S ∈ Syl_{p}(Γ). Then there is a saturated fusion system F over S containing F, and a transporter system T associated to F and containing L, such that the following hold:

(a) Ob(T ) = H =^{def} {P ≤ S | P ∩ S ∈ H}, and this set contains all subgroups of S which are F-centric and F-radical. Also, T contains L_{U} as a full subcategory.

(b) The group Γ can be identified with Aut_{T} (S) in a way so that the inclusion of L in T induces the inclusion of Γ = Aut_{L}(S) in Γ.

(c) For each γ ∈ Γ, c_{γ} = τ(γ) ∈ Aut_{typ}^{I}(L).

(d) L is normal in T (cf. Definition A.7).

(e) The space |L_{U}| is a deformation retract of |T |. The inclusion of geometric realizations |L| ⊆ |L_{U}| (≃ |T |) is homotopy equivalent to a regular covering space X ⟶ |T | with covering group G ≅ Γ/Γ.

Proof. The categories T and F will be constructed in Step 2, after preliminary constructions in Step 1. We show that T is a transporter system in Steps 3 and 4, and prove that F is saturated in Step 5. Finally we prove (d) and (e) in Step 6.

Note that S = O_{p}(Γ) ∈ Syl_{p}(Γ), since Γ/S = Aut_{L}(S)/δ_{S}(S) has order prime to p by Proposition A.4(e). For each γ ∈ Γ, let c_{γ} ∈ Aut(S) denote conjugation by γ on S = O_{p}(Γ) ⊴ Γ. By the commutativity of triangle (‡) in Definition 5.3(a), this is the restriction to S of τ(γ)_{S} ∈ Aut(Γ). Hence by Proposition 5.2, c_{γ} is fusion preserving (induces an automorphism of the category F), and τ(γ)(P) = c_{γ}(P) for all P ∈ H. To simplify notation below, we write γ(P) = τ(γ)(P) to denote this action of γ on H.

Step 1: Set L_{1} = L_{U}. Thus by Definition 5.3, Ob(L_{1}) = Ob(L) = H, and

Mor(L_{1}) = Mor(L) ×_{Γ} Γ = {[[φ, γ]] | φ ∈ Mor(L), γ ∈ Γ}.

We claim that

all morphisms in L_{1} are monomorphisms and epimorphisms.  (4)

For any [[φ, γ]], [[φ′, γ′]], and [[ψ, η]] with appropriate domain and range,

[[ψ, η]]∘[[φ, γ]] = [[ψ, η]] ∘ [[φ′, γ′]] ⟹ [[ψ ∘ τ(η)(φ), ηγ]] = [[ψ ∘ τ(η)(φ′), ηγ′]]
⟹ ∃ λ ∈ Γ, ηγ = λ^{−1}ηγ′ and ψ ∘ τ(η)(φ) = ψ ∘ τ(η)(φ′) ∘ λ
⟹ γ = (η^{−1}λη)^{−1}γ′, and φ = φ′ ∘ τ(η^{−1})(λ),

where the second equality in the last line holds since morphisms in L are monomorphisms (Proposition A.4(g)). Also, τ(η^{−1})(λ) = η^{−1}λη by the commutativity of (‡), so [[φ, γ]] = [[φ′, γ′]], and hence [[ψ, η]] is a monomorphism. The proof that morphisms are epimorphisms is similar.

Set Aut_{Γ}(S) = {c_{γ} ∈ Aut(S) | γ ∈ Γ}. Let F_{1} be the smallest fusion system over S which contains F and Aut_{Γ}(S). Define

π_{1} : L_{1} ⟶ F_{1}

to be the identity on objects, while setting π_{1}([[φ, γ]]) = π(φ) ∘ c_{γ}. The proof that this is a functor (i.e., that it preserves composition) reduces to showing that the following square commutes

\xymatrix{P \ar[r]^{π(φ)} \ar[d]_{c_{γ}} & Q \ar[d]^{c_{γ}} \\ γ(P) \ar[r]^{π(τ(γ)(φ))} & γ(Q)}

for each φ ∈ Mor_{L}(P, Q) and each γ ∈ Γ. By the above remarks, c_{γ} ∈ Aut(S, F) is the restriction to S of τ(γ)_{S}. By the last statement in Proposition 5.2, applied with α = τ(γ) and β = c_{γ}, π ∘ τ(γ) = ĉ_{γ} ∘ π, where ĉ_{γ} ∈ Aut(F) is such that ĉ_{γ}(π(φ)) = c_{γ}π(φ)c_{γ}^{−1}. Hence the square commutes. Since π_{1}(L_{1}) contains F|_{H} and Aut_{Γ}(S), and is closed under restrictions of morphisms to subgroups in H (Proposition A.4(d)), π_{1} maps onto F_{1}|_{H}.

We regard L as a subcategory of L_{1} by identifying each morphism φ ∈ Mor_{L}(P, Q) with [[φ, 1]] ∈ Mor_{L_{1}}(P, Q). By construction, π = π_{1}|_{L}. For P ≤ Q in H, the inclusion morphism ι_{P,Q} = δ_{P,Q}(1) for L is also considered as an inclusion morphism in L_{1}. The existence of restricted morphisms in L (Proposition A.4(d)) carries over easily to the existence of restricted morphisms in L_{1}, and they are unique by (4).

For all P, Q ∈ H, define

(δ_{1})_{P,Q}: N_{S}(P, Q) ⟶ Mor_{L_{1}}(P, Q)

by setting (δ_{1})_{P,Q}(s) = [[ι_{sP s^{−1},Q}, s]]. When s ∈ N_{S}(P, Q), [[ι_{sP s^{−1},Q}, s]] = [[δ_{P,Q}(s), 1]]; and thus (δ_{1})_{P,Q} extends the monomorphism δ_{P,Q} from N_{S}(P, Q) to Mor_{L}(P, Q), under the identification of L as a subcategory of L_{1}. To simplify the notation, we write δ_{1}(x) = (δ_{1})_{P,Q}(x) when P and Q are understood.

We claim that for all P, Q ∈ H, ψ ∈ Mor_{L_{1}}(P, Q), and x ∈ P,

(δ_{1})_{Q}(π_{1}(ψ)(x)) ∘ ψ = ψ ∘ (δ_{1})_{P} (x).

Set ψ = [[φ, γ]], where γ ∈ Γ and φ ∈ Mor_{L}(γ(P), Q). Then

ψ ∘ δ_{1}(x) = [[φ, γ]] ∘ [[Id_{P} , x]] = [[φ, γx]] = [[φ, c_{γ}(x)γ]] = [[φ ∘ δ(c_{γ}(x)), γ]]
= [[δ(π(φ)(c_{γ}(x))) ∘ φ, γ]] = [[δ(π_{1}(ψ)(x)) ∘ φ, γ]]
= [[δ(π_{1}(ψ)(x)), 1]] ∘ [[φ, γ]] = δ_{1}(π_{1}(ψ)(x)) ∘ ψ ,

where the fifth equality holds by axiom (C) for the linking system L.

We next show that morphisms in L_{1} have the following extension property:

∀ P, Q ∈ H, ψ ∈ Iso_{L_{1}}(P, Q), and P′, Q′ ≤ S for which P ⊴ P′, Q ⊴ Q′, and
ψδ_{1}(P′)ψ^{−1} ≤ δ_{1}(Q′), there is a unique ψ′ ∈ Mor_{L_{1}}(P′, Q′) where ψ′|_{P,Q} = ψ.

(5)

Set ψ = [[φ, γ]], where φ ∈ Mor_{L}(γ(P), Q). For all x ∈ P′,

[[φ, γ]] ∘ [[δ(x), 1]] ∘ [[φ, γ]]^{−1} = [[φ ∘ τ(γ)(δ(x)) ∘ φ^{−1}, 1]] = [[φ ∘ δ(c_{γ}(x)) ∘ φ^{−1}, 1]] ∈ δ_{1}(Q′) ,

where τ(γ)(δ(x)) = δ(c_{γ}(x)) by the commutativity of (‡). Thus φδ(γ(P′))φ^{−1} ≤ δ(Q′), so φ extends to φ′ ∈ Mor_{L}(γ(P′), Q′) by Proposition A.4(f). Set ψ′ = [[φ′, γ]]. Then ψ′|_{P,Q} = ψ since τ(γ)(ι_{P,P′}) = ι_{γ(P),γ(P′)} (i.e., τ(γ) sends inclusions to inclusions), and this proves (5).

Step 2: Throughout the rest of the proof, for each P ≤ S, we set P = P ∩ S. We next construct categories T and F_{2}, both of which have object sets H, and which contain L_{1} and the restriction of F_{1} to H, respectively. Afterwards, we let F be the fusion system over S generated by F_{2} and restrictions of morphisms.

Let T be the category with Ob(T ) = H, and where for all P, Q ∈ H,

Mor_{T} (P, Q) = {ψ ∈ Mor_{L_{1}}(P , Q) | ∀ x ∈ P, ∃ y ∈ Q such that ψ ∘ δ_{1}(x) = δ_{1}(y) ∘ ψ}. (6)

If ψ ∈ Mor(T ), then we denote the corresponding morphism in L_{1} by ψ. Let

δ_{P,Q}: N_{S}(P, Q)_{⊆N_{S}(P ,Q)} ⟶ Mor_{T} (P, Q)_{⊆Mor_{L_{1}}(P ,Q)}

be the restriction of (δ_{1})_{P ,Q}. Let F_{2} be the category with Ob(F_{2}) = H, and where

Mor_{F_{2}}(P, Q) = {φ ∈ Hom(P, Q) | ∃ ψ ∈ Mor_{L_{1}}(P , Q), where ψ ∘ δ_{1}(x) = δ_{1}(φ(x)) ∘ ψ, ∀ x ∈ P }.

Define π: T ⟶ F_{2} to be the identity on objects, and to send ψ ∈ Mor_{T} (P, Q) to the homomorphism π(ψ)(x) = y whenever ψ ∘δ_{1}(x) = δ_{1}(y) ∘ψ (uniquely defined by (6) and (4)). This is clearly a functor: it is seen to preserve composition by juxtaposing the commutative squares which define π on morphisms.

Let F be the fusion system over S generated by F_{2} and restriction of homomorphisms. Since H = Ob(F_{2}) is closed under overgroups, F_{2} is a full subcategory of F. Since L_{1} is a full subcategory of T , Hom_{F_{1}}(P, Q) = Hom_{F_{2}}(P, Q) for all P, Q ∈ H. If P, Q ≤ S are any subgroups and φ ∈ Hom_{F}(P, Q), then φ is a composite of restrictions of morphisms in F_{2}, and hence (since P ∈ Ob(F_{2}) = H implies P ∈ H) a composite of restrictions of morphisms in F_{2} (equivalently F_{1}) between subgroups in H. Thus φ ∈ Hom_{F_{1}}(P, Q); and we conclude that F_{1} is also a full subcategory of F.

Step 3: We next prove that

each P ∈ H is F-conjugate to some P′ ∈ H such that δ_{P′}(N_{S}(P′)) ∈ Syl_{p}(Aut_{T} (P′)). (7)

Fix P ∈ H. Let P be the set of all S-conjugacy classes [Q] of subgroups Q ∈ P^{F} (recall P = P ∩ S) which are fully normalized in F. (If Q is fully normalized in F, then so is every subgroup in [Q].) By Lemma 2.5, |P| is finite and prime to p.

We claim that in general, for each γ ∈ Γ and each Q, R ∈ H,

Q and R F-conjugate ⟺ γ(Q) and γ(R) F-conjugate  (8)
Q and R S-conjugate ⟺ γ(Q) and γ(R) S-conjugate.  (9)

The first holds since γ acts on L and hence on F as a group of automorphisms (Proposition 5.2), and the second since S ⊴ Γ.

Let Γ′ ⊆ Γ be the subset of those γ ∈ Γ such that γ(P) ∈ P^{F}. Then for γ_{1}, γ_{2} ∈ Γ′, γ_{1}γ_{2}(P) ∈ γ_{1}(P)^{F} by (8) and since γ_{2}(P) ∈ P^{F}, and hence γ_{1}γ_{2} ∈ Γ′. By (9) and since each γ ∈ Γ acts on S via the fusion preserving automorphism c_{γ} ∈ Aut(S) as shown above, γ permutes the S-conjugacy classes of subgroups which are fully normalized in F. Thus each element of Γ′ permutes the set P.

Fix S′ ∈ Syl_{p}(Γ′). Let η ∈ Γ be such that S′′ =^{def} ηS′η^{−1} ≤ S. Since P has order prime to p by Lemma 2.4, there is some [Q] ∈ P fixed by S′. In other words, for each γ ∈ S′, γ(Q) is S-conjugate to Q. So by (9), for each s = ηγη^{−1} ∈ S′′ (where γ ∈ S′), s(η(Q)) is S-conjugate to η(Q). Set R = η(Q). Then each coset in S′′/S contains some element s which normalizes R, i.e., the obvious homomorphism N_{S′′}(R) → S′′/S is onto with kernel N_{S}(R). Since S and S have the same identity component,

|π_{0}(N_{S}(R))| ≥ |π_{0}(N_{S′′}(R))| = |π_{0}(N_{S}(R))|·|S′′/S| = |π_{0}(N_{S}(Q))|·|S′/S| .  (10)

Since Γ′ is the subgroup of elements of Γ which send P to a subgroup in its F-conjugacy class,

|π_{0}(Aut_{T} (R))| = |π_{0}(Aut_{T} (P))| = |π_{0}(Aut_{L}(P))|·|Γ′/Γ| = |π_{0}(Aut_{L}(Q))|·|Γ′/Γ| . (11)

Since Q is fully normalized in F, S′ ∈ Syl_{p}(Γ′), and S ∈ Syl_{p}(Γ), (10) and (11) imply that δ_{R}(N_{S}(R)) is a Sylow p-subgroup of Aut_{T} (R).

Choose any ψ ∈ Iso_{T} (P , R). Then ψδ_{P} (N_{S}(P))ψ^{−1} is a p-subgroup of Aut_{T} (R). Choose χ ∈ Aut_{T} (R) such that (χψ)δ_{P} (N_{S}(P))(χψ)^{−1} ≤ δ_{R}(N_{S}(R)). By definition of the category T , χψ extends to a morphism ψ ∈ Mor_{T} (P, N_{S}(R)). Set P′ = π(ψ)(P). Then P′ = R, P′ is F-conjugate to P, and P′ ∈ H since P′ ∈ H (H is invariant under F-conjugacy). This finishes the proof of (7).

Step 4: We are now ready to show that T is a transporter system. For each P ∈ H, set

E(P) = Ker[Aut_{T} (P) ⟶^{π_{P}} Aut_{F}(P)] .

For each P, Q ∈ Ob(T ), E(P) acts on Mor_{T} (P, Q) by right composition and E(Q) by left composition. Both actions are free since all morphisms in L_{1} (hence in T ) are monomorphisms and epimorphisms by (4). We claim that π_{P,Q} is the orbit map of the action of E(P) on Mor_{T} (P, Q). Since every morphism in T (and also by definition in F) factors uniquely as the composite of an isomorphism followed by an inclusion, it suffices to prove this when P and Q are F-conjugate. It thus suffices to prove it when P = Q, and this holds by definition of E(P).

26 CARLES BROTO, RAN LEVI, AND BOB OLIVER

This proves axiom (A2). Axioms (A1) and (B) hold by construction, and (C) holds by definition of the functor π: T → F. It remains to prove axioms (I) – (III).

Fix P ∈ H such that δ_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)). By (7), every subgroup in H is Fconjugate to some such P. Write G = Aut_{T} (P), T = δ_{P} (N_{S}(P)), and P′ = δ_{P} (P) for short, where δ_{P} is injective by construction. Thus P′ ≤ T ∈ Syl_{p}(G). Fix R ∈ Syl_{p}(N_{G}(P′)), and choose α ∈ G such that αRα^{−1} ≤ T. Then αRα^{−1} ∈ Syl_{p}(N_{G}(αP′α^{−1})), and so

αP′α^{−1} ≤ αRα^{−1} = N_{T} (αP′α^{−1}).

Set Q = δ_{P}^{−1}(αP′α^{−1}) ≤ δ_{P}^{−1}(T) = N_{S}(P). Then Q is F-conjugate to P, N_{S}(Q) ≤ N_{S}(P), and so

N_{T} (αP′α^{−1}) = N_{δ_{P} (N_{S}(P))}(δ_{P} (Q)) = δ_{P} (N_{S}(Q)), and
N_{G}(αP′α^{−1}) = N_{Aut_{T} (P)}(δ_{P} (Q)) = Aut_{T} (Q) .

Thus δ_{Q}(N_{S}(Q)) ∈ Syl_{p}(Aut_{T} (Q)), and this proves axiom (I).

We next claim that

P ⊴ P′ ≤ S, Q ⊴ Q′ ≤ S, ψ ∈ Iso_{T} (P, Q), ψ(δ_{P} (P′))ψ^{−1} ≤ δ_{Q}(Q′)

⟹ ∃ ψ′ ∈ Mor_{T} (P′, Q′) with ψ′|_{P,Q} = ψ. (12)

Set ψ = ψ|_{P ,Q} = [[φ, γ]], where φ ∈ Iso_{L}(γ(P), Q) and γ ∈ Γ. By Proposition A.4(f), φ extends in a unique way to γ(P′). Hence ψ extends to some unique ψ′ ∈ Mor_{L_{1}}(P′, Q′). By definition of morphisms in T and the original hypothesis on ψ, ψ′ extends to a morphism ψ′ ∈ Mor_{T} (P′, Q′) which extends ψ. This proves (12), and thus proves axiom (II).

It remains to prove axiom (III). Fix P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ in Ob(T ) and ψ_{i} ∈ Mor_{T} (P_{i}, S) such that for all i ≥ 1, ψ_{i} = ψ_{i+1} ∘ δ_{P_{i},P_{i+1}}(1). Set P = ⋃_{i=1}^{∞} P_{i}. Let φ_{i} ∈ Mor(L) and γ_{i} ∈ Γ be such that ψ_{i} = [[φ_{i}, γ_{i}]] for each i. Since |Γ/Γ| < ∞, we can assume, after passing to a subsequence, that the γ_{i} all lie in the same coset Γγ. If γ_{i} = ν_{i}γ for some ν_{i} ∈ Γ, then ψ_{i} = [[φ_{i}ν_{i}, γ]]; we can thus assume γ_{i} = γ for each i. But then φ_{i} = φ_{i+1}|_{P_{i},P_{i+1}} for each i, so by Proposition A.4(h) applied to L, there is φ ∈ Mor_{L}(P, S) such that φ|_{P_{i},S} = φ_{i} for each i. Set ψ = [[φ, γ]] ∈ Mor_{T} (P , S); then ψ|_{P_{i}} = ψ_{i} =^{def} ψ_{i}|_{P_{i}} for each i.

Fix g ∈ P, and let i be such that g ∈ P_{i}. Set h = π(ψ_{i})(g) ∈ S. Then δ_{S}(h)∘ψ_{i} = ψ_{i} ∘δ_{P_{i}}(g) by axiom (C) for T , so (δ_{1})_{S}(h) ∘ ψ = ψ ∘ (δ_{1})_{P_{i}}(g) by (4) (the morphisms are epimorphisms in L_{1}). Hence by definition of T , there is a unique morphism ψ ∈ Mor_{T} (P, S) such that ψ|_{P} = ψ. By the uniqueness of extensions again, ψ|_{P_{i}} = ψ_{i} for each i. Step 5: We are now ready to show that F is saturated. By Theorem 4.2, it suffices to prove the following statements.

(i) The group Aut_{F}(S_{0}) is finite. (ii) For all P ≤ S, Q ≤ S_{0}, and φ ∈ Hom_{F}(P, Q), there exists φ ∈ Hom_{F}(P ·C_{S}(P)_{0}, S_{0})

such that φ|_{P} = φ. (iii) H is closed in 𝔖𝔲𝔟(S), and F is H-generated and H-saturated. (iv) For all P ∈ H and P ≤ Q ≤ P^{•}, Q ∈ H.

(v) If P ∈ F^{•} is F-centric and P ∉ H, then there is Q ∈ P^{F} such that O_{p}(Out_{F}(Q)) ∩ Out_{S}(Q) ≠ 1.

Point (i) holds by construction (and since F is saturated). By Step 4 and Proposition A.3, F is H-saturated; i.e., it satisfies the saturation axioms for subgroups in H. It is also Hgenerated by definition: each morphism in F is a composite of restrictions of morphisms between subgroups in H. Since H is closed under overgroups and F-conjugacy, H is closed under overgroups and F-conjugacy by definition, and this finishes the proofs of (iii) and (iv).

If P ≤ S, Q ≤ S_{0}, and φ ∈ Hom_{F}(P, Q), then P ≤ S since S is strongly closed in F (and S_{0} = S_{0}). By definition of F_{1}, φ = χ ∘ ψ for some χ ∈ Aut_{F}(S_{0}) (χ = π(γ)|_{S_{0}} for some γ ∈ Γ = Aut_{T} (S)) and some ψ ∈ Mor(F). Set P′ = ψ(P) ≤ S_{0}, and choose R ∈ P^{F} = P′^{F} which is receptive in F. By [BLO3, Lemma 2.4(a)], R ≤ S_{0}, and there is w ∈ Aut_{F}(S_{0}) such that w(P′) = R. Since R is receptive, w ∘ ψ extends to ψ ∈ Hom_{F}(P ·C_{S}(P)_{0}, S_{0}), so φ extends to χ ∘ w^{−1} ∘ ψ. This proves (ii).

It remains to prove (v). Let K be the set of all P ∈ F^{•} such that the saturation axioms hold for subgroups F-conjugate to P and all of their overgroups. Since H is closed under overgroups and F-conjugacy, K ⊇ H^{•}. Set K′ = F^{•}∖K, and let K′ be the set of all P = P ∩S for P ∈ K′. We will show that for all P ≤ S,

P ∈ K or P maximal in K′ ⟹ (v) holds for P  (13)

Having done that, we will prove that K′ = ∅, so K ⊇ F^{•}, and hence (using (13)) that (v) holds for all P ∈ F^{•}.

We first show that

P ∈ K or P maximal in K′ ⟹ ∃ Q ∈ P^{F} such that Q is fully normalized in F . (14)

If P is fully normalized, we are done, so assume otherwise. Let P′ be F-conjugate to P and fully normalized in F. Since F is saturated, P′ is fully automized and receptive, and so by 1.7(c), there is ρ ∈ Hom_{F}(N_{S}(P), N_{S}(P′)) such that ρ(P) = P′. Clearly, P < S, so N_{S}(P) > P. If P is maximal in K′, then by Lemma 3.5(b), the saturation axioms hold for all Q such that Q > P. So whether P ∈ K or P is maximal in K′, the saturation axioms hold for N_{S}(P), N_{S}(P′), and all subgroups of S which contain them.

Set R = N_{S}(P), and let K = {α ∈ Aut(R) | α(P) = P }. Set R′ = ρ(R) and K′ = ρK. Choose R′′ ≤ S and τ ∈ Iso_{F}(R, R′′) such that R′′ is fully ^{τ}K-normalized in F, and set P′′ = τ(P) and K′′ = ^{τ}K. Thus K′ and K′′ are the groups of automorphisms of R′ and R′′, respectively, which send P′ and P′′ to themselves. By Lemma 2.2(b), there are automorphisms χ, χ′ ∈ Aut_{F}^{K′′}(R′′), and morphisms

τ ∈ Hom_{F}(N_{S}^{K}(R), N_{S}^{K′′}(R′′)) and ρ ∈ Hom_{F}(N_{S}^{K′}(R′), N_{S}^{K′′}(R′′))

such that τ|_{R} = χτ and ρ|_{R′} = χ′τρ^{−1}.

We claim that

|N_{S}(P)| < |N_{S}^{K′}(R′)| ≤ |N_{S}^{K′′}(R′′)| ≤ |N_{S}(P′′)| .  (15)

Since P is not fully normalized in F, R′ = ρ(N_{S}(P)) < N_{S}(P′), and hence

R′ < N_{N_{S}(P′)}(R′) = N_{S}^{K′}(R′)

by Lemma 1.12. This proves the first inequality in (15). The next one holds since ρ̄ sends N_{S}^{K′}(R′) into N_{S}^{K′′}(R′′), and the last since all elements of N_{S}^{K′′}(R′′) normalize P′′. Thus P′′ is F-conjugate to P and |N_{S}(P′′)| > |N_{S}(P)|. If P′′ is not fully normalized in F, then since P^{F} contains finitely many S-conjugacy classes (Lemma 4.1), we can repeat this procedure, until we find a subgroup Q which satisfies (14).

We are now ready to prove (13). Assume P is F-centric and P ∉ H (otherwise the statement is empty), and let Q be F-conjugate to P such that Q is fully F-normalized. Thus P ∉ H = Ob(L). Since by definition, a linking system must contain all centric and radical subgroups with respect to the underlying fusion system, either P and Q are not F-centric or they are not F-radical. If Q is not F-centric, then there is g ∈ C_{S}(Q)∖Q (since Q is fully centralized). If Q is not F-radical, then O_{p}(Aut_{F}(Q)) > Inn(Q) and is contained in the Sylow subgroup Aut_{S}(Q) (Q is fully normalized), and thus there is g ∈ N_{S}(Q)∖Q such that c_{g} ∈ O_{p}(Aut_{F}(Q)). In either case,

g ∈ Q′ =^{def} {g ∈ N_{S}(Q) | c_{g} ∈ O_{p}(Aut_{F}(Q))} and g ∉ Q ,

and hence Q′ > Q. Also, Q normalizes Q′ and QQ′ > Q, so N_{QQ′}(Q) > Q, and there is x ∈ Q′∖Q such that x ∈ N_{S}(Q). For any such x, c_{x} ∉ Inn(Q) since C_{S}(Q) ≤ Q (recall P is F-centric), and c_{x}|_{Q} is in O_{p}(Aut_{F}(Q)). Also, since x ∈ S ⊴ S, [x, S] ≤ S, and so c_{x} induces

the identity on Q/Q. Thus c_{x} is in the subgroup

U = {α ∈ Aut_{F}(Q) | α|_{Q} ∈ O_{p}(Aut_{F}(Q)), α induces the identity on Q/Q} ⊴ Aut_{F}(Q).

Since the group of all α ∈ Aut(Q) which induce the identity on Q and on Q/Q is discrete p-toral by Lemma 1.13, and is contained in U with p-power index, U is a nontrivial normal discrete p-toral subgroup of Aut_{F}(Q), and U ≤ O_{p}(Aut_{F}(Q). Since x ∈ N_{S}(Q) ∖ Q, c_{x} represents a nontrivial element of Out_{S}(Q) ∩ O_{p}(Out_{F}(Q)), so (v) holds for P, and the proof of (13) is complete.

We want to show that K′ = ∅. Assume otherwise; then K′ ≠ ∅ since P ∈ K′ implies P ∈ K′. Choose Q to be maximal in K′, and choose P to be maximal among those P ∈ K′ such that P = Q. Then P is also maximal in K′. So by Lemma 3.5(b)), P is maximal among subgroups not satisfying the saturation axioms. By Lemma 4.6, this maximality of P implies that (v) does not hold for P. Since this contradicts (13), we now conclude that K′ = ∅, and hence (by (13)) that (v) holds for all P ∈ F^{•}.

Thus F is saturated. Also, (v) implies that H contains all subgroups which are F-centric and F-radical.

Step 6: By [BLO3, Corollary 3.5], F^{•} contains all subgroups which are F-centric and F-radical, so they are all contained in H by point (v) in Step 5. Point (a) holds by this together with the definition of H = Ob(T ). Point (b) holds by the definition of T in Step 2, and (c) holds by the definition of composition (of morphisms between subgroups in H) in Step 1.

Condition (i) in Definition A.7 (for the inclusion L ⊆ T ) holds by (a), and since S is strongly closed in F by construction. Condition (ii) (the Frattini condition) holds by the construction in Step 1, and (iii) (invariance of L under Aut_{T} (S)-conjugacy) holds by (c). Thus L ⊴ T .

Let r: T ⟶ T be the retraction r(P) = P and r(φ) = φ with image L_{1}. There is a natural transformation of functors r ⟶ Id_{T} which sends an object P to the inclusion ι_{P ,P}.

Hence |r| ≃ Id_{|T |}, and so |L_{U}| is a deformation retract of |T |. Also, the inclusion of |L| into |L_{U}| is homotopy equivalent to a regular covering with covering group G ≅ Γ/Γ by [OV1, Proposition A.4], and this finishes the proof of (e). □

6. The category Aut_{typ}(L)

Fix a p-local compact group (S, F, L). Let Aut_{typ}(L) be the groupoid with object set Aut_{typ}^{I}(L) and with morphisms the natural transformations. Since a natural transformation η: α ⟶ β is determined by η_{S} ∈ Aut_{L}(S) (since α(S) = β(S) = S), morphisms can be described as

Mor_{Aut_{typ}(L)}(α, β) = { χ ∈ Aut_{L}(S) | χα(φ) = β(φ)χ, all φ ∈ Mor(L) }
= { χ ∈ Aut_{L}(S) | β = c_{χ} ∘ α }.  (1)

Thus χ ∈ Aut_{L}(S) corresponds to a natural transformation η with χ = η_{S}. With this notation, composition of morphisms takes the form

\xymatrix{α \ar[rr]_{ω∘χ} \ar[r]^{χ} & β \ar[r]^{ω} & γ}

because γ = c_{ω} ∘ β = c_{ω} ∘ c_{χ} ∘ α = c_{ω∘χ} ∘ α. Here, ω ∘ χ is the composite of ω and χ in Aut_{L}(S).

Composition in Aut_{typ}^{I}(L) gives Aut_{typ}(L) the structure of a discrete strict monoidal category, where α·β = α ∘ β, and where

(α ⟶^{χ} β)·(α′ ⟶^{χ′} β′) = (α ∘ α′ ⟶_{=χ∘α(χ′)}^{β(χ′)∘χ} β ∘ β′) .  (2)

This structure makes the nerve N Aut_{typ}(L) into a simplicial group, and its geometric realization |Aut_{typ}(L)| becomes a topological group. The projection Aut_{typ}(L) ⟶ Out_{typ}(L) induces a map of simplicial groups N Aut_{typ}(L) ⟶^{pr} Out_{typ}(L), where now Out_{typ}(L), by abuse of language, denotes the discrete simplicial group with vertex set Out_{typ}(L). This projection is explicitly given by

pr(α_{0} ⟶ α_{1} ⟶ … ⟶ α_{n}) = [α_{0}] (= [α_{1}] = ⋯ = [α_{n}]) ,

and it sends π_{0}(|Aut_{typ}(L)|) isomorphically onto Out_{typ}(L).

The evaluation functor Aut_{typ}(L) × L ⟶ L induces an action of the simplicial group N Aut_{typ}(L) on the simplicial set N L as follows. In dimension 0, N_{0}Aut_{typ}(L) = Aut_{typ}^{I}(L) acts on N_{0}L = Ob(L) in the obvious way. For χ ∈ N_{1}Aut_{typ}(L) and φ ∈ Mor_{L}(P, Q) = N_{1}L,

(α ⟶^{χ} β)(P ⟶^{φ} Q) = (α(P) ⟶_{=χ_{Q}∘α(φ)}^{β(φ)∘χ_{P}} β(Q)),  (3)

and this extends naturally to higher dimensional sequences. The simplicial action induces an action of the topological group |Aut_{typ}(L)| on the space |L|.

Our aim is to describe maps BG ⟶ B|N Aut_{typ}(L)| via twisting functions (see, e.g., [Cu, Definition 3.14]). We need to show that extensions of the type constructed in Theorem 5.4 realize certain types of topological fibre bundles, and the relevant obstruction theory is encoded in the simplicial equalities that characterise the twisting functions.

For any (discrete) group G, let E(G) denote the category whose objects are the elements of G, with a unique morphism between each pair of objects. Let B(G) be the category with a single object ^{}, and with G as the automorphism group of that object. Then G acts on E(G) by translation, and the quotient category can be identified with B(G). The geometric realizations of E(G) and B(G) are the universal contractible free G-space EG, and the classifying space BG, respectively.

For a discrete group G, we will use the simplicial set N B(G)^{op} as a model for BG. This allows us to conveniently denote simplices in N_{n}B(G)^{op} by the usual bar notation

g = [g_{1}|g_{2}| ⋯ |g_{n}] =^{def} (^{}⟵^{g_{1}} ^{}⟵^{g_{2}} ^{}⋯ ^{}⟵^{g_{n}} ^{}),

where g_{i} ∈ G. We will generally omit the superscript “op” from the notation. More generally, for any small category C, we will consider the nerve of the opposite category C^{op} as a model for the nerve of C. Thus, for example, we consider face maps as

d_{0}(c_{0} ⟵^{f_{1}} c_{1} ⟵^{f_{2}} c_{2} ⋯ ⟵^{f_{n}} c_{n})= (c_{1} ⟵^{f_{2}} c_{2} ⋯ ⟵^{f_{n}} c_{n}) ,

d_{1}(c_{0} ⟵^{f_{1}} c_{1} ⟵^{f_{2}} c_{2} ⋯ ⟵^{f_{n}} c_{n})= (c_{0} ⟵^{f_{1}∘f_{2}} c_{2} ⋯ ⟵^{f_{n}} c_{n}) ,

etc.

To each simplicial group K, one associates a simplicial set W (K) (cf. [Cu, Definition 3.20] or [GJ, § V.4]), by setting W_{0}(K) = ∗,

W_{n}(K) = K_{n−1} × K_{n−2} × ⋯ K_{0} for n > 0,

and with face and degeneracy maps

d_{i}(κ_{n−1}, …, κ_{0}) = \begin{cases}(κ_{n−2}, …, κ_{0}) & if i = 0 \\ (d_{i−1}κ_{n−1}, …, (d_{0}κ_{n−i})κ_{n−i−1}, …, κ_{0}) & if 0 < i < n \\ (d_{n−1}κ_{n−1}, …, d_{1}κ_{1}) & if i = n \end{cases}

s_{i}(κ_{n−1}, …, κ_{0}) = (s_{i−1}κ_{n−1}, …, s_{0}κ_{n−i}, 1, κ_{n−i−1}, …, κ_{0}) all 0 ≤ i ≤ n

(4)

for κ_{i} ∈ K_{i}. Then |W (K)| ≃ B|K|, and so we can take \overline{W} (N Aut_{typ}(L)) as a simplicial model for classifying space of the topological group |Aut_{typ}(L)|.

A twisting function t from a simplicial set X to a simplicial group U is a collection of maps t_{n}: X_{n} → U_{n−1} satisfying certain identities formulated in [Cu, Definition 3.14]. When X = N B(G) and U = N Aut_{typ}(L), a twisting function ϕ: N B(G) ⟶ N Aut_{typ}(L) is a collection of maps

ϕ_{n}: N_{n}B(G) ⟶ N_{n−1}Aut_{typ}(L) (all n ≥ 1)

satisfying the relations

\begin{aligned}ϕ_{n−1}(d_{i}g) & = d_{i−1}ϕ_{n}(g)  for 2 ≤ i ≤ n, \\ ϕ_{n−1}(d_{1}g) & = d_{0}ϕ_{n}(g)·ϕ_{n−1}(d_{0}g) , \\ ϕ_{n+1}(s_{i}g) & = s_{i−1}ϕ_{n}(g)  for i ≥ 1, \\ ϕ_{n+1}(s_{0}g) & = 1 \end{aligned}  (5)

for all n ≥ 1 and all g ∈ N_{n}B(G). Here, 1 denotes the identity element in N_{n}Aut_{typ}(L). To a twisting function ϕ = {ϕ_{n}}_{n≥1} as above, one associates the simplicial map

φ: N B(G) ⟶ \overline{W} (N Aut_{typ}(L))  (6)

where for each g = [g_{1}| … |g_{n}] ∈ N_{n}B(G),

φ(g) = (ϕ_{n}(g), ϕ_{n−1}(d_{0}g), …, ϕ_{1}(d_{0}^{n−1}g))
= (ϕ_{n}([g_{1}| … |g_{n}]), ϕ_{n−1}([g_{2}| … |g_{n}]), …, ϕ_{1}([g_{n}])).

It is not hard to see that the simplicial equalities (5) are designed so that φ is a simplicial map. For example, φ commutes with the face maps by (5) together with the face relations

d_{0}^{j}d_{i} = \begin{cases}d_{i−j}d_{0}^{j} & if j ≤ i \\ d_{0}^{j+1} & if j ≥ i, \end{cases}

and formal manipulations.

The following lemma will be needed.

Lemma 6.1. Fix a linking triple (S, F, L), and an extension pair U = (Γ̂, τ) for L and G. Let ρ: Γ̂ ⟶ G be the surjection with kernel Γ =^{def} Aut_{L}(S), and let t_{U}: G ⟶ Γ̂ be a regular section of ρ (i.e., t_{U} is a right inverse for ρ as a map of sets, and t_{U}(1) = 1). Define maps of sets

t: G ⟶ Aut_{typ}^{I}(L)  by setting  t = τ ∘ t_{U}
v: G × G ⟶ Inn(L)  such that  t(g)t(h) = v(g, h)t(gh) ∀ g, h ∈ G (7)
χ: G × G ⟶ Aut_{L}(S) such that  t_{U}(g)t_{U}(h) = χ(g, h)t_{U}(gh) ∀ g, h ∈ G .

Then the following hold.

(a) For each g, h ∈ G, v(g, h) = c_{χ(g,h)} ∈ Inn(L).

(b) For each g ∈ G, χ(1, g) = χ(g, 1) = 1.

(c) For each g, h, k ∈ G, χ(g, h)χ(gh, k) = t(g)(χ(h, k))χ(g, hk).

Proof. Part (a) follows by the commutativity of triangle (†) in Definition 5.3(a), and (b) is immediate from the definition of χ. It remains to prove (c).

By definition,

(t_{U}(g)t_{U}(h))t_{U}(k) = χ(g, h)χ(gh, k)·t_{U}(ghk),

and

t_{U}(g)(t_{U}(h)t_{U}(k)) = ^{t_{U} (g)}χ(h, k)·χ(g, hk)·t_{U}(ghk) = t(g)(χ(h, k))·χ(g, hk)·t_{U}(ghk),

where the last equality holds because of the commutativity of triangle (‡) in Definition 5.3(a). The claim now follows by the associativity of multiplication in Γ. □

We are now ready to show that for given L and G, there is a bijective correspondence between twisting functions from N B(G) to N Aut_{typ}(L) and extension pairs for L and G.

Proposition 6.2. Fix a linking triple (S, F, L) and a finite group G.

(a) Let U = (Γ̂, τ) be an extension pair for L and G, and let t_{U}: G → Γ̂ be a regular section. Then there is a unique twisting function ϕ = ϕ_{U}: N B(G) → N Aut_{typ}(L) such that (i) for all g ∈ G, τ ∘ t_{U}(g) = ϕ_{1}([g]); and

(ii) for all g, h ∈ G, t_{U}(g)t_{U}(h) = ϕ_{2}([g|h])t_{U}(gh).

(b) Let ϕ = {ϕ_{n}} be any twisting function from N B(G) to N Aut_{typ}(L). There is an extension pair U = (Γ̂, τ), and a regular section t_{U}: G → Γ̂, such that ϕ_{U} = ϕ.

Proof. (a) Let U = (Γ̂, τ) be an extension pair for L and G, and fix a regular section t_{U}: G → Γ̂ of the natural projection. Since a simplex in the nerve of a category is determined by its 1-faces, there is at most one twisting function ϕ which satisfies (i) and (ii). We will prove that such a twisting function exists.

Define t: G → Aut_{typ}^{I}(L), v: G × G → Inn(L) and χ: G × G → Aut_{L}(S), as in (7). For g = [g_{1}| ⋯ |g_{n}] and i ≤ j, write g_{i,j} = g_{i}g_{i+1} ⋯ g_{j}. Define ϕ by setting

\begin{matrix}ϕ_{n}([g_{1}|g_{2}| … & |g_{n}]) = (t(g_{1}) ⟵^{χ(g_{1},g_{2})} t(g_{1,2})t(g_{2})^{−1}_{=v(g_{1},g_{2})^{−1}t(g_{1})} ⟵^{χ(g_{1},g_{2})^{−1}χ(g_{1},g_{2,3})} ⋯ \\ ⟵ t(g_{1,n−1})t(g_{2,n−1})^{−1}_{=v(g_{1},g_{2,n−1})^{−1}t(g_{1})} ⟵^{χ(g_{1},g_{2,n−1})^{−1}χ(g_{1},g_{2,n})} t(g_{1,n})t(g_{2,n})^{−1}_{=v(g_{1},g_{2,n})^{−1}t(g_{1})}) . \end{matrix} (8)

The equalities between the objects hold by (7) (the definition of v).

The first and third relations in (5) clearly hold for ϕ. Also, ϕ_{n}([g_{1}| ⋯ |g_{n}]) is the identity sequence if g_{1} = 1, which implies the fourth identity in (5). Thus we need only check the second relation, namely that

d_{0}ϕ_{n}([g_{1}| ⋯ |g_{n}]) · ϕ_{n−1}([g_{2}| ⋯ |g_{n}]) = ϕ_{n−1}([g_{1}g_{2}|g_{3}| ⋯ |g_{n}])  (9)

for each g_{1}, …, g_{n} ∈ G. Each side of (9) lies in N_{n−2}Aut_{typ}(L); i.e., a sequence of (n − 1) objects and (n − 2) morphisms. The objects on each side of (9) are the same, since

t(g_{1,m})t(g_{2,m})^{−1} ∘ t(g_{2,m})t(g_{3,m})^{−1} = t(g_{1,m})t(g_{3,m})^{−1}

for all m ≥ 2 (where we now set g_{3,2} = 1 and g_{2,2} = g_{2}). To show that the morphisms in the sequences coincide, we must show that

(χ(g_{1}, g_{2,m})^{−1}χ(g_{1}, g_{2,m+1}))·(χ(g_{2}, g_{3,m})^{−1}χ(g_{2}, g_{3,m+1})) = χ(g_{1,2}, g_{3,m})χ(g_{1,2}, g_{3,m+1})

for all m ≥ 2 (where again g_{3,2} = 1). By (2), the left hand side of this equation takes the form

\begin{matrix}(χ(g_{1}, g_{2,m})^{−1} & ∘ χ(g_{1}, g_{2,m+1})) ∘ (t(g_{1,m+1})t(g_{2,m+1}^{−1}))(χ(g_{2}, g_{3,m})^{−1} ∘ χ(g_{2}, g_{3,m+1})) \\ = & χ(g_{1}, g_{2,m})^{−1}χ(g_{1}, g_{2,m+1}) ∘ c_{χ(g_{1},g_{2,m+1})}^{−1}t(g_{1})(χ(g_{2}, g_{3,m})^{−1}χ(g_{2}, g_{3,m+1})) \\ = & χ(g_{1}, g_{2,m})^{−1}t(g_{1})(χ(g_{2}, g_{3,m}))^{−1} ∘ t(g_{1})(χ(g_{2}, g_{3,m+1}))χ(g_{1}, g_{2,m+1}) \\ = & (χ(g_{1}, g_{2})χ(g_{1}g_{2}, g_{3,m}))^{−1}χ(g_{1}, g_{2})χ(g_{1,2}, g_{3,m+1}) \\ = & χ(g_{1,2}, g_{3,m})^{−1}χ(g_{1,2}, g_{3,m+1}) . \end{matrix} (10)

Here, the first equality in (10) follows from (7) and the definition of v, and the third follows upon applying Lemma 6.1(c) twice. This finishes the proof that ϕ is a twisting function.

By construction, ϕ_{1}([g]) = t(g) = τ ∘ t_{U}(g), and so ϕ satisfies (i). Upon setting n = 2 in (8), we may identify ϕ_{2}([g|h]) with χ(g, h), and so (ii) holds by Lemma 6.1.

(b) Fix a twisting function ϕ, and define t: G ⟶ Aut_{typ}^{I}(L) by t(g) =^{def} ϕ_{1}([g]). Then t(1) = 1.

For each g, h ∈ G, the formulas in (5) for faces of ϕ_{2}([g|h]) take the form

d_{1}ϕ_{2}([g|h]) = t(g) and d_{0}ϕ_{2}([g|h]) = t(gh)·t(h)^{−1} .

So there is χ(g, h) ∈ N_{1}Aut_{typ}(L) = Aut_{L}(S) such that

ϕ_{2}([g|h]) = (t(g) ⟵^{χ(g,h)} t(gh)t(h)^{−1}) .

By (1), t(g) = c_{χ(g,h)} ∘ t(gh)t(h)^{−1}, and hence

t(g)t(h) = c_{χ(g,h)}·t(gh)  (11)

Also, by the degeneracy relations in (5), for each g ∈ G, χ(g, 1) = 1 = χ(1, g).

Similarly, for each g, h, k ∈ G, ϕ_{3}([g|h|k]) has faces

\begin{aligned}d_{2}ϕ_{3}([g|h|k]) & = ϕ_{2}([g|h]) , \\ d_{1}ϕ_{3}([g|h|k]) & = ϕ_{2}([g|hk]) , and \\ d_{0}ϕ_{3}([g|h|k]) & = ϕ_{2}([gh|k])·ϕ_{2}([h|k])^{−1} \end{aligned}  (12)

by (5). Hence ϕ_{3}([g|h|k]) takes the form

ϕ_{3}([g|h|k]) = (t(g) ⟵^{χ(g,h)} t(gh)t(h)^{−1} ⟵^{χ_{1}} t(ghk)t(hk)^{−1})

for some χ_{1} ∈ Aut_{L}(S). The second formula in (12) implies that χ(g, h) ∘ χ_{1} = χ(g, hk), and hence χ_{1} = χ(g, h)^{−1}χ(g, hk). The third formula in (12) now implies that

(t(gh)t(h)^{−1} ⟵^{χ(g,h)^{−1}χ(g,hk)} t(ghk)t(hk)^{−1})·(t(h) ⟵^{χ(h,k)} t(hk)t(k)^{−1})

= (t(gh) ⟵^{χ(gh,k)} t(ghk)t(k)^{−1}).

Hence by definition of the monoidal structure on Aut_{typ}(L) (see (2)),

χ(gh, k) = χ(g, h)^{−1}χ(g, hk)·(t(ghk)t(hk)^{−1})(χ(h, k)).

By (11), c_{χ(g,hk)} ∘ t(ghk)t(hk)^{−1} = t(g). Hence

χ(g, h)χ(gh, k) = χ(g, hk)·(c_{χ(g,hk)}^{−1} ∘ t(g))(χ(h, k))
= t(g)(χ(h, k))χ(g, hk) .  (13)

Now define Γ̂ = Aut_{L}(S) × G, with group multiplication

(a, g)·(b, h) = (a·t(g)(b)·χ(g, h), gh)

for each g, h ∈ G and each a, b ∈ Aut_{L}(S). Define τ: Γ̂ ⟶ Aut_{typ}^{I}(L) by setting τ(a, g) = c_{a} ∘ t(g). For each g, h, k ∈ G and each a, b, c ∈ Aut_{L}(S),

((a, g)·(b, h))·(c, k) = (a·t(g)(b)·χ(g, h)·t(gh)(c)·χ(gh, k), ghk)
= (a·t(g)(b)·t(g)t(h)(c)·χ(g, h)·χ(gh, k), ghk)
= (a·t(g)(b)·t(g)t(h)(c)·t(g)(χ(h, k))·χ(g, hk), ghk)
= (a, g)·(b·t(h)(c)·χ(h, k), hk)
= (a, g)·((b, h)·(c, k)) ,

where the second and third equalities follow from (11) and (13) respectively. Thus multiplication in Γ̂ is associative, and Γ̂ is a group, with the obvious identity and inverses. Similarly,

τ((a, g)·(b, h)) = τ(a·t(g)(b)·χ(g, h), gh) = c_{a} ∘ c_{t(g)(b)} ∘ c_{χ(g,h)} ∘ t(gh)
= c_{a} ∘ t(g)c_{b}t(g)^{−1} ∘ c_{χ(g,h)}t(gh) = c_{a} ∘ t(g)c_{b}t(g)^{−1} ∘ t(g)t(h)
= τ(a, g)τ(b, h) ,

and so τ is a homomorphism.

The triangle (†) in Definition 5.3)(a) commutes by definition of τ. Triangle (‡) commutes since

(a, g)(b, 1)(a, g)^{−1} = (at(g)(b)a^{−1}, 1) = (τ(a, g)(b), 1) ∈ Γ

for each (a, g) ∈ Γ̂ and each b ∈ Aut_{L}(S). This shows that U =^{def} (Γ̂, τ) is an extension pair for L and G.

Finally, for g ∈ G, set t_{U}(g) = (1, g) ∈ Γ̂. Then t_{U} is a regular section of the obvious projection ρ: Γ̂ → G. By the definitions, τ(t_{U}(g)) = t(g) = ϕ_{1}([g]), while for g, h ∈ G,

t_{U}(g)t_{U}(h) = (1, g)·(1, h) = (χ(g, h), gh) = (χ(g, h), 1)·(1, gh).

Upon identifying (χ(g, h), 1) ∈ Γ̂ with ϕ_{2}([g|h]), we obtain (i) and (ii). □

A twisting function ϕ: N B(G) ⟶ N Aut_{typ}(L) determines a map BG → B|N Aut_{typ}(L)| (see (6)), and hence determines a fibre bundle over BG with fibre |L| and structure group |Aut_{typ}(L)|. This pullback bundle is the realization of a simplicial set E(ϕ) = |L| ×_{ϕ} BG: a twisted cartesian product which is described as follows (cf. [Cu, (6.4)]).

Definition 6.3. For any twisting function ϕ: N B(G) → N Aut_{typ}(L), let E(ϕ) be the simplicial set with n-simplices E(ϕ)_{n} = N_{n}L × N_{n}B(G), and with face and degeneracy maps:

d_{i}(ξ, g) = \begin{cases}(ϕ_{n}(g)^{−1}·d_{0}ξ, d_{0}g) & if i = 0 \\ (d_{i}ξ, d_{i}g) & if i > 0 \end{cases}
s_{i}(ξ, g) = (s_{i}ξ, s_{i}g)  for all i

for all ξ ∈ N_{n}L and all g ∈ N_{n}B(G). Let p_{ϕ}: E(ϕ) ⟶ N B(G) be the natural projection which maps a pair (ξ, g) ∈ E(ϕ) to g.

By [Cu, § 6.4], in the situation of Definition 6.3, p_{ϕ} induces a fibre bundle |p_{ϕ}|: |E(ϕ)| → BG with fibre |L| and structure group |Aut_{typ}(L)|. We want to identify the nerve of the extension L_{U} of Definition 5.3 with E(ϕ) for the associated twisting function ϕ. Before we can do this, one more technical lemma is required.

Lemma 6.4. Let X be a simplicial set such that for each n ≥ 2, the map

D_{n} =: X_{n} ⟶^{((d_{2})^{n−1},d_{0})} {(x_{1}, x_{2}) ∈ X_{1} × X_{n−1} | d_{0}x_{1} = (d_{1})^{n−1}(x_{2}) ∈ X_{0}}

is a bijection. Then there is a category X with Ob(X) = X_{0} and Mor(X) = X_{1}, where f ∈ X_{1} is a morphism from d_{0}f to d_{1}f and Id_{x} = s_{0}x for x ∈ X_{0}, in which composition is defined as follows. If f_{1}, f_{2} ∈ X_{1} are composable morphisms (i.e., if d_{0}f_{1} = d_{1}f_{2}), then their composite is defined by setting f_{1} ∘ f_{2} = d_{1}x, where x ∈ X_{2} is the unique element such that D_{2}(x) = (f_{1}, f_{2}). In other words, f_{1} ∘ f_{2} = d_{1} ∘ D_{2}^{−1}(f_{1}, f_{2}). Furthermore, N X ≅ X as simplicial sets.

Proof. For each 0 ≤ i < n, let e_{i}^{n}: X_{n} ⟶ X_{1} be the “edge map” induced by the morphism [1] ⟶ [n] in Δ with image {i, i + 1}. Thus, for example, e_{0}^{2} = d_{2}, e_{1}^{2} = d_{0}, e_{0}^{n} = (d_{2})^{n−1} for n > 2, etc.

For each n ≥ 2, set

N_{n}X = {(f_{0}, …, f_{n−1}) ∈ (X_{1})^{n} | d_{0}f_{i} = d_{1}f_{i+1} for all 0 ≤ i ≤ n − 2}:

the set of n-tuples of composable morphisms. Set

E_{n} = (e_{0}^{n}, e_{1}^{n}, …, e_{n−1}^{n}): X_{n} ⟶ N_{n}X ⊆ (X_{1})^{n}.

It is easy to check that E_{n} = (Id × D_{2}) ∘ (Id × D_{3}) ∘ ⋯ ∘ D_{n}, and so E_{n} is a bijection since the D_{i} are. In particular, for each (f_{1}, f_{2}, f_{3}) ∈ N_{3}X, there is a unique y ∈ X_{3} such that E_{3}(y) = (f_{1}, f_{2}, f_{3}) = (d_{2}^{2}y, d_{2}d_{0}y, d_{0}^{2}y) = (d_{2}d_{3}y, d_{0}d_{3}y, d_{0}d_{1}y), and

(f_{1} ∘ f_{2}) ∘ f_{3} = d_{1}d_{3}y_{=d_{2}d_{1}y} ∘ f_{3}_{=d_{0}d_{1}y} = d_{1}d_{1}y = d_{1}d_{2}y = f_{1}_{=d_{2}d_{2}y} ∘ d_{1}d_{0}y_{=d_{0}d_{2}y} = f_{1} ∘ (f_{2} ∘ f_{3}).

The first equality follows since D_{2}(d_{3}y) = (d_{2}d_{3}y, d_{0}d_{3}y) = (f_{1}, f_{2}), and the fifth since D_{2}(d_{0}y) = (d_{0}^{2}y, d_{0}d_{0}y) = (f_{2}, f_{3}), the second and fourth equalities hold by definition of D_{2}, and the third is a simplicial identity. This shows that composition in X is associative, and hence that X is a category.

Set E_{0} = Id_{X_{0}}, E_{1} = Id_{X_{1}}, and E = {E_{n}}: X ⟶ N X. By construction, E commutes with face maps on X_{n} for n ≤ 2. We claim that d_{i} ∘ E_{n} = E_{n−1} ∘ d_{i} for n ≥ 3 and 0 ≤ i ≤ n. This is clear from the definition of E_{n}, when i = 0 or i = n, since in these two cases d_{0} and d_{n} on N_{n}X are restrictions of the obvious projections p_{0}, p_{n}: (X_{1})^{n} ⟶ (X_{1})^{n−1}. So assume 0 < i < n, and consider the map

T: X_{n} ⟶^{(e_{0}^{n},...,e_{i−2}^{n},t,e_{i+1}^{n},...,e_{n−1}^{n})} (X_{1})^{i−1} × X_{2} × (X_{1})^{n−i−1} ,

where t is induced by the morphism [2] ⟶ [n] with image {i − 1, i, i + 1}. Then E_{n} = (Id, (d_{2}, d_{0}), Id) ∘ T, while E_{n−1} ∘ d_{i} = (Id, d_{1}, Id) ∘ T. In other words, for each x ∈ X_{n} with E_{n}(x) = (f_{1}, …, f_{n}), we have E_{n−1}(d_{i}x) = (f_{1}, …, f_{i−1}, f_{i} ∘ f_{i+1}, …, f_{n}) = d_{i}(E_{n}(x)).

This proves that E commutes with face maps, and it is easily seen (by the choice of identity morphisms) to commute with degeneracies. Since all of the E_{i} are bijections, E is an isomorphism of simplicial sets. □

For any extension pair U = (Γ̂, τ) for L and G, we let pr: L_{U} ⟶ B(G) be the functor defined by setting pr([[φ, γ]]) = ρ(γ).

Proposition 6.5. Fix a linking triple (S, F, L), a finite group G, and an extension pair U = (Γ̂, τ) for L and G with a regular section t_{U}: G → Γ̂. Let ϕ = ϕ_{U} be the associated twisting function, as in Proposition 6.2(a). Then N L_{U} ≅ E(ϕ), via a simplicial isomorphism

which commutes with the projections to N B(G). In particular, |L_{U}| ⟶^{|pr_{U} |} BG is a fibre bundle over BG with fibre |L| and structure group |Aut_{typ}(L)|.

Proof. We first claim, for each n ≥ 2, that the map

D_{n} = (d_{2}^{n−1}, d_{0}): E(ϕ)_{n} ⟶ {(η_{1}, η_{2}) ∈ E(ϕ)_{1} × E(ϕ)_{n−1} | d_{0}η_{1} = d_{1}^{n−1}η_{2} ∈ E(ϕ)_{0}}

is a bijection. For ξ = (P_{0} ⟵^{φ_{1}} P_{1} ← ⋯ ← P_{n}) and g = [g_{1}| ⋯ |g_{n}],

D_{n}(ξ, g) = ((P_{0} ⟵^{φ_{1}} P_{1}, [g_{1}]) , (ϕ_{n}(g)^{−1}·d_{0}ξ, [g_{2}| ⋯ |g_{n}])) ,

and

d_{1}^{n−1}(ϕ_{n}(g)^{−1}·d_{0}ξ, [g_{2}| ⋯ |g_{n}]) = (ϕ_{1}(d_{2}^{n−1}g)^{−1}·d_{1}^{n−1}d_{0}ξ, ∗)

= (ϕ_{1}([g_{1}])^{−1}P_{1}, ∗) = d_{0}(P_{0} ⟵^{φ_{1}} P_{1}, [g_{1}]).

For fixed g, D_{n} restricts to a function from N_{n}L to the set of pairs (ξ_{1}, ξ_{2}) ∈ N_{1}L × N_{n−1}L such that

ϕ_{1}([g_{1}])^{−1}(d_{0}ξ_{1}) = d_{1}^{n−1}ξ_{2}.  (14)

This function is in fact a bijection, since by (14), the last term of ξ_{2} is the first term of ϕ_{1}([g_{1}])^{−1}(d_{0}ξ_{1}), and since ϕ_{1}([g_{1}]) is invertible. Since g = [g_{1}| ⋯ |g_{n}] is determined by [g_{1}] and d_{0}g = [g_{2}| ⋯ |g_{n}], it follows that D_{n} is a bijection.

By Lemma 6.4, E(ϕ) ≅ N E(ϕ), where E(ϕ) is the category with

Ob(E(ϕ)) = E(ϕ)_{0} = Ob(L) × {∗} and Mor(E(ϕ)) = E(ϕ)_{1} = Mor(L) × G ,

with source and target defined by d_{0} and d_{1}, respectively, with Id_{(P,∗)} = s_{0}(P, ∗) = (Id_{P} , 1), and with composition defined by d_{1} ∘ D_{2}^{−1}. Let t and χ be the functions associated to U and t_{U} via (7) (we do not use v here). For each φ ∈ Mor_{L}(P, Q) and g ∈ G,

d_{0}(Q ⟵^{φ} P, [g]) = (ϕ_{1}([g])^{−1}(P), ∗) = (t(g)^{−1}(P), ∗) ,
d_{1}(Q ⟵^{φ} P, [g]) = (Q, ∗) ,

so (φ, g) ∈ Mor_{E(ϕ)}((t(g)^{−1}(P), ∗), (Q, ∗)).

We next describe composition in E(ϕ). For each

η = (R ⟵^{φ} Q ⟵^{λ} P , [g|h]) ∈ E(ϕ)_{2},

d_{1}η = (φ ∘ λ, [gh]) and d_{2}η = (φ, [g]), while

d_{0}η = (ϕ_{2}([g|h])^{−1}(Q ⟵^{λ} P), [g])

= ((t(g) ⟵^{χ(g,h)} t(gh)t(h)^{−1})^{−1}·(Q ⟵^{λ} P ), [g])

= (t(g)^{−1}(Q) ⟵^{ψ} t(h)t(gh)^{−1}(P), [g])

where by formula (3) for the action of Mor(Aut_{typ}(L)) on Mor(L),

λ = (t(gh)t(h)^{−1} ⟶^{χ(g,h)} t(g))·ψ = t(g)(ψ) ∘ χ(g, h) .

In other words, composition in E(ϕ) satisfies

(φ, g) ∘ (ψ, h) = (φ ∘ λ, gh) = (φ ∘ t(g)(ψ) ∘ χ(g, h), gh) .  (15)

It remains to show that E(ϕ) ≅ L_{U}. By construction, each morphism in L_{U} has the form [[φ, t_{U}(g)]] for some unique φ and g. Define ω: L_{U} ⟶ E(ϕ) by setting ω(P) = (P, ∗) and ω([[φ, t_{U}(g)]]) = (φ, g). By (15), this preserves composition, and hence is an isomorphism of categories. □

7. The proof of Theorem A

Before proving Theorem A, we need one more result, which allows us to compare fibrations with fibre |L|_{p}^{∧} and fibre bundles with fibre |L|.

Proposition 7.1. For each p-local compact group (S, F, L) and each finite group G, there is a bijection Φ from the set of equivalence classes of fibre bundles over BG with fibre |L| and structure group |Aut_{typ}(L)| to the set of equivalence classes of fibrations over BG with fibre homotopy equivalent to |L|_{p}^{∧}: a bijection which sends the class of a fibre bundle to the equivalence class of its fibrewise p-completion.

Proof. For any space X, let Aut(X) denote the space of its self homotopy equivalences. Let

Ω: |Aut_{typ}(L)| ⟶ Aut(|L|) and Ω̂: |Aut_{typ}(L)| ⟶ Aut(|L|_{p}^{∧})

be the homomorphisms induced by the evaluation functor Aut_{typ}(L) × L ⟶ L and by p-completion.

By [BGM, Theorem 5.6], equivalence classes of fibrations over BG with fibre |L|_{p}^{∧} are in one-to-one correspondence with the set [BG, BAut(|L|_{p}^{∧})]_{∗} of homotopy classes of pointed maps. Also, equivalence classes of |Aut_{typ}(L)|-bundles over BG with fibre |L| are in one-to-one correspondence with the set of homotopy classes of pointed maps [BG, B|Aut_{typ}(L)|]_{∗}.

If a map f: BG ⟶ B|Aut_{typ}(L)| classifies an |Aut_{typ}(L)|-bundle ξ_{f} with fibre |L|, then BΩ̂ ∘ f classifies the fibrewise p-completion ξ̂_{f} of ξ_{f}. So we must show that the map

Φ: [BG, B|Aut_{typ}(L)|]_{∗} ⟶^{BΩ̂∘−} [BG, BAut(|L|_{p}^{∧})]_{∗}

is a bijection.

By [BLO3, Theorem 7.1], Ω̂ induces a homotopy equivalence after (componentwise) pcompletion. Hence BAut(|L|_{p}^{∧}) is the fibrewise p-completion of BAut_{typ}(L) over BOut(|L|_{p}^{∧}) ≅ BOut_{typ}(L), and there is a map of fibration sequences

\xymatrix{K(Z, 2) \ar[r] \ar[d]^{κ_{p}} & B|Aut_{typ}(L)| \ar[r] \ar[d]^{BΩ̂} & BOut_{typ}(L) \ar[d]^{≃} \\ K(Z, 2)_{p}^{∧} \ar[r] & BAut(|L|_{p}^{∧}) \ar[r] & BOut(|L|_{p}^{∧}) .}

Here, Z = Aut_{Aut_{typ}(L)}(Id), and by (1) in Section 6, it can be identified as a subgroup of Z(S). (In fact, Z = Z(F): the center of the fusion system F.) Hence it is an abelian discrete p-toral group of the form Z ≅ (ℤ/p^{∞})^{r} × A for some finite abelian p-group A.

By [BK, Proposition VI.5.1], K(ℤ/p^{∞}, 2)_{p}^{∧} ≃ K(ℤ_{p}, 3), K(ℚ_{p}, 2)_{p}^{∧} ≃ ∗, and K(ℤ_{p}, 2) is p-complete. Hence by the fibre completion lemma [BK, Lemma II.4.8], there is a map of fibration sequences

[figure: K(ℤ_{p}, 2); ≅; K(ℚ_{p}, 2); K(ℤ/p^{∞}, 2); κ; K(ℤ_{p}, 2); ∗; K(ℤ_{p}, 3) ,]

and so hofibre(κ) ≃ K(ℚ_{p}, 2). Thus K(Z, 2)_{p}^{∧} ≃ K((ℤ_{p})^{r}, 3) × K(A, 2), and the homotopy fibre of BΩ̂ is equivalent to hofibre(κ_{p}) ≃ K((ℚ_{p})^{r}, 2).

Since G is finite, H^{i}(G; (ℚ_{p})^{r}) = 0 for all i ≥ 1. Thus Φ is a bijection by obstruction theory. □

Remark 7.2. In particular, we have shown that each class of fibrations over BG with fibre |L|_{p}^{∧} has a representative which is actually an |Aut_{typ}(L)|-bundle. In other words, the structure group of a fibration can always be reduced to |Aut_{typ}(L)|.

We are now ready to prove Theorem A, in the following slightly more precise form.

Theorem 7.3. Assume f: X ⟶ Y is a finite regular covering space with covering group G, where X ≃ |L|_{p}^{∧} is the classifying space of a p-local compact group (S, F, L). Then Y _{p}^{∧} is the classifying space of a p-local compact group (S, F, L). Furthermore, there is a transporter system T associated to F and L such that L ⊴ T , and such that Aut_{T} (S)/Aut_{L}(S) ≅ G.

Proof. By Proposition 7.1, there is a fibre bundle |L| ⟶ E ⟶ BG with structure group |Aut_{typ}(L)| whose fibrewise p-completion is equivalent up to homotopy to the fibration X ⟶ X ×_{G} EG ⟶ BG. This bundle is classified by a map φ from BG to B|Aut_{typ}(L)| = |W Aut_{typ}(L)|. Since W Aut_{typ}(L) is fibrant (cf. [GJ, Corollary V.6.8]), we can assume that φ is the realization of a simplicial map, and hence is determined by a twisting function ϕ.

By Proposition 6.2(b), ϕ = ϕ_{U} for some extension pair U = (Γ̂, τ) with regular section t_{U}: G → Γ̂. By Proposition 6.5, |L_{U}| ≅ E as bundles over BG. In particular, there is a mod p equivalence from |L_{U}| to X ×_{G} EG ≃ Y.

By Theorem 5.4, there is a saturated fusion system F over a discrete p-toral group S, and a transporter system T associated to F, such that L ⊴ T , Aut_{T} (S)/Aut_{L}(S) ≅ G, and such that |T | contains |L_{U}| as deformation retract. Hence |T |_{p}^{∧} ≃ |L_{U}|_{p}^{∧} ≃ Y _{p}^{∧}. Let T_{1} ⊆ T be the full subcategory whose objects are the objects of T which are F-centric. Then Ob(T_{1}) is invariant under F-conjugacy, closed under overgroups, and contains all subgroups which are F-centric and F-radical. Hence |T_{1}|_{p}^{∧} ≃ |T |_{p}^{∧} by Proposition A.9(a). By Proposition A.6, there is a linking system L_{1} associated to T_{1} and to F, with Ob(L_{1}) = Ob(T_{1}), and such that |L_{1}|_{p}^{∧} ≃ |T_{1}|_{p}^{∧}. By Proposition A.12, there is a centric linking system L associated to F which contains L_{1} as a full subcategory, and |L|_{p}^{∧} ≃ |L_{1}|_{p}^{∧} by Corollary A.10. So |L|_{p}^{∧} ≃ Y _{p}^{∧}, and this finishes the proof of the theorem. □

Appendix A. Transporter systems over discrete p-toral groups

Transporter systems are a generalization of linking systems, which were first defined (over finite p-groups) in [OV1]. We need them here in order to state our main theorem on extensions of fusion and linking systems (Theorem 5.4) in sufficient generality. In the first half of this section, we define and prove the basic properties of transporter systems over discrete p-toral groups; especially those properties needed to prove and apply Theorem 5.4.

Afterwards, we give some conditions under which an inclusion of transporter or linking systems (one a full subcategory of the other) induces an equivalence or mod p equivalence of geometric realizations. We also prove that every linking system all of whose objects are centric can be embedded in a centric linking system. When it is a question of adding only finitely many conjugacy classes, these results can be proven using arguments similar to those already used in [BCGLO1] and [OV1] for linking and transporter systems over finite p-groups. What is new here (and makes the proofs harder) is the necessity of handling infinitely many classes at a time.

Let G be a group, and let H be a family of subgroups of G which is invariant under G-conjugacy and overgroups. The transporter system of G with respect to H is the category T_{H}(G) with object set H, and which has morphism sets

Mor_{T_{H}(G)}(P, Q) = T_{G}(P, Q) =^{def} {x ∈ G | xP x^{−1} ≤ Q}

(the transporter set) for each pair of subgroups P, Q ∈ H.

Definition A.1. Let F be a fusion system over a discrete p-toral group S. A transporter system associated to a fusion system F is a nonempty category T such that Ob(T ) ⊆ Ob(F) is closed under F-conjugacy and overgroups, together with a pair of functors

T_{Ob(T )}(S) ⟶^{ε} T ⟶^{ρ} F ,

satisfying the following conditions:

(A1) The functor ε is the identity on objects and the functor ρ is the inclusion on objects.

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

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

acts freely on Mor_{T} (P, Q) by right composition, and ρ_{P,Q}: Mor_{T} (P, Q) → Hom_{F}(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} (g)} & Q \ar[d]^{ε_{Q}(ρ(φ)(g))} \\ P \ar[r]^{φ} & Q}

commutes in T .

(I) Each F-conjugacy class of subgroups in Ob(T ) contains a subgroup P such that ε_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)); i.e., such that [Aut_{T} (P):ε_{P} (N_{S}(P))] is finite and prime to p.

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

(III) Assume P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ in Ob(T ) and ψ_{i} ∈ Mor_{T} (P_{i}, S) are such that for all i ≥ 1, ψ_{i} = ψ_{i+1} ∘ ε_{P_{i},P_{i+1}}(1). Set P = ⋃_{i=1}^{∞} P_{i}. Then there is ψ ∈ Mor_{T} (P, S) such that

ψ ∘ ε_{P_{i},P} (1) = ψ_{i} for each i.

When P ≤ Q are both in Ob(T ), we write ι_{P,Q} = ε_{P,Q}(1), considered to be the “inclusion” of P into Q. By axiom (B), ρ sends ι_{P,Q} ∈ Mor_{T} (P, Q) to incl_{P}^{Q} ∈ Hom_{F}(P, Q) (the inclusion in the usual sense).

Proposition A.2. The following hold for any transporter system T associated to a fusion system F over a discrete p-toral group S.

(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) For every morphism ψ ∈ Mor_{T} (P, Q), and every P_{∗}, Q_{∗} ∈ Ob(T ) such that P_{∗} ≤ P, Q_{∗} ≤ Q, and ρ(ψ)(P_{∗}) ≤ Q_{∗}, there is a unique morphism ψ_{∗} ∈ Mor_{T} (P_{∗}, Q_{∗}) such that ψ ∘ ι_{P,P_{∗}} = ι_{Q,Q_{∗}} ∘ ψ_{∗}.

(c) For each ψ ∈ Mor(T ), ψ is an isomorphism in T if and only if ρ(ψ) is an isomorphism in F.

(d) All morphisms in T are monomorphisms and epimorphisms in the categorical sense.

Proof. Point (a) follows from [OV1, Lemma A.7(a)], and (b) is a special case of (a). All morphisms in T are monomorphisms by [OV1, Lemma A.7(b)] and since morphisms in F are monomorphisms.

If ψ ∈ Mor_{T} (P, Q) is such that ρ(ψ) ∈ Iso_{F}(P, Q), then by (a), there are ψ′ ∈ Mor_{T} (Q, P) and ψ′′ ∈ Mor_{T} (P, Q) such that ψ ∘ ψ′ = Id_{Q} and ψ′ ∘ ψ′′ = Id_{P}. Then ψ = ψψ′ψ′′ = ψ′′ is an isomorphism in T with ψ′ as inverse. This proves (c).

It remains to prove that all morphisms in T are epimorphisms. Fix ψ ∈ Mor_{T} (P, Q) and φ_{1}, φ_{2} ∈ Mor_{T} (Q, R) such that φ_{1} ∘ ψ = φ_{2} ∘ ψ; we must show that φ_{1} = φ_{2}. Since ψ is the composite of an isomorphism followed by an inclusion by (b) and (c), it suffices to prove this when P ≤ Q and ψ = ι_{P,Q} is the inclusion.

Assume we can show that ρ(φ_{1}) = ρ(φ_{2}). By axiom (A2), 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 (a), and α = Id_{Q} since E(Q) acts freely on Mor_{T} (P, Q) (axiom (A2)). Thus φ_{1} = φ_{2}.

To complete the proof, we need to show that ρ(φ_{1}) = ρ(φ_{2}). Assume otherwise, and let P < Q be the subgroup of all g ∈ Q such that ρ(φ_{1})(g) = ρ(φ_{2})(g). Then P < N_{Q}(P) by Lemma 1.12, and ρ(φ_{1})|_{N_{Q}(P)} ≠ ρ(φ_{2})|_{N_{Q}(P)} by definition of P. So upon replacing P by P

and Q by N_{Q}(P), we can arrange that P ⊴ Q.

Set P′ = ρ(φ_{1})(P) = ρ(φ_{2})(P). By (b), 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 for each i = 1, 2:

\xymatrix{Q \ar[r]^{φ_{i}} \ar[d]_{ε_{Q}(x)} & R \ar[d]^{ε_{R}(y_{i})} & P \ar[r]^{β} \ar[d]_{ε_{P} (x)} & P′ \ar[d]^{ε_{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} (x) sends P to itself because P ⊴ Q′. Note also that ε_{P} (x) is the restriction of ε_{Q′}(x) since ε is a functor (and since ι_{U,V} = ε_{U,V} (1) for all U ≤ V in Ob(T )). Hence the second square commutes by the uniqueness of restriction morphisms (point (b)). Thus ε_{P′}(y_{1}) = ε_{P′}(y_{2}) = β ∘ ε_{P} (x) ∘ β^{−1}. Since ε_{P′} is injective (axiom (B)), this shows that y_{1} = y_{2}. Since this holds for all x ∈ Q, ρ(φ_{1}) = ρ(φ_{2}), contradicting our assumption that they are distinct. □

As usual, in the situation of Proposition A.2(b), we write ψ_{∗} = ψ|_{P_{∗},Q_{∗}} ∈ Mor_{T} (P_{∗}, Q_{∗}), regarded as the restriction of ψ.

Proposition A.3. For any fusion system F over a discrete p-toral group S, and any transporter system T associated to F, F is Ob(T )-saturated.

Proof. Set H = Ob(T ) for short. By axiom (I) for T , each subgroup in H is F-conjugate to some P such that ε_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{T} (P)). Hence

Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)) and ε_{P} (C_{S}(P)) ∈ Syl_{p}(E(P))  (1)

where as usual, E(P) = Ker[Aut_{T} (P) ⟶^{ρ_{P}} Aut_{F}(P)]. In particular, P is fully automized in F.

We claim that P is also receptive. Fix any Q ∈ P^{F} and φ ∈ Iso_{F}(Q, P). By axiom (A2), there is ψ ∈ Iso_{T} (Q, P) such that ρ_{Q,P} (ψ) = φ. Let N_{φ} ≤ N_{S}(Q) be the subgroup of all g ∈ N_{S}(Q) such that φc_{g}φ^{−1} ∈ Aut_{S}(P). Then ψε_{Q}(N_{φ})ψ^{−1} ≤ ε_{P} (N_{S}(P))·E(P). Since ε_{P} (C_{S}(P)) ∈ Syl_{p}(E(P)), ε_{P} (N_{S}(P)) ∈ Syl_{p}(ε_{P} (N_{S}(P))·E(P)). So there is χ ∈ E(P) such that

(χψ)ε_{Q}(N_{φ})(χψ)^{−1} ≤ ε_{P} (N_{S}(P)) .

Axiom (II) now implies that there is ψ ∈ Mor_{T} (N_{φ}, S) such that ψ ∘ ι_{Q,N_{φ}} = ι_{P,S} ∘ χψ, and so ρ(ψ) ∈ Hom_{F}(N_{φ}, S) is an extension of φ = ρ(ψ) = ρ(χψ).

It remains to prove that axiom (III) (for a fusion system) holds for all subgroups in H. So assume P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ are subgroups in H, P = ⋃_{i=1}^{∞} P_{i}, and φ ∈ Hom(P, S) is such that φ_{i} =^{def} φ|_{P_{i}} ∈ Hom_{F}(P_{i}, S) for each i. For each i, E(P_{i}) contains a discrete p-toral group with index prime to p, and thus is an extension of a discrete p-torus by a finite group. Also, restriction defines a homomorphism from E(P_{i+1}) to E(P_{i}), and this is injective by Proposition A.2(d). We can thus regard the E(P_{i}) as a decreasing sequence of discrete-p-toral-by-finite groups, and any such sequence becomes constant, since discrete p-toral groups are artinian (see [BLO3, Proposition 1.2]). In other words, there is N such that for all j > i ≥ N, the restriction of E(P_{j}) to E(P_{i}) is an isomorphism.

For each i ≤ N, choose ψ′_{i} ∈ Hom_{T} (P_{i}, S) such that φ_{i} = ρ_{P_{i},S}(ψ′_{i}). Set ψ_{N} = ψ′_{N}. By (A2), for each i > N, there is χ_{i} ∈ E(P_{N}) such that ψ_{N} = ψ′_{i}|_{P_{N} ,S} ∘χ_{i}. We just saw that there is χ_{i} ∈ E(P_{i}) such that χ_{i}|_{P_{N}} = χ_{i}. So if we set ψ_{i} = ψ′_{i} ∘χ_{i}, then ψ_{i}|_{P_{N} ,S} = ψ′_{i}|_{P_{N} ,S} ∘χ_{i} = ψ_{N}. Since morphisms in T are epimorphisms by Proposition A.2(d), ψ_{i+1}|_{P_{i},S} = ψ_{i} for each i > N, and by axiom (III) for T , there is ψ ∈ Mor_{T} (P, S) such that ψ|_{P_{i},S} = ψ_{i} for each i ≥ N. Then ρ(ψ)|_{P_{i},S} = φ_{i} for each i, so φ = ρ(ψ) ∈ Hom_{F}(P, S). □

We next show that linking systems over saturated fusion systems are transporter systems and characterize linking systems among transporter systems. This is a generalization to linking systems over discrete p-toral groups of [O1, Proposition 4]. We will first list the properties of linking systems that we need as a separate proposition that might be useful for future reference and then state the result as a corollary.

Proposition A.4. The following hold for any linking system L associated to a saturated fusion system F over a discrete p-toral group S.

(a) For each P, Q ∈ Ob(L), the subgroup E(P) =^{def} Ker[Aut_{L}(P) ⟶ Aut_{F}(P)] acts freely on Mor_{L}(P, Q) via right composition, and π_{P,Q} induces a bijection

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

(b) The functor δ is injective on all morphism sets.

(c) The action of E(Q) on Hom_{F}(P, Q) via left composition is free.

(d) For every morphism ψ ∈ Mor_{L}(P, Q), and every P_{∗}, Q_{∗} ∈ Ob(L) such that P_{∗} ≤ P, Q_{∗} ≤ Q, and π(ψ)(P_{∗}) ≤ Q_{∗}, there is a unique morphism ψ|_{P_{∗},Q_{∗}} ∈ Mor_{L}(P_{∗}, Q_{∗}) (the “restriction” of ψ) such that ψ ∘ ι_{P_{∗},P} = ι_{Q_{∗},Q} ∘ ψ|_{P_{∗},Q_{∗}}.

(e) If P ∈ Ob(L) is fully normalized in F, then δ_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{L}(P)).

(f) Let P ⊴ P ≤ S and Q ≤ Q ≤ S be objects in L. Let ψ ∈ Mor_{L}(P, Q) be such that for each g ∈ P, there is h ∈ Q satisfying ι_{Q,Q} ∘ ψ ∘ δ_{P} (g) = δ_{Q,Q}(h) ∘ ψ. Then there is a

unique morphism ψ ∈ Mor_{L}(P , Q) such that ψ|_{P,Q} = ψ.

(g) All morphisms in L are monomorphisms and epimorphisms in the categorical sense.

(h) Assume P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ in Ob(L) and ψ_{i} ∈ Mor_{L}(P_{i}, S) are such that for all i ≥ 1, ψ_{i} = ψ_{i+1}|_{P_{i},S}. Set P = ⋃_{i=1}^{∞} P_{i}. Then there is ψ ∈ Mor_{L}(P, S) such that ψ|_{P_{i},S} = ψ_{i} for each i.

Proof. Points (a) and (b) are exactly the same as points (a) and (c), respectively, in [O1, Proposition 4], and the proofs go through unchanged. The proof of (c) is contained in that of [O1, Proposition 4(f)], again with no modification necessary. We prove the remaining points.

(d) This is a special case of [BLO3, Lemma 4.3(a)] (which is Proposition A.2(a) for linking systems).

(e) For each P ∈ Ob(L) which is fully centralized in F,

Aut_{F}(P) ≅ Aut_{L}(P)/δ_{P} (C_{S}(P)) and Aut_{S}(P) ≅ N_{S}(P)/C_{S}(P) .

Hence [Aut_{F}(P):Aut_{S}(P)] = [Aut_{L}(P):δ_{P} (N_{S}(P))], since δ_{P} is injective. If P ∈ Ob(L) is fully normalized in F, then Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)), and so δ_{P} (N_{S}(P)) ∈ Syl_{p}(Aut_{L}(P)).

(f) The proof of existence of an extension ψ is identical in our case to the proof of the corresponding statement [O1, Propostion 4(e)]. It remains to prove uniqueness.

Assume ψ_{1}, ψ_{2} ∈ Iso_{L}(P , Q) are two extensions of ψ. We must show that ψ_{1} = ψ_{2}. It suffices to do this when ψ is an isomorphism, and also (after composing by an isomorphism, if necessary) when P is fully centralized in F.

Fix x ∈ P, set y_{i} = π(ψ_{i})(x), and consider the following two squares of morphisms in L:

\xymatrix{P \ar[r]^{^{ψ^{i}}} \ar[d]_{δ_{P} (x)} & Q \ar[d]^{δ_{Q}(y_{i})} & P \ar[r]^{ψ}_{≅} \ar[d]_{δ_{P} (x)} & Q \ar[d]^{δ_{Q}(y_{i})} \\ P \ar[r]^{_{ψ_{i}}} & Q & P \ar[r]^{ψ}_{≅} & Q}

The first square commutes by axiom (C), and the second square is defined to be a restriction of the first. Note that δ_{P} (x) is the restriction of δ_{P} (x) since δ is a functor (and since ι_{U,V} = δ_{U,V} (1) for all U ≤ V in Ob(L)). Hence the second square commutes by the uniqueness of restriction morphisms (point (d)). Thus δ_{Q}(y_{1}) = ψ ∘ δ_{P} (x) ∘ ψ^{−1} = δ_{Q}(y_{2}). Since δ_{Q} is injective by (b), this shows that y_{1} = y_{2}. Since this holds for all x ∈ P, π(ψ_{1}) = π(ψ_{2}).

By axiom (A) (and since P is fully centralized), ψ_{2} = ψ_{1} ∘ δ_{P} (a) for some a ∈ C_{S}(P). Hence

ι_{Q,Q} ∘ ψ = ψ_{2}|_{P,Q} = (ψ_{1} ∘ δ_{P} (a))|_{P,Q} = ψ_{1} ∘ δ_{P,P} (a) = ψ_{1}|_{P,Q} ∘ δ_{P} (a) = ι_{Q,Q} ∘ ψ ∘ δ_{P} (a),

so δ_{P} (a) = 1 since E(P) acts freely on Mor_{L}(P, Q) by (a) (note that a ∈ C_{S}(P)). Then a = 1 since δ_{P} is injective by (b), and so ψ_{1} = ψ_{2}.

(g) By [OV1, Lemma A.6], for each φ ∈ Mor(L), φ is an isomorphism in L if and only if π(φ) is an isomorphism in F. Hence by (d), each morphism in L is the composite of an isomorphism followed by an inclusion. So it suffices to prove that inclusions are monomorphisms and epimorphisms. That they are monomorphisms follows from the uniqueness of restrictions in (d).

By the uniqueness of extensions in (f), any inclusion ι_{P,Q} for P ⊴ Q is an epimorphism. Since a composite of epimorphisms is an epimorphism, this proves that ι_{P,Q} is an epimorphism if P < Q with finite index.

Assume this is not true in general. Then there are ψ_{1}, ψ_{2} ∈ Iso_{L}(P, Q) and P_{∗} < P such that ψ_{1} ≠ ψ_{2} and ψ_{1}|_{P_{∗}} = ψ_{2}|_{P_{∗}}. Choose P_{∗} < P_{1} ≤ P which is minimal among subgroups of P containing P_{∗} for which ψ_{1}|_{P_{1}} ≠ ψ_{2}|_{P_{1}} (S is artinian by [BLO3, Proposition 1.2]). There is no proper subgroup P_{2} < P_{1} of finite index which contains P_{∗} (otherwise ψ_{1}|_{P_{2}} = ψ_{2}|_{P_{2}} and the result follows from (f)), so P_{1} is the union of its proper subgroups which contain P_{∗}. In particular, π(ψ_{1})|_{P_{1}} = π(ψ_{2})|_{P_{1}}. The rest of the argument to show that ψ_{1}|_{P_{1}} = ψ_{2}|_{P_{1}} (thus giving a contradition) goes through exactly as in the last paragraph of the proof of (f). (Note that we can easily arrange for P to be fully centralized.)

(h) Assume P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ in Ob(L) and ψ_{i} ∈ Mor_{L}(P_{i}, S) are such that for each i ≥ 1, ψ_{i} = ψ_{i+1}|_{P_{i},S}. Set P = ⋃_{i=1}^{∞} P_{i}, and set φ_{i} = π(ψ_{i}) ∈ Hom_{F}(P_{i}, S) for each i. Then φ_{i+1}|_{P_{i}} = φ_{i} for each i, so the union of the φ_{i} is a homomorphism φ ∈ Hom(P, S). By axiom (III) in Definition 1.4, φ ∈ Hom_{F}(P, S).

By Lemma 3.2(d), there is N such that P_{i}^{•} = P^{•} for each i ≥ N. Let Q_{N} ∈ (P_{N})^{F} be fully centralized in F, fix ν ∈ Iso_{F}(P_{N}, Q_{N}), and set Q = ν^{•}(P). Thus E(Q_{N}) = δ_{Q_{N}} (C_{S}(Q_{N})), each element of C_{S}(Q_{N}) also centralizes Q_{N}^{•} by the uniqueness of extensions to Q_{N}^{•}, so each element of E(Q_{N}) extends to an element of E(Q), which implies that each element of E(P_{N}) extends to an element of E(P).

Choose ψ′ ∈ Mor_{L}(P, S) such that π(ψ′) = φ. By axiom (A), there is χ_{N} ∈ E(P_{N}) such that ψ_{N} = ψ′|_{P_{N} ,S} ∘ χ_{N}. Let χ ∈ E(P) be such that χ|_{P_{N}} = χ_{N}, and set ψ = ψ′ ∘ χ. Then ψ|_{P_{N} ,S} = ψ′|_{P_{N} ,S} ∘χ_{N} = ψ_{N}, and ψ|_{P_{i},S} = ψ_{i} for each i: via composition with inclusions when i ≤ N, and by (g) (morphisms in L are epimorphisms) when i > N. □

We are now ready to show that all linking systems are transporter systems. Corollary A.5. Fix a saturated fusion system F over a discrete p-toral group S. Then each linking system L associated to F is also a transporter system. Conversely, a transporter system T associated to F which contains all F-centric F-radical subgroups as objects, and such that the kernel subgroups E(P) are all discrete p-toral groups, is a linking system.

Proof. Assume L is a linking system associated to F. Axiom (A2) in Definition A.1 follows from Proposition A.4(a,c), (B) and (C) follow from the corresponding axioms in Definition 1.9, and (I), (II), and (III) follow from points (e), (f), and (h), respectively, in Proposition A.4. Thus L is a transporter system.

Assume now that T is a transporter system associated to F such that Ob(T ) contains all F-centric F-radical subgroups of S. By axiom (A2), ρ_{P,Q} sends Mor_{T} (P, Q) surjectively onto Hom_{F}(P, Q) for each P, Q ∈ Ob(T ), so every object of T is isomorphic in T to an object which is fully centralized. If P is fully centralized, then ε_{P} sends C_{S}(P) injectively to a Sylow p-subgroup of E(P), and hence E(P) = ε_{P} (C_{S}(P)) in that case since we are assuming that E(P) is discrete p-toral. It follows that T is a linking system associated to F. □ Proposition A.6. Let F be a saturated fusion system over a discrete p-toral group S. Let T be a transporter system associated to F, all of whose objects are F-centric, which contains all F-centric F-radical subgroups of S. Then for every P ∈ Ob(T ), E(P) ≅ E_{0}(P) × Z(P) for some E_{0}(P) ⊴ E(P) which is finite of order prime to p and normal in Aut_{T} (P). There is a linking system L associated to F, defined by setting Ob(L) = Ob(T ) and

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

and the natural functor T −↠ L induces a mod p equivalence between the geometric realizations and hence a homotopy equivalence |T |_{p}^{∧} ≃ |L|_{p}^{∧}.

Proof. By axiom (C), for all P ∈ Ob(T ), E(P) commutes with ε_{P} (P) in the group Aut_{T} (P). Hence ε_{P} (Z(P)) = ε_{P} (P) ∩ E(P) is central in E(P), and it has finite index prime to p by axiom (I) and since P is F-centric. So E(P) ≅ Z(P)×E_{0}(P) for some unique E_{0}(P) ⊴ E(P) which is finite of order prime to p.

It is now straightforward to check that L, when defined as above, is a quotient category of T (i.e., composition is well defined). Axioms (A1), (A2), (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. The induced map of spaces |T | ⟶ |L| is a mod p equivalence (and hence |T |_{p}^{∧} ≃ |L|_{p}^{∧}) by [BLO1, Lemma 1.3]. □

We next define normal transporter subsystems. Recall that for any fusion system F over a discrete p-toral group S, a normal subgroup S ⊴ S is weakly closed in F if for each g ∈ S, g^{F} ⊆ S. In other words, no element of S is F-conjugate to any element of S∖S.

Definition A.7. Fix a pair of saturated fusion systems F ⊆ F over discrete p-toral groups S ⊴ S, and let T ⊆ T be associated transporter systems. Then T is normal in T (T ⊴ T ) if

(i) S is strongly closed in F and Ob(T ) = {P ∩ S | P ∈ Ob(T )};

(ii) for all P, Q ∈ Ob(T ) and ψ ∈ Mor_{T} (P, Q), there are morphisms γ ∈ Aut_{T} (S) and ψ_{∗} ∈ Mor_{T} (γ(P), Q) such that ψ = ψ_{∗} ∘ γ|_{P,γ(P)}; and

(iii) for all P, Q ∈ Ob(T ), ψ ∈ Mor_{T} (P, Q), and γ ∈ Aut_{T} (S), γ|_{Q,γ(Q)} ∘ ψ ∘ γ|_{P,γ(P)}^{−1} is in Mor_{T} (γ(P), γ(Q)).

Here, in (ii) and (iii), we write γ(P) = ρ(γ)(P) and γ(Q) = ρ(γ)(Q) for short. In this situation, we define

T /T = Aut_{T} (S)/Aut_{T} (S) .

Let T be a transporter system. A subgroup P ∈ Ob(T ) is defined to be T -radical if O_{p}(Aut_{T} (P)) = ε_{P} (P). When T is a linking system associated to F, then P is T -radical if and only if it is F-centric and F-radical. If T = T_{S}(G) for a finite group G and S ∈ Syl_{p}(G), then P ≤ S is T -radical if and only if it is a radical p-subgroup of G in the usual sense.

Lemma A.8. Let F be a saturated fusion system over a discrete p-toral group S, and let T be a transporter system associated to F. Let P ∈ Ob(T ) be a minimal object, let P be the F-conjugacy class of P, and let T_{0} ⊆ T the full subcategory with object set Ob(T )∖P. Assume that

• P is fully normalized,

• N_{S}(P)/P is finite, and

• P is either not F-centric or not T -radical.

Then the inclusion of |T_{0}| into |T | induces an isomorphism in mod p cohomology.

Proof. Let Φ: T^{op} ⟶ ℤ_{(p)}-mod be the functor which sends objects in P to 𝔽_{p} and other objects to 0, regarded as a subfunctor of the constant functor \underline{𝔽}_{p}. By the minimality of P, and since the quotient functor \underline{𝔽}_{p}/Φ vanishes on P, it follows that

H^{∗}(T , \underline{𝔽}_{p}/Φ) ≅ H^{∗}(T_{0}, \underline{𝔽}_{p}) ≅ H^{∗}(|T_{0}|, 𝔽_{p}),

where the second isomorphism holds because \underline{𝔽}_{p} is constant on T_{0}. Upon applying H^{∗}(T , −) to the short exact sequence of functors associated to the inclusion of Φ in \underline{𝔽}_{p}, and using the above isomorphism, we conclude that H^{∗}(T , Φ) ≅ H^{∗}(|T |, |T_{0}|, 𝔽_{p}).

Let O(T ) denote the orbit category associated to T : the category with the same objects as T , and with morphism sets Mor_{O(T )}(P, Q) = Mor_{T} (P, Q)/Q, where Q ≅ ε_{Q}(Q) acts freely by left composition. The projection T → O(T ) is target regular in the sense of [OV1, Definition A.5(b)], so by [OV1, Proposition A.11], there is a spectral sequence

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

Since N_{S}(P)/P is finite, and [Aut_{T} (P):ε_{P} (N_{S}(P))] < ∞ by axiom (I) (recall P is fully normalized), Aut_{T} (P)/ε_{P} (P) is also finite. By [BLO3, Proposition 5.4], and since Φ(Q) = 0 for Q ∉ P,

H^{i}(O(T ); H^{j}(−; Φ(−))) ≅ Λ^{i}(Aut_{T} (P)/ε_{P} (P); H^{j}(P; Φ(P)))
≅ Λ^{i}(Aut_{T} (P)/ε_{P} (P); H^{j}(P; \underline{𝔽}_{p}))

for each i and j. If P is not F-centric (and since it is fully normalized), there is g ∈ C_{S}(P)∖P, and the class of ε_{P} (g) ≠ 1 in Aut_{T} (P)/ε_{P} (P) acts trivially on H^{j}(P; \underline{𝔽}_{p}). If P is not T -radical, then O_{p}(Aut_{T} (P)/ε_{P} (P)) ≠ 1. In either case, by [JMO, Proposition 6.1(ii)], Λ^{i}(Aut_{T} (P)/ε_{P} (P); H^{j}(P; \underline{𝔽}_{p})) = 0 for each i and j, so

H^{∗}(|T |, |T_{0}|; 𝔽_{p}) ≅ H^{∗}(T ; Φ) = 0.

Thus the inclusion |T_{0}| → |T | induces an isomorphism on mod p cohomology as claimed. □

Lemma A.8 gives us a tool for proving that under certain conditions, adding or subtracting one conjugacy class in a transporter system does not change the (mod p) homotopy type of its realization. However, more is needed when dealing with infinitely many conjugacy classes. One problem when doing this is that the “bullet functor” P ↦ P^{•} need not lift to a functor on a transporter system. This is why we need to assume the existence of a normal linking subsystem in the following proposition.

Proposition A.9. Let F be a saturated fusion system over a discrete p-toral group S, and let T be a transporter system associated to F. Let T_{0} ⊆ T be a full subcategory which is a transporter subsystem associated to a full subcategory F_{0} ⊆ F. Assume there is a normal linking subsystem L ⊴ T over a subgroup S ≤ S of finite index, and let F ⊆ F be the corresponding fusion subsystem. Assume also that either

(a) all objects in L are F-centric and Ob(T_{0}) ⊇ Ob(T ) ∩ Ob(F^{c}); or

(b) each object in Ob(T )∖Ob(T_{0}) is F-centric and not T -radical.

Then the inclusion of |T_{0}| in |T | is a mod p homology equivalence.

Proof. For each P ≤ S, we write P = P ∩ S, and set P^{∨} = P P^{•} (where (−)^{•} means the bullet construction for F). By Lemma 3.2(b), N_{S}(P) ≤ N_{S}(P^{•}), and since P ⊴ P, it follows that P normalizes P^{•} (so P^{∨} is a subgroup). Also, since P^{•} ≤ S, P^{∨} = P ·P^{•} = P^{•}, and so (P^{∨})^{∨} = P^{∨}.

For each P, Q ∈ Ob(T ) and each ψ ∈ Mor_{T} (P, Q), let ψ = ψ|_{P ,Q}. Then ψ extends to

ψ ∈ Mor_{T} (P, Q) and to ψ^{•} ∈ Mor_{T} (P^{•}, Q^{•}) (Lemma 3.5(a)), and hence by axiom (II) to ψ^{∨} ∈ Mor_{T} (P^{∨}, Q^{∨}). By the uniqueness of these extensions, ψ^{∨}|_{P} = ψ, and thus (−)^{∨} defines an idempotent functor from T to itself.

Let T^{∨} ⊆ T be the full subcategory whose objects are those P such that P^{∨} = P (and use the same notation for the corresponding subcategory of any sub-transporter system of T ). For each P ≤ S, P^{∨}/P^{•} is a subgroup of order at most |S/S| in N_{S}(P^{•})/P^{•}, and by [BLO3, Lemma 1.4(a)], there are only finitely many conjugacy classes of such subgroups. Since F^{•} has finitely many S-conjugacy classes of objects, T^{∨} also has finitely many S-conjugacy classes of objects.

Let ℌ be the set of all transporter subsystems T′ ⊆ T such that

• T′ ⊇ T_{0},

• T′ is a full subcategory of T , and • the inclusion |T_{0}| ⊆ |T′| is a mod p homology equivalence in case (a), or a homotopy

equivalence in case (b).

We must show that T ∈ ℌ.

Assume otherwise, and choose T_{1} ∈ ℌ for which Ob(T_{1}) ∩ Ob(T^{∨}) contains the largest possible number of F-conjugacy classes. Let T_{2} ⊆ T be the full subcategory with Ob(T_{2}) = {P ∈ Ob(T ) | P^{∨} ∈ Ob(T_{1})}. By the above discussion, |T_{1}^{∨}| = |T_{2}^{∨}| is a strong deformation retract of |T_{1}| and of |T_{2}|, so |T_{1}| ≃ |T_{2}|, and T_{2} ∈ ℌ.

Since T_{2} ⫋ T by assumption, Ob(T^{∨}) ⊈ Ob(T_{2}). Let P be maximal among objects in T^{∨} not in T_{2}. By definition of Ob(T_{2}), P is maximal among all objects of T not in T_{2}. Let T_{3} ⊆ T be the full subcategory with Ob(T_{3}) = Ob(T_{2}) ∪ P^{F}.

Now, N_{S}(P)/P C_{S}(P) ≅ Out_{S}(P) is finite by [BLO3, Lemma 2.5]. In case (a), P C_{S}(P)/P is finite since P is F-centric and [S:S] < ∞, while C_{S}(P) ≤ P in case (b). So |N_{S}(P)/P | < ∞ in either case.

By assumption, P is either not F-centric (case (a)) or not T -radical (case (b)). So by Lemma A.8, the inclusion of |T_{2}| into |T_{3}| is a mod p equivalence. Hence T_{3} ∈ ℌ, contradicting our maximality assumption on T_{1}. □

Corollary A.10. Let (S, F, L) be a linking triple, and let L_{0} ⊆ L be a full subcategory which is a linking subsystem associated to some F_{0} ⊆ F. Then the inclusion of |L_{0}| into |L| is a mod p homology equivalence.

Proof. Since L is a linking system, a subgroup P ≤ S is L-radical exactly when it is F-centric and F-radical. So by definition, L_{0} contains all L-radical subgroups. Hence the lemma is a special case of Proposition A.9(b). □

It remains to look at the problem of constructing a centric linking system which contains as full subcategory a linking system over a smaller set of objects. As usual, we first check what happens when we add one conjugacy class of objects.

Lemma A.11. Let F be a saturated fusion system over a discrete p-toral group S, and let F^{c} be the full subcategory of F-centric objects. Let F_{0} ⊆ F_{1} ⊆ F^{c} be full subcategories such that Ob(F_{0}) and Ob(F_{1}) are invariant under F-conjugacy and closed under overgroups, F_{0} contains all F-centric F-radical subgroups of S, and Ob(F_{1}) = Ob(F_{0}) ∪ P for some Fconjugacy class P. If L_{0} is a linking system associated to F_{0}, then there is a linking system L_{1} associated to F_{1} such that L_{0} is isomorphic as a linking system to the full subcategory of L_{1} with same set of objects as L_{0}.

Proof. This follows by the same proof as in [AKO, Proposition III.4.8] (Steps 1–3), together with Lemma 2.4.

Alternatively, let Φ be the functor: Φ(P) = Z(P) when P ∈ P, and Φ(P) = 0 when P ∈ Ob(F_{0}). For P ∈ P, Λ^{∗}(Out_{F}(P); Φ(P)) = 0 by [JMO, Proposition 6.1(ii)] and since O_{p}(Out_{F}(P)) ≠ 1. Hence by [BLO3, Proposition 5.4], H^{3}(O(F_{1}); Φ) = 0. By an argument similar to that used to prove [BLO2, Proposition 3.1], the obstruction to extending L_{0} to L_{1} lies in H^{3}(O(F_{1}); Φ), and hence L_{1} does exist. □

This is now generalized as follows.

Proposition A.12. Let F be a saturated fusion system over a discrete p-toral group S, and let F^{c} be the full subcategory of F-centric objects. Let F_{0} ⊆ F^{c} be a full subcategory such that Ob(F_{0}) is invariant under F-conjugacy and closed under overgroups, and contains all F-centric F-radical subgroups of S. If L_{0} is a linking system associated to F_{0}, then there is a centric linking system L associated to F (associated to F^{c}) which contains L_{0} as a linking subsystem.

Proof. Let ℌ be the set of all families of subgroups H ⊆ Ob(F^{c}) such that

• H ⊇ Ob(F_{0}),

• H is invariant under F-conjugacy and closed under overgroups, and

• there is a linking system with object set H which contains L_{0} as a linking subsystem.

We must show that Ob(F^{c}) ∈ ℌ.

Choose H_{1} ∈ ℌ for which H_{1} ∩ F^{•} contains the largest possible number of F-conjugacy classes. Set H_{2} = {P ∈ Ob(F^{c}) | P^{•} ∈ H_{1}}, and let F_{1} ⊆ F_{2} ⊆ F be the full subcategories with Ob(F_{i}) = H_{i}. Let L_{2} be the pullback of L_{1} and F_{2} via the functors

L_{1} ⟶^{π_{1}} F_{1} ⟵^{(−)^{•}} F_{2} .

Then L_{2} is a linking system associated to F_{2} (recall that Z(P) = Z(P^{•})), and hence H_{2} ∈ ℌ.

Assume Ob((F^{c})^{•})∖H_{1} ≠ ∅, and let P be maximal among subgroups in this set. Then P is maximal among subgroups in Ob(F^{c})∖H_{2}, so H_{2} ∪ P^{F} ∈ ℌ by Lemma A.11, and this contradicts the choice of H_{1}. Hence Ob((F^{c})^{•}) ⊆ H_{1}, and so H_{2} = Ob(F^{c}) ∈ ℌ. □

References

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Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

E-mail address: broto@mat.uab.es

Institute of Mathematics, University of Aberdeen, Fraser Noble Building 138, Aberdeen AB24 3UE, U.K.

E-mail address: r.levi@abdn.ac.uk

LAGA, Institut Galilée, Université Paris 13, Av. J-B Clément, 93430 Villetaneuse, France

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