DISCRETE MODELS FOR THE p-LOCAL HOMOTOPY THEORY OF COMPACT LIE GROUPS AND p-COMPACT GROUPS

CARLES BROTO, RAN LEVI, AND BOB OLIVER

Abstract. We define and study a certain class of spaces which includes p-completed classifying spaces of compact Lie groups, classifying spaces of p-compact groups, and p-completed classifying spaces of certain locally finite discrete groups. These spaces are determined by fusion and linking systems over “discrete p-toral groups” — extensions of (ℤ/p^{∞})^{r} by finite p-groups — in the same way that classifying spaces of p-local finite groups as defined in [BLO2] are determined by fusion and linking systems over finite p-groups. We call these structures “p-local compact groups”.

In our earlier paper [BLO2], we defined and studied a certain class of spaces which in many ways behave like p-completed classifying spaces of finite groups. These spaces occur as “classifying spaces” of certain algebraic objects called p-local finite groups. The purpose of this paper is to generalize the concept of p-local finite groups to what we call p-local compact groups. The motivation for introducing this family comes from the observation that p-completed classifying spaces of finite and compact Lie groups, as well as classifying spaces of p-compact groups [DW], share many similar homotopy theoretic properties, but earlier studies of these properties usually required different techniques for each case. Moreover, while p-completed classifying spaces of finite and, more generally, compact Lie groups arise from the algebraic and geometric structure of the groups in question, p-compact groups are purely homotopy theoretic objects. Unfortunately, many of the techniques used in the study of p-compact groups fail for p-completed classifying spaces of general compact Lie groups. With the approach presented here, we propose a framework general enough to include p-completed classifying spaces of arbitrary compact Lie groups as well as p-compact groups.

The new idea here is to replace fusion systems over finite p-groups, as handled in [BLO2], by fusion systems over discrete p-toral groups. A discrete p-toral group is a group which contains a discrete p-torus (a group of the form (ℤ/p^{∞})^{r} for finite r ≥ 0) as a normal subgroup of p-power index. A p-local compact group consists of a triple (S, F, L), where S is a discrete p-toral group, F is a saturated fusion system over S (a collection of fusion data between subgroups of S arranged in the form of a category and satisfying certain axioms), and L is a centric linking system associated to F (a category whose objects are a certain distinguished subcollection of the object of F, and of which the corresponding full subcategory of F is a quotient category). The linking system L allows us to define the classifying space of this p-local compact group to be the p-completed nerve |L|_{p}^{∧}. If S is a finite p-group, then the theory reduces to the case of p-local finite groups as studied in [BLO2].

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 57T10.

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

C. Broto is partially supported by MEC grant MTM2004–06686.

R. Levi is partially supported by EPSRC grant MA022 RGA0756.

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

All three of the authors were partially supported by the Mittag-Leffler Institute in Sweden, and also by the EU grant nr. HPRN-CT-1999-00119.

We hope that working with this setup will make it possible to prove results of interest in a uniform fashion for the entire family. In this paper (Theorem 7.1), we give a combinatorial description of the space of self equivalences of |L|_{p}^{∧} in terms of automorphisms of the category L, and a description of the group Out(|L|_{p}^{∧}) of homotopy classes of self equivalences in terms of “fusion preserving automorphisms” of S. We also show that a p-local compact group (S, F, L) is determined up to isomorphism by the homotopy type of its classifying space |L|_{p}^{∧}. One future goal is to show that the mod p cohomology of the classifying space |L|_{p}^{∧} of a p-local compact group (S, F, L) can always be described in terms of the fusion system F, as a ring of “stable elements” in the cohomology of S. Other goals are to define connected p-local compact groups, and understand their properties and their relation to connected p-compact groups; and to characterize algebraically (connected) p-compact groups among all (connected) p-local compact groups. Finally, a more general question which is still open is whether the p-completion of the classifying space of every finite loop space is the classifying space of a p-local compact group.

As one might expect, passing from a finite to an infinite setup introduces an array of problems one must deal with in order to produce a coherent theory. Some of the basic properties of fusion systems over discrete p-toral groups are analogous or even identical to the finite case, whereas other aspects are more delicate. Once the definition of a saturated fusion system over a discrete p-toral group is given and their basic properties are studied, one defines associated centric linking systems and p-local compact groups in a fashion more or less identical to the finite case. However, while in the finite case, any finite group G gives rise automatically to a saturated fusion system and an associated centric linking system, the corresponding construction for compact Lie groups is less obvious. Similar complications present themselves when dealing with the fusion system and the centric linking system associated to a p-compact group. It is for that reason that the only aims of this paper are to establish the setup, study some basic properties, and prove that the classifying spaces which are the obvious candidates to give rise to p-local compact groups indeed do so.

We proceed by describing the contents of the paper in some detail. In Section 1, we define and list some properties of discrete p-toral groups. We show why this class of groups is a natural one to consider for our purposes, and study some of its useful properties. Then in Section 2, we define saturated fusion systems over discrete p-toral groups. The definitions in this section are very similar to those given in [BLO2] for the finite case, but some modifications are needed due to having given up finiteness.

Much of the work on p-local finite groups makes implicit use of the fact that the categories one works with are finite. If S is an infinite discrete p-toral group, then any fusion system over it will have infinitely many objects. In Section 3 we show that any saturated fusion system F over a discrete p-toral group S contains a full subcategory with finitely many objects, which in the appropriate sense determines F completely. More precisely, we show that F contains only finitely many objects which are both centric and radical, and then prove the appropriate analog of Alperin’s fusion theorem. The latter, roughly speaking, says that in a saturated fusion system, every morphism can be factored into a sequence of morphisms each of which is the restriction of an automorphism of a centric radical subgroup.

Linking systems associated to fusion systems over discrete p-toral groups are defined in Section 4. In fact, the definition is identical to that used when working over a finite p-group, and the proof that the nerve |L| of a linking system is p-good is essentially identical to that in the finite case. The connection between linking systems associated to a given fusion system F and rigidifications of the homotopy functor P ↦ BP on the orbit category O^{c}(F) is then studied.

Higher limits over the orbit category of a fusion system are investigated in Section 5. We first describe how to reduce the general problem to one of higher limits over a finite subcategory, and then show how those can be computed with the help of the graded groups Λ^{∗}(Γ; M) introduced in [JMO]. These general results are then applied to prove the acyclicity of certain explicit functors whose higher limits appear later as obstruction groups.

Spaces of maps Map(BQ, |L|_{p}^{∧}) are studied in Section 6, when Q is a discrete p-toral group and |L|_{p}^{∧} is the classifying space of a p-local compact group, and the space of self equivalences of |L|_{p}^{∧} is handled in Section 7. In both cases, the descriptions we obtain in this new situation (in Theorems 6.3 and 7.1) are the obvious generalizations of those obtained in [BLO2] for linking systems over finite p-groups. We also prove (Theorem 7.4) that a p-local compact group is determined by the homotopy type of its classifying space: if (S, F, L) and (S′, F′, L′) are p-local compact groups such that |L|_{p}^{∧} ≃ |L′|_{p}^{∧}, then they are isomorphic as triples of groups and categories.

We finish with three sections of examples: certain infinite locally finite groups in Section 8, including linear torsion groups; compact Lie groups in Section 9; and pcompact groups in Section 10. In all cases, we show that the groups in question fit into our theory: they have saturated fusion systems and associated linking systems, defined in a unique way (unique up to isomorphism at least), and the classifying spaces of the resulting p-local compact groups are homotopy equivalent to the p-completed classifying spaces of the groups in the usual sense.

The first and third authors would like to thank the University of Aberdeen for it’s hospitality during several visits; in particular, this work began when we got together there in January 2002. All three authors would like to thank the Mittag-Leffler Institute, where this work was finished over a period of several months.

1. Discrete p-toral groups

When attempting to generalize the theory of p-local finite groups to certain infinite groups, the first problem is to decide which groups should replace the finite p-groups over which we studied fusion systems in [BLO2]. The following is the class of groups we have chosen for this purpose. Let ℤ/p^{∞} ≅ ℤ[\frac{1}{p}]/ℤ denote the union of the cyclic p-groups ℤ/p^{n} under the obvious inclusions.

Definition 1.1. A discrete p-toral group is a group P, with normal subgroup P_{0} ◁ P, such that P_{0} is isomorphic to a finite product of copies of ℤ/p^{∞}, 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) =^{def} P/P_{0}: the group of components of P.

The identity component P_{0} of a discrete p-toral group P can be characterized as the subset of all infinitely p-divisible elements in P, and also as the minimal subgroup of finite index in P. Define rk(P) = k if P_{0} ≅ (ℤ/p^{∞})^{k}, and set

|P | =^{def} (rk(P), |π_{0}(P)|) = (rk(P), |P/P_{0}|) .

We regard the order of a discrete p-toral group as an element of ℕ^{2} with the lexicographical ordering. Thus |P | ≤ |P′| if and only if rk(P) < rk(P′), or rk(P) = rk(P′) and |π_{0}(P)| ≤ |π_{0}(P′)|. In particular, P′ ≤ P implies |P′| ≤ |P |, with equality only if P′ = P.

The obvious motivation for choosing this class is the role they play as “Sylow psubgroups” in compact Lie groups and p-compact groups. But in fact, it seems difficult to construct fusion systems with interesting properties over any larger class of subgroups. The reason for this is that discrete p-toral groups are characterized by certain finiteness properties, which are needed in order for fusion systems over them to be manageable, and for related homotopy theoretic phenomena to be controled by p-local information.

A group G is locally finite if every finitely generated subgroup of G is finite, and is a locally finite p-group if every finitely generated subgroup of G is a finite p-group. The class of locally finite (p-)groups is closed under subgroups and quotient groups. It is also closed under group extensions, since finite index subgroups of finitely generated groups are again finitely generated.

A group G is artinian (satisfies the minimum condition in the terminology of [W]) if every non-empty set of subgroups of G, partially ordered by inclusion, has a minimal element. Equivalently, G is artinian if its subgroups satisfy the descending chain condition. The class of artinian groups is closed under taking subgroups, quotients, and extensions. Every artinian group is a torsion group (since an infinite cyclic group is not artinian). If G is artinian and φ ∈ Inj(G, G) is an injective endomorphism of G, then φ is an automorphism, since otherwise {φ^{n}(G)} would be an infinite descending chain. This is just one example of why it will be important that the groups we work with are artinian; the descending chain condition will be used in other ways later.

It is an open question whether every artinian group is locally finite (see [KW, pp. 31–32] for a discussion of this). If one restricts attention to groups all of whose elements have p-power order for some fixed prime p, then artinian groups are known to be locally finite if p = 2 [KW, Theorem 1.F.6], but this seems to be unknown for odd primes. However, any counterexample to these questions would probably be far too wild for our purposes. Hence it is natural to restrict attention to locally finite groups, and since we are working with local structure at a prime p, to locally finite p-groups. The next proposition tells us that in fact, this restricts us to the class of discrete p-toral groups. It is included only as a way to help motivate this choice of groups to work with.

Proposition 1.2. A group is a discrete p-toral group if and only if it is artinian and a locally finite p-group.

Proof. The group ℤ/p^{∞} is clearly a locally finite p-group and artinian. Since both of these properties are preserved under extensions of groups, they are satisfied by every discrete p-toral group.

Conversely, assume that G is artinian and a locally finite p-group. By [KW, Theorem 5.8], every locally finite artinian group is a Černikov group; in particular, it contains a normal abelian subgroup with finite index. By [Fu, Theorems 25.1 & 3.1], every abelian artinian group is a finite product of groups of the form ℤ/q^{m} where q is a prime and m ≤ ∞. Thus G is an extension of the form

1 ⟶ A ⟶ G ⟶ π ⟶ 1,

where π is a finite p-group, and A is a finite product of groups ℤ/p^{m} for m ≤ ∞. The subgroup of A generated by the factors ℤ/p^{∞} is the subgroup of infinitely p-divisible elements, thus a characteristic subgroup of A, and a normal subgroup of G of p-power index. It follows that G is a discrete p-toral group. □

We next note some of the other properties which make discrete p-toral groups convenient to work with.

Lemma 1.3. Any subgroup or quotient group of a discrete p-toral group is a discrete p-toral group. Any extension of one discrete p-toral group by another is a discrete p-toral group.

Proof. These statements are easily checked directly. They also follow at once from Proposition 1.2, since the classes of locally finite p-groups and artinian groups are both closed under these operations. □

Clearly, the main difficulty when working with infinite discrete p-toral groups, instead of finite p-groups, is that they have infinitely many subgroups and infinite automorphism groups. We next investigate what finiteness properties these groups do have.

Lemma 1.4. The following hold for each discrete p-toral group P.

(a) For each n ≥ 0, P contains finitely many conjugacy classes of subgroups of order p^{n}.

(b) P contains finitely many conjugacy classes of elementary abelian p-subgroups.

Proof. Clearly, for each n, P_{0} contains finitely many subgroups of order p^{n}, since they are all contained inside the p^{n}-torsion subgroup of P_{0} which is finite. So to prove (a), it suffices, for each finite subgroup A ≤ P_{0} and each subgroup B = B̃/P_{0} ≤ P/P_{0}, to show that there are finitely many P -conjugacy classes of subgroups Q ≤ P such that Q ∩ P_{0} = A and QP_{0} = B̃. Let Q be the set of all such subgroups, and assume Q ≠ ∅. Then Q ∈ Q if and only if Q/A ∩ P_{0}/A = 1 and QP_{0}/P_{0} = B; and this implies that A ◁ QP_{0} = B̃ and that Q/A is the image of a splitting of the extension

1 → P_{0}/A → B̃/A → B → 1.

In other words, Q is in one-to-one correspondence with the set of splittings of this extension. The set of P_{0}-conjugacy classes of such splittings (if there are any) is in one-to-one correspondence with the elements of H^{1}(B; P_{0}/A) (see [Bw, Proposition IV.2.3]). Since this cohomology group is finite, so is the set of conjugacy classes of such extensions.

This proves point (a). Point (b) follows from (a), together with the observation that for any elementary abelian subgroup E ≤ P, rk(E) ≤ rk(P) + rk_{p}(P/P_{0}). □

We next check what can be said about finiteness in automorphism groups.

Proposition 1.5. Let P be a discrete p-toral group.

(a) Any torsion subgroup of Aut(P) is an extension of an abelian group by a finite group.

(b) Any torsion subgroup of Out(P) is finite.

(c) For each Q ≤ P, Out_{P} (Q) is a finite p-group.

Proof. Assume first that P ≅ (ℤ/p^{∞})^{r}: a discrete p-torus of rank r ≥ 0. Then Aut(P) ≅ GL_{r}(ℤ̂_{p}), and it is well known that the subgroup (1 + p^{2}M_{r}(ℤ̂_{p}))^{×} of matrices which are congruent modulo p^{2} to the identity is torsion free. This follows, for example, from the inverse bijections

(1 + p^{2}M_{r}(ℤ̂_{p}))^{×} \overrightarrow{⟵}_{exp}^{log} p^{2}M_{r}(ℤ̂_{p})

defined by the usual power series: while log is not a homomorphism, it does satisfy the relation log(X^{r}) = r log(X). So if H is a torsion subgroup of Aut(P) (equivalently, of GL_{r}(ℤ̂_{p})), then the composite

H ⟶ GL_{r}(ℤ̂_{p}) ⟶^{/p^{2}} GL_{r}(ℤ/p^{2})

is injective, and thus H is finite.

Now let P be an arbitrary discrete p-toral group with connected component P_{0} and group of components π = P/P_{0}. There is an exact sequence

0 ⟶ H^{1}(π; P_{0}) ⟶ Aut(P)/ Aut_{P_{0}}(P) ⟶ Aut(P_{0}) × Aut(π)

(cf. [Sz, 2.8.7]), where Aut_{P_{0}}(P) = {c_{x} ∈ Aut(P) | x ∈ P_{0}}. We have just seen that every torsion subgroup of Aut(P_{0}) is finite, and H^{1}(π; P_{0}) and Aut(π) are clearly finite. Hence every torsion subgroup of Aut(P)/ Inn_{P_{0}}(P) is finite. This proves (b); and also proves (a) (every torsion subgroup of Aut(P) is an extension of an abelian group by a finite group) since Aut_{P_{0}}(P) is abelian. Point (c) follows immediately from (b), since P is a torsion group all of whose elements have p-power order. □

In the next section (in Definition 2.2), we will need some more precise bounds on the size of normalizers and centralizers.

Lemma 1.6. Let S be any discrete p-toral group, and set N = |π_{0}(S)|^{rk(S)+1}. Then for all P ≤ S,

|π_{0}(C_{S}(P))| ≤ N,  |π_{0}(N_{S}(P)/P)| ≤ N,  and  |π_{0}(N_{S}(P))| ≤ N·|π_{0}(P)|.

Proof. Set T = S_{0} for short, and set Q = P T/T. Let N_{Q}: T → T be the norm map for the Q action: N_{Q}(x) = ∏_{gT ∈Q} gxg^{−1}. The image of N_{Q} is connected and centralizes P, and thus Im(N_{Q}) ≤ C_{S}(P)_{0} = C_{T} (P)_{0}. If x ∈ C_{T} (P), then

x^{|Q|} = N_{Q}(x) ∈ C_{T} (P)_{0}.

Thus every element in C_{T} (P)/C_{T} (P)_{0} has order dividing |Q|, and it follows that

|π_{0}(C_{T} (P))| = |C_{T} (P)/C_{T} (P)_{0}| ≤ |Q|^{rk(S)} ≤ |π_{0}(S)|^{rk(S)}.

Thus |π_{0}(C_{S}(P))| ≤ |π_{0}(C_{T} (P))|·|S/T | ≤ N.

If x ∈ N_{T} (P), then

x^{|Q|} = N_{Q}(x)· ∏_{gT ∈Q}[x, g] ∈ C_{T} (P)_{0}·P ≤ N_{T} (P)_{0}·P.

Thus

|N_{T} (P)/(N_{T} (P)_{0}·(T ∩ P))| ≤ |Q|^{rk(S)} ≤ |π_{0}(S)|^{rk(S)},

and hence |π_{0}(N_{S}(P)/P)| ≤ N, by the same arguments as those used for π_{0}(C_{S}(P)). The last inequality is now immediate. □

Note that discrete p-toral groups are all solvable, but (in contrast to finite p-groups) need not be nilpotent. For instance, the infinite dihedral group, a split extension of ℤ/2^{∞} by ℤ/2, is a discrete 2-toral group which is not nilpotent (since the nilpotency class of D_{2^{n}} is n − 1).

The following lemma contains some generalizations of a standard theorem about automorphisms of finite p-groups: if α ∈ Aut(P) is the identity on Q ◁ P and on P/Q, then it has p-power order.

Lemma 1.7. The following hold for any discrete p-toral group P and any automorphism α ∈ Aut(P).

(a) Assume, for some Q ◁ P, that α|_{Q} = Id_{Q} and α ≡ Id (mod Q). Then every α-orbit in P is finite of p-power order. If, in addition, [P : Q] < ∞, then α has finite order.

(b) α has finite order if and only if α|_{P_{0}} has finite order.

(c) Set P_{(1)} = {g ∈ P_{0} | g^{p} = 1}. If α|_{P_{(1)}} = Id and α ≡ Id (mod P_{0}), then each orbit of α acting on P has p-power order.

Proof. (a) The proof is identical to the proof for finite p-groups (see [Go, Theorem 5.3.2]), and in fact applies whenever all elements of Q have p-power order. For any g ∈ P, α(g) = gx for some x ∈ Q (since α ≡ Id (mod Q)), and α(x) = x since α|_{Q} = Id. Thus α^{n}(g) = gx^{n} for all n, and α^{p^{k}}(g) = g if p^{k} = |x|. Since the order of {α^{i}(g)} depends only on the coset gQ, this also shows that |α| is finite (and a power of p) if P/Q is finite.

(b) If α|_{P_{0}} has finite order, then there is n ≥ 1 such that α^{n}|_{P_{0}} = Id and α^{n} ≡ Id (mod P_{0}). Then α^{n} has finite order by (a), so α also has finite order.

(c) For each m ≥ 1, let P_{(m)} ≤ P_{0} be the p^{m}-torsion in P_{0}. Fix g ∈ P, and set x = g^{−1}α(g), p^{k} = |x|, and Q = ⟨g, P_{(k)}⟩. The P_{(m)} are all α-invariant, and so Q is also α-invariant since g^{−1}α(g) ∈ P_{(k)}. Also, α acts via the identity on P_{(1)} by assumption, hence on P_{(i)}/P_{(i−1)} for all 1 ≤ i ≤ k, and also on Q/P_{(k)}. So by (a) (and since Q is a finite group), α|_{Q} has p-power order. In particular, the α-orbit of g has p-power order. □

The next lemma is another easy generalization of a standard result about finite p-groups.

Lemma 1.8. If P ≨ Q are distinct discrete p-toral groups, then P ≨ N_{Q}(P).

Proof. When [Q:P] < ∞, this follows by the same proof as for finite p-groups. More precisely, when Q/P is finite, the action of P on Q/P (defined by x(gP) = xgP for x ∈ P and g ∈ Q) factors through a finite quotient group P/N of P. Also, P/N is a p-group since P is a p-torsion group. Thus

|N_{Q}(P)/P | = |(Q/P)^{P/N}| ≡ |Q/P | ≡ 0 (mod p),

and so N_{Q}(P)/P ≠ 1.

Now assume that [Q:P] is infinite; i.e., that P_{0} ≨ Q_{0}. For each n, set A_{n} = {x ∈ Q_{0} | x^{p^{n}} = 1}. Then A_{n} ◁ Q, and in particular is normalized by P. For n large enough, A_{n} ≰ P, so P ≨ P A_{n} ≤ Q, P ≨ N_{P A_{n}}(P) since [P A_{n}:P] < ∞, and thus P ≨ N_{Q}(P). □

We will also need the following well known result about finite subgroups of discrete p-toral groups.

Lemma 1.9. For any discrete p-toral group P, there is a finite subgroup Q ≤ P such that P = QP_{0}. There is also an increasing sequence Q_{1} ≤ Q_{2} ≤ Q_{3} ≤ ⋯ of finite subgroups of P such that P = ⋃_{n=1}^{∞} Q_{n}. More generally, for any finite subgroup K ≤ Aut(P), the Q_{i} can be chosen to be K-invariant.

Proof. Fix any (finite) set X of coset representatives for P_{0} in P, and set Q = ⟨α(g) | α ∈ K, g ∈ X⟩. Then Q is K-invariant, Q is finite since P is locally finite, and P = QP_{0} by construction. For each n ≥ 1, let P_{n} ≤ P_{0} be the p^{n}-torsion subgroup, and set Q_{n} = QP_{n}. Then the Q_{n} are also finite and K-invariant, and P = ⋃_{n=1}^{∞} Q_{n}. □

To finish the section, we consider maps between the p-completed classifying spaces of discrete p-toral groups. This following lemma is implicit in [DW] and [DW2] (the spaces in question are classifying spaces of p-compact groups). But it does not seem to be stated explicitly anywhere there.

Lemma 1.10. For any pair P, Q of discrete p-toral groups,

B: Rep(P, Q) ⟶ [BP _{p}^{∧}, BQ_{p}^{∧}]

is a bijection. In particular, any homotopy equivalence BP _{p}^{∧} ⟶^{≃} BQ_{p}^{∧} is induced by an isomorphism P ≅ Q. Also, for any homomorphism ρ: P ⟶ Q, the homomorphism C_{Q}(ρ(P)) × P ⟶^{(incl,ρ)} Q induces a homotopy equivalence

BC_{Q}(ρ(P))_{p}^{∧} ⟶^{≃} Map(BP _{p}^{∧}, BQ_{p}^{∧})_{Bρ}.

Proof. For any pair G, H of discrete groups,

[BG, BH] ≅ Rep(G, H) and Map(BG, BH)_{Bρ} ≃ BC_{H}(ρ(G))

for each ρ ∈ Hom(G, H). See, for example, [BKi, Proposition 7.1] for a proof.

By [DW2, Proposition 3.1], the homotopy fiber of the map BQ ⟶ BQ_{p}^{∧} is a K(V, 1) for some ℚ̂_{p}-vector space V. Using this, together with standard obstruction theory and the fact that H̃^{∗}(BQ; ℚ) = 0, one checks that

[BP _{p}^{∧}, BQ_{p}^{∧}] ≅ [BP, BQ] ≅ Rep(P, Q).

Now fix some ρ ∈ Hom(P, Q). By [DW, Propositions 5.1 & 6.22], Map(BP _{p}^{∧}, BQ_{p}^{∧})_{Bρ} is the classifying space of some p-compact group X, and in particular is p-complete. Since Map(BP, BQ)_{Bρ} ≃ C_{Q}(ρ(P)) (P and Q are both discrete), we will be done upon showing that the completion map

Map(BP, BQ)_{Bρ} ⟶ Map(BP, BQ_{p}^{∧})_{Bρ}  (1)

is a mod p homology equivalence.

Fix a sequence of finite subgroups P_{1} ≤ P_{2} ≤ ⋯ whose union is P. Since Q is artinian, C_{Q}(ρ(P_{n})) = C_{Q}(ρ(P)) for n sufficiently large. Also, Map(BP, BQ_{p}^{∧})_{Bρ} is the homotopy inverse limit of the mapping spaces Map(BP_{n}, BQ_{p}^{∧})_{Bρ}. So if (1) is a mod p equivalence upon replacing P by P_{n} for each n, it is also a mod p equivalence for P. In other words, it suffices to prove this when P is a finite p-group.

Let X be the homotopy fiber of the completion map BQ ⟶ BQ_{p}^{∧}. As noted above, X is a K(V, 1) where V is a rational vector space. Since the map from Map(BP, BQ) to Map(BP, BQ_{p}^{∧}) is a bijection on components, the homotopy fiber of the map in (1) is X^{hP} for a proxy action of P on X (in the sense of [DW]) induced by ρ.

Consider the fibration sequence

X^{hP} ⟶ Map(BP, X_{hP})_{{1}} ⟶^{pr ∘−} Map(BP, BP)_{Id},

where pr denotes the projection of X_{hP} to BP, and the total space is the set of all maps f: BP ⟶ X_{hP} such that pr ∘f ≃ Id. Since X_{hP} is the total space of a fibration over BP with fiber X, it is a K(π, 1) where V ◁ π and π/V ≅ P. Since P is a finite p-group and V is a rational vector space, this extension splits, and the splitting is unique up to conjugacy by elements of V.

It follows that

[BP, X_{hP}]_{≅Rep(P,π)} ⟶ [BP, BP]_{≅Rep(P,P)}

is a bijection. Also, the induced map

π_{1}(Map(BP, X_{hP})_{{1}})_{≅C_{π}(P)} ⟶ π_{1}(Map(BP, BP)_{Id})_{≅Z(P)}

is surjective, and its kernel V^{P} (where the action of P on V is induced by the action on X) is a rational vector space.

Thus X^{hP} ≃ K(V^{P} , 1). It follows that X^{hP} is mod p acyclic, and hence that (1) is a mod p equivalence. This finishes the proof. □

2. Fusion systems over discrete p-toral groups

We now define saturated fusion systems over dicrete p-toral groups and study their basic properties. The definitions are almost identical to those in the finite case ([BLO2, §1]).

Definition 2.1. A fusion system F over a discrete p-toral group S is a category whose objects are the subgroups of S, and 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 ≤ S.

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

Two subgroups P, P′ ≤ S are called F-conjugate if Iso_{F}(P, P′) ≠ ∅.

Definition 2.2. 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}(P′)| for all P′ ≤ S which is F-conjugate to P.

• A subgroup P ≤ S is fully normalized in F if |N_{S}(P)| ≥ |N_{S}(P′)| for all P′ ≤ S which is F-conjugate to P.

• 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) | φc_{g}φ^{−1} ∈ 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).

By Lemma 1.6, there is a global upper bound for |π_{0}(C_{S}(P))| and |π_{0}(N_{S}(P))|, taken over all subgroups P of any given S. In particular, for any given subgroup P ≤ S, |C_{S}(P′)| and |N_{S}(P′)| take on maximal values among all P′ which are F-conjugate to P. This proves that the conjugacy class of P always contains fully centralized subgroups and fully normalized subgroups.

It is very convenient, in the above definition, to be working with a class of groups where the concept of “order” of subgroups is defined. However, there are other ways to define fully normalized and fully centralized subgroups in a fusion system, and hence to define saturation; and this property was not a factor in our decision to restrict attention to fusion systems over discrete p-toral groups. The crucial properties of these groups, which seem to be needed frequently when developing the theory, are that they are artinian and locally finite.

When F is a saturated fusion system over the discrete p-toral subgroup S, then by (I), Out_{F}(P) = Aut_{F}(P)/ Inn(P) is finite for fully normalized P ≤ S, and hence for all P ≤ S. Since Inn(P) is discrete p-toral (being a quotient group of P), Aut_{F}(P) inherits many of the properties of discrete p-toral groups. In particular, it is artinian, locally finite, and contains a unique conjugacy class of maximal discrete p-toral subgroups. This condition that Out_{F}(P) be finite does simplify slightly the definition of a saturated fusion system, but it is in fact unnecessary, as is shown by the following proposition.

Proposition 2.3. Let F be a fusion system over the discrete p-toral group S. Assume that axiom (II) in Definition 2.2 holds, and that (I) holds for all finite fully normalized subgroups of S. Then Out_{F}(P) is finite for all P ≤ S.

Proof. Fix P ≤ S. For all m ≥ 1, set P_{(m)} = {g ∈ P_{0} | g^{p^{m}} = 1}. By Proposition 1.5(b), to show that Out_{F}(P) is finite, it suffices to show that Aut_{F}(P) is a torsion group.

Fix α ∈ Aut_{F}(P). We want to show that α has finite order; by Lemma 1.7(b), it suffices to do this when P = P_{0} is connected. After replacing α by α^{n} for some appropriate n ≥ 1, we can assume that α|_{P_{(1)}} = Id. Then by Lemma 1.7(c), α_{m} =^{def} α|_{P_{(m)}} has p-power order for all m. For each m, there is φ_{m} ∈ Hom_{F}(P_{(m)}, S) such that φ_{m}(P_{(m)}) is fully normalized, and by (I), φ_{m}(P_{(m)}) is fully centralized, and φ_{m} can be chosen such that φ_{m}α_{m}φ_{m}^{−1} ∈ Aut_{S}(φ_{m}(P_{(m)})). Also, φ_{m} can be extended to φ_{m} ∈ Hom_{F}(S_{0}, S) by (II), so φ_{m}(P_{(m)}) ≤ S_{0}, and hence | Aut_{S}(φ_{m}(P_{(m)}))| ≤ |S/S_{0}|. Thus (α_{m})^{|S/S_{0}|} = Id_{P_{(m)}} for each m, so α^{|S/S_{0}|} = Id_{P}, and α has finite order. □

In fact, one can show that in the definition of a saturated fusion system, it suffices to require that (I) holds for all finite fully normalized subgroups P ≤ S; it then follows that (I) holds for all fully normalized subgroups.

When F is a (saturated) fusion system over a discrete p-toral group S, we think of the identity component S_{0} as the “maximal torus” of the fusion system, and think of Aut_{F}(S_{0}) as its “Weyl group”. The following lemma describes how morphisms between subgroups of the maximal torus are controlled by the Weyl group.

Lemma 2.4. Let F be a saturated fusion system over a discrete p-toral group S with connected component T = S_{0}. Then the following hold for all P ≤ T.

(a) For every P′ ≤ S which is F-conjugate to P and fully centralized in F, P′ ≤ T, and there exists some w ∈ Aut_{F}(T) such that w|_{P} ∈ Iso_{F}(P, P′).

(b) Every φ ∈ Hom_{F}(P, T) is the restriction of some w ∈ Aut_{F}(T).

Proof. We first prove the following statement.

(c) For each φ ∈ Hom_{F}(P, S) such that P′ =^{def} φ(P) is fully centralized in F, there exists w ∈ Aut_{F}(T) such that w|_{P} = φ.

By assumption, P ≤ T ≤ C_{S}(P). By condition (II) in Definition 2.2, there is φ ∈ Hom_{F}(C_{S}(P), S) such that φ|_{P} = φ. Then φ(T) ≤ T since T is connected (infinitely p-divisible), and so φ(T) = T since T is artinian. Thus w =^{def} φ|_{T} ∈ Aut_{F}(T) is such that w|_{P} = φ. This proves (c), and also proves (a) since P′ = w(P) ≤ T.

Now fix any φ ∈ Hom_{F}(P, T). Let Q be a fully centralized subgroup of S in the F-conjugacy class of P and φ(P), and choose ψ ∈ Iso_{F}(φ(P), Q). By (c), there are elements u, v ∈ Aut_{F}(T) such that u|_{P} = ψ ∘ φ and v|_{φ(P)} = ψ. So if we set w = v^{−1}u, then w|_{P} = φ. □

By Proposition 2.3, Out_{F}(P) is finite for every subgroup P ≤ S. The following lemma extends this statement.

Lemma 2.5. Let F be a saturated fusion system over a discrete p-toral group S. Then for all P, Q ≤ S, the set Rep_{F}(P, Q) =^{def} Inn(Q)\ Hom_{F}(P, Q) is finite.

Proof. As just noted, Out_{F}(P) is finite for all P ≤ S. Also, if φ, φ′ ∈ Hom_{F}(P, Q) and Im(φ) = Im(φ′), then φ′ = φ ∘ α for some α ∈ Aut_{F}(P) by condition (b) in Definition 2.1. So there is a bijection

Rep_{F}(P, Q)/ Out_{F}(P) ⟶^{≅} {P′ ≤ Q | P′ F-conjugate to P }/(Q-conjugacy), (1)

which sends the class of a homomorphism to the conjugacy class of its image.

By Lemma 2.4, the F-conjugacy class (P_{0}) of P_{0} is just its orbit under the action of Aut_{F}(S_{0}), and hence a finite set. By Lemma 1.4(a), for any given Q ∈ (P_{0}), there are only finitely many N_{S}(Q)/Q-conjugacy classes of subgroups of order |P/P_{0}| in N_{S}(Q)/Q. Hence there are only finitely many S-conjugacy classes of subgroups P′ ≤ S which are F-conjugate to P and such that P′_{0} = Q. This shows that the target set in (1) is finite, and hence that Rep_{F}(P, Q) is also finite. □

The definitions of centric and radical subgroups in a fusion system over a discrete p-toral group are essentially the same as those in the finite case.

Definition 2.6. Let F be a fusion system over a discrete p-toral group S. A subgroup P ≤ S is called F-centric if P and all its F-conjugates contain their S-centralizers. A subgroup P ≤ S is called F-radical if O_{p}(Out_{F}(P)) = 1; i.e., if Out_{F}(P) contains no nontrivial normal p-subgroup.

Notice that any F-centric subgroup is fully centralized. Conversely, if P ≤ S is fully centralized and centric in S; that is, Z(P) = C_{S}(P), then it is F-centric. The next proposition says that the set of F-centric subgroups is closed under overgroups.

Proposition 2.7. Let F be a saturated fusion system over the discrete p-toral group S, and let P ≤ Q ≤ S be such that P is F-centric. Then Q is also F-centric.

Proof. Fix any Q′ which is F-conjugate to Q, choose φ ∈ Iso_{F}(Q, Q′), and set P′ = φ(P). Then

C_{S}(Q′) ≤ C_{S}(P′) ≤ P′ ≤ Q′,

where the second inequality holds since P is F-centric. So Q is also F-centric. □

The next proposition gives another important property of F-centric subgroups; one which is much less obvious.

Proposition 2.8. Let F be a saturated fusion system over the discrete p-toral group S. Then for each P ≤ Q ≤ S such that P is F-centric, and each φ, φ′ ∈ Hom_{F}(Q, S) such that φ|_{P} = φ′|_{P}, there is some g ∈ Z(P) such that φ = φ′ ∘ c_{g}.

Proof. The hypothesis implies that φ ∘ φ′^{−1}|_{φ′(P)} = Id_{φ′(P)}, and we must show that φ ∘ φ′^{−1} = Id_{φ′(Q)}. It thus suffices to prove, for P ≤ Q ≤ S and φ ∈ Hom_{F}(Q, S) where P is F-centric, that φ|_{P} = Id_{P} implies φ = c_{g} for some g ∈ Z(P).

Assume first that P ◁ Q. Then for each x ∈ Q, c_{φ(x)}|_{P} = c_{x}|_{P}. Thus φ(x) ≡ x (mod C_{S}(P)), and C_{S}(P) ≤ P since P is F-centric. In particular, this shows that φ(Q) = Q, and thus that φ ∈ Aut_{F}(Q). It also shows that φ induces the identity on Q/P. Since Q/P has finite order, φ has p-power order by Lemma 1.7(a).

Without loss of generality, we can replace Q by any other subgroup in its Fconjugacy class. In particular, we can assume that Q is fully normalized, and hence that Out_{S}(Q) ∈ Syl_{p}(Out_{F}(Q)). So every p-subgroup of Aut_{F}(Q) is conjugate to a subgroup of Aut_{S}(Q). Thus there is χ ∈ Aut_{F}(Q) such that χ ∘ φ ∘ χ^{−1} = c_{y} for some y ∈ N_{S}(Q). Since φ|_{P} = Id_{P}, c_{y} acts as the identity on φ(P), which is also F-centric, hence y ∈ C_{S}(φ(P)) = φ(Z(P)). Set x = χ^{−1}(y); then φ = c_{x}.

Now assume P is not normal in Q. Let Q be the set of subgroups Q′ ≤ Q containing P such that φ|_{Q′} = c_{g}|_{Q′} for some g ∈ Z(P). If P ≤ Q′ ≨ Q and Q′ ∈ Q, then N_{Q}(Q′) ≩ Q′ by Lemma 1.8, and N_{Q}(Q′) ∈ Q since the proposition holds for the normal pair Q′ ◁ N_{Q}(Q′). Hence if Q contains a maximal element, it must be Q itself.

Let Q_{1} ≤ Q_{2} ≤ ⋯ be any increasing chain in Q, and set Q_{∞} = ⋃_{n=1}^{∞} Q_{n}. Let g_{n} ∈ Z(P) be such that φ|_{Q_{n}} = c_{g_{n}}|_{Q_{n}}. Since P is F-centric, so are the Q_{n}, and thus Z(Q_{1}) ≥ Z(Q_{2}) ≥ ⋯ is a decreasing sequence of subgroups. Since S is artinian, there is some k such that Z(Q_{n}) = Z(Q_{k}) for all n ≥ k. This shows that g_{n} ≡ g_{k} (mod Z(Q_{k})) for all n ≥ k, hence that φ|_{Q_{∞}} = c_{g_{k}}|_{Q_{∞}}, and hence that Q_{∞} ∈ Q. Thus by Zorn’s lemma, Q contains a maximal element, so Q ∈ Q, and this finishes the proof. □

3. A finite retract of a saturated fusion system

A fusion system F over a discrete p-toral group S generally has infinitely many isomorphism classes of objects. In this section, we construct a subcategory F^{•} of F with only finitely many isomorphism classes of objects, together with a retraction functor from F to F^{•} which is a left adjoint to the inclusion. This means that in many cases, it will suffice to work over the “finite” subcategory F^{•} rather than the full fusion system F. As a first application, we show that Ob(F^{•}) contains all F-centric F-radical subgroups, and hence that there are only finitely many conjugacy classes of such subgroups. A second application is Alperin’s fusion theorem in this setting: restriction to F^{•} allows us to repeat the same inductive argument as that used for fusion systems over a finite p-group.

Following the group theorists’ usual notation, whenever Γ is a group of automorphisms of a group G and H ≤ G, we write

C_{Γ}(H) = {γ ∈ Γ | γ|_{H} = Id_{H}}.

The following definitions were motivated by some constructions of Benson [Be], which he in fact used to prove a version of Alperin’s fusion theorem for compact Lie groups.

Definition 3.1. Let F be a saturated fusion system over a discrete p-toral group S, let T = S_{0} be the identity component of S, and set W = Aut_{F}(T) = Out_{F}(T) (the “Weyl group”). Set

p^{m} = exp(S/T) =^{def} min{p^{k} | x^{p^{k}} ∈ T ∀ x ∈ S}.

(a) For each P ≤ T, set

I(P) = T^{C_{W} (P)} = {t ∈ T | w(t) = t ∀ w ∈ W such that w|_{P} = Id_{P} };

and let I(P)_{0} be the identity component of I(P).

(b) For each P ≤ S, let P^{[m]} = ⟨g^{p^{m}} | g ∈ P ⟩ ≤ T, and set

P^{•} = P ·I(P^{[m]})_{0} =^{def} {gt | g ∈ P, t ∈ I(P^{[m]})_{0}}.

(c) Set

H(F) = {I(P)_{0} | P ≤ T }, and H^{•}(F) = {P^{•} | P ≤ S};

and let F^{•} ⊆ F be the full subcategory with object set H^{•}(F).

Thus for P ≤ T, I(P) is the maximal subgroup of T such that for all w ∈ W, w|_{P} = Id if and only w|_{I(P)} = Id. In particular, for all v and w in W, v|_{P} = w|_{P} if and only if v|_{I(P)} = w|_{I(P)}. Together with Lemma 2.4(b), this implies that every φ ∈ Hom_{F}(P, T) extends to a unique I(φ) ∈ Hom_{F}(I(P), T), which is obtained by first extending φ to T and then restricting to I(P). In other words, every F-isomorphism φ: P → Q between subgroups of T extends to a unique F-isomorphism I(φ): I(P) → I(Q).

For an arbitrary subgroup P ≤ S, P^{[m]} is a subgroup of T, and the above arguments apply. Since P^{[m]} ◁ P, any x ∈ P normalizes P^{[m]}, and hence also normalizes I(P^{[m]}). Thus P normalizes I(P^{[m]})_{0}, and this shows that the subset P^{•} =^{def} P ·I(P^{[m]})_{0} is a group.

More generally, for any k ≥ m, we could define subgroups P^{•k} ≥ P for each P ≤ S by setting P^{•k} = P ·I(P^{[k]}). This can be different from P^{•}, but P ↦ P^{•k} has all of the same properties which we prove here for P^{•}. However, the only way in which this generalization might be needed would be if we wanted to compare these “bullet functors” for two different fusion systems over two different discrete p-toral groups, and that will not be needed in this paper.

Lemma 3.2. The following hold for every saturated fusion system F over a discrete p-toral group S.

(a) The set H(F) is finite, and the set H^{•}(F) contains finitely many S-conjugacy classes of subgroups of S.

(b) For all P ≤ S, (P^{•})^{•} = P^{•}.

(c) If P ≤ Q ≤ S, then P^{•} ≤ Q^{•}.

(d) If P ≤ S is F-centric, then Z(P^{•}) = Z(P).

Proof. Let T = S_{0} ◁ S be the identity component, and set W = Aut_{F}(T) and p^{m} = exp(S/T). Note that for any P ≤ Q ≤ T, C_{W} (P) ≥ C_{W} (Q), and hence I(P) ≤ I(Q). Also, C_{W} (I(P)) = C_{W} (P) by definition, and hence I(I(P)) = I(P).

(a) By definition, each subgroup in H(F) has the form I(P)_{0} = (T^{K})_{0} for some P ≤ T, where K = C_{W} (P) ≤ W. Since the finite group W = Out_{F}(T) has a finite number of subgroups, this shows that H(F) is finite. Also, for any P ≤ S, P_{0} ≤ P^{[m]} ≤ I(P^{[m]}), and so (P^{•})_{0} = I(P^{[m]})_{0} ∈ H(F). In particular, there are only finitely many possibilities for identity components of subgroups in H^{•}(F).

Fix P ≤ S, and set K = C_{W} (P^{[m]}). Since P^{[m]} is generated by all p^{m}-powers in P (and p^{m} = exp(S/T)), P^{[m]} ≤ T and

[P:P^{[m]}] = [P:(P ∩T)]·[(P ∩T):P^{[m]}] ≤ |S/T |·p^{m· rk(T)}.

Here, the last inequality holds since (P ∩ T)/P^{[m]} is abelian with exponent at most p^{m} and rank at most rk(T). Also, since P^{[m]}·I(P^{[m]})_{0} = P^{[m]}·(T^{K})_{0} ≤ T^{K},

|π_{0}(P^{•})| = |π_{0}(P ·(T^{K})_{0})| ≤ |π_{0}(P^{[m]}·(T^{K})_{0})|·|P/P^{[m]}|

≤ |π_{0}(T^{K})|·|P/P^{[m]}| ≤ |π_{0}(T^{K})|·|S/T |·p^{m· rk(T)}.

We have already seen that (T^{K})_{0} is the identity component of P^{•}, and we have just shown that the number of components of P^{•} is bounded by an integer which depends only on K (and on S). Since N_{S}((T^{K})_{0})/(T^{K})_{0} has only finitely many conjugacy classes of finite subgroups of any given order (Lemma 1.4(a)), this shows that there are only finitely many conjugacy classes of subgroups in H^{•}(F) corresponding to any given K ≤ W; and thus (since W is finite) only finitely many conjugacy classes of subgroups in H^{•}(F).

(b) Fix P ≤ S. Since P normalizes I(P^{[m]})_{0}, for any g ∈ P and any x ∈ I(P^{[m]})_{0}, (gx)^{p^{m}} ∈ g^{p^{m}}·I(P^{[m]})_{0} ≤ P^{[m]}·I(P^{[m]})_{0}. This proves the second inequality on the following line:

P^{[m]} ≤ (P^{•})^{[m]} ≤ P^{[m]}·I(P^{[m]})_{0} ≤ I(P^{[m]}),

and the others are clear. Since I(−) is idempotent and preserves order, this shows that I((P^{•})^{[m]}) = I(P^{[m]}). Hence (P^{•})^{•} = P^{•}·I(P^{[m]})_{0} = P^{•}.

(c) If P ≤ Q, then P^{[m]} ≤ Q^{[m]}, so I(P^{[m]}) ≤ I(Q^{[m]}), and hence P^{•} ≤ Q^{•}.

(d) For any P ≤ S, P ≤ P^{•}. Thus if P is F-centric, then so is P^{•}, and Z(P^{•}) ≤ Z(P). To see that this is an equality, it suffices to show that every element in Z(P) commutes with I(P^{[m]}). For all x ∈ Z(P), c_{x} (as an element of W = Aut_{F}(T)) lies in C_{W} (P^{[m]}), hence commutes with all elements of I(P^{[m]}) = T^{C_{W} (P^{[m]})}, and in particular with all elements of I(P^{[m]})_{0}. □

We are now ready to prove the main, crucial, property of these subgroups P^{•}.

Proposition 3.3. Let F be a saturated fusion system over a discrete p-toral group S. Fix P, Q ≤ S and φ ∈ Hom_{F}(P, Q). Then φ extends to a unique homomorphism φ^{•} ∈ Hom_{F}(P^{•}, Q^{•}); and this makes P ↦ P^{•} into a functor from F to itself.

Proof. The functoriality of P ↦ P^{•} and φ ↦ φ^{•} (i.e., the fact that (Id_{P})^{•} = Id_{P^{•}} and (ψ ∘ φ)^{•} = ψ^{•} ∘ φ^{•}) follows immediately from the existence and uniqueness of these extensions. So this is what we need to prove.

As usual, we set T = S_{0} and W = Aut_{F}(T). For all Q ≤ T, C_{W} (Q) = C_{W} (I(Q)) by definition of I(−). This will be used frequently throughout the proof.

We first check that there is at most one morphism φ^{•} which extends φ. Assume that ψ, ψ′ ∈ Hom_{F}(P^{•}, Q^{•}) are two such extensions. By Lemma 2.4(b), there are elements w, w′ ∈ W such that ψ|_{P^{[m]}·I(P^{[m]})_{0}} = w|_{P^{[m]}·I(P^{[m]})_{0}} and ψ′|_{P^{[m]}·I(P^{[m]})_{0}} = w′|_{P^{[m]}·I(P^{[m]})_{0}}. Since w|_{P^{[m]}} = w′|_{P^{[m]}}, w^{−1}w′ ∈ C_{W} (P^{[m]}) = C_{W} (I(P^{[m]})), so w|_{I(P^{[m]})} = w′|_{I(P^{[m]})} as well. It follows that ψ = ψ′, since they take the same values on P and on I(P^{[m]})_{0}.

It remains to prove the existence of φ^{•}. By Lemma 3.2(c), it suffices to prove this when φ ∈ Iso_{F}(P, Q). Recall that P^{•} = P ·I(P^{[m]})_{0}. Fix u ∈ W = Aut_{F}(T) such that u|_{P^{[m]}} = φ|_{P^{[m]}}. Define φ^{•} by setting, for all g ∈ P and all x ∈ I(P^{[m]})_{0},

φ^{•}(gx) = φ(g)u(x).

After two preliminary steps, we show in Step 3 that φ^{•} is well defined and a homomorphism, and in Step 4 that it is a morphism in F.

Step 1: Fix A, A′ ≤ T, and w ∈ W such that w(A) = A′. We show here that

A ≤ B ≤ I(A), ψ ∈ Hom_{F}(B, T), ψ|_{A} = w|_{A} ⟹ ψ = w|_{B};  (1)

and also that

A ≤ B ≤ A·I(A)_{0}, ψ ∈ Hom_{F}(B, S), ψ|_{A} = w|_{A} ⟹ ψ(B) ≤ T and ψ = w|_{B}. (2)

If ψ(B) ≤ T, then ψ = w′|_{B} for some w′ ∈ W by Lemma 2.4(b), w^{−1}w′ ∈ C_{W} (A) = C_{W} (I(A)), and thus ψ = w′|_{B} = w|_{B}. This proves (1).

Now assume B ≤ A·I(A)_{0}. By Lemma 2.4(a), there is w′ ∈ W such that w′(B) is fully centralized in F. It thus suffices to prove (2) when B is fully centralized. Set B′ = ψ(B) for short.

Now, B′ ≥ A′ and B′ is abelian. So for all x ∈ B′, if we regard c_{x} as an element of W = Aut_{F}(T), then c_{x} ∈ C_{W} (A′) = C_{W} (I(A′)). Thus I(A′) = w(I(A)) ≤ C_{S}(B′). By axiom (II) (and since B = ψ^{−1}(B′) is fully centralized), ψ^{−1} extends to an F-morphism defined on B′·C_{S}(B′), and in particular to a morphism β ∈ Hom_{F}(B′·I(A′), S). Since β|_{A′} = w^{−1}|_{A′} and β(I(A′)_{0}) ≤ T, β|_{A′·I(A′)_{0}} = w^{−1}|_{A′·I(A′)_{0}} by (1).

Thus for all x ∈ B′, β(x) = ψ^{−1}(x) ∈ B ≤ A·I(A)_{0} = β(A′·I(A′)_{0}). Since β is injective, this shows that x ∈ A′·I(A′)_{0} ≤ T. So B′ ≤ T, and (2) now follows from (1).

Step 2: We next show that for all x ∈ I(P^{[m]}) and all g ∈ P, the following identity holds:

u(gxg^{−1}) = φ(g)u(x)φ(g)^{−1};  (3)

or equivalently that c_{φ(g)}^{−1} ∘u ∘c_{g}(x) = u(x). Set w = c_{φ(g)}^{−1} ∘u ∘c_{g} ∈ W for short. Then (3) holds for x ∈ P^{[m]} since φ|_{P^{[m]}} = u|_{P^{[m]}}, and thus w|_{P^{[m]}} = u|_{P^{[m]}}. So w|_{I(P^{[m]})} = u|_{I(P^{[m]})} by (1), and this proves (3) for all x ∈ I(P^{[m]}).

Step 3: Recall that we defined φ^{•}(gx) = φ(g)u(x) for all g ∈ P and x ∈ I(P^{[m]})_{0}. By assumption, φ|_{P^{[m]}} = u|_{P^{[m]}}. Hence the restrictions of φ and u to P^{[m]}·(P ∩I(P_{0}^{[m]})) are equal by (2), and this shows that φ^{•} is well defined.

For all g, g′ ∈ P and all x, x′ ∈ I(P^{[m]})_{0},

φ^{•}((gx)(g′x′)) = φ(gg′)·u(g^{′−1}xg′x′) = φ(gg′)·(φ(g′)^{−1}u(x)φ(g′))·u(x′)
= φ(g)u(x)φ(g′)u(x′) = φ^{•}(gx)·φ^{•}(g′x′),

where the second equality follows from Step 2. Thus φ^{•} is a homomorphism.

Step 4: It remains to show that φ^{•} ∈ Iso_{F}(P^{•}, Q^{•}); i.e., that φ^{•} is a morphism in the category F. By condition (III) in Definition 2.2, together with Zorn’s lemma, there is a maximal subgroup P′ ≤ P^{•} containing P such that φ^{•}|_{P′} ∈ Hom_{F}(P′, Q^{•}). Assume P′ ≨ P^{•}; and set φ′ = φ^{•}|_{P′} and P′′ = N_{P^{•}}(P′) ≩ P′. By condition (II) in Definition 2.2, φ′ extends to some morphism ψ ∈ Hom_{F}(P′′, S) (the existence of the homomorphism φ^{•} shows that N_{φ′} ≥ P′′). By (2) again, the restrictions of ψ, u, and φ^{•} to P′′ ∩ (P^{[m]}·I(P^{[m]})_{0}) are equal. Since P′′ = P ·(P′′ ∩ I(P^{[m]})_{0}), this shows that ψ = φ^{•}|_{P′′}. This contradicts the maximality assumption about P′; so P′ = P^{•}, and we are done. □

Note in particular that by Lemma 3.2(c), the functor F ⟶^{(−)^{•}} F^{•} of Proposition 3.3 sends inclusions of subgroups to inclusions.

Corollary 3.4. The functor (−)^{•} is a left adjoint to the inclusion of F^{•} as a full subcategory of F.

Proof. Fix any P in F and any Q in F^{•}. Since Q = Q^{•} by Lemma 3.2(b), every φ ∈ Hom_{F}(P, Q) extends to a unique φ^{•} ∈ Hom_{F}(P^{•}, Q) by Proposition 3.3. The restriction map

Hom_{F}(P^{•}, Q) ⟶^{Res} Hom_{F}(P, Q)

is thus a bijection, and this proves adjointness. □

Corollary 3.4 will later be extended to orbit and linking categories associated to F and F^{•}.

Corollary 3.5. Let F be a saturated fusion system over a discrete p-toral group S. Then all F-centric F-radical subgroups of S are in H^{•}(F), and in particular there are only finitely many conjugacy classes of such subgroups.

Proof. Assume P is F-centric and F-radical. We claim that I(P^{[m]})_{0} ≤ P, and thus that P = P^{•} ∈ H^{•}(F).

Assume otherwise. Then P^{•} ≩ P, and hence N_{P^{•}}(P) ≩ P by Lemma 1.8. Thus N_{P^{•}}(P)/P ≠ 1, and since P is F-centric, this group can be identified with a p-subgroup of Out_{F}(P). By Proposition 3.3, any α ∈ Aut_{F}(P) extends to an automorphism of P^{•}, and in particular to an automorphism of N_{P^{•}}(P). This shows that N_{P^{•}}(P)/P ◁ Out_{F}(P), which contradicts the assumption that P is F-radical.

The last statement now follows since H^{•}(F) contains only finitely many conjugacy classes by Lemma 3.2(a). □

As a third consequence of Proposition 3.3, we now prove Alperin’s fusion theorem in our context. This theorem was originally formulated for finite groups in [Al], and then for saturated fusion systems over finite p-groups by Puig [Pu] (see also [BLO2, Theorem A.10]). Our approach here (and our definition of P^{•}) is modelled on Benson’s proof of the theorem for fusion in compact Lie groups [Be].

Theorem 3.6 (Alperin’s fusion theorem). Let F be a saturated fusion system over a discrete p-toral group S. Then for each φ ∈ Iso_{F}(P, P′), there exist sequences of subgroups of S

P = P_{0}, P_{1}, …, P_{k} = P′  and  Q_{1}, Q_{2}, …, Q_{k},

and elements φ_{i} ∈ Aut_{F}(Q_{i}), such that

(a) Q_{i} is fully normalized in F, F-radical, and F-centric for each i;

(b) P_{i−1}, P_{i} ≤ Q_{i} and φ_{i}(P_{i−1}) = P_{i} for each i; and

(c) φ = φ_{k} ∘ φ_{k−1} ∘ ⋯ ∘ φ_{1}.

Proof. For each P ≤ S, let ν(P) be the number of F-conjugacy classes of subgroups in H^{•}(F) which contain P. We prove the theorem by induction on ν(P). Using Proposition 3.3, we can assume that P, P′ ∈ H^{•}(F). The claim is clear when ν(P) = 1 (i.e., P = S).

Assume P ≨ S. Let P′′ ≤ S be any subgroup which is F-conjugate to P and fully normalized in F, and fix ψ ∈ Iso_{F}(P, P′′). The theorem holds for φ ∈ Iso_{F}(P, P′) if it holds for ψ and for ψ ∘ φ^{−1} ∈ Iso_{F}(P′, P′′). So we are reduced to proving the theorem when the target group P′ is fully normalized in F.

Since P′ is fully normalized, the p-subgroup φ∘Aut_{S}(P)∘φ^{−1} of Aut_{F}(P′) is conjugate to a subgroup of Aut_{S}(P′). Let χ ∈ Aut_{F}(P′) be such that (χ∘φ)∘Aut_{S}(P)∘(χ∘φ)^{−1} ≤ Aut_{S}(P′). By condition (II) in Definition 2.2, there is φ ∈ Hom_{F}(N_{S}(P), S) such that φ|_{P} = χ ∘ φ. Since N_{S}(P) ≩ P (since P ≨ S) and P ∈ H^{•}(F), ν(N_{S}(P)) < ν(P), and the theorem holds for φ (as an isomorphism to its image) by the induction hypothesis. So it holds for φ if and only if it holds for χ. Hence it now remains only to prove it when P = P′ is fully normalized in F, P ∈ H^{•}(F), and φ ∈ Aut_{F}(P).

In particular, P is fully centralized in F. So if P is not F-centric, then by condition (II) in Definition 2.2, φ extends to an automorphism φ ∈ Aut_{F}(C_{S}(P)·P). Since ν(C_{S}(P)·P) < ν(P), the theorem holds for φ by the induction hypothesis.

Now assume that P is not F-radical. Let K ≤ Aut_{F}(P) be the subgroup such that K/ Inn(P) = O_{p}(Out_{F}(P)) ≠ 1. Since P is fully normalized in F, Out_{S}(P) ∈ Syl_{p}(Out_{F}(P)), and so K ≤ Aut_{S}(P). In particular,

N_{S}^{K}(P) =^{def} {g ∈ N_{S}(P) | c_{g}|_{P} ∈ K} ≩ P

since K ≩ Inn(P). Also, for each g ∈ N_{S}^{K}(P), φc_{g}φ^{−1} ∈ K (since K ◁ Aut_{F}(P)), and hence φc_{g}φ^{−1} = c_{h} for some h ∈ N_{S}^{K}(P). So by condition (II) in Definition 2.2, φ extends to an automorphism of N_{S}^{K}(P) ≩ P, and the theorem again holds for φ by the induction hypothesis.

Finally, if φ ∈ Aut_{F}(P) and P ∈ H^{•}(F) is a fully normalized F-centric F-radical subgroup of S, then the theorem holds for trivial reasons. □

4. Linking systems over discrete p-toral groups

We are now ready to define linking systems associated to a fusion system over a discrete p-toral group, and to study the relationship between linking systems and certain finite full subcategories.

Definition 4.1. Let F be a fusion system over the discrete p-toral group S. A centric linking system associated to F is a category L whose objects are the F-centric subgroups of S, together with a functor

π: L ⟶ F^{c},

and “distinguished” monomorphisms P ⟶^{δ_{P}} Aut_{L}(P) for each F-centric subgroup P ≤ S, which satisfy the following conditions.

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

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

(B) For each F-centric subgroup P ≤ S and each g ∈ P, π sends δ_{P} (g) ∈ Aut_{L}(P) to c_{g} ∈ Aut_{F}(P).

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

\begin{matrix}P & \overline{^{f} →} & Q \\ P & ↓^{δ_{P} (g)} & \overline{^{f} →} & Q. & ↓^{δ_{Q}(π(f)(g))} \end{matrix}

More generally, if F_{0} ⊆ F^{c} is any subcategory, then a linking system associated to F_{0} is a category L_{0}, together with a functor L_{0} ⟶^{π_{0}} F_{0} and distinguished monomorphisms P ⟶^{δ_{P}} Aut_{L_{0}}(P) for P ∈ Ob(F_{0}) = Ob(L_{0}), which satisfy conditions (A), (B), and (C) above.

It is now clear, by analogy with the finite case, how to define p-local compact groups.

Definition 4.2. A p-local compact group is a triple (S, F, L), where S is a discrete ptoral group, F is a saturated fusion system over S, and L is a linking system associated to F. The classifying space of such a triple (S, F, L) is the p-completed nerve |L|_{p}^{∧}.

The following very basic lemma about linking systems extends [BLO2, Lemma 1.10] to this situation.

Lemma 4.3. Fix a p-local compact group (S, F, L), and let π: L ⟶ F^{c} be the projection. Fix F-centric subgroups P, Q, R in S. Then the following hold.

(a) Fix any sequence P ⟶^{φ} Q ⟶^{ψ} R of morphisms in F^{c}, and let ψ̃ ∈ π_{Q,R}^{−1}(ψ) and ψ̃φ ∈ π_{P,R}^{−1}(ψφ) be arbitrary liftings. Then there is a unique morphism φ̃ ∈ Mor_{L}(P, Q) such that

ψ̃ ∘ φ̃ = ψ̃φ;  (1)

and furthermore π_{P,Q}(φ̃) = φ.

(b) If φ̃, φ̃′ ∈ Mor_{L}(P, Q) are such that the homomorphisms φ =^{def} π_{P,Q}(φ̃) and φ′ =^{def} π_{P,Q}(φ̃′) are conjugate (differ by an element of Inn(Q)), then there is a unique element g ∈ Q such that φ̃′ = δ_{Q}(g) ∘ φ̃ in Mor_{L}(P, Q).

Proof. Part (a) is an easy application of axiom (A) for a linking system. Part (b) is first reduced to the case where φ = φ′ using axiom (B), and this case then follows from (A) and (C). For more detail, see the proof of [BLO2, Lemma 1.10]. □

We next show that the nerve of a linking system is p-good, and hence that the classifying space of a p-local compact group is p-complete.

Proposition 4.4. Let (S, F, L) be any p-local compact group at the prime p. Then |L| is p-good. Also, the composite

S ⟶^{π_{1}(θ)} π_{1}(|L|) ⟶ π_{1}(|L|_{p}^{∧}),

induced by the inclusion BS ⟶^{θ} |L|, factors through a surjection π_{0}(S) −↠ π_{1}(|L|_{p}^{∧}).

Proof. For each F-centric subgroup P ≤ S, fix a morphism ι_{P} ∈ Mor_{L}(P, S) which lifts the inclusion (and set ι_{S} = Id_{S}). By Lemma 4.3(a), for each P ≤ Q ≤ S, there is a unique morphism ι_{P}^{Q} ∈ Mor_{L}(P, Q) such that ι_{Q} ∘ ι_{P}^{Q} = ι_{P}.

Regard the vertex S as the basepoint of |L|. Define

ω: Mor(L) ⟶ π_{1}(|L|)

by sending each φ ∈ Mor_{L}(P, Q) to the loop formed by the edges ι_{P}, φ, and ι_{Q} (in that order). Clearly, ω(ψ ∘ φ) = ω(ψ)·ω(φ) whenever ψ and φ are composable, and ω(ι_{P}^{Q}) = ω(ι_{P}) = 1 for all P ≤ Q ≤ S. Also, π_{1}(|L|) is generated by Im(ω) since any loop in |L| can be split up as a composite of loops of the above form.

By Theorem 3.6 (Alperin’s fusion theorem), each morphism in F, and hence each morphism in L, is (up to inclusions) a composite of automorphisms of fully normalized F-centric subgroups. Thus π_{1}(|L|) is generated by the subgroups ω(Aut_{L}(P)) for all fully normalized F-centric P ≤ S.

Let K ◁ π_{1}(|L|) be the subgroup generated by all infinitely p-divisible elements. For each fully normalized F-centric P ≤ S, Aut_{L}(P) is generated by its Sylow subgroup N_{S}(P) together with elements of order prime to p. Hence π_{1}(|L|) is generated by K together with the subgroups ω(N_{S}(P)); and ω(N_{S}(P)) ≤ ω(S) for each P. This shows that ω sends S surjectively onto π_{1}(|L|)/K, and hence (since the identity component of S is infinitely divisible) factors through a surjection of π_{0}(S) onto π_{1}(|L|)/K. In particular, this quotient group is a finite p-group.

Set π = π_{1}(|L|)/K for short. Since K is generated by infinitely p-divisible elements, the same is true of its abelianization, and hence H_{1}(K; 𝔽_{p}) = 0. Thus, K is p-perfect. Let X be the cover of |L| with fundamental group K. Then X is p-good and X_{p}^{∧} is simply connected since π_{1}(X) is p-perfect [BK, VII.3.2]. Also, since π is a finite p-group, it acts nilpotently on H_{i}(X; 𝔽_{p}) for all i. Hence X_{p}^{∧} ⟶ |L|_{p}^{∧} ⟶ Bπ is a fibration sequence and |L|_{p}^{∧} is p-complete by [BK, II.5.1]. So |L| is p-good, and π_{1}(|L|_{p}^{∧}) ≅ π is a quotient group of π_{0}(S). □

Recall, from Section 3, that for any saturated fusion system F, we defined a finite subcategory F^{•} such that the inclusion F^{•} ⊆ F has a left adjoint (−)^{•}. We next show that we can do the same on the level of linking systems.

Proposition 4.5. Let F be a saturated fusion system over a discrete p-toral group S, and let F^{c•} ⊆ F^{c} be the full subcategory whose objects are the F-centric subgroups contained in H^{•}(F).

(a) Let L be a centric linking system associated to F, and let L^{•} ⊆ L be the full subcategory with Ob(L^{•}) = Ob(F^{c•}). Then the inclusion L^{•} ↪ L has a left adjoint, which sends P to P^{•} for each F-centric P ≤ S. In particular, the inclusion |L^{•}| ⊆ |L| is a homotopy equivalence.

(b) Let L^{•} be a linking system associated to F^{c•}. Let L be the category whose objects are the F-centric subgroups of S, and where

Mor_{L}(P, Q) = {φ ∈ Mor_{L^{•}}(P^{•}, Q^{•}) | π^{•}(φ)(P) ≤ Q};

and let δ_{P} : P ⟶ Aut_{L}(P) be the restriction of Δ_{P^{•}}. In other words, L is the pullback category in the following square:

\begin{matrix}L & \overline{→} & L^{•} \\ F^{c} & ↓^{π} & \overline{→^{(−)^{•}}} & F^{c•}. & ↓^{π^{•}} \end{matrix}

Then L is a centric linking system associated to F.

Proof. (a) For each F-centric subgroup P ≤ S, fix a morphism ι_{P} ∈ Mor_{L}(P, S) such that π(ι_{P}) is the inclusion (and such that ι_{S} = Id_{S}). For any pair of F-centric subgroups P ≤ Q ≤ S, the same group Z(P) acts freely and transitively on the sets of morphisms in L covering the inclusions P ⊆ Q and P ⊆ S, and hence there is a unique morphism ι_{P}^{Q} ∈ Mor_{L}(P, Q) such that ι_{Q} ∘ ι_{P}^{Q} = ι_{P}.

Now let φ ∈ Hom_{F}(P, Q) be any morphism in F^{c}. By Proposition 3.3, φ has a unique extension to φ^{•} ∈ Hom_{F}(P^{•}, Q^{•}). Also, by Lemma 3.2(d), Z(P^{•}) = Z(P). Hence by condition (A) in the definition of a linking system, restriction sends the morphisms in π^{−1}(φ^{•}) bijectively to the morphisms in π^{−1}(φ). Thus for any ψ ∈ Mor_{L}(P, Q) such that π(ψ) = φ, there is a unique “extension” ψ^{•} ∈ Mor_{L}(P^{•}, Q^{•}) of ψ; i.e., a unique morphism such that ψ^{•} ∘ ι_{P}^{P^{•}} = ι_{Q}^{Q^{•}} ∘ ψ.

Thus, if we define θ: L ⟶ L^{•} by setting θ(P) = P^{•} and θ(ψ) = ψ^{•}, then θ is well defined. This also shows that Mor_{L}(P, Q) = Mor_{L}(P^{•}, Q) when Q = Q^{•}, and thus that θ is a left adjoint functor to the inclusion. Since the inclusion has a left adjoint, it follows that it induces a homotopy equivalence |L^{•}| ≃ |L|.

(b) Since Z(P) = Z(P^{•}) for all F-centric P ≤ S (Lemma 3.2(d) again), axiom (A) for L follows from the same axiom applied to L^{•}. Axioms (B) and (C) for L follow immediately from axioms (B) and (C) for L^{•} by restriction. □

We finish the section with a description of the relation between linking systems associated to a given fusion system F_{0}, and rigidifications of the homotopy functor B: O^{c}(F_{0}) ⟶ hoTop defined by setting B(P) = BP. Each linking system L_{0} induces a rigidification of B, which in turn defines a decomposition of |L_{0}| as a homotopy colimit. More precisely, by a “rigidification of the homotopy functor B” in the following proposition is meant a functor B̃: O(F_{0}) ⟶ Toptogether with a natural homotopy equivalence of functors (in hoTop) from B to ho ∘ B̃; i.e., a natural transformation of functors to hoTop which defines a homotopy equivalence BP ⟶ B̃(P) for each P. A natural homotopy equivalence of rigidifications from B̃ to B̃′ is a natural transformation

B̃ ⟶^{κ} B̃′

of functors to Topsuch that ho(κ) commutes with the functors from B. Two rigidifications B̃_{1} and B̃_{2} are equivalent if there is a third rigidification B̃_{0} and natural homotopy equivalences B̃_{1} ⟶ B̃_{0} ⟵ B̃_{2}; this is seen to be an equivalence relation by taking pushouts.

By a linking system L_{0} in the following proposition is always meant the category L_{0} together with the projection to the associated fusion system and the distinguished monomorphisms. Hence an isomorphism of linking systems means an isomorphism of the categories which is natural with respect to these other structures.

Proposition 4.6. Fix a saturated fusion system F over a discrete p-toral group S, and let F_{0} ⊆ F^{c} be any full subcategory. Then there are mutually inverse bijections

\{ \begin{matrix}linking systems \\ associated to F_{0} \\ up to isomorphism \end{matrix} \} \overrightarrow{⟵}_{ls}^{ke} \{ \begin{matrix}rigidifications O(F_{0}) → Top \\ of the homotopy functor B \\ up to natural homotopy equivalence \end{matrix} \} .

More precisely, the following hold for any linking system L_{0} associated to F_{0} and any rigidification B̃ of the homotopy functor B on O(F_{0}).

(a) The left homotopy Kan extension ke(L_{0}) of the constant functor L_{0} ⟶^{∗} Topalong the projection π̃_{0} : L_{0} ⟶ O(F_{0}) is a rigidification of B, and there is a homotopy equivalence

|L_{0}| ≃ \underrightarrow{hocolim}_{O(F_{0})}(ke(L_{0})) .  (1)

(b) There is a linking system ls(B̃) associated to F_{0}, and a natural homotopy equiva-

lence of functors

ke(ls(B̃)) ⟶^{≃} B̃.

Furthermore, if B̃′ is another rigidification of B, any natural homotopy equivalence of rigidifications κ: B̃ ⟶ B̃′ induces an isomorphim κ_{♯}: ls(B̃) ⟶ ls(B̃′) of linking systems.

(c) There is an isomorphism L_{0} ≅ ls(ke(L_{0})) of linking systems associated to F_{0}.

We define ke([L_{0}]) = [ke(L_{0})] for each L_{0}, and ls([B̃]) = [ls(B̃)] for each B̃.

Proof. The left homotopy Kan extension is natural with respect to isomorphisms L_{0} ⟶ L′_{0} of linking systems. Thus ke sends isomorphic systems to natural homotopy equivalent functors O(F_{0}) ⟶ Top, these are rigidifications of B by (a), and hence ke is well defined. Point (b) implies that ls is well defined, and it also implies that ls ∘ ke is the identity. Finally, (c) implies that ke ∘ ls is the identity. Hence the Proposition follows once we prove (a), (b), and (c).

(a) Fix L_{0}, and set B̃ = ke(L_{0}) for short. Recall that we write Rep_{F}(P, Q) = Mor_{O(F)}(P, Q). By definition, for each P in F_{0}, B̃(P) is the nerve (homotopy colimit of the point functor) of the overcategory π̃_{0}↓P, whose objects are pairs (Q, α) for Q in L_{0} and α ∈ Rep_{F}(Q, P), and where

Mor_{π̃_{0}↓P} ((Q, α), (R, β))= {φ ∈ Mor_{L}(Q, R) | α = β ∘ π̃_{0}(φ)}.  (2)

Since |L_{0}| ≅ \underrightarrow{hocolim}_{L_{0}}(∗), (1) holds by [HV, Theorem 5.5].

It remains to show that B̃ is a rigidification of the homotopy functor B. Fix a section σ̃: Mor(O(F_{0})) ⟶ Mor(L_{0}) of π̃_{0} which sends identity morphisms to identity morphisms. For each P, let B(P) be the category with one object o_{P} and morphism group P (so |B(P)| ≅ BP), and define functors

B(P) ⟶^{θ_{P}} π̃_{0}↓P ⟶^{Ψ_{P}} B(P)

as follows. Let θ_{P} (o_{P}) = (P, Id), and θ_{P} (g) = δ_{P} (g) (as a morphism in π̃_{0}↓P using (2)) for all g ∈ P. Set Ψ_{P} (Q, α) = o_{P}; and let Ψ_{P} send each morphism φ ∈ Mor_{π̃_{0}↓P} ((Q, α), (R, β)) to the unique element g ∈ P (unique by Lemma 4.3(b)) such that the following square commutes:

\begin{matrix}Q & \overline{^{φ} →} & R \\ P & ↓^{σ̃(α)} & \overline{→^{δ_{P} (g)}} & P . & ↓^{σ̃(β)} \end{matrix}

Clearly, Ψ_{P} ∘ θ_{P} = Id_{B(P)}. As for the other composite, define f: Id ⟶ θ_{P} ∘ Ψ_{P} by sending each object (Q, α) to the morphism σ̃(α) ∈ Mor_{L}(Q, P). This is clearly a natural transformation of functors, and thus

B̃(P) = |π̃_{0}↓P | ≃ |B(P)| ≃ BP.

To finish the proof that B̃ is a rigidification of the homotopy functor B, we must show, for any φ ∈ Hom_{F}(P, Q), that the following square commutes up to natural transformation:

\begin{matrix}B(P) & \overline{^{θ_{P}} →} & π̃_{0}↓P \\ B(Q) & ↓^{Bφ} & \overline{^{θ_{Q}} →} & π̃_{0}↓Q. & ↓^{φ∘−} \end{matrix}

Here, [φ] ∈ Rep_{F}(P, Q) denotes the class of φ. This means constructing a natural transformation F_{1} ⟶^{Φ} F_{2} of functors B(P) ⟶ π̃_{0}↓Q, where F_{1} = ([φ] ∘ −) ∘ θ_{P} and F_{2} = θ_{Q} ∘ Bφ are given by the formulas

F_{1}(o_{P}) = (P, [φ]), F_{1}(g) = δ_{P} (g), and F_{2}(o_{P}) = (Q, Id), F_{2}(g) = δ_{Q}(φ(g)).

Let φ̃ ∈ Mor_{L}(P, Q) be any lifting of φ. Then by condition (C), Φ can be defined by sending the object o_{P} to the morphism φ̃ ∈ Mor_{π̃_{0}↓P} ((P, [φ]), (Q, Id)).

(b) We first fix some notation. For any space X and any x, x′ ∈ X, π_{1}(X; x, x′) denotes the set of homotopy classes of paths in X (relative endpoints) from x to x′. For any u ∈ π_{1}(X; x, x′), u_{∗} denotes the induced isomorphism from π_{1}(X, x) to π_{1}(X, x′). Also, for any map of spaces f : X → Y, f_{∗} denotes the induced map from π_{1}(X; x, x′) to π_{1}(Y; f(x), f(x′)).

Now fix a rigidification B̃: O(F_{0}) → Top; we want to define a linking system L_{0} = ls(B̃) associated to F. Since B̃ is a rigidification of the homotopy functor B, we are given homotopy equivalences BP ⟶^{ϵ_{P}} B̃(P) such that the following square commutes up to homotopy for each φ ∈ Hom_{F}(P, Q):

\begin{matrix}BP & \overline{^{ϵ_{P}} →} & B̃(P) \\ BQ & ↓^{Bφ} & \overline{^{ϵ_{Q}} →} & B̃(Q) . & ↓^{B̃([φ])} \end{matrix}

Here, [φ] ∈ Rep_{F}(P, Q) denotes the class of φ (mod Inn(Q)). For each P in F_{0}, let ∗_{P} ∈ B̃(P) be the image under ϵ_{P} of the base point of BP, and let

γ_{P} : P ⟶^{≅} π_{1}(B̃(P), ∗_{P})

be the isomorphism induced by ϵ_{P} on fundamental groups.

Let L_{0} = ls(B̃) be the category with Ob(L_{0}) = Ob(F_{0}), and with

Mor_{L_{0}}(P, Q) = {(φ, u) | φ ∈ Rep_{F}(P, Q), u ∈ π_{1}(B̃(Q); B̃φ(∗_{P}), ∗_{Q})}.

Composition is defined by setting

(ψ, v) ∘ (φ, u) = (ψφ, v · B̃ψ_{∗}(u)),

where paths are composed from right to left. Let π_{0} : L_{0} ⟶ F_{0} be the functor which is the identity on objects, and which sends (φ, u) ∈ Mor_{L_{0}}(P, Q) to the composite

P ⟶_{≅}^{γ_{P}} π_{1}(B̃(P), ∗_{P}) ⟶^{B̃φ_{∗}} π_{1}(B̃(Q), B̃φ(∗_{P})) ⟶^{u_{∗}} π_{1}(B̃(Q), ∗_{Q}) ⟶_{≅}^{γ_{Q}^{−1}} Q.

Also, for each P, define

δ_{P} : P ⟶ Aut_{L_{0}}(P) by setting δ_{P} (g) = (Id_{P} , γ_{P} (g)).

Axioms (A), (B), and (C) for a centric linking system are easily seen to hold for L_{0}. For example, (C) follows as an immediate consequence of the definition of π_{0}.

Now set B_{1} = ke(L_{0}) = ke(ls(B̃)): the left homotopy Kan extension along the projection π̃_{0} : L_{0} ⟶ O(F_{0}) of the constant point functor on L_{0}. Thus B_{1}(P) = |B_{1}(P)| for each P, where B_{1}(P) is the category with objects the pairs (Q, α) for α ∈ Rep_{F}(Q, P), and with morphism sets

Mor_{B_{1}(P)}((Q, α), (R, β)) = {φ̂ ∈ Mor_{L_{0}}(Q, R) | α = β ∘ π̃_{0}(φ̂)}

= {(φ, u) | φ ∈ Rep_{F}(Q, R), α = β ∘ φ, u ∈ π_{1}(B̃(R); B̃φ(∗_{Q}), ∗_{R})}.

We define a natural homotopy equivalence of functors Ψ: B_{1} ⟶ B̃ as follows. For all P, maps Ψ_{P} : B_{1}(P) → B̃(P) are defined inductively, one skeleton at a time, (and simultaneously for all P) as follows.

• Each vertex (Q, α) in B_{1}(P) = |B_{1}(P)| is sent to B̃(α)(∗_{Q}) ∈ B̃(P).

• For each edge σ = ((Q, φ) ⟶^{(φ,u)} (P, Id)) in B_{1}(P), where

φ ∈ Rep_{F}(Q, P) and u ∈ π_{1}(B̃(P); B̃φ(∗_{Q}), ∗_{P}),

Φ_{P} |_{σ} = û for some path û in the homotopy class of u.

• For each edge σ = ((Q, φ) ⟶^{(φ,u)} (R, β)) in B_{1}(P), where β ≠ Id_{P}, write σ′ =

((Q, φ) ⟶^{(φ,u)} (R, Id)) (an edge in B_{1}(R)), and set Φ_{P} |_{σ} = B̃(β) ∘ (Ψ_{R}|_{σ′}).

• Consider a simplex of dimension m ≥ 2 in B_{1}(P) of the form

σ = ((Q_{0}, α_{0}) ⟶ (Q_{1}, α_{1}) ⟶ ⋯ ⟶ (Q_{m}, α_{m})).

If (Q_{m}, α_{m}) = (P, Id), then let Ψ_{P} |_{σ} be any singular simplex in B̃(P) whose boundary is as already defined. Otherwise, let σ′ be the unique simplex in B_{1}(Q_{m}) representing a chain ending in (Q_{m}, Id) such that σ = B_{1}(α_{m})(σ′), and set Ψ_{P} |_{σ} = B̃(α_{m}) ∘ (Ψ_{Q_{m}}|_{σ′}).

Since B_{1}(P) ≃ B̃(P) ≃ BP (where P is given the discrete topology), the above construction is always possible, and defines a homotopy equivalence. It induces the identity on fundamental groups, under their given identifications with P. By construction, the Φ_{P} form a natural morphism of functors Ψ from B_{1} to B̃.

Let (B̃′, {ϵ′_{P} }) be another rigidification of B, and let κ: B̃ ⟶ B̃′ be a natural homotopy equivalence of rigidifications. We have already chosen our basepoint ∗_{P} = ϵ_{P} (∗), where ∗ ∈ BP is a fixed basepoint, and we now set ∗′_{P} = ϵ′_{P} (∗). Fix, for each P, a homotopy H_{P} between κ_{P} ∘ ϵ_{P} and ϵ′_{P}. The restriction of H_{P} to the base point of BP provides a canonical path in B̃′(P) from κ_{P} (∗_{P}) to ∗′_{P}, whose homotopy class we denote w_{P} ∈ π_{1}(P̃; κ_{P} (∗_{P}), ∗′_{P}). We now define

κ_{♯}: L_{0} ⟶ L′_{0}

to be the identity on objects, and for (φ, u) ∈ Mor_{L_{0}}(P, Q),

κ_{♯}(φ, u) = (φ, w_{P} · κ_{Q}_{∗}(u) · B̃′φ_{∗}(w_{P})^{−1}) .  (1)

It is straightforward to show that κ is a well defined isomorphim of linking systems; i.e., an isomorphism of categories which is natural with respect to the projections to F_{0} and the distinguished monomorphisms.

(c) Now assume that L_{0} is given; it remains to construct an isomorphism L_{0} ≅ ls(ke(L_{0})) of linking systems associated to F_{0}. Set B̃ = ke(L_{0}) and L_{1} = ls(B̃) for short. By definition, L_{0} and L_{1} have the same objects, and a morphism in L_{1} from P to Q is a pair (φ, u), where φ ∈ Rep_{F}(P, Q) and u ∈ π_{1}(B̃(Q); B̃φ(∗_{P}), ∗_{Q}). Also, B̃(P) = |π̃_{0}↓P | where π̃_{0} is the projection of L_{0} onto O(F_{0}); in particular, we choose ∗_{P} to be the vertex of (P, Id). Define Ψ: L_{0} → L_{1} by sending each object to itself, and by sending α ∈ Mor_{L_{0}}(P, Q) to (π̃_{0}(α), [α]), where [α] is the homotopy class of α, regarded as an edge in |π̃_{0}↓Q| from (P, π̃_{0}(α)) = π̃_{0}α(∗_{P}) to (Q, Id) = ∗_{Q}. This is easily checked to be an isomorphism of categories, and to commute with the distinguished monomorphisms and the projections to F_{0}. □

5. Higher limits over orbit categories

If F is any fusion system over a discrete p-toral group S, then O(F) will denote its orbit category: the category whose objects are the subgroups of S, and where

Mor_{O(F)}(P, Q) = Rep_{F}(P, Q) =^{def} Inn(Q)\ Hom_{F}(P, Q).

Also, we write O^{c}(F) = O(F^{c}) to denote the full subcategory of O(F) whose objects are the F-centric subgroups of S; and more generally write O(F_{0}) to denote the full subcategory of O(F) corresponding to any full subcategory F_{0} of F.

By Lemma 2.5, the morphism sets in the orbit category are all finite. There is a canonical projection functor F → O(F) which is the identity on objects and the natural projection Hom_{F}(P, Q) → Rep_{F}(P, Q) on morphisms.

Throughout this section, when C is a category, we frequently write C-mod to denote the category of functors C^{op} ⟶ Ab. This notation will not be used in the statements of results here, but it is used in several of the proofs.

Lemma 5.1. Let F be a saturated fusion system over a discrete p-toral group S, and let F_{0} ⊆ F be any full subcategory such that P ∈ Ob(F_{0}) implies P^{•} ∈ Ob(F_{0}). Set F_{0}^{•} = F_{0} ∩ F^{•}. Then there are well defined functors

O^{c}(F) \overrightarrow{⟵}_{incl}^{(−)^{•}} O(F^{c•}) ,

where (−)^{•} sends P to P^{•} and [φ] to [φ^{•}]. Also, (−)^{•} is a left adjoint to the inclusion.

Proof. This follows from Corollary 3.4. The only thing to check is that (−)^{•} is well defined on morphisms in the orbit category. If φ_{1}, φ_{2} ∈ Hom_{F}(P, Q) represent the same morphism in the orbit category, then φ_{1} = c_{g} ∘ φ_{2} for some g ∈ Q, so φ_{1}^{•} = c_{g} ∘ φ_{2}^{•} by functoriality, and hence [φ_{1}^{•}] = [φ_{2}^{•}] in Rep_{F}(P^{•}, Q^{•}). □

The following proposition shows that the problem of describing higher limits over the orbit categories we are considering can always be reduced to one over a finite subcategory.

Proposition 5.2. Let F be a saturated fusion system over a discrete p-toral group S. Let F_{0} ⊆ F be any full subcategory such that P ∈ Ob(F_{0}) implies P^{•} ∈ Ob(F_{0}), and set F_{0}^{•} = F_{0} ∩ F^{•}. Then for any functor F: O(F_{0})^{op} ⟶ ℤ_{(p)}-mod, restriction to F_{0}^{•} induces an isomorphism

\underleftarrow{lim}^{∗}_{O(F_{0})}(F) ≅ \underleftarrow{lim}^{∗}_{O(F_{0}^{•})}(F |_{O(F_{0}^{•})}) .

Proof. Consider the functors

O(F_{0})-mod \overrightarrow{⟵}_{T}^{R} O(F_{0}^{•})-mod ,

where R is given by restriction and T by composition with the functor (−)^{•}. Then T is a left adjoint to R, since (−)^{•} is a left adjoint to the inclusion by Lemma 5.1. Also, T and R are both exact functors, and R sends injectives to injectives since it is right adjoint to an exact functor.

Let \underline{ℤ} be the constant functor on O(F_{0}^{•}) which sends all objects to ℤ. Then T (\underline{ℤ}) is the constant functor on O(F_{0}), and hence for any functor F on O(F_{0}),

\underleftarrow{lim}_{O(F_{0})}(F) = Hom_{O(F_{0})-mod} (T (\underline{ℤ}), F) ≅ Hom_{O(F_{0}^{•})-mod} (\underline{ℤ}, R(F)) = \underleftarrow{lim}_{O(F_{0}^{•})}(R(F)) .

Since R is exact and sends injectives to injectives, it sends injective resolutions to injective resolutions, and thus induces an isomorphism between higher limits over O(F_{0}) and over O(F_{0}^{•}). □

We next want to show that the techniques which we have already developped for handling higher limits over orbit categories in the finite case [BLO2, §3] also apply in this new situation. The proof of this is similar to the proof in [BLO2] of the analogous result for fusion systems over finite p-groups, and is in fact a special case of a very general result which we prove here.

For any group Γ (not necessarily finite), and any set H of subgroups of Γ, we define O_{H}(Γ) to be the corresponding orbit category of Γ: the category with Ob(O_{H}(Γ)) = H, and with morphism sets

Mor_{O_{H}(Γ)}(H, H′) = H′\N_{Γ}(H, H′) ≅ Map_{Γ}(Γ/H, Γ/H′).

Here, N_{Γ}(H, H′) is the transporter set:

N_{Γ}(H, H′) = {g ∈ Γ | gHg^{−1} ≤ H′}.

If 1 ∈ H, then for any ℤ[Γ]-module M, we define

Λ_{H}^{∗}(Γ; M) = \underleftarrow{lim}^{∗}_{O_{H}(Γ)}(F_{M}),

where F_{M}: O_{H}(Γ)^{op} ⟶ Ab is the functor F_{M}(H) = 0 if H ≠ 1 and F_{M}(1) = M.

It is important to distinguish between the orbit category of a group and the orbit category of a fusion system. When G is a finite group and S ∈ Syl_{p}(G), the orbit category of the fusion system F_{S}(G) is not the same as the orbit category O_{S}(G) (the orbit category of G with objects the subgroups of S).

Proposition 5.3. Fix a category C, a group Γ, a set H of subgroups of Γ such that 1 ∈ H, and a functor

α: O_{H}(Γ) ⟶ C.

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

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

(b) For each d ∈ Ob(C) such that d ≅ c̸_{0}, all isotropy subgroups of the Γ-action on Mor_{C}(c_{0}, d) are nontrivial and conjugate to subgroups in H.

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

(d) For any H ∈ H, any d ∈ Ob(C), and any φ ∈ Mor_{C}(c_{0}, d) which is H-invariant, there is some φ ∈ Mor_{C}(α(H), d) such that φ = φ ∘ α(incl_{1}^{H}).

Let

Φ : C^{op} ⟶ Ab

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

\underleftarrow{lim}_{C}^{∗}(Φ) ⟶_{≅}^{α^{∗}} \underleftarrow{lim}^{∗}_{O_{H}(Γ)}(Φ ∘ α) = Λ_{H}^{∗}(Γ; Φ(c_{0}))

is an isomorphism.

Proof. Consider the functors

O_{H}(Γ)-mod \overleftarrow{⟶}_{R_{α}}^{α^{∗}} C-mod,

where α^{∗} is composition with α^{op}, and R_{α} is the right Kan extension of α^{op}. Specifically, for d ∈ Ob(C), let α↓d be the overcategory whose objects are pairs (H, φ) for φ ∈ Mor_{C}(α(H), d), and where a morphism from (H, φ) to (K, ψ) is a morphism χ ∈ Mor_{O_{H}(Γ)}(H, K) such that ψ ∘ α(χ) = φ. Let κ_{d}: α↓d ⟶ O_{H}(Γ) be the forgetful functor. Then (α↓d)^{op} = d↓α^{op} (the undercategory), and for F: O_{H}(Γ)^{op} ⟶ Ab, R_{α}(F) is defined by setting

R_{α}(F)(d) = \underleftarrow{lim}_{(α↓d)^{op}}(F ∘ κ_{d}^{op}).

On morphisms, R_{α}(F) sends f ∈ Mor_{C}(d, d′) to the morphism induced by the functor (α↓d ⟶^{f ∘−} α↓d′). By [McL, §X.3, Theorem 1], R_{α} is right adjoint to α^{∗}. In particular, since α^{∗} preserves exact sequences, R_{α} sends injectives to injectives.

Fix H ∈ H and d ∈ Ob(C). Consider the map

μ: Mor_{C}(α(H), d) ⟶ Mor_{C}(c_{0}, d)

defined by composition with the “inclusion” morphism α(incl_{1}^{H}). This map is injective by (c), and Im(μ) ⊇ Mor_{C}(c_{0}, d)^{H} by (d). Also, Im(μ) is contained in Mor_{C}(c_{0}, d)^{H} since incl_{1}^{H} ∘x = incl_{1}^{H} for all x ∈ H. Thus μ induces a bijection

Mor_{C}(α(H), d) ⟶_{≅}^{μ_{0}} Mor_{C}(c_{0}, d)^{H} .  (1)

Fix representatives {φ_{i}^{d}}_{i∈I_{d}} for the Γ-orbits in Mor_{C}(c_{0}, d), and let Γ_{i}^{d} ≤ Γ be the stabilizer subgroup of φ_{i}^{d}. By (b), we can choose the φ_{i}^{d} such that Γ_{i}^{d} ∈ H for all i. By (1), each φ_{i}^{d} has a unique “extension” to ψ_{i}^{d} ∈ Mor_{C}(α(Γ_{i}^{d}), d); i.e., there is a unique ψ_{i}^{d} such that φ_{i}^{d} = ψ_{i}^{d} ∘ α(incl_{1}^{Γ_{i}^{d}}). Also, for any (H, χ) in α↓d, there is a unique i ∈ I_{d} and a unique morphism χ_{0} ∈ Mor_{O_{H}(Γ)}(H, Γ_{i}^{d}) such that χ = ψ_{i}^{d} ∘ χ_{0}. So each object (Γ_{i}^{d}, ψ_{i}^{d}) is a final object in its connected component of the overcategory α↓d. Thus for any F in O_{H}(Γ)-mod,

R_{α}(F)(d) ≅ ∏_{i∈I_{d}} F (Γ_{i}^{d}).  (2)

In particular, R_{α} is an exact functor.

Let \underline{ℤ} denote the constant functor on C^{op} which sends each object to ℤ and each morphism to the identity. Then α^{∗}\underline{ℤ} is the constant functor on O_{H}(Γ)^{op}. If F : C^{op} → Ab is any functor, then

\underleftarrow{lim}_{C}(F) ≅ Hom_{C-mod} (\underline{ℤ}, F);

and similarly for functors in O_{H}(Γ)-mod.

Assume H ∈ H is such that α(H) ≅ α(1) = c_{0}. Since all endomorphisms of c_{0} are automorphisms (by (a)), Mor_{C}(c_{0}, α(H)) contains only isomorphisms, and in particular α(incl_{1}^{H}) is an isomorphism. Also, incl_{1}^{H} ∘x = incl_{1}^{H} for all x ∈ H, so α(x) = Id_{c_{0}} for all x ∈ H. By (a) again, this implies that H = 1.

The functor α^{∗}Φ = Φ ∘ α^{op} : O_{H}(Γ)^{op} → ℤ_{(p)}-mod thus sends the object 1 to Φ(c_{0}) (with the given action of Γ), and sends all other objects to 0. Then R_{α} sends an injective resolution I_{∗} of α^{∗}Φ to an injective resolution R_{α}(I_{∗}) of R_{α}(α^{∗}Φ). It follows that

Λ_{H}^{∗}(Γ; Φ(c_{0})) =^{def} \underleftarrow{lim}^{∗}_{O_{H}(Γ)}(α^{∗}Φ) ≅ H^{∗}(Mor_{O_{H}(Γ)-mod} (α^{∗}\underline{ℤ}, I_{∗}))

≅ H^{∗}(Mor_{C-mod} (\underline{ℤ}, R_{α}(I_{∗}))) ≅ \underleftarrow{lim}_{C}^{∗}(R_{α}(α^{∗}Φ)).

It remains only to show that R_{α}(α^{∗}Φ) ≅ Φ. For each d ∈ Ob(C), if d ≅ c̸_{0}, then Mor_{C}(c_{0}, d) is a disjoint union of orbits Γ/Γ_{i}^{d}, where 1 ≠ Γ_{i}^{d} ∈ H by (b). So by (2),

R_{α}(α^{∗}Φ)(d) = R_{α}(Φ ∘ α)(d) ≅ ∏_{i} Φ(α(H_{i})) = 0;

where the last equality holds since we already showed that H ≠ 1 implies α(H) ≅ c_{0}. If d ≅ c_{0}, then Mor_{C}(c_{0}, d) consists of one free orbit of Γ (by (a)), and hence R_{α}(α^{∗}Φ)(d) ≅ Φ(α(1)) ≅ Φ(c_{0}). This finishes the proof that R_{α}(α^{∗}Φ) ≅ Φ. □

Our first application of Proposition 5.3 is to the case where C is the orbit category of a saturated fusion system over a discrete p-toral group. As in [JMO] and [BLO2], when Γ is finite and H is the set of p-subgroups of Γ (or the set of subgroups of a given Sylow p-subgroup), we write Λ^{∗}(Γ; M) = Λ_{H}^{∗}(Γ; M) (and the prime p is understood).

Proposition 5.4. Let F be a saturated fusion system over S. Let

Φ : O^{c}(F)^{op} ⟶ ℤ_{(p)}-mod

be any functor which vanishes except on the isomorphism class of some fixed F-centric subgroup Q ≤ S. Then

\underleftarrow{lim}^{∗}_{O^{c}(F)}(Φ) ≅ Λ^{∗}(Out_{F}(Q); Φ(Q)).

Proof. It suffices to do this when Q is fully normalized. Set Γ = Out_{F}(Q) and Σ = Out_{S}(Q) ∈ Syl_{p}(Γ), and let H be the set of subgroups of Σ. Since Σ ≅ N_{S}(Q)/Q, each subgroup of Σ has the form Out_{P} (Q) for some unique P ≤ N_{S}(Q) containing Q. Define

α: O_{Σ}(Γ) ⟶ O^{c}(F)

on objects by setting α(Out_{P} (Q)) = P for Q ≤ P ≤ N_{S}(Q). If φ ∈ Aut_{F}(Q) is such that [φ] ∈ N_{Γ}(Out_{P} (Q), Out_{P′}(Q)) (the set of elements which conjugate Out_{P} (Q) into Out_{P′}(Q)), then φ can be extended to some φ ∈ Hom_{F}(P, P′) by axiom (II), the class of φ in the orbit category is uniquely determined by φ by Proposition 2.8, and α sends the class of [φ] to the class of φ.

We apply Proposition 5.3 to this functor α. Condition (a) is clear, (c) holds for O^{c}(F) by Proposition 2.8, and (d) holds by axiom (II) of a saturated fusion system. As for (b), since every morphism in F is the composite of an isomorphism followed by an inclusion, it suffices to prove that the stabilizer in Γ of an inclusion incl_{Q}^{P} ∈ Hom_{F}(Q, P), where Q ≨ P, is a nontrivial p-subgroup. But the stabilizer is Out_{P} (Q) ≅ N_{P} (Q)/Q, which is nontrivial by Lemma 1.8. All of the hypotheses of Proposition 5.3 thus hold, and the result follows. □

Using the terminology of [BLO2], we say that a category C has bounded limits at p if there is k > 0 such that for any functor Φ: C^{op} ⟶ ℤ_{(p)}-mod, \underleftarrow{lim}^{i}(Φ) = 0 for all i > k. The following is a first corollary of Proposition 5.4.

Corollary 5.5. Let F be a saturated fusion system over a discrete p-toral group S, and let F_{0} ⊆ F^{c} be a full subcategory such that P ∈ Ob(F_{0}) implies P^{•} ∈ Ob(F_{0}). Then the orbit category O(F_{0}) has bounded limits at p.

Proof. By Proposition 5.2, it suffices to prove this when F_{0} ⊆ F^{•}; in particular, when F_{0} has only finitely many isomorphism classes. By [JMO2, Proposition 4.11], for each finite group Γ, there is some k_{Γ} such that Λ^{i}(Γ; M) = 0 for all ℤ_{(p)}[Γ]-modules M and all i > k_{Γ}. Let k be the maximum of the k_{Out_{F} (P)} for all P ∈ Ob(F_{0}). Then by Proposition 5.4, for each functor Φ : O(F_{0})^{op} ⟶ ℤ_{(p)}-mod which vanishes except on one orbit type, \underleftarrow{lim}^{i}(Φ) = 0 for i > k. The same result for an arbitrary p-local functor Φ on O(F_{0}) now follows from the exact sequences of higher limits associated to short exact sequences of functors. □

In practice, when computing higher limits over orbit categories O^{c}(F), it is useful to combine Propositions 5.2 and 5.4, as illustrated by the following corollary.

Corollary 5.6. Let F be a saturated fusion system over a discrete p-toral group S. Let F: O^{c}(F)^{op} → ℤ_{(p)}-mod be a functor with the property that for each F-centric subgroup P ∈ H^{•}(F), Λ^{∗}(Out_{F}(P); F (P)) = 0. Then \underleftarrow{lim}^{∗}(F) = 0.

Proof. Let F_{0} : O^{c}(F^{•})^{op} → ℤ_{(p)}-mod be the restriction of F. By Proposition 5.2,

\underleftarrow{lim}^{∗}_{O^{c}(F)}(F) ≅ \underleftarrow{lim}^{∗}_{O^{c}(F^{•})}(F_{0}).

Assume first that F_{0} vanishes except on the conjugacy class of one subgroup P ∈ H^{•}(F). Let F′ be the functor on O^{c}(F) which takes the same value on the conjugacy class of P and vanishes on all other subgroups. Then

\underleftarrow{lim}^{∗}_{O^{c}(F^{•})}(F_{0}) ≅ \underleftarrow{lim}^{∗}_{O^{c}(F)}(F′) ≅ Λ^{∗}(Out_{F}(P); F (P))

by Propositions 5.2 and 5.4, and this is zero by assumption.

By Lemma 3.2(a), the category O^{c}(F^{•}) contains only finitely many isomorphism classes. Hence there is a sequence

0 = Φ_{0} ⊆ Φ_{1} ⊆ ⋯ ⊆ Φ_{k} = F_{0}

of subfunctors defined on O^{c}(F^{•}), with the property that for each i, Φ_{i}/Φ_{i−1} vanishes except on the conjugacy class of one subgroup P, and (Φ_{i}/Φ_{i−1})(P) ≅ F (P). We have just seen that \underleftarrow{lim}^{∗}(Φ_{i}/Φ_{i−1}) = 0 for all i; and hence \underleftarrow{lim}^{∗}(F_{0}) = 0 by the relative long exact sequences of higher limits. □

The following lemma will be useful in showing that certain functors on the orbit category are acyclic. As usual, when F is a fusion system over S, a subgroup P ≤ S will be called weakly closed in F if it is the only subgroup in its F-conjugacy class.

Lemma 5.7. Let F be any saturated fusion system over a discrete p-toral group S, and let Q ◁ S be any F-centric subgroup which is weakly closed in F. Set Γ = Out_{F}(Q), and let F_{≥Q} ⊆ F^{c} be the full subcategory whose objects are the subgroups which contain Q. Define the functor

Θ: O(F_{≥Q})^{op} ⟶ O_{p}(Γ)

by sending an object P to Out_{P} (Q) ≤ Γ, and by sending a morphism φ ∈ Rep_{F}(P, P′) to the class of φ|_{Q} ∈ N_{Γ}(Θ(P), Θ(P′)). Then for any pair of functors

F: O^{c}(F)^{op} ⟶ ℤ_{(p)}-mod  and  Φ: O_{p}(Γ)^{op} ⟶ ℤ_{(p)}-mod

such that Φ ∘ Θ ≅ F |_{O(F_{≥Q})}, and such that Out_{Q}(P) ≅ N_{P Q}(P)/P acts trivially on F (P) for all P ≤ S,

\underleftarrow{lim}^{∗}_{O^{c}(F)}(F) ≅ \underleftarrow{lim}^{∗}_{O_{p}(Γ)}(Φ).

Proof. Define a functor

F′: O^{c}(F)^{op} ⟶ ℤ_{(p)}-mod

by setting F′(P) = F (P) if P ≥ Q and F′(P) = 0 otherwise. Regard F′ as a quotient functor of F, and set F′′ = Ker[F −↠ F′].

If P ≤ S is F-centric and P ≱ Q, then Out_{Q}(P) ≅ N_{P Q}(P)/P ≠ 1, and by assumption this group acts trivially on F (P) ≅ F′′(P). Hence the kernel of the action of Out_{F}(P) on F′′(P) has order a multiple of p, and so Λ^{∗}(Out_{F}(P); F′′(P)) = 0 by [JMO, Proposition 5.5]. Thus \underleftarrow{lim}^{∗}(F′′) = 0 by Corollary 5.6, and hence

\underleftarrow{lim}^{∗}_{O^{c}(F)}(F) ≅ \underleftarrow{lim}^{∗}_{O^{c}(F)}(F′).

Recall that Γ = Out_{F}(Q). Since Q is fully normalized in F (it is the unique subgroup in its F-conjugacy class), Θ(S) = Out_{S}(Q) ∈ Syl_{p}(Γ). Also, Θ defines a bijection between subgroups of Θ(S) ≅ S/Q and subgroups of S which contain Q. For all Q ≤ P, P′ ≤ S,

Rep_{F}(P, P′) ⟶^{(−)|_{Q}} Mor_{O_{p}(Γ)}(Θ(P), Θ(P′))

is injective by Proposition 2.8. If g ∈ N_{Γ}(Θ(P), Θ(P′)) is any element in the transporter, and g = [φ] for φ ∈ Aut_{F}(Q), then for all x ∈ P there is y ∈ P′ such that φc_{x}φ^{−1} = c_{y} as automorphisms of Q. Hence by condition (II) in Definition 2.2, φ extends to a homomorphism φ ∈ Hom_{F}(P, P′), and Θ sends [φ] ∈ Rep_{F}(P, P′) to the class of g.

This proves that Θ induces bijections on all morphism sets, and thus is an equivalence of categories. Hence if Φ is such that Φ ∘ Θ ≅ F |_{O(F_{≥Q})}, then

\underleftarrow{lim}^{∗}_{O_{p}(Γ)}(Φ) ≅ \underleftarrow{lim}^{∗}_{O(F_{≥Q})}(F |_{O(F_{≥Q})}) ≅ \underleftarrow{lim}^{∗}_{O^{c}(F)}(F′) ≅ \underleftarrow{lim}^{∗}_{O^{c}(F)}(F).  □

This can now be applied to prove the acyclicity of certain explicit functors.

Proposition 5.8. Let F be any saturated fusion system over a discrete p-toral group S. Define

F_{1}, F_{2} : O^{c}(F)^{op} ⟶ ℤ_{(p)}-mod

on objects by setting F_{1}(P) = Z(P)_{0} and F_{2}(P) = π_{2}(B(Z(P))_{p}^{∧}). On morphisms, each F_{i} sends the class of φ ∈ Hom_{F}(P, P′) to the homomorphism induced by the inclusion of Z(P′) into Z(φ(P)) followed by φ^{−1}|_{Z(φ(P))}. Then F_{1} and F_{2} are both acyclic.

Proof. Set T = S_{0} (the “maximal torus” in F), Q = C_{S}(T) ◁ S, and Γ = Out_{F}(Q). Then Q is F-centric, and is weakly closed in F since T is. Let

Θ: O(F_{≥Q}) ⟶ O_{p}(Γ)

be the functor of Lemma 5.7. For each p-subgroup Π ≤ Γ, regarded as a group of automorphisms of Q, let N_{Π} be the norm map for the action of Π on T; i.e., N_{Π}(t) = ∏_{γ∈Π} γ(t) for t ∈ T. Define

Φ_{1}(Π) = N_{Π}(T) and Φ_{2}(Π) = Hom(ℤ/p^{∞}, T)^{Π}.

These define functors Φ_{i}: O_{p}(Γ)^{op} → ℤ_{(p)}-mod.

For each P ≤ S which contains Q, N_{P/Q}(T) is connected (i.e., infinitely p-divisible), and has finite index in Z(P) since Z(P) ∩ T = T^{P} and T^{P} /N_{P/Q}(T) has exponent at most |P/Q|. Hence N_{P/Q}(T) is equal to the identity component Z(P)_{0}, and we have

F_{1}(P) = Z(P)_{0} = N_{P/Q}(T) = Φ_{1}(Θ(P)).

In general, for any discrete p-toral group P,

π_{2}(BP _{p}^{∧}) = [S^{2}, BP _{p}^{∧}] ≅ [BS^{1}, BP _{p}^{∧}] ≅ Hom(ℤ/p^{∞}, P).

Here, the last equivalence follows from Lemma 1.10, while the middle one follows by obstruction theory (since π_{i}(BP _{p}^{∧}) = 0 for i > 2). Hence for any P ≤ S which contains Q,

F_{2}(P) = π_{2}(BZ(P)_{p}^{∧}) ≅ Hom(ℤ/p^{∞}, Z(P)) ≅ Hom(ℤ/p^{∞}, T)^{P/Q} = Φ_{2}(Θ(P)).

Thus Φ_{i} ∘ Θ ≅ F_{i}|_{O(F_{≥Q})} (for i = 1, 2). Also, for each P ≤ S, Out_{Q}(P) acts trivially on F_{i}(P) for i = 1, 2 since Q centralizes Z(P)_{0} ≤ T. So by Lemma 5.7,

\underleftarrow{lim}^{∗}_{O^{c}(F)}(F_{i}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(Γ)}(Φ_{i}).

The functors Φ_{1} and Φ_{2} are both Mackey functors on O_{p}(Γ) (see [JM, Proposition 5.14] or [JMO, Proposition 5.2]), and hence are acyclic. □

As in Section 4, when F is a saturated fusion system over S, we let B denote the homotopy functor B(P) = BP, and by extension let B_{p}^{∧} denote the functor B_{p}^{∧}(P) = BP _{p}^{∧}. The following proposition is a first application of Proposition 5.8. It shows that there is a bijective correspondence between rigidifications of these two functors.

Proposition 5.9. Let F be a saturated fusion system over a discrete p-toral group S, and let F_{0} ⊆ F^{c} be any full subcategory which contains F^{c•}. Let B̂: O(F_{0}) ⟶ Topbe any rigidification of the homotopy functor B_{p}^{∧}. Then there is a functor B̃: O(F_{0}) ⟶ Top such that B̃(P) ≃ BP for all P, together with a natural transformation of functors B̃ → B̂ which is a homotopy equivalence after p-completion. Moreover, there is a bijection between equivalence classes of rigidifications of B and equivalence classes of rigidifications of B_{p}^{∧}.

Proof. Let χ: B ⟶ B_{p}^{∧} be the natural transformation of homotopy functors which sends BP to BP _{p}^{∧} by the canonical map. We want to apply Theorem A.3, which is a relative version of the Dwyer-Kan theorem [DK] for rigidifying centric homotopy diagrams. We first check that χ is relatively centric in the sense of Theorem A.3. This means showing, for each φ ∈ Mor_{O(F_{0})}(P, Q), that the square

Map(BP, BP)_{Id}  \overline{^{Bφ∘−}→} Map(BP, BQ)_{Bφ}

Map(BP, BP _{p}^{∧})_{χ(P)} ↓^{χ(P)∘−}  \overline{→^{Bφ∘−}} Map(BP, BQ_{p}^{∧})_{χ(Q)∘Bφ} ↓^{χ(Q)∘−}

is a homotopy pullback. By a classical result, the top row is a homotopy equivalence, and both mapping spaces have the homotopy type of BZ(P) (cf. [BKi, Proposition 7.1]). By Lemma 1.10, the second row is also a homotopy equivalence, and both mapping spaces have the homotopy type of BZ(P)_{p}^{∧}. So the square is a homotopy pullback.

For each i ≥ 1, let β_{i}: O(F_{0})^{op} ⟶ Ab be the functor defined in Theorem A.3, where for each P,

β_{i}(P) = π_{i}(hofiber(Map(BP, BP)_{Id}_{BZ(P)} ⟶^{χ(P)∘−} Map(BP, BP _{p}^{∧})_{χ(P)}_{BZ(P)_{p}^{∧}})).

By [DW2, Proposition 3.1], this homotopy fiber is a K(V, 1) for some ℚ̂_{p}-vector space V. In particular, the fiber is connected, β_{1}(P) is abelian for all P, and β_{i} = 0 for all i ≥ 2. Also, by the homotopy exact sequence for the fibration, there is a short exact sequence of functors

0 ⟶ F_{2} ⟶ β_{1} ⟶ F_{1} ⟶ 0,

where F_{1} and F_{2} are the functors of Proposition 5.8. By Proposition 5.2, for all i ≥ 1 and j = 1, 2,

\underleftarrow{lim}^{i}_{O(F_{0})}(F_{j}) ≅ \underleftarrow{lim}^{i}_{O(F^{•})}(F_{j}) ≅ \underleftarrow{lim}^{i}_{O^{c}(F)}(F_{j}),

where the last group vanishes by Proposition 5.8. Thus \underleftarrow{lim}^{i}(β_{1}) = 0 for all i ≥ 1. The proposition now follows directly from Theorem A.3. □

In Section 8, we will also need to work with higher limits over orbit categories of certain infinite groups. For any (discrete) group G, let O_{dpt}(G) denote the orbit category of G whose objects are the discrete p-toral subgroups of G; and define (for any ℤ[G]-module M),

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

We are now ready to give a second application of Proposition 5.3.

Lemma 5.10. Fix a group G, a discrete p-toral subgroup Q ≤ G, and a functor Φ: O_{dpt}(G)^{op} → Ab with the property that Φ(P) = 0 except when P is G-conjugate to Q. Let Φ′: O_{dpt}(N_{G}(Q)/Q)^{op} → Ab be the functor Φ′(P/Q) = Φ(P). Then

\underleftarrow{lim}^{∗}_{O_{dpt}(G)}(Φ) ≅ \underleftarrow{lim}^{∗}_{O_{dpt}(N_{G}(Q)/Q)}(Φ′) ≅ Λ_{dpt}^{∗}(N_{G}(Q)/Q; Φ(Q)) .  (1)

Proof. We apply Proposition 5.3, where C = O_{dpt}(G), Γ = N_{G}(Q)/Q, and H is the set of discrete p-toral subgroups of Γ. A functor

α: O_{dpt}(Γ) ⟶ O_{dpt}(G)

is defined by setting α(P/Q) = P, and by sending the set (P′/Q)\N_{Γ}(P/Q, P′/Q) to P′\N_{G}(P, P′) in the obvious way.

The hypotheses of Proposition 5.3 follow easily from the definition of the orbit categories, and so the isomorphisms between higher limits follow from the proposition. □

The following very general lemma will help in certain cases to reduce computations of higher limits to those taken over finite subcategories.

Lemma 5.11. Let C be a (small) category, and let C_{1} ⊆ C_{2} ⊆ ⋯ be an increasing sequence of subcategories of C whose union is C. Let F: C^{op} ⟶ Ab be a functor such that for each k,

\underleftarrow{lim}_{i}^{1}(\underleftarrow{lim}_{C_{i}}^{k}(F |_{C_{i}})) = 0.

Then the homomorphism

\underleftarrow{lim}_{C}^{k}(F) ⟶^{≅} \underleftarrow{lim}_{i}(\underleftarrow{lim}_{C_{i}}^{k}(F |_{C_{i}}))

induced by the restrictions is an isomorphism for all k.

Proof. For any category D and any functor Φ: D^{op} → Ab, \underleftarrow{lim}^{∗}(Φ) is the homology of the chain complex (C^{∗}(D; Φ), d), defined by setting

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

where the product is taken over composable n-tuples of morphisms in D, and where

d(ξ)(c_{0} ⟶^{α} c_{1} → ⋯ → c_{n+1}) = α^{∗}ξ(c_{1} → ⋯ → c_{n+1}) + ∑_{i=1}^{n+1} ξ(c_{0} → ⋯ ĉ_{i} ⋯ → c_{n+1}).

See, for example, [GZ, Appendix II, Proposition 3.3] or [Ol, Lemma 2]. If D_{0} ⊆ D is a subcategory, then the restriction homomorphism from \underleftarrow{lim}_{D}^{∗}(Φ) to \underleftarrow{lim}_{D_{0}}^{∗}(Φ|_{D_{0}}) is induced

by the obvious surjections C^{∗}(D; Φ) −↠ C^{∗}(D_{0}; Φ).

In the above situation, the chain complex (C^{∗}(C; F), d) is the limit of an inverse system of chain complexes (C^{∗}(C_{i}; F |_{C_{i}}), d) with surjections, where the inverse system of homology groups of these chain complexes has vanishing \underleftarrow{lim}^{1}(−). Since \underleftarrow{lim}^{1}(−) vanishes for a (countable directed) inverse system with surjections, we conclude that the cohomology of (C^{∗}(C; F), d) is isomorphic to the inverse limit of the cohomology of the complexes (C^{∗}(C_{i}; F |_{C_{i}}), d). □

The next lemma describes how, in some cases, the computation of Λ_{dpt}^{∗}(G; M) can be reduced to the case where G is finite. When G is a finite group and M is a ℤ[G]- module, we let Λ^{∗}(G; M) denote the Λ-functor taken with respect to p-subgroups of G.

Lemma 5.12. Let G be a locally finite group. Assume there is a discrete p-toral subgroup S ≤ G such that every discrete p-toral subgroup of G is conjugate to a subgroup of S. Fix a ℤ[G]-module M, and assume that for some finite subgroup H_{0} ≤ G, Λ^{∗}(H; M) = 0 for all finite subgroups H ≤ G which contain H_{0}. Then Λ_{dpt}^{∗}(G; M) = 0. In particular, Λ_{dpt}^{∗}(G; M) = 0 if M is a ℤ_{(p)}[G]-module and the kernel of the action of G on M contains an element of order p.

Proof. By [JMO, Proposition 5.5], for any finite group H and any ℤ_{(p)}[H]-module M such that the kernel of the H-action on M has order a multiple of p, Λ^{∗}(H; M) = 0. Hence the last statement follows as a special case of the first.

Fix a Sylow p-subgroup S ∈ Syl_{p}(G), and let O_{S}(G) ⊆ O_{dpt}(G) be the full subcategory whose objects are the subgroups of S. Since each discrete p-toral subgroups of G is G-conjugate to a subgroup of S, these categories are equivalent, and so we can work over O_{S}(G) instead. Define

F_{M}: C^{op} ⟶ Ab by setting F_{M}(P) = \begin{cases}M & if P = 1 \\ 0 & if P ≠ 1. \end{cases}

By definition, Λ_{dpt}^{∗}(G; M) = \underleftarrow{lim}^{∗}(F_{M}), and we must show that this vanishes in all degrees.

Step 1: To simplify the notation, we write C = O_{S}(G), and let C_{0} ⊆ C be the full subcategory whose objects are the finite subgroups of S. For each subgroup Q ≤ S and each abelian group A, let ℑ_{Q}^{A} in C-mod be the functor

ℑ_{Q}^{A}(P) = Map(Mor_{C}(Q, P), A) ≅ ∏_{Mor_{C}(Q,P)} A.

For any F in C-mod, Hom_{C-mod} (F, ℑ_{Q}^{A}) ≅ Hom_{ℤ}(F (Q), A). Hence ℑ_{Q}^{A} is injective if A is injective as an abelian group, and each functor on C injects into a product of such injectives. Also, when Q is finite,

\underleftarrow{lim}_{C_{0}}(ℑ_{Q}^{A}|_{C_{0}}) ≅ \underleftarrow{lim}_{C}(ℑ_{Q}^{A}) ≅ A

(where the second isomorphism holds for arbitrary Q ≤ S).

Choose a sequence of functors

0 ⟶ F_{M} ⟶^{d_{0}} ℑ_{0} ⟶^{d_{1}} ℑ_{1} ⟶^{d_{2}} ⋯,  (1)

where each ℑ_{k} is a product of injective functors ℑ_{Q}^{A} for finite subgroups Q ≤ S and injective abelian groups A, and where (1) is exact after restriction to C_{0}. We claim that this is an injective resolution of F_{M}. In other words, the sequence

0 ⟶ F_{M}(P) ⟶ ℑ_{0}(P) ⟶ ℑ_{1}(P) ⟶ ⋯,  (2)

is exact for all finite P ≤ S, and we want to show it is exact for all P ≤ S. Fix an infinite subgroup P ≤ S, and choose finite subgroups P_{1} ≤ P_{2} ≤ ⋯ such that P = ⋃_{j=1}^{∞} P_{j} (Lemma 1.9). Then F_{M}(P) = 0 = \underleftarrow{lim}_{j} F_{M}(P_{j}). For all finite Q ≤ S and

all A,

ℑ_{Q}^{A}(P) = Map(Mor_{C}(Q, P), A) = \underleftarrow{lim}_{j}(Map(Mor_{C}(Q, P_{j}), A))

since Mor_{C}(Q, P) is the union of the Mor_{C}(Q, P_{j}); and furthermore this is an inverse system of surjections. Hence (2) is the inverse limit of the corresponding exact sequences for the P_{j}, all restriction maps ℑ_{k}(P_{j+1}) ⟶ ℑ_{k}(P_{j}) are surjective, and so (2) is also exact. Thus

Λ_{dpt}^{∗}(G; M) = \underleftarrow{lim}_{C}^{∗}(F_{M}) ≅ H^{∗}(\underleftarrow{lim}_{C}(ℑ_{k}), d_{k}) ≅ H^{∗}(\underleftarrow{lim}_{C_{0}}(ℑ_{k}|_{C_{0}}), d_{k}) ≅ \underleftarrow{lim}_{C_{0}}^{∗}(F_{M}|_{C_{0}}). (3)

Step 2: Fix a sequence S_{1} ≤ S_{2} ≤ S_{3} ≤ ⋯ of finite subgroups of S such that S = ⋃_{j=1}^{∞} S_{j} (Lemma 1.9). We first construct inductively a sequence of finite subgroups H_{1} ≤ H_{2} ≤ ⋯ of G containing H_{0} such that for each j ≥ 1, H_{j} ≥ S_{j}, and O_{p}(H_{j}) contains the full subcategory with object set the p-subgroups of H_{j−1}. Fix j ≥ 1, and assume that H_{j−1} has been constructed. Let C_{j} be the full subcategory of O_{p}^{fin}(G) whose objects are the p-subgroups of ⟨H_{j−1}, S_{j}⟩ (a finite group since G is locally finite). Choose a finite set of morphisms in C_{j} which generate it, let X_{j} ⊆ G be a finite set of elements which induce those morphisms, and set H_{j} = ⟨X_{j}⟩. Since G is locally finite, H_{j} is a finite subgroup. By construction, O_{p}(H_{j}) ⊇ C_{j}; and hence contains both O(S_{j}) and the full subcategory with the same objects as O_{p}(H_{j−1}).

Set C′ =^{def} ⋃_{j=1}^{∞} O_{p}(H_{j}). This is a full subcategory of O_{p}^{fin}(G) which contains all finite subgroups of S as objects. In particular, C′ is equivalent to C_{0}, and hence \underleftarrow{lim}_{C_{0}}^{∗}(F_{M}) ≅

\underleftarrow{lim}_{C′}^{∗}(F_{M}|_{C′}). Since \underleftarrow{lim}^{∗}_{O_{p}(H_{j})}(F_{M}|_{O_{p}(H_{j})}) = 0 for all j, \underleftarrow{lim}_{C′}^{∗}(F_{M}|_{C′}) = 0 by Lemma 5.11. □

6. Mapping spaces

We now look at the spaces of maps from BQ to |L|_{p}^{∧}, when Q is a discrete p-toral group and L is a linking system. In general, for any p-local compact group (S, F, L) and any discrete p-toral group Q, we define

Rep(Q, L) = Hom(Q, S)/∼,

where ∼ is the equivalence relation defined by setting ρ ∼ ρ′ if there is some χ ∈ Hom_{F}(ρ(Q), ρ′(Q)) such that ρ′ = χ∘ρ. We want to show that [BQ, |L|_{p}^{∧}] ≅ Rep(Q, L).

The following lemma will be needed to reduce this to the case where Q is finite. The functor (−)^{•} of Section 3 plays an important role when doing this.

Lemma 6.1. Fix a discrete p-toral group Q, and let Q_{1} ≤ Q_{2} ≤ ⋯ ≤ Q be a sequence of finite subgroups such that Q = ⋃_{n=1}^{∞} Q_{n}. Let (S, F, L) be a p-local compact group. Then the following hold.

(a) The natural map

R: Rep(Q, L) ⟶^{≅} \underleftarrow{lim}_{n} Rep(Q_{n}, L) ,

induced by restriction, is a bijection.

(b) Assume Q ≤ S. Then for n large enough, Q_{n}^{•} = Q^{•} ≥ Q, and hence restriction induces a bijection Hom_{F}(Q, P) ≅ Hom_{F}(Q_{n}, P) for all P ∈ Ob(F^{•}).

Proof. In general, for any homomorphism φ ∈ Hom(H, K), we let [φ] denote its class in Rep(H, K).

(a) Assume first that φ, ψ ∈ Hom(Q, S) are such that R([φ]) = R([ψ]). Thus φ|_{Q_{n}} and ψ|_{Q_{n}} are F-conjugate for each n; i.e., ψ|_{Q_{n}} = α_{n} ∘ φ|_{Q_{n}} for some unique α_{n} ∈ Iso_{F}(φ(Q_{n}), ψ(Q_{n})). In particular, Ker(φ)∩Q_{n} = Ker(ψ)∩Q_{n} for each n, so Ker(φ) = Ker(ψ), and ψ = α ∘ φ for some unique α ∈ Iso(φ(Q), ψ(Q)). Then α|_{Q_{n}} = α_{n} is in F for each n, so α ∈ Iso_{F}(φ(Q), ψ(Q)) by axiom (III), and [ψ] = [φ] ∈ Rep(Q, F).

This proves the injectivity of R, and it remains to prove surjectivity. Fix some

\{[φ_{n}]\}_{n≥1} ∈ \underleftarrow{lim}_{n} Rep(Q_{n}, L).

Thus for each n, φ_{n} ∈ Hom(Q_{n}, S), and φ_{n+1}|_{Q_{n}} is F-conjugate to φ_{n}. By Lemma 3.2(a), the set {φ_{n}(Q_{n})^{•} | n ≥ 1} contains finitely many conjugacy classes. Since for all n, φ_{n}(Q_{n}) is F-conjugate to a subgroup of φ_{n+1}(Q_{n+1}), φ_{n}(Q_{n})^{•} is F-conjugate to a subgroup of φ_{n+1}(Q_{n+1})^{•} by Lemma 3.2(b) and Proposition 3.3. Hence for some m, φ_{n}(Q_{n})^{•} is F-conjugate to φ_{m}(Q_{m})^{•} for all n ≥ m.

We now construct inductively homomorphisms φ′_{n} ∈ Hom(Q_{n}, S) for all n > m such that [φ′_{n}] = [φ_{n}] in Rep(Q_{n}, L), and φ′_{n}|_{Q_{n−1}} = φ′_{n−1}. Assume φ′_{n−1} has been constructed, and set α_{n} = φ_{n} ∘ φ′_{n−1}^{−1} ∈ Hom_{F}(φ′_{n−1}(Q_{n−1}), φ_{n}(Q_{n})). By Proposition 3.3 again, this extends to a unique morphism α_{n}^{•} ∈ Hom_{F}(φ′_{n−1}(Q_{n−1})^{•}, φ_{n}(Q_{n})^{•}), which must be an isomorphism since it is injective and the two groups are abstractly isomorphic and artinian. Set φ′_{n} = (α_{n}^{•})^{−1} ∘ φ_{n}; then φ′_{n}|_{Q_{n−1}} = φ′_{n−1}. Let φ ∈ Hom(Q, S) be the union of the φ′_{n}; then [φ] ∈ R^{−1}({[φ_{n}]}), and this proves the surjectivity of R.

(b) Now assume Q ≤ S. By Lemma 3.2(a,b), for all n, Q_{n}^{•} ≤ Q_{n+1}^{•} ≤ Q^{•}, and the set {Q_{n}^{•} | n ≥ 1} is finite. Hence Q_{n}^{•} ≥ Q for n sufficiently large, and this implies Q_{n}^{•} = Q^{•}. If P = P^{•} ≤ S, then every φ ∈ Hom_{F}(Q_{n}, P) extends to a unique φ^{•} ∈ Hom_{F}(Q_{n}^{•}, P) by Proposition 3.3, and thus Hom_{F}(Q, P) ≅ Hom_{F}(Q_{n}, P) whenever Q_{n}^{•} = Q^{•}. □

For any linking system L and any discrete p-toral group Q, we let L^{Q} be the category whose objects are the pairs (P, α) for P ∈ Ob(L) and α ∈ Hom(Q, P), and where

Mor_{L^{Q}}((P, α), (P′, α′)) = {φ ∈ Mor_{L}(P, P′) | α′ = π(φ) ∘ α ∈ Hom(Q, P′)}.

We next show that Map(BQ, |L|_{p}^{∧}) ≃ |L^{Q}|_{p}^{∧} in this situation.

Proposition 6.2. Fix a p-local compact group (S, F, L) and a discrete p-toral group Q. Let F_{0} ⊆ F^{c} be any full subcategory which contains all F-centric F-radical subgroups of S, and such that P ∈ Ob(F_{0}) implies P^{•} ∈ Ob(F_{0}). Let L_{0} ⊆ L and L_{0}^{Q} ⊆ L^{Q} be the full subcategories where Ob(L_{0}) = Ob(F_{0}), and Ob(L_{0}^{Q}) is the set of pairs (P, α) ∈ Ob(L^{Q}) such that P ∈ Ob(L_{0}). Then there is a bijection

π_{0}(|L_{0}^{Q}|) ⟶^{≅} Rep(Q, L)  (1)

which sends a vertex (P, α) to the class of α as a homomorphism to S. If, furthermore, we define Φ: L_{0}^{Q} × B(Q) ⟶ L_{0} by setting

Φ((P, α), o_{Q}) = P  and  Φ((P, α) ⟶^{φ} (P′, α′) , x) = φ ∘ δ_{P} (α(x)) ,

then the map

|Φ|′: |L_{0}^{Q}|_{p}^{∧} ⟶ Map(BQ, |L_{0}|_{p}^{∧})  (2)

adjoint to |Φ| is a homotopy equivalence.

Proof. Every vertex (P, α) in |L_{0}^{Q}| is connected by an edge to the vertex (S, incl_{P}^{S} ∘α). Furthermore, by the assumption that F_{0} contains all F-centric F-radical subgroups, together with Alperin’s fusion theorem (Theorem 3.6), two vertices (S, α) and (S, α′) in |L_{0}^{Q}| are in the same connected component if and only if α and α′ represent the same element of Rep(Q, L). This proves (1).

Since Φ(φ, x) = φ ∘ δ_{P} (α(x)) = δ_{P′}(α′(x)) ∘ φ by condition (C), Φ is a well defined functor. It remains to prove the homotopy equivalence (2). Step 1, where we handle the case Q is finite, is essentially the same as the corresponding proof in [BLO2]. In Step 2, we extend this to the general case.

By assumption, for each P ∈ Ob(L_{0}), P^{•} ∈ Ob(L_{0}). So the functor (−)^{•} of Proposition 4.5 restricts to a functor from L_{0} to L_{0}^{•}, and also induces a functor from L_{0}^{Q} to L_{0}^{•Q}. All of these are left adjoint to the inclusion functors, and hence induce homotopy equivalences between their geometric realizations. Thus, without loss of generality, we can assume that L_{0} = L_{0}^{•}; i.e., that P = P^{•} for all P in L_{0}. This assumption will be needed at the end of each of Steps 1 and 2 below.

Step 1: Assume that Q is a finite p-group. Let O(F_{0}) ⊆ O^{c}(F) be the full subcategory with Ob(O(F_{0})) = Ob(F_{0}) = Ob(L_{0}), and let π̃: L_{0} ⟶ O(F_{0}) be the projection functor. Let π̃_{Q}: L_{0}^{Q} ⟶ O(F_{0}) be the functor π̃_{Q}(P, α) = P and π̃_{Q}(φ) = π̃(φ). Let

B̃_{Q}, B̃: O(F_{0}) ⟶ Top

be the left homotopy Kan extensions over π̃_{Q} and π̃, respectively, of the constant functors ∗. Then

|L_{0}| ≃ \underrightarrow{hocolim}_{O(F_{0})}(B̃) and |L_{0}^{Q}| ≃ \underrightarrow{hocolim}_{O(F_{0})}(B̃_{Q})  (3)

(cf. [HV, Theorem 5.5]).

For each P in O(F_{0}), B̃(P) is the nerve of the overcategory π̃↓P, whose objects are the pairs (R, χ) for R ∈ Ob(L_{0}) = Ob(O(F_{0})) and χ ∈ Rep_{F}(R, P), and where

Mor_{π̃↓P} ((R, χ), (R′, χ′)) = {φ ∈ Mor_{L_{0}}(R, R′) | χ = χ′ ∘ π̃(φ)}.

Let B′(P) be the full subcategory of π̃↓P with the unique object (P, Id), and with morphisms the group of all δ_{P} (g) for g ∈ P.

Similarly, B̃_{Q}(P) is the nerve of the category π̃_{Q}↓P, whose objects are the triples (R, α, χ) for R ∈ Ob(L_{0}) = Ob(O(F_{0})), α ∈ Hom(Q, R), and χ ∈ Rep_{F}(R, P); and where

Mor_{π̃_{Q}↓P} ((R, α, χ), (R′, α′, χ′)) = {φ ∈ Mor_{L_{0}}(R, R′) | α′ = π(φ) ∘ α, χ = χ′ ∘ π̃(φ)}.

Let B′_{Q}(P) be the full subcategory of π̃_{Q}↓P with objects the triples (P, α, Id) for α ∈ Hom(Q, P).

Fix a section σ̃: Mor(O(F_{0})) ⟶ Mor(L_{0}) which sends identity morphisms to identity morphisms. Retractions

π̃↓P ⟶^{Ψ} B′(P) and π̃_{Q}↓P ⟶^{Ψ_{Q}} B′_{Q}(P)

are defined by setting

Ψ(R, χ) = (P, Id) and Ψ_{Q}(R, α, χ) = (P, πσ̃(χ) ∘ α, Id);

and by sending φ in Mor_{π̃↓P} ((R, χ), (R′, χ′)) or Mor_{π̃_{Q}↓P} ((R, α, χ), (R′, α′, χ′)) to the automorphism δ_{P} (g) ∈ Aut_{L_{0}}(P), where g ∈ P is the unique element such that σ̃(χ′) ∘ φ = δ_{P} (g) ∘ σ̃(χ) in Mor_{L_{0}}(R, P) (Lemma 4.3(b)). There are natural transformations

Id_{π̃↓P} ⟶ incl ∘Ψ and Id_{π̃_{Q}↓P} ⟶ incl ∘Ψ_{Q}

of functors which send an object (R, χ) to χ ∈ Mor_{π̃↓P} ((R, χ), (P, Id)) and similarly for an object (R, α, χ). This shows that |B′(P)| ⊆ |π̃↓P | and |B′_{Q}(P)| ⊆ |π̃_{Q}↓P | are deformation retracts.

We have now shown that for all P ∈ Ob(L_{0}),

B̃(P) ≃ |B′(P)| ≃ BP  and B̃_{Q}(P) ≃ |B′_{Q}(P)| .  (4)

All morphisms in B′_{Q}(P) are isomorphisms, two objects (P, α, Id) and (P, α′, Id) are isomorphic if and only if α and α′ are conjugate in P, and the automorphism group of (P, α, Id) is isomorphic to C_{P} (αQ). Thus

B̃_{Q}(P) ≃ ∐_{α∈Rep(Q,P)} BC_{P} (αQ).  (5)

Let B̃_{p}^{∧} and B̃_{Q}_{p}^{∧} be the p-completions of B̃ and B̃_{Q}; i.e., (B̃_{p}^{∧})(P) = (B̃(P))_{p}^{∧} and (B̃_{Q}_{p}^{∧})(P) = (B̃_{Q}(P))_{p}^{∧}. By (3), and since the spaces B̃(P) and B̃_{Q}(P) are all p-good by (4) and (5),

|L_{0}|_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O(F_{0})}(B̃_{p}^{∧}))_{p}^{∧} and |L_{0}^{Q}|_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O(F_{0})}(B̃_{Q}_{p}^{∧}))_{p}^{∧} .

Consider the commutative triangle

\xymatrix{L_{0}^{Q} × B(Q) \ar[rr]^{Φ} \ar@{-->}[dr]_{_{π̃_{Q}_{∘}pr_{1}}} &  & L_{0} \ar[dl]^{π̃} \\  & O(F_{0}) .}

The left homotopy Kan extension over π̃_{Q} ∘ pr_{1} of the constant functor ∗ is the functor B̃_{Q} × BQ, and so the triangle induces a natural transformation of functors

Φ′: B̃_{Q} × BQ ⟶ B̃.

The map Φ̃: B̃_{Q} ⟶ Map(BQ, B̃) adjoint to Φ′ is also a natural transformation of functors from O(F_{0}) to Top, and induces a commutative diagram

(\underrightarrow{hocolim}_{O(F_{0})}(B̃_{Q})_{p}^{∧})_{p}^{∧} hocolim(Φ̃) → (\underrightarrow{hocolim}_{O(F_{0})} Map(BQ, B̃_{p}^{∧}))_{p}^{∧} →^{ω} Map(BQ, \underrightarrow{hocolim}_{O(F_{0})}(B̃)_{p}^{∧})

|L_{0}^{Q}|_{p}^{∧} ↓^{≃}  \overline{^{|Φ|′}  →} Map(BQ, |L_{0}|_{p}^{∧}) . ↓^{≃}

For each P ≤ S and Q_{0} ≤ Q, Lemma 1.10 (together with (4)) implies that each component of Map(BQ_{0}, B(P)_{p}^{∧}) has the form BC_{P} (ρ(Q_{0}))_{p}^{∧} for some ρ ∈ Hom(Q_{0}, P). So all such mapping spaces are p-complete and have finite mod p cohomology in each degree. Also, O(F_{0}) is a finite category (it has finitely many isomorphism classes of objects by Lemma 3.2(a) and has finite morphism sets by Lemma 2.5), and it has bounded limits at p by Corollary 5.5. Hence ω is a homotopy equivalence by [BLO2, Proposition 4.2].

It remains only to show that Φ̃(P) is a homotopy equivalence for each P ∈ Ob(L_{0}). By (4), this means showing that Φ̃(P) restricts to a homotopy equivalence

Φ̃′(P): |B′_{Q}(P)| ⟶ Map(BQ, |B′(P)|) .

Since |B′(P)| ≅ BP, and since Φ̃′(P) is induced by the homomorphisms (incl ·α) from C_{P} (α(Q)) × Q to P, this follows from (5).

Step 2: Now let Q be an arbitrary p-toral group. Let Q_{1} ≤ Q_{2} ≤ ⋯ Q be an increasing sequence of finite subgroups whose union is Q (Lemma 1.9). Then

π_{0}(|L_{0}^{Q}|) ≅ \underleftarrow{lim}_{n} π_{0}(|L_{0}^{Q_{n}}|) ≅ \underleftarrow{lim}_{n}[BQ_{n}, |L_{0}|_{p}^{∧}] :  (6)

the first bijection holds by Lemma 6.1 and (1), and the second by Step 1.

Fix φ ∈ Hom(Q, S), and set φ_{n} = φ|_{Q_{n}}. Let Map(BQ, |L_{0}|_{p}^{∧})φ̂ be the space of maps f: BQ ⟶ |L_{0}|_{p}^{∧} such that f|_{BQ_{n}} ≃ Bφ_{n} for each n. (This contains the connected component of Bφ, but could, a priori, contain other components.) Let (L_{0}^{Q})_{φ} ⊆ L_{0}^{Q} and (L_{0}^{Q_{n}})_{φ} ⊆ L_{0}^{Q_{n}} be the full subcategories with objects those (P, α) such that α is F-conjugate to φ or to φ_{n}, respectively. Thus |(L_{0}^{Q})_{φ}| is the connected component of |L_{0}^{Q}| which contains (S, φ), and |(L_{0}^{Q_{n}})_{φ}| is the connected component which contains (S, φ_{n}).

Consider the following commutative diagram, for all n ≥ 1:

\begin{aligned}|(L_{0}^{Q})_{φ}|_{p}^{∧} & \overline{→} & Map(BQ, |L_{0}|_{p}^{∧})φ̂ \\ |(L_{0}^{Q_{n}})_{φ}|_{p}^{∧} & ↓ & →^{≃} & Map(BQ_{n}, |L_{0}|_{p}^{∧})_{φ_{n}}. & ↓ \end{aligned}  (7)

We want to show that the top row is a homotopy equivalence; the proposition then follows by taking the union of such maps as φ runs through representatives of all elements of Rep(Q, L). The bottom row is a homotopy equivalence by Step 1. So we will be done if we can show that the vertical maps are homotopy equivalences for n large enough.

By Lemma 6.1(b), there is some m such that for all n ≥ m, φ(Q_{n})^{•} = φ(Q)^{•}, and restriction induces a bijection Rep_{F}(φ(Q), P) ≅ Rep_{F}(φ(Q_{n}), P) for all P ∈ Ob(L_{0}). (Recall that we are assuming L_{0} = L_{0}^{•}.) This implies that |(L_{0}^{Q})_{φ}| ≅ |(L_{0}^{Q_{n}})_{φ}| for all n ≥ m. Hence the components Map(BQ_{n}, |L_{0}|_{p}^{∧})_{Bφ_{n}} are all homotopy equivalent for n ≥ m by Step 1, so Map(BQ, |L|_{p}^{∧})φ̂ ≃ Map(BQ_{n}, |L|_{p}^{∧})_{Bφ_{n}} for n ≥ m, and this proves that the vertical maps in (7) are equivalences. □

The following theorem gives a more explicit description of the set [BQ, |L|_{p}^{∧}] of homotopy classes of maps, as well as of the individual components in certain cases.

Theorem 6.3. Let (S, F, L) be a p-local compact group, and let θ: BS ⟶ |L|_{p}^{∧} be the natural inclusion followed by completion. Then the following hold, for any discrete p-toral group Q.

(a) The natural map

Rep(Q, L) ⟶^{≅} [BQ, |L|_{p}^{∧}]

is a bijection. Thus each map BQ ⟶ |L|_{p}^{∧} is homotopic to θ ∘ Bρ for some ρ ∈ Hom(Q, S). If ρ, ρ′ ∈ Hom(Q, S) are such that θ ∘ Bρ ≃ θ ∘ Bρ′ as maps from BQ to |L|_{p}^{∧}, then there is χ ∈ Hom_{F}(ρ(Q), ρ′(Q)) such that ρ′ = χ ∘ ρ.

(b) For each ρ ∈ Hom(Q, S) such that ρ(Q) is F-centric, the composite

BZ(ρ(Q)) × BQ ⟶^{incl ·Bρ} BS ⟶^{θ} |L|_{p}^{∧}

induces a homotopy equivalence

BZ(ρ(Q))_{p}^{∧} ⟶^{≃} Map(BQ, |L|_{p}^{∧})_{θ∘Bρ}.

(c) The evaluation map induces a homotopy equivalence

Map(BQ, |L|_{p}^{∧})_{triv} ≃ |L|_{p}^{∧}.

Proof. We refer to the category L^{Q}, and to the homotopy equivalence

|Φ|′: |L^{Q}|_{p}^{∧} ⟶^{≃} Map(BQ, |L|_{p}^{∧})

of Proposition 6.2. Point (a) is an immediate consequence of point (1) in the proposition, and (c) holds since the component of L_{Q} which contains the objects (P, 1) is equivalent to L.

If ρ ∈ Hom(Q, S) is such that ρ(Q) is F-centric, then the connected component of |L_{Q}| which contains the vertex (ρ(Q), ρ) contains as deformation retract the nerve of the full subcategory with that as its only object. Since Aut_{L_{Q}}(ρ(Q), ρ) ≅ Z(ρ(Q)), this component has the homotopy type of BZ(ρ(Q)), which proves point (b). □

7. Equivalences of classifying spaces

We next describe the monoid Aut(|L|_{p}^{∧}) of self homotopy equivalences of |L|_{p}^{∧} (Theorem 7.1); and also show that p-local compact groups which have homotopy equivalent classifying spaces are themselves isomorphic (Theorem 7.4). There is some overlap between the proofs in this section and those of the corresponding results for p-local finite groups in [BLO2, Sections 8 & 7]; but they differ in some key respects, mostly due to the fact that we do not have a way to recover the category L from the space |L|_{p}^{∧} via a functor from spaces to categories.

We first recall some notation used in [BLO1] and [BLO2]. For any space X, Aut(X) denotes the monoid of self homotopy equivalences of X, and Out(X) = π_{0}(Aut(X)) is the group of homotopy classes of self equivalences. For any discrete category C, Aut(C) is the category whose objects are the self equivalences of C and whose morphisms are the natural isomorphisms between self equivalences, and Out(C) = π_{0}(|Aut(C)|) is the group of isomorphism classes of self equivalences. We consider Aut(C) as a discrete strict monoidal category, in the sense that composition defines a strictly associative functor

Aut(−) × Aut(−) ⟶ Aut(−)

with strict identity. The nerve of Aut(C) is thus a simplicial monoid, and its realization |Aut(C)| is a topological monoid.

Consider the evaluation functor

ev: Aut(C) × C ⟶ C

which sends a pair of objects (Ψ, c) to Ψ(c) ∈ Ob(C), and which is defined on morphisms by setting

ev(Ψ ⟶^{χ} Ψ′, c ⟶^{φ} d) = (Ψ(c) ⟶_{=χ(d)∘Ψ(φ)}^{Ψ′(φ)∘χ(c)} Ψ′(d)).

Upon taking geometric realizations, this defines a map of spaces from |Aut(C)| × |C| to |C|, which is adjoint to a homomorphism of topological groups

Ω_{C}: |Aut(C)| ⟶ Aut(|C|).

Recall that part of the structure of a centric linking system L associated to a fusion system is a homomorphism P ⟶^{δ_{P}} Aut_{L}(P) for each P in L. We write P_{δ} = Im(δ_{P}), which we think of as a “distinguished subgroup” of Aut_{L}(P) which can be identified with P. For the purposes of this paper, an equivalence of categories L ⟶^{Ψ} L will be called isotypical if for each P, Ψ_{P,P} sends the subgroup P_{δ} ≤ Aut_{L}(P) to the subgroup Ψ(P)_{δ} ≤ Aut_{L}(Ψ(P)). Let Aut_{typ}(L) be the full subcategory of Aut(L) whose objects are the isotypical equivalences, and set Out_{typ}(L) = π_{0}(|Aut_{typ}(L)|).

By [BLO2, Lemma 8.2], when L is a linking system over a finite p-group, an equivalence Ψ: L ⟶ L is isotypical if and only if the triangle involving Ψ and the forgetful functor from L to groups commutes up to natural isomorphism. The same proof applies for linking systems over discrete p-toral groups, although we won’t be using that here.

Clearly, any equivalence which is naturally isomorphic to an isotypical equivalence is itself isotypical, and any inverse to an isotypical equivalence (inverse up to natural isomorphism of functors) is also isotypical. The subcategory Aut_{typ}(L) is thus a union of connected components of Aut(L), and Out_{typ}(L) is a subgroup of Out(L).

The main result of this section is the following theorem:

Theorem 7.1. Fix a p-local compact group (S, F, L), and set Ω = Ω_{L}. Then the composite

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

induces a homotopy equivalence of topological monoids from |Aut_{typ}(L)|_{p}^{∧} to Aut(|L|_{p}^{∧}). In particular, if we let π_{i}(BZ_{p}^{∧}) denote the functor O^{c}(F)^{op} ⟶ Ab which sends P to π_{i}(BZ(P)_{p}^{∧}) (each i ≥ 1), then

Out(|L|_{p}^{∧}) ≅ Out_{typ}(L) , π_{i}(Aut(|L|_{p}^{∧})) ≅ \underleftarrow{lim}^{0}_{O^{c}(F)}(π_{i}(BZ_{p}^{∧})) for i = 1, 2 ,

and π_{i}(Aut(|L|_{p}^{∧})) = 0 for i ≥ 3.

Proof. We prove the isomorphism between groups of components in Step 2, and the homotopy equivalence between the individual components in Step 3. In Step 1, we outline the general procedure for describing the mapping space Aut(|L|_{p}^{∧}).

Assume we have fixed inclusion morphisms ι_{P} ∈ Mor_{L}(P, S) for each P. If Ψ is an isotypical self equivalence of L, then clearly Ψ(S) = S, and hence Ψ_{S,S} is an automorphism of Aut_{L}(S) which sends S_{δ} (= Im(δ_{S})) to itself. Set

ψ = δ_{S}^{−1} ∘ Φ_{S,S}|_{S_{δ}} ∘ δ_{S} ∈ Aut(S).

For each P ∈ Ob(L), axiom (C) and the functoriality of Ψ imply that the following diagram commutes for all g ∈ P:

\begin{matrix}Ψ(P) & \overline{^{Ψ(ι_{P})} →} & S \\ Ψ(P) & ↓^{Ψ(δ_{P} (g)) ∈Ψ(P)_{δ}} & \overline{^{Ψ(ι_{P})} →} & S. & ↓^{Ψ(δ_{S}(g))=δ_{S}(ψ(g))} \end{matrix}

Hence π(Ψ(ι_{P}))(Ψ(P)) = ψ(P) (by axiom (C) again). So Ψ(ι_{P}) = ι_{ψ(P)} ∘ α_{P} for a unique α_{P} ∈ Iso_{L}(Ψ(P), ψ(P)) by Lemma 4.3(a). Thus Ψ is naturally isomorphic to an automorphism Ψ′ of L such that Ψ′_{S,S} = Ψ_{S,S}, and Ψ′(P) = ψ(P) and Ψ′_{P,S}(ι_{P}) = ι_{ψ(P)} for each P. This shows that every object in Aut_{typ}(L) is isomorphic to an isotypical automorphism of L which sends inclusions to inclusions, and from now on we restrict attention to such automorphisms.

Step 1: Consider the decomposition

pr : \underrightarrow{hocolim}_{O^{c}(F)}(B̃) ⟶^{≃} |L|

of Proposition 4.6(a), where B̃: O^{c}(F) ⟶ Top is a rigidification of the homotopy functor P ↦ BP. In the following constructions, we regard \underrightarrow{hocolim}(B̃) as the union of skeleta:

\underrightarrow{hocolim}_{O^{c}(F)}^{(n)}(B̃) = (∐_{i=0}^{n} ∐_{P_{0}→···→P_{n}} B̃(P_{0}) × D^{i})/∼

where we divide out by the usual face and degeneracy relations.

Define functors Z, Z_{0} : O^{c}(F)^{op} ⟶ Ab and BZ_{p}^{∧}: O^{c}(F)^{op} ⟶ Topby setting

Z(P) = Z(P),  Z_{0}(P) = Z(P)_{0},  and BZ_{p}^{∧}(P) = BZ(P)_{p}^{∧},

and by sending [φ] ∈ Mor_{O^{c}(F)}(P, Q) to φ^{−1}|_{Z(Q)} or B(φ^{−1}|_{Z(Q)})_{p}^{∧}. For any element

f = (f_{P} )_{P ∈O^{c}(F)} ∈ \underleftarrow{lim}_{O^{c}(F)}[B−, |L|_{p}^{∧}],

let Map(|L|_{p}^{∧}, |L|_{p}^{∧})_{f} be the union of the components of the mapping space which restrict to f. By [Wo], the obstructions to this space being nonempty lie in the groups

\underleftarrow{lim}^{i+1}_{O^{c}(F)}(π_{i}(Map(B−, |L|_{p}^{∧})_{f_{−}})) ≅ \underleftarrow{lim}^{i+1}_{O^{c}(F)}(π_{i}(BZ_{p}^{∧}))

for i ≥ 1; the functor vanishes for i > 2, and the higher limits vanish for i = 2 by Proposition 5.8. Also, if Map(|L|, |L|_{p}^{∧})_{f} ≠ ∅, then the filtration of the mapping space

Map(|L|_{p}^{∧}, |L|_{p}^{∧}) ≃ Map(\underrightarrow{hocolim}_{O^{c}(F)}(B̃), |L|_{p}^{∧})

by the skeleta of the homotopy colimit defines a spectral sequence with E^{2}-term

E_{−i,j}^{2} = \underleftarrow{lim}^{i}_{O^{c}(F)}(π_{j}(Map(B−, |L|_{p}^{∧})_{f_{−}})),

which converges to π_{j−i}(Map(|L|, |L|_{p}^{∧})_{f}).

By Theorem 6.3(b),

π_{j}(Map(B−, |L|_{p}^{∧})_{f_{−}}) ≅ π_{j}(BZ(−)_{p}^{∧}) ≅ \begin{cases}Z/Z_{0} & if j = 1 \\ π_{2}(BZ_{p}^{∧}) & if j = 2 \\ 0 & if j ≥ 3. \end{cases}

Since π_{2}(BZ_{p}^{∧}) is acyclic by Proposition 5.8, the only obstruction to Map(|L|_{p}^{∧}, |L|_{p}^{∧})_{f} being nonempty lies in \underleftarrow{lim}^{2}(Z/Z_{0}); while the spectral sequence takes the form

E_{−i,j}^{2} ≅ \begin{cases}\underleftarrow{lim}^{i}(Z/Z_{0}) & if j = 1 \\ \underleftarrow{lim}^{0}(π_{2}(BZ_{p}^{∧})) & if (i, j) = (0, 2) \\ 0 & otherwise. \end{cases}

Step 2: Let Aut_{fus}(S) be the group of fusion preserving automorphisms of S; i.e., the group of those α ∈ Aut(S) which induce an automorphism of the fusion system F by sending P to α(P) and φ ∈ Hom_{F}(P, Q) to (α|_{Q}) ∘ φ ∘ (α|_{P})^{−1} ∈ Hom_{F}(α(P), α(Q)).

The proof that Out(|L|_{p}^{∧}) ≅ Out_{typ}(L) is based on the following diagram:

1  \overline{→} \underleftarrow{lim}^{1}(Z) \overline{^{λ′}→} Out_{typ}(L) \overline{^{μ′} →} Out_{fus}(S) \overline{^{ω′} →} \underleftarrow{lim}^{2}(Z)

1  \overline{→} \underleftarrow{lim}^{1}(Z/Z_{0}) ↓^{ω_{1} ≅}  \overline{→^{λ}} Out(|L|_{p}^{∧}) ↓^{π_{0}(Ω_{p}^{∧})}  \overline{→^{μ}} \underleftarrow{lim} IRep(−, F) ↓^{ω_{2} ≅}  →^{ω} \underleftarrow{lim}^{2}(Z/Z_{0}) . ↓^{ω_{3} ≅}

(1)

Here, IRep(P, F) ⊆ Rep(P, F) denotes the set of classes of injective homomorphisms. All limits are taken over O^{c}(F), and ω_{1} and ω_{3} are induced by the natural surjection of functors from Z onto Z/Z_{0}. They are isomorphisms since \underleftarrow{lim}^{i}(Z_{0}) = 0 for all i ≥ 1 (Proposition 5.8). Also, ω_{2} is induced by the inclusion of Out_{fus}(S) = Aut_{fus}(S)/ Aut_{F}(S) into IRep(S, F) = Aut(S)/ Aut_{F}(S), and Im(ω_{2}) = \underleftarrow{lim} IRep(−, F) (thus ω_{2} is a bijection) by definition of fusion preserving. It remains to define the two rows, and prove that they are exact and the diagram commutes. It will then follow immediately that π_{0}(Ω_{p}^{∧}) is an isomorphism. Note that this does not require us to know that \underleftarrow{lim} IRep(−, F) is a group or that ω′ is a homomorphism; only that Im(μ) = ω^{−1}(0), Im(μ′) = ω′^{−1}(0), and the inverse image under μ of each element in the target is a coset of Im(λ).

We first consider the top row, where μ′ is defined by restricting an isotypical equivalence of L to the image of δ_{S}. Any fusion preserving automorphism α ∈ Aut_{fus}(S) defines an isotypical automorphism α of F, and ω′(α) is the obstruction of [BLO2, Proposition 3.1] to lifting α to an automorphism of L. (The proof in [BLO2] applies without change to the case of a linking system over a discrete p-toral group.) Finally, the description of Ker(μ′) is identical to that shown in [BLO1, Theorem 6.2]. More specifically, a reduced 1-cocycle ε ∈ Z^{1}(O^{c}(F); Z) sends each morphism [φ] ∈ Mor_{O^{c}(F)}(P, Q) to ε(φ) ∈ Z(P) (where ε(Id_{P}) = 1), and λ′([ε]) is represented by the automorphism A_{ε} ∈ Aut(L) defined by setting A_{ε}(P) = P for all P, and A_{ε}(ψ) = ψ ∘ δ_{P} (ε([π(ψ)]))^{−1} for all ψ ∈ Mor_{L}(P, Q). This proves the exactness of the top row.

As for the bottom row in (1), let μ be the homomorphism defined by restriction:

μ: Out(|L|_{p}^{∧}) ⟶^{Res} [BS, |L|_{p}^{∧}] ≅ IRep(S, F).

We want to compare

Map(|L|_{p}^{∧}, |L|_{p}^{∧}) ≃ Map(\underrightarrow{hocolim}_{O^{c}(F)}(B̃), |L|_{p}^{∧})

with

\underleftarrow{lim}_{O^{c}(F)}[B(−), |L|_{p}^{∧}] ≅ \underleftarrow{lim}_{O^{c}(F)} IRep(−, F).

By Step 1, the only obstruction to extending any given α in this last set to an automorphism of |L|_{p}^{∧} lies in \underleftarrow{lim}^{2}(Z/Z_{0}), while if there are liftings, then the set of homotopy classes is in bijective correspondence with \underleftarrow{lim}^{1}(Z/Z_{0}). This proves the exactness of the bottom row in the sense explained above.

The second square in (1) clearly commutes. To prove that the first square commutes, fix some ε ∈ Z^{1}(O^{c}(F); Z). Then λ′([ε]) = [A_{ε}] where A_{ε} ∈ Aut(L) is the automorphism defined above; and |A_{ε}| ∈ Aut(|L|) sends each BP ⊆ B(Aut_{L}(P)) ⊆ |L| to |L| by the inclusion. For each φ ∈ Hom_{F^{c}}(P, Q), let C_{φ} ⊆ L be the subcategory with two objects P and Q, whose morphisms are those morphisms in L which get sent to [Id_{P}], [Id_{Q}], or [φ] in O^{c}(F). Then |C_{φ}| ⊆ |L| is homeomorphic to the mapping cylinder of Bφ: BP ⟶ BQ; and |A_{ε}| sends |C_{φ}| to itself by a map which differs from the identity via a loop in Map(BP, BQ)_{Bφ} ≃ BZ(P) which represents ε([φ]) ∈ Z(P). After taking the p-completion, this shows that [|A_{ε}|_{p}^{∧}] = λ([ε̄]), where ε̄ ∈ Z^{1}(O^{c}(F); Z/Z_{0}) is the class of ε modulo Z_{0}. This proves that the first square in (1) commutes.

Fix α ∈ Out_{fus}(S), and let α̂ be the automorphism of the fusion system F induced by α. Choose maps

Mor_{L}(P, Q) ⟶^{α_{P,Q}^{∗}} Mor_{L}(α(P), α(Q))

which lift those defined by α̂; then ω′(α) is the class of the 2-cocycle β ∈ Z^{2}(O^{c}(F); Z) which measures the deviation of the α_{P,Q}^{∗} from defining a functor. These same liftings α_{P,Q}^{∗} allow us to define a map of spaces

α^{∗}: \underrightarrow{hocolim}_{O^{c}(F)}^{(1)}(B̃) ⟶ |L|,

and the obstruction to extending this to \underrightarrow{hocolim}^{(2)}(B̃) is precisely the class of the same 2-cocycle β: but regarded as a 2-cocycle with coefficients in

Z/Z_{0} ≅ π_{1}(Map(B−, |L|_{p}^{∧})_{α}).

This proves that the third square commutes, and finishes the proof that π_{0}(Ω_{p}^{∧}) is an isomorphism.

Step 3: Set Z(F) = \underleftarrow{lim}(Z), regarded as a subgroup of S. Let

λ: B(Z(F)) × L ⟶ L

be the functor which sends (x, P ⟶^{φ} Q) to φ ∘ δ_{P} (x). This is adjoint to a functor from B(Z(F)) to Aut(L), which in turn induces a map η′: BZ(F) ⟶ |Aut(L)|_{Id} upon taking geometric realizations. On the other hand, if we first take geometric realizations, then p-complete, and then take the adjoint, we get a map η from BZ(F)_{p}^{∧} to Aut(|L|_{p}^{∧})_{Id}. These maps now fit together in the following commutative square:

BZ(F) \overline{\begin{matrix}η′ \\ ≃ \end{matrix} →} |Aut(L)|_{Id}

BZ(F)_{p}^{∧} ↓^{(−)_{p}^{∧}}  \overline{\begin{matrix}η \\ ≃ \end{matrix}→} Aut(|L|_{p}^{∧})_{Id} . ↓^{Ω}

(2)

Since we are restricting attention to automorphisms of L (as opposed to working with all equivalences), Aut(L) is a groupoid, and so π_{1}(|Aut(L)|) is the group of natural isomorphisms of functors from Id_{L} to itself. A natural equivalence α sends each object P to an element α(P) ∈ Aut_{L}(P), such that for each φ ∈ Mor_{L}(P, Q), φ ∘ α(P) = α(Q) ∘ φ. In particular, upon restricting to the case P = Q and φ ∈ δ_{P} (P), we see that π(α(P)) = Id_{P} for each P, and thus α(P) ∈ δ_{P} (Z(P)) ≅ Z(P). The other relations are equivalent to requiring that α ∈ \underleftarrow{lim}^{0}(Z) = Z(F). This proves that π_{1}(|Aut(L)|) ≅ Z(F); and since |Aut(L)|_{Id} is aspherical, shows that η′ is a homotopy equivalence.

The E^{2}-term of the spectral sequence for maps defined on a homotopy colimit was described in Step 1: it vanishes except for the row coming from \underleftarrow{lim}^{∗}(Z/Z_{0}), and the position E_{0,2}^{2} ≅ \underleftarrow{lim}^{0}(π_{2}(BZ_{p}^{∧})). Hence from the spectral sequence, one sees immediately that for i ≥ 1,

π_{i}(Aut(|L|_{p}^{∧})) ≅ \underleftarrow{lim}_{O^{c}(F)}(π_{i}(BZ_{p}^{∧})) ≅ π_{i}(BZ(F)_{p}^{∧}).

By naturality, these isomorphisms are induced by η, and thus η is a homotopy equivalence.

It now follows from (2) and from Step 1 that Ω_{p}^{∧} induces a homotopy equivalence Aut(|L|_{p}^{∧}) ≃ |Aut_{typ}(L)|_{p}^{∧}. □

We also note here the following result, which was shown while proving Theorem 7.1.

Proposition 7.2. For any p-local compact group (S, F, L), there is an exact sequence

0 ⟶ \underleftarrow{lim}^{1}_{O^{c}(F)}(Z/Z_{0}) ⟶ Out(|L|_{p}^{∧}) ⟶ Out_{fus}(S) ⟶ \underleftarrow{lim}^{2}_{O^{c}(F)}(Z/Z_{0}),

where Z_{0} ⊆ Z: O^{c}(F)^{op} ⟶ Ab are the functors Z(P) = Z(P) and Z_{0}(P) = Z(P)_{0}.

In Section 9, we will show that for any compact Lie group G, there is a p-local compact group (S, F, L) = (S, F_{S}(G), L_{S}^{c}(G)) such that |L|_{p}^{∧} ≃ BG_{p}^{∧}. Hence when G is connected, the exact sequence of Proposition 7.2 gives a new way to describe Out(BG_{p}^{∧}), which is different from but closely related to the descriptions in [JMO] and [JMO3].

We now turn our attention to maps between p-completed nerves of different linking systems. We first look at the case where the linking systems in question are associated to the same fusion system. As usual, when we talk about an isomorphism of linking systems, we mean an isomorphism of categories which is natural with respect to the projections to the fusion system and with respect to the distinguished monomorphisms.

Lemma 7.3. Let F be a saturated fusion system over a discrete p-toral group S, and let F_{0} ⊆ F^{c} be any full subcategory which contains F^{c•}. Let L_{0} and L′_{0} be two linking systems associated to F_{0}. Assume that there is a map f: |L_{0}|_{p}^{∧} ⟶ |L′_{0}|_{p}^{∧} such that the triangle

BS
[figure: θ; θ′]
|L_{0}|_{p}^{∧}  \overline{^{f} →} |L′_{0}|_{p}^{∧}

is homotopy commutative. Here, θ and θ′ are the maps induced by the inclusion of B(S) into L_{0} or L′_{0}. Then L_{0} and L′_{0} are isomorphic linking systems associated to F_{0}. Furthermore, we can choose an isomorphism L_{0} ⟶^{≅} L′_{0} of linking systems that induces f on p-completed nerves.

Proof. Let ke(L_{0}) and ke(L′_{0}) be the left homotopy Kan extensions of the constant point functors along the projections π̃_{0} : L_{0} ⟶ O(F_{0}) and π̃′_{0} : L′_{0} ⟶ O(F_{0}) respectively. Let κ_{P} : ke(L_{0})(P) ⟶ |L_{0}|_{p}^{∧} be induced by the forgetful functor from π̃_{0}↓P to L_{0}, and similarly for κ′_{P} : ke(L_{0})(P) ⟶ |L′_{0}|_{p}^{∧}. Then θ and θ′ factor through κ_{S} and κ′_{S}, and we have a homotopy commutative diagram

\xymatrix{ & ke(L_{0})(P) \ar[r] & ke(L_{0})(S) \ar[r]^{κ_{S}} & |L_{0}|_{p}^{∧} \ar[dd]^{f} \\ BP \ar[r] \ar[ur]^{≃} \ar[dr]_{≃} & BS \ar[ur]^{≃} \ar[dr]_{≃} \ar[urr]_{θ} \ar[drr]^{θ′} \\  & ke(L′_{0})(P) \ar[r] & ke(L′_{0})(S) \ar[r]^{κ′_{S}} & |L′_{0}|_{p}^{∧}.}

Hence the maps f_{P} : ke(L_{0})(P) ⟶ |L′_{0}|_{p}^{∧} and f′_{P} : ke(L′_{0})(P) ⟶ |L′_{0}|_{p}^{∧}, defined as the obvious composites shown in the above diagram satisfy:

(a) The composites BP ⟶^{≃} ke(L_{0})(P) ⟶^{f_{P}} |L′_{0}|_{p}^{∧} and BP ⟶^{≃} ke(L′_{0})(P) ⟶^{f′_{P}} |L′_{0}|_{p}^{∧}

are homotopic, and are centric after p-completion by Theorem 6.3(b).

(b) f_{Q} ∘ ke(L_{0})(φ) ≃ f_{P} and f′_{Q} ∘ ke(L′_{0})(φ) ≃ f′_{P} for each morphism φ: P ⟶ Q of O(F_{0}).

Thus ke(L_{0})_{p}^{∧} and ke(L′_{0})_{p}^{∧} are equivalent rigidifications of B_{p}^{∧} by Corollary A.5; and so ke(L_{0}) and ke(L′_{0}) are equivalent rigidifications of B by Proposition 5.9. Hence by Proposition 4.6, L_{0} and L′_{0} are isomorphic linking systems associated to F_{0}.

More precisely, there is a third rigidification B̃ of B, and a commutative diagram of natural transformations between functors O(F_{0}) ⟶ Topof the following form:

\begin{aligned}ke(L_{0}) & & \overline{^{ψ} →} & B̃ & ← ^{ψ′} & ke(L′_{0}) \\ |L_{0}|_{p}^{∧} & ↓ & \overline{^{f} →} & |L′_{0}|_{p}^{∧} & \overline{\begin{matrix}f_{1} \\ ≃ \end{matrix} →} & X & ↓ & ← \begin{matrix}f_{2} \\ ≃ \end{matrix} & |L′_{0}|_{p}^{∧}. & ↓ \end{aligned}

Here, ψ(P) and ψ′(P) are homotopy equivalences for each P; X is some space homotopy equivalent to |L′_{0}|_{p}^{∧}; all functors in the bottom row of the diagram are constant functors on O(F_{0}) (sending all objects to the given space and all morphisms to the identity); and f_{1} and f_{2} are homotopy equivalences. Upon taking homotopy colimits of the functors in the top row, we get the homotopy commutative diagram:

\underrightarrow{hocolim}_{O(F_{0})}(ke(L_{0}))  \overline{^{≃}  →} \underrightarrow{hocolim}_{O(F_{0})}(B̃) ←^{≃} \underrightarrow{hocolim}_{O(F_{0})}(ke(L′_{0}))

|L_{0}|_{p}^{∧} ↓^{≃}  \overline{^{f} →} |L′_{0}|_{p}^{∧}  \overline{\begin{matrix}f_{1} \\ ≃ \end{matrix} →} X ↓ ←  \begin{matrix}f_{2} \\ ≃ \end{matrix}  |L′_{0}|_{p}^{∧}. ↓^{≃}

Here, the left and right vertical maps are homotopy equivalences by Proposition 4.6(a). This proves that f is a homotopy equivalence. The last statement (an isomorphism L_{0} ≅ L′_{0} can be chosen to induce f) now follows since by Theorem 7.1, every homotopy equivalence from |L′_{0}|_{p}^{∧} to itself is induced by some self equivalence of L′_{0}. □

An isomorphism (S, F, L) ⟶ (S′, F′, L′) of p-local compact groups consists of a triple (α, α_{F}, α_{L}), where

S ⟶^{α} S′,  F ⟶^{α_{F}} F′,  and L ⟶^{α_{L}} L′

are isomorphisms of groups and categories which satisfy the following compatibility conditions:

(a) α_{F}(P) = α_{L}(P) = α(P) for all P ≤ S;

(b) α_{F} and α_{L} commute with the projections π: L ⟶ F and π′: L′ ⟶ F′; and

(c) α_{L} commutes with the distinguished monomorphisms δ_{P} : P ⟶ Aut_{L}(P) and δ′_{P} : P ⟶ Aut_{L′}(P).

We are now ready to show that the isomorphism class of a p-local compact group is determined by the homotopy type of its classifying space. This was shown for p-local finite groups in [BLO2, Theorem 7.4].

Theorem 7.4. If (S, F, L) and (S′, F′, L′) are two p-local compact groups such that |L|_{p}^{∧} ≃ |L′|_{p}^{∧}, then (S, F, L) and (S′, F′, L′) are isomorphic as p-local compact groups.

Proof. If |L|_{p}^{∧} ⟶_{≃}^{f} |L′|_{p}^{∧} is a homotopy equivalence, then by Theorem 6.3(a), there are homomorphisms α ∈ Hom(S, S′) and α′ ∈ Hom(S′, S) such that the squares

\begin{matrix}BS & \overline{^{Bα} →} & BS′ & \overline{^{Bα′} →} & BS \\ |L|_{p}^{∧} & ↓^{θ} & \overline{^{f} →} & |L′|_{p}^{∧} & ↓^{θ′} & \overline{^{f′} →} & |L|_{p}^{∧} & ↓^{θ} \end{matrix}

commute up to homotopy, where f′ is any homotopy inverse to f. The composites α′ ∘ α and α ∘ α′ are F-conjugate to Id_{S} and Id_{S′} by Theorem 6.3(a) again, and thus α is an isomorphism.

By yet another application of Theorem 6.3(a), for any P, Q ≤ S,

Hom_{F}(P, Q) = {φ ∈ Inj(P, Q) | θ|_{BQ} ∘ Bφ ≃ θ|_{BP} }.

From this, and the corresponding result for Hom_{F′}(α(P), α(Q)), we see that α induces an isomorphism of categories from F to F′.

Upon replacing S′ and F′ by S and F, we can now assume that L and L′ are two linking systems associated to F, for which there is a homotopy equivalence |L|_{p}^{∧} ⟶^{f} |L′|_{p}^{∧} such that f ∘ θ ≃ θ′. Then L ≅ L′ (as linking systems associated to F) by Lemma 7.3. □

8. Fusion and linking systems of infinite groups

We now want to find some general conditions on an infinite group G which guarantee that we can associate to G a p-local compact group (S, F_{S}(G), L_{S}^{c}(G)) such that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}. This will be done in as much generality as possible. For example, we prove the saturation of the fusion system F_{S}(G) in sufficient generality so that the result also applies to the case where G is a compact Lie group.

At the end of the section, to show that the theory we have built up does contain some interesting examples, we show that it applies in particular to all linear torsion groups in characteristic different from p.

We say that a group G “has Sylow p-subgroups” if there is a discrete p-toral subgroup S ≤ G which contains all discrete p-toral subgroups of G up to conjugacy. For any such G, we let Syl_{p}(G) be the set of such maximal discrete p-toral subgroups.

Lemma 8.1. Fix a group G, a normal discrete p-toral subgroup Q ◁ G, and a subgroup K ≤ G such that G = QK. Assume that K has Sylow p-subgroups. Then G has Sylow p-subgroups, and

Syl_{p}(G) = {QS | S ∈ Syl_{p}(K)}.

Proof. Let Syl′_{p}(G) = {QS | S ∈ Syl_{p}(K)}. All subgroups in Syl′_{p}(G) are G-conjugate since all subgroups in Syl_{p}(K) are K-conjugate. If P ≤ G is an arbitrary discrete p-toral subgroup, then QP is also discrete p-toral (since Q and QP/Q are discrete p-toral), and

QP = QK ∩ QP = Q·(K ∩ QP).

Thus P ≤ QP ≤ QS ∈ Syl′_{p}(G) for any S ∈ Syl_{p}(K) which contains K ∩ QP. This shows that G has Sylow p-subgroups, and that they are precisely the subgroups in Syl′_{p}(G). □

We first establish some general conditions on an infinite group G with Sylow psubgroups, which imply that F_{S}(G) is a saturated fusion system for S ∈ Syl_{p}(G). The following technical lemma will be needed when doing this.

Lemma 8.2. Fix a group G, and normal subgroups N, Q ◁ G, with the following properties:

(a) Q is a discrete p-toral group.

(b) G/QN is a finite group.

(c) For each H ≤ G such that H ≥ N and H/N is finite, H has Sylow p-subgroups.

(d) each coset gN ∈ G/N contains at least one element of finite order.

Then G has Sylow p-subgroups. For any discrete p-toral subgroup P ≤ G, P ∈ Syl_{p}(G) if and only if P ≥ Q, P ∩ N ∈ Syl_{p}(N), and P N/QN ∈ Syl_{p}(G/QN).

Proof. Fix any G′ ≤ G such that G′ ≥ QN and G′/QN ∈ Syl_{p}(G/QN). For every discrete p-toral subgroup P ≤ G, P QN/QN is conjugate to a subgroup of G′/QN, hence P is G-conjugate to a subgroup of G′. Hence G has Sylow p-subgroups if G′ does, and in that case, Syl_{p}(G′) is the set of subgroups of G′ which are in Syl_{p}(G). It thus suffices to prove the lemma when G = G′; i.e., when G/QN is a finite p-group; and we assume this from now on.

Step 1: Assume first that Q = 1, and thus that |G/N| is a finite p-group. Then G has Sylow p-subgroups by (c). Throughout this step, we fix some S ∈ Syl_{p}(G). We first prove that NS = G (hence NS/N ∈ Syl_{p}(G/N)) and S ∩ N ∈ Syl_{p}(N). Afterwards, we prove the converse: P ∩ N ∈ Syl_{p}(N) and NP = G imply P is G-conjugate to S, and hence P ∈ Syl_{p}(G).

If NS ≨ G, then NS/N ≨ G/N, where the latter is a finite p-group. Since every proper subgroup of a p-group is contained in a proper normal subgroup, there is a proper normal subgroup N̂ ◁ G which contains NS. By (d), there is an element g ∈ G∖N̂ of finite order. Write |g| = mp^{k} where p ∤ m, and set g′ = g^{m}. Then g′ ∈ G∖N̂ since N̂ has p-power index, and |g′| = p^{k}. This means that ⟨g′⟩ is a finite p-subgroup of G which is not conjugate to a subgroup of S, which contradicts the assumption that S ∈ Syl_{p}(G). Thus NS = G.

For all S′ ∈ Syl_{p}(N), there are elements x ∈ G and y ∈ N such that xS′x^{−1} ≤ S ∩ N and y(S ∩ N)y^{−1} ≤ S′. Thus (yx)S′(yx)^{−1} ≤ S′, and this must be an equality since S′ is artinian. It follows that S ∩ N = xS′x^{−1} ∈ Syl_{p}(N).

Now let P ≤ G be any subgroup such that P ∩N ∈ Syl_{p}(N) and P N = G. Fix x ∈ G such that xP x^{−1} ≤ S. Then (xP x^{−1})N = xP Nx^{−1} = G, xP x^{−1} ∩N = x(P ∩N)x^{−1} ≤ S ∩ N, and this last must be an equality since P ∩ N ∈ Syl_{p}(N). It follows that

|G/N| = |xP x^{−1}·N/N| = |xP x^{−1}/(xP x^{−1} ∩ N)| ≤ |S/(S ∩ N)| = |SN/N| ≤ |G/N|;

so these are all equalities, and P = x^{−1}Sx ∈ Syl_{p}(G).

Step 2: Now consider the general case. By assumption, G/N is an extension of the discrete p-toral group QN/N by the finite p-group G/QN, and hence is discrete p-toral. So by Lemma 1.9, there is a finite p-subgroup G_{0}/N ≤ G/N such that (G_{0}/N)·(QN/N) = G/N, and thus QG_{0} = G (since G_{0} ≥ N). Then G_{0} has Sylow p-subgroups by (c). Hence G has Sylow p-subgroups by Lemma 8.1. Also, by Step 1 applied to the pair N ◁ G_{0} (recall G_{0}/N is a p-group),

P ∈ Syl_{p}(G_{0}) ⟺ P ∩ N ∈ Syl_{p}(N) and P N = G_{0}.

Let P ≤ G be any discrete p-toral subgroup which contains Q, and set P_{0} = P ∩ G_{0}. In general, for any A, B ≤ G and C ◁ G with C ≤ A, C·(A ∩ B) = A ∩ CB. Thus

\begin{aligned}QP_{0} & = Q·(P ∩ G_{0}) = P ∩ QG_{0} = P ∩ G = P \\ P_{0}N & = (P ∩ G_{0})·N = P N ∩ G_{0} . \end{aligned}  (1)

Also, by Lemma 8.1 again, Syl_{p}(G) = {QS | S ∈ Syl_{p}(G_{0})}. Hence

P ∈ Syl_{p}(G) ⟺ P_{0} = P ∩ G_{0} ∈ Syl_{p}(G_{0})
⟺ P_{0} ∩ N ∈ Syl_{p}(N) and P_{0}N = G_{0}  (Step 1)
⟺ P ∩ N ∈ Syl_{p}(N) and P N = G;

where the last equivalence holds by (1) and since P_{0} ∩ N = P ∩ (G_{0} ∩ N) = P ∩ N. □

Let G be any group which has Sylow p-subgroups. For any S ∈ Syl_{p}(G), we let F_{S}(G) be the fusion system over S with objects the subgroups of S and morphisms

Hom_{F_{S}(G)}(P, Q) = Hom_{G}(P, Q) .

Proposition 8.3. Let G be a group for which the following conditions hold:

(a) For each discrete p-toral subgroup P ≤ G, each element of Aut_{G}(P) is conjugation by some x ∈ N_{G}(P) of finite order.

(b) For each discrete p-toral subgroup P ≤ G, and each finite subgroup H/C_{G}(P) ≤ N_{G}(P)/C_{G}(P), H has Sylow p-subgroups.

(c) For each increasing sequence P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ of discrete p-toral subgroups of G, there is some k such that C_{G}(P_{n}) = C_{G}(P_{k}) for all n ≥ k.

Then for each S ∈ Syl_{p}(G), F_{S}(G) is a saturated fusion system. Furthermore, the following hold for each subgroup P ≤ S.

(1) C_{G}(P) has Sylow p-subgroups, and P is fully centralized in F_{S}(G) if and only if C_{S}(P) ∈ Syl_{p}(C_{G}(P)).

(2) N_{G}(P) has Sylow p-subgroups, and P is fully normalized in F_{S}(G) if and only if N_{S}(P) ∈ Syl_{p}(N_{G}(P)).

Proof. Note first that G has Sylow p-subgroups by (b), applied with P = 1.

Fix S ∈ Syl_{p}(G), and let P ≤ S be any subgroup. By (a), Aut_{G}(P) is a torsion group, so Out_{G}(P) is a torsion group, and hence is finite by Proposition 1.5(b). We first claim that

P ·C_{G}(P) ≤ Γ ≤ N_{G}(P) ⟹ Γ has Sylow p-subgroups;  (3)

and that if we set S_{0} = S ∩ Γ, then

S_{0} ∈ Syl_{p}(Γ) ⟺ C_{S}(P) ∈ Syl_{p}(C_{G}(P)) and \frac{S_{0}·C_{G}(P)}{P ·C_{G}(P)} ∈ Syl_{p}(Γ/P ·C_{G}(P)). (4)

Points (3) and (4) follow from Lemma 8.2, applied with N = C_{G}(P) and Q = P. Conditions (c) and (d) of Lemma 8.2 follow from conditions (b) and (a) above. Note that Γ/QN is finite since Out_{G}(P) ≅ N_{G}(P)/QN is finite.

We next prove (1) and (2). For all P ≤ S, (3) (applied with Γ = N_{G}(P)) implies that there is Q ∈ Syl_{p}(N_{G}(P)) such that N_{S}(P) ≤ Q ≤ N_{G}(P). Choose g ∈ G such that gQg^{−1} ≤ S; then gQg^{−1} is a Sylow p-subgroup of gN_{G}(P)g^{−1} = N_{G}(gP g^{−1}). Since gQg^{−1} ≤ S, gQg^{−1} = S ∩ N_{G}(gP g^{−1}) = N_{S}(gP g^{−1}). Hence N_{S}(gP g^{−1}) is a Sylow psubgroup of N_{G}(gP g^{−1}). If P is fully normalized, then |N_{S}(P)| ≥ |N_{S}(gP g^{−1})| = |Q|. Since N_{S}(P) ≤ Q, this implies that N_{S}(P) = Q ∈ Syl_{p}(N_{G}(P)).

Conversely, suppose that N_{S}(P) ∈ Syl_{p}(N_{G}(P)). Choose g ∈ G such that gP g^{−1} ≤ S and is fully normalized in F_{S}(G). Then N_{S}(gP g^{−1}) ∈ Syl_{p}(N_{G}(gP g^{−1})), so N_{S}(P) ≅ N_{S}(gP g^{−1}) since N_{G}(P) ≅ N_{G}(gP g^{−1}), and P is also fully normalized.

This proves (2). The proof of (1) (the condition for P to be fully centralized) is similar, except that C_{G}(P) has Sylow p-subgroups by (b).

We now prove that F_{S}(G) is saturated.

(I) Assume that P ≤ S is fully normalized in F_{S}(G). We have already seen that Out_{G}(P) is finite (since it is a torsion group by (a)). Also, N_{S}(P) ∈ Syl_{p}(N_{G}(P)) by (2). So by (4), applied with Γ = N_{G}(P), C_{S}(P) ∈ Syl_{p}(C_{G}(P)) (hence P is fully centralized by (1)), and Out_{S}(P) ∈ Syl_{p}(Out_{G}(P)).

(II) Let P ≤ S be an arbitrary subgroup, and let g ∈ G be such that P′ =^{def} gP g^{−1} ≤ S is fully centralized. Set Γ = N_{S}(P′)·C_{G}(P′), and define

N = {x ∈ N_{S}(P) | c_{g} ∘ c_{x} ∘ c_{g}^{−1} ∈ Aut_{S}(P′)} = {x ∈ N_{S}(P) | gxg^{−1} ∈ Γ} .

Then C_{S}(P′) ∈ Syl_{p}(C_{G}(P′)) by (1), and so by (4), N_{S}(P′) is a Sylow p-subgroup of Γ = N_{S}(P′)·C_{G}(P′) (S_{0}·C_{G}(P) = Γ in the notation of (4)). Since gNg^{−1} is a discrete ptoral subgroup of Γ, it is Γ-conjugate to a subgroup of N_{S}(P′). Thus there are elements x ∈ N_{S}(P′) and y ∈ C_{G}(P′) such that (xyg)N(xyg)^{−1} ≤ N_{S}(P′). Then (yg)N(yg)^{−1} ≤ N_{S}(P′) ≤ S, and c_{yg} ∈ Hom_{G}(N, S) is an extension of c_{g} ∈ Hom_{G}(P, S).

(III) Let P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ be a sequence of subgroups of S, and set P_{∞} = ⋃_{n=1}^{∞} P_{n}. Assume φ ∈ Hom(P_{∞}, S) is a monomorphism such that φ|_{P_{n}} ∈ Hom_{G}(P_{n}, S) for each n. Fix elements g_{n} ∈ G for each n such that φ(x) = g_{n}xg_{n}^{−1}, for x ∈ P_{n}. Then for all 1 ≤ k < n, g_{n}^{−1}g_{k} ∈ C_{G}(P_{k}).

By (c), there is k such that C_{G}(P_{k}) = C_{G}(P_{n}) = C_{G}(P_{∞}) for all n ≥ k. Hence for all n ≥ k and all x ∈ P_{n}, φ(x) = g_{n}xg_{n}^{−1} = g_{k}xg_{k}^{−1}, and thus φ = c_{g_{k}} ∈ Hom_{G}(P_{∞}, S). □

In general, for any group G, we define a p-centric subgroup of G to be a discrete p-toral subgroup P ≤ G such that Z(P) is the unique Sylow p-subgroup of C_{G}(P) (i.e., every discrete p-toral subgroup of C_{G}(P) is contained in Z(P)). Equivalently, P is p-centric if and only if C_{G}(P)/Z(P) has no elements of order p.

Proposition 8.4. Let G be any group which has Sylow p-subgroups, and fix S ∈ Syl_{p}(G). Then a subgroup P ≤ S is F_{S}(G)-centric if and only if P is p-centric in G.

Proof. Assume that P is p-centric in G; i.e., that Z(P) ∈ Syl_{p}(C_{G}(P)). For every g ∈ G such that gP g^{−1} ≤ S, C_{S}(gP g^{−1}) is a discrete p-toral subgroup of C_{G}(gP g^{−1}) = gC_{G}(P)g^{−1}, and Z(gP g^{−1}) = gZ(P)g^{−1} is a Sylow p-subgroup (hence the unique one) of gC_{G}(P)g^{−1}. It follows that Z(gP g^{−1}) = C_{S}(gP g^{−1}) for all such g, and so P is F_{S}(G)-centric.

Conversely, suppose that P ≤ S is F_{S}(G)-centric. Let Q be any discrete p-toral subgroup of C_{G}(P). Then QP is a discrete p-toral subgroup, and hence there is an element g ∈ G such that g(QP)g^{−1} ≤ S. Thus gP g^{−1} ≤ S, and gQg^{−1} ≤ S ∩ C_{G}(gP g^{−1}) = C_{S}(gP g^{−1}). Since P is F_{S}(G)-centric, this shows that gQg^{−1} ≤ Z(gP g^{−1}), and thus that Q ≤ Z(P). In other words, every discrete p-toral subgroup of C_{G}(P) is contained in Z(P), and so P is p-centric in G. □

We now restrict attention to locally finite groups. For any such group G, for the purposes of this section, we define O^{p}(G) ◁ G be the subgroup generated by all elements of order prime to p. This clearly generalizes the usual definition of O^{p}(G) for finite G (although it is not the only generalization).

Proposition 8.5. If G is locally finite, then a discrete p-toral subgroup P ≤ G is p-centric if and only if C_{G}(P) = Z(P) × O^{p}(C_{G}(P)) and all elements of O^{p}(C_{G}(P)) have order prime to p.

Proof. By the above definition, a discrete p-toral subgroup P ≤ G is p-centric if and only if C_{G}(P)/Z(P) has no elements of order p. So if P is not p-centric, then either O^{p}(C_{G}(P)) has p-torsion, or C_{G}(P) is not generated by Z(P) and O^{p}(C_{G}(P)).

Conversely, assume that P is p-centric, and thus that C_{G}(P)/Z(P) has no p-torsion. Consider the universal coefficient exact sequence

0 ⟶ Ext(H_{1}(C_{G}(P)/Z(P)), Z(P)) ⟶ H^{2}(C_{G}(P)/Z(P); Z(P))

⟶ Hom(H_{2}(C_{G}(P)/Z(P)), Z(P)) ⟶ 0.

By assumption, all elements of C_{G}(P)/Z(P) have order prime to p, all elements of Z(P) have p-power order, and both groups are locally finite. Hence for i = 1, 2, H_{i}(C_{G}(P)/Z(P)) is a direct limit of finite abelian groups of order prime to p, and thus a torsion group all of whose elements have order prime to p. This shows that all terms in the above sequence vanish. Hence the central extension

1 ⟶ Z(P) ⟶ C_{G}(P) ⟶ C_{G}(P)/Z(P) ⟶ 1

splits, so C_{G}(P) ≅ Z(P) × (C_{G}(P)/Z(P)), and all elements of the group O^{p}(C_{G}(P)) ≅ C_{G}(P)/Z(P) have order prime to p. □

When working with fusion systems over discrete p-toral groups and their orbit categories, we are able to reduce certain problems to ones involving finite categories using the functor (−)^{•} constructed in Section 3. This is not a functor on the orbit category of a group, and so we need a different way to make such reductions. For any group G with Sylow p-subgroups, we let 𝔛 = 𝔛(G) denote the set of all subgroups of G which are intersections of (nonempty) subsets of Syl_{p}(G). Since discrete p-toral groups are artinian, it makes no difference whether we require finite intersections or allow infinite intersections.

Lemma 8.6. Let G be a group such that for each discrete p-toral subgroup P ≤ G, N_{G}(P) has Sylow p-subgroups. Assume, for every increasing sequence P_{(1)} ≤ P_{(2)} ≤ P_{(3)} ≤ ⋯ of discrete p-toral subgroups of G, that the union of the P_{(i)} is again a discrete p-toral group, and that there is some k such that C_{G}(P_{(n)}) = C_{G}(P_{(k)}) for all n ≥ k. Then the set 𝔛(G) contains finitely many G-conjugacy classes.

Proof. For each discrete p-toral subgroup P ≤ G, we let P^{∘} ≥ P denote the intersection of all Sylow p-subgroups of G which contain P. We first prove

(1) For each discrete p-toral subgroup P ≤ G, there is a finite subgroup P′ ≤ P such that P′^{∘} = P^{∘}.

To see this, set p^{n} = exp(π_{0}(S)) for S ∈ Syl_{p}(G). The discrete p-torus S_{0} is the union of an increasing sequence of finite p-subgroups, and since centralizers stabilize by assumption, there is a finite subgroup Q ≤ P_{0} such that C_{G}(Q) = C_{G}(P_{0}). Set Q′ = {x ∈ P_{0} | x^{p^{n}} ∈ Q}: also a finite p-subgroup. By Lemma 1.9, there is a finite subgroup P′ ≤ P such that P′ ≥ Q′ and P′P_{0} = P.

Fix S ∈ Syl_{p}(G) which contains P′. Then S_{0} ≥ Q (since S ≥ Q′), and hence S_{0} ≤ C_{G}(Q) = C_{G}(P_{0}). Since S_{0} is a maximal discrete p-torus in G and S_{0}·P_{0} is also a discrete p-torus, this implies that S_{0} ≥ P_{0}. Hence S ≥ P′P_{0} = P. Since this holds for all S ∈ Syl_{p}(G) which contains P′, we have shown that P′^{∘} = P^{∘}; and this finishes the proof of (1).

Let 𝔛(G) be the set of G-conjugacy classes of subgroups in 𝔛(G). We let (P) denote the conjugacy class of the subgroup P, and make 𝔛(G) into a poset by setting (P) ≤ (Q) if P ≤ xQx^{−1} for some x ∈ G. Let P ⊆ 𝔛(G) be the set of all classes (P) which are contained in infinitely many other classes. We claim that P = ∅. Since 𝔛(G) contains a smallest element which is contained in all the others (the class of the intersection of all Sylow p-subgroups of G), P = ∅ implies that 𝔛(G) is finite, which is what we want to prove.

Assume otherwise: assume P ≠ ∅. We claim that P has a maximal element. For any totally ordered subset P_{0} of P, upon restricting to those subgroups of maximal rank, we obtain a sequence of subgroups P_{(1)} ≤ P_{(2)} ≤ P_{(3)} ≤ ⋯ whose conjugacy classes are cofinal in P_{0}. If this sequence is finite, then P_{0} clearly has a maximal element. Otherwise, set P_{(∞)} = ⋃_{i=1}^{∞} P_{(i)}, and let P′ ≤ P_{(∞)} be a finite subgroup such that P′^{∘} = P_{(∞)} (apply (1)). Then P′ ≤ P_{(k)} for some k, and so (P_{(k)}) = (P_{(∞)}) is a maximal element in P_{0}.

Thus by Zorn’s lemma, P contains a maximal element (Q), and clearly Q ∉ Syl_{p}(G). Since N_{G}(Q) has Sylow p-subgroups, there is some S ∈ Syl_{p}(G) such that every p-toral subgroup of G containing Q with index p is G-conjugate to a subgroup of N_{S}(Q); and hence by Lemma 1.4, there are finitely many G-conjugacy classes of such subgroups. Hence since (Q) is contained in infinitely many classes in 𝔛(G), the same holds for (Q′) for some Q′ ≤ G such that Q ◁ Q′ with index p. Then (Q′^{∘}) ∈ P, which contradicts the maximality assumption about Q. So P contains no maximal element, hence must be empty, and so 𝔛(G) has finitely many G-conjugacy classes. □

Now, for any discrete group G which has Sylow p-subgroups, let L_{p}^{c}(G) be the category whose objects are the p-centric subgroups of G, and where

Mor_{L_{p}^{c}(G)}(P, Q) = N_{G}(P, Q)/O^{p}(C_{G}(P)).

For any S ∈ Syl_{p}(G), let L_{S}^{c}(G) ⊆ L_{p}^{c}(G) be the equivalent full subcategory whose objects are the subgroups of S which are p-centric in G.

It will be convenient, throughout the rest of this section, to use the term “p-group” to mean any group each of whose elements has p-power order. It is not hard to show that if G is locally finite, and has Sylow p-subgroups in the sense described above, then every p-subgroup of G is a discrete p-toral subgroup. Hence there is no loss of generality to assume this in the following theorem.

Theorem 8.7. Let G be any group which satisfies the following conditions:

(a) G is locally finite.

(b) Each p-subgroup of G is a discrete p-toral group.

(c) For any increasing sequence A_{(1)} ≤ A_{(2)} ≤ A_{(3)} ≤ ⋯ of finite abelian p-subgroups of G, there is some k such that C_{G}(A_{(n)}) = C_{G}(A_{(k)}) for all n ≥ k.

Then G has a unique conjugacy class Syl_{p}(G) of maximal discrete p-toral subgroups. For any S ∈ Syl_{p}(G), (S, F_{S}(G), L_{S}^{c}(G)) is a p-local compact group, with classifying space |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.

Proof. We first apply Proposition 8.3 to show that F_{S}(G) is a saturated fusion system over S. Once this has been checked, then it easily follows that L_{S}^{c}(G) is a centric linking system associated to F_{S}(G): condition (A) in Definition 4.1 holds by Propositions 8.5 and 8.4, and conditions (B) and (C) are immediate. It then will remain only to show that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.

By [KW, Theorem 3.4], conditions (a) and (c) above imply that all maximal psubgroups of G are conjugate, and hence (by (b)) that G has Sylow p-subgroups. Since these three conditions are carried over to subgroups of G, this also shows that each subgroup of G has Sylow p-subgroups. This proves condition (b) in Proposition 8.3, and condition (a) holds since G is locally finite.

It remains to prove condition (c) in Proposition 8.3, which we state here as:

(d) For any increasing sequence P_{(1)} ≤ P_{(2)} ≤ P_{(3)} ≤ ⋯ of discrete p-toral subgroups of G, there is some k such that C_{G}(P_{(n)}) = C_{G}(P_{(k)}) for all n ≥ k.

To see this, fix any such sequence, and let P_{(∞)} be its union. Let A = (P_{(∞)})_{0} be the identity component, and set A_{(i)} = A ∩ P_{(i)} for all i. Let r be such that P_{(i)} surjects onto π_{0}(P_{(∞)}) for all i ≥ r; equivalently, P_{(i)}·A = P_{(∞)} for all i ≥ r. For each i, let A′_{i} ≤ A_{(i)} be the finite subgroup of elements of order at most p^{i}. Then A = ⋃_{i=1}^{∞} A′_{i}; and so by (c) there is r such that C_{G}(A) = C_{G}(A′_{r}). Hence C_{G}(A_{(i)}) = C_{G}(A_{(r)}) for all i ≥ r (since A′_{r} ≤ A_{(r)} ≤ A_{(i)} ≤ A). We can assume that r is chosen large enough so that P_{(r)} surjects onto P_{(∞)}/A; i.e., such that P_{(r)}·A = P_{(∞)}. Then for all i ≥ r, P_{(i)} = A_{(i)}·P_{(r)},

C_{G}(P_{(i)}) = C_{G}(A_{(i)}) ∩ C_{G}(P_{(r)}) = C_{G}(A_{(r)}) ∩ C_{G}(P_{(r)}) = C_{G}(P_{(r)}),

and this finishes the proof of (d).

We have now shown that the hypotheses of Proposition 8.3 hold, and thus that F_{S}(G) is a saturated fusion system over S. We have already seen that L_{S}^{c}(G) is a linking system associated to F_{S}(G), and it remains only to show that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.

As in Section 5, for any discrete group G, we let O_{p}(G) be the category whose objects are the discrete p-toral subgroups P ≤ G, and where

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

Let O_{𝔛}(G) ⊆ O_{p}(G) be the full subcategory with object set 𝔛 = 𝔛(G): the set of all intersections of subgroups in Syl_{p}(G). For each discrete p-toral subgroup P ≤ G, we let P^{∘} ∈ 𝔛 denote the intersection of all subgroups in Syl_{p}(G) which contain P. Clearly, for any P and Q, N_{G}(P, Q) ⊆ N_{G}(P^{∘}, Q^{∘}), and so this defines a functor (−)^{∘} from O_{p}(G) to O_{𝔛}(G). Since N_{G}(P^{∘}, Q) = N_{G}(P, Q) when Q ∈ 𝔛(G), the two functors

O_{𝔛}(G) \overrightarrow{⟵}_{(−)^{∘}}^{incl} O_{p}(G)  (1)

are adjoint.

Step 1: Let I and Φ be the following functors from O_{p}(G) to (G-)spaces:

I(P) = G/P and Φ(P) = EG ×_{G} I(P) ≅ EG/P.

Then for any full subcategory C ⊆ O_{p}(G),

\underrightarrow{hocolim}_{C}(I) = (∐_{n=0}^{∞} ∐_{G/P_{0}→···→G/P_{n}} G/P_{0} × Δ^{n})/∼

is the nerve of the category whose objects are the cosets gP for all P ∈ Ob(C), and with a unique morphism gP → hQ exactly when gP g^{−1} ≤ hQh^{−1}. When C = O_{𝔛}(G), this category has as initial object the intersection of all Sylow p-subgroups of G, and hence \underrightarrow{hocolim}_{O_{𝔛}(G)}(I) is contractible. Since the Borel construction commutes with homotopy colimits in this situation (being itself a special case of a homotopy colimit),

\underrightarrow{hocolim}_{O_{𝔛}(G)}(Φ) ≅ EG ×_{G} (\underrightarrow{hocolim}_{O_{𝔛}(G)}(I)) ≃ BG.  (2)

Step 2: Fix some Q ∈ 𝔛 which is not p-centric. For each i ≥ 0, consider the functor

F_{i}^{[Q]} : O_{p}(G)^{op} → Ab where F_{i}^{[Q]}(P) = \begin{cases}H^{i}(BP; 𝔽_{p}) & if P is G-conjugate to Q \\ 0 & otherwise. \end{cases}

The subgroup C_{G}(Q)·Q/Q ≅ C_{G}(Q)/Z(Q) of Aut_{O_{p}(G)}(Q) = N(Q)/Q acts trivially on F_{i}^{[Q]}(Q), and contains an element of order p since Q is not p-centric. Hence by Lemmas 5.10 and 5.12,

\underleftarrow{lim}^{∗}_{O_{𝔛}(G)}(F_{i}^{[Q]}) ≅ \underleftarrow{lim}^{∗}_{O_{p}(G)}(F_{i}^{[Q]}) ≅ Λ^{∗}(N_{G}(Q)/Q; F_{i}^{[Q]}(P)) = 0 for all i,

where the first isomorphism follows from the adjoint functors (1).

Step 3: Now let O_{𝔛}^{c}(G) ⊆ O_{𝔛}(G) be the full subcategory with objects the p-centric subgroups which lie in 𝔛. Let P_{1}, P_{2}, …, P_{m} ≤ S be representatives for those Gconjugacy classes in 𝔛(G) which are not p-centric (a finite set by Lemma 8.6). We assume these are ordered such that |P_{i}| ≤ |P_{i+1}| for each i.

For each i ≥ 0, consider the functor F_{i}: O_{p}(G)^{op} → Ab, defined by setting F_{i}(P) = H^{i}(BP; 𝔽_{p}) for all P. For all k = 0, …, m, define functors

F_{i,k}: O_{𝔛}(G)^{op} → Ab by setting F_{i,k}(P) = \begin{cases}0 & if P ∼_{G} P_{j}, some j ≥ k \\ F_{i}(P) & otherwise. \end{cases}

Here, “∼_{G}” means “G-conjugate”, and these are all defined to be quotient functors of F_{i}|_{O_{𝔛}(G)}. In particular, F_{i,0} = F_{i}|_{O_{𝔛}(G)} and F_{i,m} = F_{i}|_{O_{𝔛}^{c}(G)}. Also, for all k,

Ker[F_{i,k} −↠ F_{i,k+1}] ≅ F_{i}^{[P_{k}]}|_{O_{𝔛}(G)},

and the higher limits of this last functor vanish by Step 2. So there are isomorphisms

\underleftarrow{lim}^{∗}_{O_{𝔛}(G)}(F_{i}) = \underleftarrow{lim}^{∗}_{O_{𝔛}(G)}(F_{i,0}) ≅ \underleftarrow{lim}^{∗}_{O_{𝔛}(G)}(F_{i,1}) ≅ ⋯ ≅ \underleftarrow{lim}^{∗}_{O_{𝔛}(G)}(F_{i,m}) ≅ \underleftarrow{lim}^{∗}_{O_{𝔛}^{c}(G)}(F_{i});

whose composite is induced by restriction from O_{𝔛}(G) to O_{𝔛}^{c}(G).

The spectral sequence for the cohomology of a homotopy colimit now implies that the inclusion of O_{𝔛}^{c}(G) into O_{𝔛}(G) induces a mod p homology isomorphism of homotopy colimits of Φ, and hence a homotopy equivalence

(\underrightarrow{hocolim}_{O_{𝔛}^{c}(G)}(Φ))_{p}^{∧} ⟶^{≃} (\underrightarrow{hocolim}_{O_{𝔛}(G)}(Φ))_{p}^{∧} .  (3)

Also, the adjoint functors in (1) restrict to adjoint functors between O_{𝔛}^{c}(G) and O_{p}^{c}(G), and hence induce a homotopy equivalence

(\underrightarrow{hocolim}_{O_{𝔛}^{c}(G)}(Φ))_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O_{p}^{c}(G)}(Φ))_{p}^{∧} .  (4)

Step 4: Let T_{p}^{c}(G) be the centric transporter category for G: the category whose objects are the p-centric subgroups of G, and where the set of morphisms from P to Q is the transporter N_{G}(P, Q). By exactly the same argument as in [BLO1, Lemma 1.2],

\underrightarrow{hocolim}_{O_{p}^{c}(G)}(Φ) ≃ |T_{p}^{c}(G)|.  (5)

The canonical projection functor T_{p}^{c}(G) → L_{p}^{c}(G) satisfies all of the hypotheses of the functor in [BLO1, Lemma 1.3], except that we only know that

K(P) =^{def} Ker[Aut_{T_{p}^{c}(G)}(P) −↠ Aut_{L_{p}^{c}(G)}(P)] = O^{p}(C_{G}(P))

is a locally finite group all of whose elements have order prime to p (not necessarily a finite group). But this suffices to ensure that coinvariants preserve exact sequences of ℤ_{(p)}[K_{P}]-modules, which is the only way this property of K_{P} is used in the proof of [BLO1, Lemma 1.3]. Hence the induced map

|T_{p}^{c}(G)| ⟶ |L_{p}^{c}(G)|.

is a mod p homology equivalence. Together with (2), (3), (4), and (5), this shows that

|L_{S}^{c}(G)|_{p}^{∧} ≃ |L_{p}^{c}(G)|_{p}^{∧} ≃ |T_{p}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.  □

We now finish the section by exhibiting a more concrete class of groups which satisfy the hypotheses of Theorem 8.7. A linear torsion group is a torsion subgroup of GL_{n}(k), for any positive integer n and any (commutative) field k. These are also referred to as “periodic linear groups”, since their elements are all periodic transformations (automorphisms of finite order) of a finite dimensional vector space.

The following facts about linear torsion groups are the starting point of our work here.

Proposition 8.8. The following hold for every field k, and every linear torsion group G ≤ GL_{n}(k).

(a) G is locally finite.

(b) For p ≠ char(k), every p-subgroup of G is a discrete p-toral group.

Proof. Point (a) is a theorem of Schur, and is shown in [W, Corollary 4.9]. By [W, 2.6], every locally finite p-subgroup of GL_{n}(k) is artinian (when p ≠ char(k)), and hence is discrete p-toral by Proposition 1.2. □

In order to apply Theorem 8.7, it remains only to check that centralizers of discrete p-toral subgroups of linear torsion groups stabilize in the sense of Theorem 8.7.

Proposition 8.9. Let A_{1} ≤ A_{2} ≤ A_{3} ≤ … be an increasing sequence of finite abelian p-subgroups of a linear torsion group G ≤ GL_{n}(k), where char(k) ≠ p. Then there is r such that C_{G}(A_{i}) = C_{G}(A_{r}) for all i ≥ r.

Proof. Upon replacing k by its algebraic closure if necessary, we can assume that k is algebraically closed. Hence any representation over k of a finite abelian p-group A splits as a sum of 1-dimensional irreducible representations. Moreover, if A ≤ GL_{n}(k), and k^{n} = U_{1} ⊕ ⋯ ⊕ U_{m} is the unique decomposition with the property that each U_{i} is a sum of irreducible modules with the same character and different U_{i} correspond to different characters of A, then

C_{GL_{n}(k)}(A) ≅ ∏_{i=1}^{m} Aut_{k}(U_{i}) ≅ ∏_{i=1}^{m} GL_{d_{i}}(k).  (d_{i} = dim(U_{i}))

From this observation, it is clear that for any increasing sequence of such subgroups A_{i}, the centralizers C_{GL_{n}(k)}(A_{i}) stabilize for i sufficiently large, and hence the stabilizers C_{G}(A_{i}) also stabilize. □

Propositions 8.8 and 8.9 show that linear torsion groups satisfy all of the hypotheses of Theorem 8.7. So as an immediate consequence, we get:

Theorem 8.10. Fix a linear torsion group G, a prime p different than the defining characteristic of G, and a Sylow subgroup S ∈ Syl_{p}(G). Then (S, F_{S}(G), L_{S}^{c}(G)) is a p-local compact group, with classifying space |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}. □

9. Compact Lie groups

Throughout this section, we fix a compact Lie group G and a prime p. Our main result is to show that G defines a p-local compact group whose classifying space has the homotopy type of BG_{p}^{∧}.

A compact Lie group P is called p-toral if its identity component is a torus and if its group of components is a p-group. The closure P of a discrete p-toral subgroup P ≤ G is a p-toral group, since P_{0} is abelian and connected, hence a torus, and has p-power index in P. We will generally denote p-toral groups (including tori) by P , Q, T , etc., to distinguish them from discrete p-toral groups P, Q, T, etc. Our first task is to identify the maximal (discrete) p-toral subgroups of G.

Definition 9.1. (a) For any p-toral group P, Syl_{p}(P) denotes the set of discrete ptoral subgroups P ≤ P such that P ·P_{0} = P and P contains all p-power torsion in P_{0}.

(b) A discrete p-toral subgroup P ≤ G is snugly embedded if P ∈ Syl_{p}(P).

(c) Syl_{p}(G) denotes the set of all p-toral subgroups S ≤ G such that the identity component S_{0} is a maximal torus of G and S/S_{0} ∈ Syl_{p}(N(S_{0})/S_{0}).

(d) Syl_{p}(G) denotes the set of all discrete p-toral subgroups P ≤ G such that P ∈ Syl_{p}(G) and P ∈ Syl_{p}(P).

For any discrete p-toral subgroup P ≤ G, P_{0} is a torus, as noted above, and has finite index in P. Hence P_{0} = (P)_{0}, and π_{0}(P) ≅ P/P_{0}. So P is snugly embedded in G if and only if P_{0} is snugly embedded, and this holds exactly when rk(P) = rk(P). As an example of a subgroup which is not snugly embedded, one can construct a rank one subgroup P ≅ ℤ/p^{∞} which is densely embedded in a torus (S^{1})^{r} for r > 1.

Clearly, when rk(P) < rk(P), we cannot expect BP _{p}^{∧} and BP _{p}^{∧} to have the same homotopy type. But we do get a homotopy equivalence when P is snugly embedded.

Proposition 9.2. If P ≤ G is snugly embedded, then the inclusion of P in P induces a homotopy equivalence BP _{p}^{∧} ≃ BP _{p}^{∧}.

Proof. This means showing that the inclusion of BP into BP induces an isomorphism on mod p cohomology. See, for example, [Fe, Proposition 2.3]. □

The following proposition is well known. It says that Syl_{p}(G) is the set of maximal ptoral subgroups of G, that Syl_{p}(G) is the set of maximal discrete p-toral subgroups of G, and that each of these sets contains exactly one G-conjugacy class. Note in particular the case where G = P is p-toral: there is a unique conjugacy class of discrete p-toral subgroups snugly embedded in P , and every discrete p-toral subgroup of P is contained in a snugly embedded subgroup.

Proposition 9.3. The following hold for any compact Lie group G and any p-toral group P.

(a) Any two subgroups in Syl_{p}(G) are G-conjugate, and each p-toral subgroup P ≤ G is contained in some subgroup S ∈ Syl_{p}(G).

(b) Any two subgroups in Syl_{p}(G) are G-conjugate, and each discrete p-toral subgroup P ≤ G is contained in some subgroup S ∈ Syl_{p}(G).

Proof. (a) The subgroups in Syl_{p}(G) are clearly all conjugate to each other, since all maximal tori in G are conjugate. For any S ∈ Syl_{p}(G) with identity component the maximal torus T = S_{0}, χ(G/N(T )) = 1 (see [Br, Proposition 0.6.3]), and hence χ(G/S ) is prime to p. If Q is an arbitrary p-toral subgroup of G, then χ((G/S )^{Q}) is congruent mod p to χ(G/S ), so (G/S )^{Q} ≠ ∅, and hence Q ≤ gS g^{−1} ∈ Syl_{p}(G) for some g ∈ G.

(b) Assume first that G = P is p-toral. Set T = P_{0}, and let T ≤ T be the subgroup of elements of p-power torsion. By definition, Syl_{p}(P ) is the set of all subgroups P ≥ T such that P/T is the image of a splitting of the extension

1 ⟶ T /T ⟶ P /T ⟶ P /T ⟶ 1.  (1)

The cohomology groups H^{i}(P /T ; T /T) vanish for all i > 0, since P /T is a p-group and T /T is uniquely p-divisible. Hence the extension (1) is split, and any two splittings are conjugate by an element of T /T. Thus Syl_{p}(P ) ≠ ∅, and its elements are conjugate to each other by elements of T .

Now let Q ≤ P be an arbitrary discrete p-toral subgroup. Then QT is also a discrete p-toral subgroup (since T ◁ P ), and QT/T is the image of a splitting of the extension of T /T by QT /T . We have seen that any two such splittings are conjugate by elements of T /T, and hence they all extend to splittings of the extension by P /T . In other words, there is a subgroup P ∈ Syl_{p}(P ) which contains QT, and hence contains Q.

Now let G be an arbitrary compact Lie group. For any S, S′ ∈ Syl_{p}(G), S is G-

conjugate to S′ by (a), so S = gS′g^{−1} for some g ∈ G, and S, gS′g^{−1} ∈ Syl_{p}(S). We have just shown that all subgroups in Syl_{p}(S) are conjugate, and hence S and S′ are conjugate. If P ≤ G is an arbitrary discrete p-toral subgroup, then its closure P is a p-toral subgroup, and hence contained in some maximal subgroup S ∈ Syl_{p}(G) by (a) again. So there is some S ∈ Syl_{p}(S ) ⊆ Syl_{p}(G) which contains P. □

We next need some information about the outer automorphisms of (discrete) p-toral subgroups of G.

Lemma 9.4. The following hold for all discrete p-toral subgroups P, Q ≤ G.

(a) If P ≤ Q, then Out_{Q}(P) is a finite p-group, and Out_{Q}(P) = Out_{Q}(P) if Q is snugly embedded in G. In particular, Out_{Q}(Q) = 1 if Q is snugly embedded in G.

(b) Out_{G}(P) and Out_{G}(P) are both finite.

(c) If Q is snugly embedded, then the natural map

Rep_{G}(P, Q) ⟶^{≅} Rep_{G}(P , Q)

is a bijection.

Proof. (a) Choose Q′ ≤ Q such that Q′ = Q and Q′ is snugly embedded. Then Out_{Q′}(P) is a finite p-group by Proposition 1.5(c). The first statement thus follows from the second.

Now assume Q is snugly embedded. We must show that Out_{Q}(P) = Out_{Q}(P); or equivalently that Aut_{Q}(P) = Aut_{Q}(P). Fix x ∈ N_{Q}(P), and set |P | = p^{k}.

Let Q/Q be the set of left cosets gQ for g ∈ Q, and let (Q/Q)^{P} be the fixed point set of the left P -action. Then for g ∈ Q, gQ ∈ (Q/Q)^{P} if and only if g^{−1}P g ≤ Q. In particular, xQ ∈ (Q/Q)^{P} since x normalizes P and P ≤ Q. Since Q/Q = Q_{0}/Q_{0} and the latter group is uniquely p-divisible (since Q is snugly embedded), there is y ∈ Q_{0} such that y^{p^{k}} ∈ xQ and yQ ∈ (Q/Q)^{P}.

Set

ŷ = ∏_{a∈P}(aya^{−1}) = y^{p^{k}}· ∏_{a∈P}((y^{−1}ay)·a^{−1}) ∈ y^{p^{k}}Q = xQ;

where the inclusion holds since P ≤ Q and y^{−1}P y ≤ Q. Then ŷ ∈ C_{Q}(P). Since x was arbitrary, this proves that N_{Q}(P) = C_{Q}(P)·N_{Q}(P), and finishes the proof that Aut_{Q}(P) = Aut_{Q}(P). In the case P = Q, this shows that Out_{Q}(Q) = Out_{Q}(Q) = 1.

(b) The kernel of the homomorphism

Out_{G}(P) ⟶ Out_{G}(P)  (1)

is Out_{P} (P). By (a), this is always finite, and is trivial if P is snugly embedded. If P is snugly embedded, i.e., if P ∈ Syl_{p}(P), then N_{G}(P) = P ·N_{G}(P) (any subgroup of P which is G-conjugate to P is also P -conjugate to P), and hence the map in (1) is also surjective. Thus in this case, Out_{G}(P) ≅ Out_{G}(P) ≅ N_{G}(P)/P ·C_{G}(P) is a compact Lie group, all torsion subgroups of which are finite by Theorem 1.5. Thus Out_{G}(P) is finite (since otherwise it would contain S^{1}). If P is an arbitrary discrete p-toral subgroup of G, then the kernel and the image of the map in (1) are finite, and hence Out_{G}(P) is also finite in this case.

(c) Assume P, Q ≤ S, where Q is snugly embedded. We must show that the map from Rep_{G}(P, Q) to Rep_{G}(P , Q) which sends a homomorphism to its unique continuous extension is a bijection. For any φ ∈ Hom_{G}(P , Q), φ(P) is Q-conjugate to a subgroup of Q ∈ Syl_{p}(Q), and hence φ is Q-conjugate to a homomorphism which sends P into Q. This proves surjectivity. To prove injectivity, fix φ_{1}, φ_{2} ∈ Hom_{G}(P, Q) which induce the same class in Rep_{G}(P , Q), and set P_{i} = Im(φ_{i}). Then φ_{2} = χ ∘ φ_{1} for some χ ∈ Iso_{Q}(P_{1}, P_{2}). We must show that χ ∈ Iso_{Q}(P_{1}, P_{2}), and it suffices to do this when χ ∈ Iso_{Q_{0}}(P_{1}, P_{2}). In this case, P_{1}Q_{0} = P_{2}Q_{0}, so χ extends to χ ∈ Aut_{Q_{0}}(P_{1}Q_{0}). Also, P_{1}Q_{0} is snugly embedded since Q is, so Out_{Q_{0}}(P_{1}Q_{0}) = 1 by (b), and hence χ is

conjugation by an element of Q_{0} ∩ P_{1}Q_{0} = Q_{0}. □

The fusion system of a compact Lie group is defined exactly as in Section 8. For any S ∈ Syl_{p}(G), F_{S}(G) is the fusion system over S where for P, Q ≤ S,

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

is the set of homomorphisms from P to Q induced by conjugation by elements of G. Here, as usual,

N_{G}(P, Q) = {x ∈ G | xP x^{−1} ≤ Q}

denotes the transporter set.

Lemma 9.5. For each maximal discrete p-toral subgroup S ∈ Syl_{p}(G), F_{S}(G) is a saturated fusion system over S. Also, a subgroup P ≤ S is fully centralized in F_{S}(G) if and only if C_{S}(P) ∈ Syl_{p}(C_{G}(P)).

Proof. We must show that conditions (a), (b), and (c) of Proposition 8.3 all hold. For each discrete p-toral subgroup P ≤ G, Out_{G}(P) is finite by Lemma 9.4(b), so Aut_{G}(P) ≅ N_{G}(P)/C_{G}(P) is a torsion group. Hence for each g ∈ N_{G}(P), ⟨g⟩·C_{G}(P) is a finite extension of C_{G}(P), thus a closed subgroup, and so the coset gC_{G}(P) contains elements of finite order. Also, for each finite subgroup H/C_{G}(P) in N_{G}(P)/C_{G}(P), H is a closed subgroup of G, and hence has Sylow p-subgroups in the sense of Section 8. If P_{1} ≤ P_{2} ≤ P_{3} ≤ ⋯ is an increasing sequence of discrete p-toral subgroups of G, then the centralizers C_{G}(P_{i}) form a decreasing sequence of closed subgroups of G, and hence is constant for i sufficiently large.

Thus Proposition 8.3 applies: for any S ∈ Syl_{p}(G), F_{S}(G) is a saturated fusion system over S, and a subgroup P ≤ S is fully centralized in F_{S}(G) if and only if C_{S}(P) ∈ Syl_{p}(C_{G}(P)). □

Recall from Section 8 that a discrete p-toral subgroup P ≤ G is called p-centric in G if Z(P) ∈ Syl_{p}(C_{G}(P)). By analogy with this definition, a p-toral subgroup P ≤ G is called p-centric if Z(P ) ∈ Syl_{p}(C_{G}(P )). We next note some conditions which characterize p-toral and discrete p-toral subgroups of G which are p-centric.

Lemma 9.6. The following hold for any discrete p-toral subgroup P ≤ G.

(a) If P is p-centric in G, then N_{G}(P)/P is finite, and C_{G}(P)/Z(P) is finite of order prime to p.

(b) If P is p-centric in G, then P is p-centric in G.

(c) If P is p-centric in G and P is snugly embedded, then P is p-centric in G.

Proof. (a) Assume P is p-centric in G, and consider the groups

C_{G}(P )/Z(P ) ≅ P ·C_{G}(P )/P  and  Out_{G}(P ) ≅ N_{G}(P )/(P ·C_{G}(P )).

The first group is finite of order prime to p since Z(P ) is a maximal p-toral subgroup of C_{G}(P ) which is also central. The second group is finite by Lemma 9.4(b). Hence N_{G}(P )/P is also finite.

(b) If P is p-centric in G, then Z(P) ≤ Z(P) is a maximal p-toral subgroup in C_{G}(P) = C_{G}(P), and hence P is also p-centric in G.

(c) Assume P ∈ Syl_{p}(P ). If x ∈ Z(P ) has p-power order, then since [x, P_{0}] = 1, the only elements of p-power order in xP_{0} are those in xP_{0}. Since some element of xP_{0} lies in P and has p-power order, this shows that x ∈ P, and hence that x ∈ Z(P). In other words, Z(P) ∈ Syl_{p}(Z(P )). So if P is p-centric in G, then Z(P) is a maximal discrete p-toral subgroup of C_{G}(P) = C_{G}(P ), and hence P is also p-centric in G. □

We want to apply Proposition 4.6, to construct a centric linking system L_{S}^{c}(G) associated to F_{S}(G), and to show that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}. This means constructing a rigidification of the homotopy functor B: P ↦ BP; which by Proposition 5.9 is equivalent to constructing a rigidification of the homotopy functor B_{p}^{∧}: P ↦ BP _{p}^{∧}. This last is closely related to the homotopy decomposition of BG constructed in [JMO].

For any S ∈ Syl_{p}(G), we let O_{S}(G) denote the category whose objects are the p-toral subgroups of S , and where

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

Define B : O_{S}(G) ⟶ Topby setting

B (P ) = EG/P and B (P ⟶^{Qx} Q) = (EG/P ⟶^{·x^{−1}} EG/Q).

Let

Φ: \underrightarrow{hocolim}_{O_{S} (G)}(B ) ⟶ EG/G = BG

be the map induced by the obvious surjections from B (P ) = EG/P onto BG = EG/G.

Lemma 9.7. Fix a maximal p-toral subgroup S ∈ Syl_{p}(G). Let O_{S}^{c}(G) ⊆ O_{S}(G) be the full subcategory whose objects are those p-toral subgroups of S which are p-centric in G, and let

B_{c}: O_{S}^{c}(G) ⟶ Top and Φ_{c}: \underrightarrow{hocolim}_{O_{S}^{c}(G)}(B_{c}) ⟶ BG

be the restrictions of B and Φ, respectively. Then Φ_{c} is a mod p homology equivalence.

Proof. Define

X = {P ≤ S p-toral | |N_{G}(P )/P | < ∞, O_{p}(N_{G}(P )/P ) = 1 or P is p-centric} .

Let O_{S}^{X}(G) ⊆ O_{S}(G) be the full subcategory with object set X, and let

B_{X}: O_{S}^{X}(G) ⟶ Top and Φ_{X}: \underrightarrow{hocolim}_{O_{S}^{X}(G)}(B_{X}) ⟶ BG

be the restrictions of B and Φ.

By [JMO, Theorem 1.4], Φ_{X} is a mod p homology equivalence. So to prove the proposition, we must show that the inclusion of \underrightarrow{hocolim}(B_{c}) in \underrightarrow{hocolim}(B_{X}) is a mod p homology equivalence. Set F = H^{∗}(B_{X}(−); 𝔽_{p}), regarded as a functor on O_{S}^{X}(G)^{op}. Let F_{0} ⊆ F be the subfunctor defined by setting F_{0}(P ) = 0 if P is p-centric in G and F_{0}(P ) = F (P ) otherwise. We claim that \underleftarrow{lim}^{∗}(F_{0}) = 0. Assuming this, we see that

\underleftarrow{lim}^{∗}_{O_{S}^{X}(G)}(H^{∗}(B_{X}(−); 𝔽_{p})) ≅ \underleftarrow{lim}^{∗}_{O_{S}^{X}(G)}(F/F_{0}) ≅ \underleftarrow{lim}^{∗}_{O_{S}^{c}(G)}(H^{∗}(B_{c}(−); 𝔽_{p}));

the last step since there are no morphisms from any object of the subcategory to any object not in the subcategory. This shows that the spectral sequences for the cohomology of \underrightarrow{hocolim}(B_{X}) and \underrightarrow{hocolim}(B_{c}) have isomorphic E_{2}-terms, and hence that the inclusion is a mod p homology equivalence.

It remains to prove that \underleftarrow{lim}^{∗}(F_{0}) = 0. By [JMO, Proposition 1.6], X_{0} contains finitely many G-conjugacy classes. Hence by [JMO, Proposition 5.4] and an appropriate finite filtration of F_{0}, it suffices to prove that Λ^{∗}(N_{G}(P )/P ; H^{∗}(BP ; 𝔽_{p})) = 0 for each p-toral subgroup P in O_{S}^{X}(G) which is not p-centric. For each such P , C_{G}(P )·P /P ≅ C_{G}(P )/Z(P ) is a finite group of order a multiple of p which acts trivially on H^{∗}(EG/P ; 𝔽_{p}) ≅ H^{∗}(BP ; 𝔽_{p}), and hence Λ^{∗}(N_{G}(P )/P ; H^{∗}(BP ; 𝔽_{p})) = 0 by [JMO, Proposition 5.5]. □

We are now ready to construct a rigidification of the homotopy functor B_{p}^{∧}.

Proposition 9.8. Fix a maximal discrete p-toral subgroup S ∈ Syl_{p}(G), and set F = F_{S}(G) for short. Let F^{cs} ⊆ F^{c} be the full subcategory of subgroups P ≤ S which are p-centric in G and snugly embedded, and let O^{cs}(F) ⊆ O^{c}(F) be its orbit category. Then there is a functor

B̂: O^{cs}(F) ⟶ Top

which is a rigidification of the homotopy functor B_{p}^{∧}, and a homotopy equivalence

Φ̂: (\underrightarrow{hocolim}_{O^{cs}(F)}(B̂))_{p}^{∧} ⟶ BG_{p}^{∧}.

Proof. Set S = S ∈ Syl_{p}(G). We will construct orbit categories and functors as indicated in the following diagram:

\xymatrix{O^{cs}(F) \ar[r]^{cl} \ar[dr]_{B_{cs}} & O_{S}^{c}(G) \ar[d]^{_{B_{c}}} & O_{S}^{c}(G) \ar@{->>}[l]_{pr} \ar[dl]^{_{B_{c}}} \\  & _{Top,}}  (1)

together with mod p homology equivalences

\xymatrix{\underrightarrow{hocolim}(B_{cs}) \ar[r]^{^{cl^{∗}}} \ar[dr]_{Φ_{cs}} & \underrightarrow{hocolim}(B_{c}) \ar[r]^{ν}_{≃} \ar[d]^{_{Φ_{c}}} & \underrightarrow{hocolim}(B_{c}) \ar[dl]^{Φ_{c}} \\  & BG.}  (2)

We then set B̂ = (B_{cs})_{p}^{∧}.

The category O_{S}^{c}(G), together with B_{c} and Φ_{c}, were already constructed in Lemma 9.7. We construct O_{S}^{c}(G), B_{c}, and Φ_{c} in Step 1 (and prove the properties we need); and then do the same for O^{cs}(F), B_{cs}, and Φ_{cs} in Step 2.

Step 1: Define O_{S}^{c}(G) by setting

Ob(O_{S}^{c}(G)) = Ob(O_{S}^{c}(G)) = {P ≤ S | P p-toral and p-centric in G},

and

Mor_{O_{S}^{c} (G)}(P , Q) = Q\N_{G}(P , Q)/C_{G}(P ) ≅ Rep_{G}(P , Q).

Let B_{c} be the left homotopy Kan extension of B_{c} along the projection functor. Let Φ_{c} be the composite of Φ_{c} with the standard homotopy equivalence

ν: \underrightarrow{hocolim}_{O_{S}^{c} (G)}(B_{c}) ⟶^{≃} \underrightarrow{hocolim}_{O_{S}^{c}(G)}(B_{c})

of [HV, Proposition 5.5]. Thus Φ_{c} is a mod p homology equivalence by Lemma 9.7, and it remains only to show that B_{c} is a rigidification of B_{p}^{∧} (after p-completion).

This means showing that the natural morphism of functors

B_{c} ⟶ B_{c} ∘ pr

(natural up to homotopy) is a mod p homology equivalence on all objects. By definition, for each P ,

B_{c}(P ) = \underrightarrow{hocolim}_{pr ↓P}(B_{c} ∘ ξ).

Here, pr ↓P is the overcategory whose objects are the morphisms Q ^{α}→ P in O_{S}^{c}(G), and where a morphism from (Q, α) to (R, β) is a morphism φ ∈ Mor_{O_{S}^{c}(G)}(Q, R) such that α = β ∘ pr(φ). Also, ξ is the forgetful functor from pr ↓P to O_{S}^{c}(G).

Consider the spectral sequence

E_{2}^{i,j} ≅ \underleftarrow{lim}^{i}_{pr ↓P}(H^{j}(B_{c} ∘ ξ(−); 𝔽_{p})) ⟹ H^{i+j}(B_{c}(P ); 𝔽_{p}) .  (3)

For each Q in O_{S}^{c}(G), set

K(Q) = Ker[Aut_{O_{S}^{c}(G)}(Q) ⟶^{pr_{P}} Aut_{O_{S}^{c} (G)}(Q)] ≅ C_{G}(Q)/Z(Q).

This is a finite group of order prime to p, and it acts trivially on H^{∗}(EG/Q). Since K(Q) acts trivially on the homology H^{∗}(B_{c} ∘ ξ(Q, α); 𝔽_{p}) = H^{∗}(EG/Q; 𝔽_{p}) for each object (Q, α), this functor factors through the overcategory Id ↓P . The projection of pr ↓P onto Id ↓P satisfies the hypotheses of [BLO1, Lemma 1.3] (in particular, the target category is obtained from the source by dividing out by these automorphism groups K(Q) of order prime to p), and hence

\underleftarrow{lim}^{i}_{pr ↓P}(H^{∗}(B_{c} ∘ ξ(−); 𝔽_{p})) ≅ \underleftarrow{lim}^{i}_{Id ↓P}(H^{∗}(B_{c} ∘ ξ(−); 𝔽_{p})) ≅ \begin{cases}H^{∗}(B_{c}(P ); 𝔽_{p}) & if i = 0 \\ 0 & if i > 0. \end{cases}

Here, the last isomorphism holds since Id ↓P has final object (P , Id). The spectral sequence (3) thus collapses, and hence H^{∗}(B_{c}(P ); 𝔽_{p}) ≅ H^{∗}(B_{c}(P ); 𝔽_{p}) (and the isomorphism is induced by the natural inclusion of B_{c}(P ) into B_{c}(P )).

Step 2: The “closure functor”

O^{cs}(F) ⟶^{cl} O_{S}^{c}(G),

is defined to send P to P. It induces a bijection between isomorphism classes of objects by definition of F^{cs} and Lemma 9.6(b,c), and induces bijections on morphism sets by Lemma 9.4(c). So this is an equivalence of categories.

Set B_{cs} = B_{c} ∘ cl. Since B_{c} is (after p-completion) a rigidification of the homotopy functor P ↦ BP _{p}^{∧} by Step 1, and since BP _{p}^{∧} ≃ BP _{p}^{∧} when P is snugly embedded (Proposition 9.2), B_{cs} is a rigidification of the homotopy functor B_{p}^{∧}: P ↦ BP _{p}^{∧} (again, up to p-completion).

Now let Φ_{cs} be the composite of Φ_{c} with the map

\underrightarrow{hocolim}_{O^{cs}(F)}(B_{cs}) ⟶^{cl_{∗}} \underrightarrow{hocolim}_{O_{S}^{c} (G)}(B_{c})

induced by cl. Then \underrightarrow{hocolim}(B_{c}) ≃ \underrightarrow{hocolim}(B_{cs}) since cl is an equivalence of categories, and thus Φ_{cs} is a mod p homology equivalence since Φ_{c} is by Step 1.

Now set B̂ = (B_{cs})_{p}^{∧} and let

Φ̂: (\underrightarrow{hocolim}_{O^{cs}(F)}(B̂))_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O^{cs}(F)}(B_{cs}))_{p}^{∧} ⟶^{(Φ_{cs})_{p}^{∧}} BG_{p}^{∧}

be the completion of Φ_{cs}. Then B̂ is a rigidification of the homotopy functor B_{p}^{∧} (see Proposition 9.2), and Φ̂ is a homotopy equivalence. □

We also need the following result about snugly embedded subgroups:

Lemma 9.9. For each discrete p-toral subgroup P ≤ G, P^{•} is snugly embedded.

Proof. Fix S ∈ Syl_{p}(G), and set T = S_{0} and p^{m} = exp(S/T). By Definition 3.1, P^{•} = P ·I(Q), where Q = P^{[m]} = ⟨g^{p^{m}} | g ∈ P ⟩ ≥ P_{0}, and I(Q) = T^{C_{W} (Q)}. Then I(Q) contains all elements of p-power order in I(Q) = T^{C_{W} (Q)}, and hence is snugly embedded. Since [P^{•} : I(Q)] is finite, P^{•} is also snugly embedded. □

We are now ready to prove the main result.

Theorem 9.10. Fix a compact Lie group G and a maximal discrete p-toral subgroup S ∈ Syl_{p}(G). Then there exists a centric linking system L_{S}^{c}(G) associated to F_{S}(G) such that (S, F_{S}(G), L_{S}(G)) is a p-local compact group with classifying space |L_{S}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.

Proof. Set F = F_{S}(G) for short; a saturated fusion system by Lemma 9.5. Let F^{cs} ⊆ F^{c} be the full subcategory with objects the set of all P ≤ S which are p-centric and snugly embedded in G, and let O^{cs}(F) be its orbit category. By Lemma 9.9, F^{cs} ⊇ F^{c•}.

By Proposition 9.8, there is a functor

B̂: O^{cs}(F) ⟶ Top

which is a rigidification of the homotopy functor B_{p}^{∧}, and a homotopy equivalence

Φ̂: (\underrightarrow{hocolim}_{O^{cs}(F)}(B̂))_{p}^{∧} ⟶ BG_{p}^{∧}.

By Proposition 5.9, there is a functor B̃: O^{cs}(F) ⟶ Topwhich is a rigidification of the homotopy functor B, and a natural transformation of functors χ: B̃ ⟶ B̂ such that χ(P) is homotopic to the completion map for each P. By Proposition 4.6, there is a centric linking system L_{S}^{cs}(G) associated to F^{cs} whose nerve has the homotopy type of \underrightarrow{hocolim}(B̃), and thus

|L_{S}^{cs}(G)|_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O^{cs}(F)}(B̃))_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O^{cs}(F)}(B̂))_{p}^{∧} ≃ BG_{p}^{∧}.

Now define F^{c} ⟶^{Ψ} F^{cs} by setting Ψ(P) = P ·(P_{0})_{(p)}, where (P_{0})_{(p)} denotes the subgroup of elements of p-power order in the torus P_{0}. By Lemma 9.4(c), for each P ∈ Ob(F^{c}) and Q ∈ Ob(F^{cs}),

Rep_{G}(P, Q) ≅ Rep_{G}(P , Q) ≅ Rep_{G}(Ψ(P), Q),

and thus Ψ is left adjoint to the inclusion. Also, for each P, C_{G}(Ψ(P)) = C_{G}(P) = C_{G}(P), and hence Z(Ψ(P)) = Z(P). So if we define L_{S}^{c}(G) to be the pullback of L_{S}^{cs}(G) and F^{c} over F^{cs}, then it is a centric linking system, and |L_{S}^{c}(G)| ≃ |L_{S}^{cs}(G)|. (Compare this argument with the proof of Proposition 4.5(b).) □

The above construction of the linking system of G has the disadvantage that it seems rather arbitrary. We know, by Theorem 7.4, that there is (up to isomorphism) at most one linking system L_{S}^{c}(G) such that |L_{S}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}, but we would really like to have a more obvious algebraic connection between L_{S}^{c}(G) and the group G itself. We end this section by showing that L_{S}^{c}(G) can, in fact, be obtained as a subquotient of the transporter category of G — although not in a completely canonical way.

Fix a compact Lie group G, and choose S ∈ Syl_{p}(G). The transporter category T_{S}^{c}(G) of G over S is the category whose objects are the subgroups of S that are pcentric in G, and where Mor_{T_{S}^{c}(G)}(P, Q) = N_{G}(P, Q), for each pair of objects P and Q of T_{S}^{c}(G). Let C_{G}: O^{c}(F)^{op} ⟶ Ab be the functor which sends P to its centralizer. For any subfunctor Φ ⊆ C_{G}, T_{S}^{c}(G)/Φ denotes the quotient category with the same objects as T_{S}^{c}(G), and where

Mor_{T_{S}^{c}(G)/Φ}(P, Q) = Mor_{T_{S}^{c}(G)}(P, Q)/Φ(P) = N_{G}(P, Q)/Φ(P).

For example, in this notation, F_{S}^{c}(G) = T_{S}^{c}(G)/C_{G}.

For each P ∈ Ob(F^{c}), there is a central extension

1 ⟶ Z(P) ⟶ C_{G}(P) ⟶ C_{G}(P)/Z(P) ⟶ 1,

where Z(P) is abelian and p-toral and C_{G}(P)/Z(P) is finite of order prime to p (by definition of p-centric). Hence the set of elements of C_{G}(P) of finite order prime to p forms a subgroup, which we denote here ν_{p′}(P). Also, Z(P) and ν_{p′}(P) are both normal subgroups of C_{G}(P), and the quotient group C_{G}(P)/(Z(P) × ν_{p′}(P)) is a ℚvector space. As earlier, we write Z(P) = Z(P), and regard Z, ν_{p′}, and Z × ν_{p′} as subfunctors of C_{G}.

Lemma 9.11. The extension

T_{S}^{c}(G)/(Z × ν_{p′}) ⟶^{pr} F^{c}

is split, by a splitting which sends Out_{P} (P) to P/Z(P) for each object P; and such a splitting is unique up to natural isomorphism of functors.

Proof. For all P, Q ∈ Ob(F^{c}), choose maps

σ_{P,Q}: Hom_{F}(P, Q) ⟶ N_{G}(P, Q)/(Z(P) × ν_{p′}(P)) = Mor_{T_{S}^{c}(G)/(Z×ν_{p′})}(P, Q)

which split the natural projection. This can be done in such a way that for each φ ∈ Hom_{F}(P, Q) and each g ∈ Q, σ_{P,Q}(c_{g} ∘ φ) = [g] ∘ σ_{P,Q}(φ) (define it first on orbit representatives for the action of Inn(Q) and then extend it appropriately). Also, when Q = P, we let σ_{P,P} (Id_{P}) be the class of 1 ∈ N_{G}(P).

The “deviation” of {σ_{P,Q}} from being a functor is a 2-cocycle with values in the functor C_{G}/(Z × ν_{p′}), and the assumption that they commute with the Inn(Q)-actions implies that we get a cocycle over the orbit category O^{c}(F). If, furthermore, this cocycle is a coboundary, then the σ_{P,Q} can be replaced by maps σ′_{P,Q} which define a splitting functor. The obstruction to the existence of such a splitting thus lies in \underleftarrow{lim}^{2}_{O^{c}(F)}(C_{G}/(Z × ν_{p′})). In a similar (but simpler) way, the obstruction to uniqueness is

seen to lie in \underleftarrow{lim}^{1}_{O^{c}(F)}(C_{G}/(Z × ν_{p′})).

We will show that both of these groups vanish, using Lemma 5.7 (and an argument similar to that used to prove Proposition 5.8). Let F be the functor C_{G}/(Z × ν_{p′}). As in the proof of Proposition 5.8, set T = S_{0}, Q = C_{S}(T), and Γ = Out_{F}(Q). Set M = T /(torsion), regarded as a ℚ[Γ]-module. Let

Φ: O_{p}(Γ)^{op} ⟶ ℤ_{(p)}-mod

be the functor Φ(Π) = M^{Π} for all p-subgroups Π ≤ Γ. Then F (P) ≅ Φ(Out_{P} (Q)) (functorially) for all P ≤ S containing Q, and Out_{Q}(P) acts trivially on F (P) ≅ Z(P)/(torsion) for each P. The hypotheses of Lemma 5.7 thus hold, and so

\underleftarrow{lim}^{i}_{O^{c}(F)}(F) ≅ \underleftarrow{lim}^{i}_{O_{p}(Γ)}(Φ)

for all i. Since Φ is a Mackey functor, these groups vanish for all i ≥ 1 by [JM, Proposition 5.14]. □

We are now ready to construct a more explicit linking system L_{S}^{c}(G), and prove it is isomorphic to the one already constructed in Theorem 9.10.

Proposition 9.12. Let G be a compact Lie group, and choose S ∈ Syl_{p}(G). Fix a splitting s of T_{S}^{c}(G)/(Z × ν_{p′}) ⟶^{pr} F^{c}, and define L_{S}^{c}(G) to be the pullback category in the following pullback diagram

\begin{matrix}L_{S}^{c}(G) & \overline{^{s̄} →} & T_{S}^{c}(G)/ν_{p′} \\ F_{S}^{c}(G) & ↓^{π} & \overline{^{s}→} & T_{S}^{c}(G)/(Z × ν_{p′}) . & ↓^{pr} \end{matrix}

Then L_{S}^{c}(G) is a centric linking system associated to F_{S}^{c}(G), and is isomorphic to the centric linking system of Theorem 9.10. In particular, L_{S}^{c}(G) ⟶^{s̄} T_{S}^{c}(G)/ν_{p′} describes the linking system L_{S}^{c}(G) as a subquotient of the transporter catgeory.

Proof. We will first show that the pullback category L_{S}^{c}(G) is a centric linking system associated to F_{S}(G). Since s and pr are the identity on objects, we can as well assume that the pullback category L_{S}^{c}(G) has the same objects, and that s̄ and π are the identity on objects. Then for any pair of objects P, Q ≤ S p-centric in G, we have

Mor_{L_{S}^{c}(G)}(P, Q) =

{ (φ, ψ) | φ ∈ Mor_{F_{S}^{c}(G)}(P, Q), ψ ∈ Mor_{T_{S}^{c}(G)/ν_{p′}}(P, Q), and π(φ) = pr(ψ) }

Now, for each P ≤ S which is p-centric in G, we have P ≤ ℕ_{G}(P)/ν_{p′}(P) and then we can define distinguished homomorphisms

δ_{P} : P ⟶ Aut_{L_{S}^{c}(G)}(P)

by setting δ_{P} (g) = (c_{g}, g). Conditions (A), (B), and (C) in the definition of a centric linking system are easily checked.

Next we will find a map |L_{S}^{c}(G)| ⟶ BG_{p}^{∧} that commutes with the respective natural maps from BS. To do this, we first lift L_{S}^{c}(G) to a subcategory L̃_{S}^{c}(G) of the transporter category T_{S}^{c}(G), defined via the pullback square:

\begin{matrix}L̃_{S}^{c}(G) & \overline{^{incl} →} & T_{S}^{c}(G) \\ L_{S}^{c}(G) & ↓ & \overline{^{s̄}→} & T_{S}^{c}(G)/ν_{p′} & ↓ \end{matrix}

We will then construct the maps in the following commutative diagram:

\xymatrix{ & BS \ar[dl] \ar[d] \ar[dr] \\ |L_{S}^{c}(G)|_{p}^{∧} & |L̃_{S}^{c}(G)|_{p}^{∧} \ar[l]^{≃} \ar[r] & BG_{p}^{∧}.}

We proceed in two steps.

(a) A map |L̃_{S}^{c}(G)| ⟶ |L_{S}^{c}(G)| commuting with the respective natural maps from BS is induced by the functor L̃_{S}^{c}(G) ⟶ L_{S}^{c}(G). We will show that it is a mod p homology equivalence.

By definition of L̃_{S}^{c}(G), for all P, Q ≤ S centric, we have that ν′_{p}(P) acts freely on Mor_{L̃_{S}^{c}(G)}(P, Q) and the orbit set if Mor_{L_{S}^{c}(G)}(P, Q). In particular,

ν′_{p}(P) = Ker [Aut_{L̃_{S}^{c}(G)}(P) ⟶ Aut_{L_{S}^{c}(G)}(P)] .

Recall that ν_{p′}(P) is the subgroup of elements of C_{G}(P) of finite order prime to p. It sits in an extension ν_{p′}(P)^{0} ⟶ ν_{p′}(P) ⟶ C′_{G}(P), where ν_{p′}(P)^{0} is the set of elements of the maximal torus of \overline{Z(P)} of finite order prime to p and C′_{G}(P) = C_{G}(P)/\overline{Z(P)}. Therefore ν_{p′}(P) is locally finite and can be written as union ⋃_{m≥0} ν_{p′}(P)^{m} of finite groups of order prime to p. A generalized version of [BLO1, Lemma 1.3] now applies to the constant functor defined on L_{S}^{c}(G) and the result follows.

In fact, [BLO1, Lemma 1.3] generalizes to allow that the kernels K(−) be countable increasing unions of finite groups of order prime to p. Proving this requires showing, for any such K, that H_{0}(K; −) is an exact functor on the category of ℤ_{(p)}[K]-modules. But if K is the union of a sequence of subgroups K_{1} ≤ K_{2} ≤ ⋯ each of which is finite, then H_{0}(K_{i}; −) is exact for each i, H_{0}(K; M) ≅ \underrightarrow{colim}_{i} H_{0}(K_{i}; M) for each M, and hence H_{0}(K; −) is exact since direct limits of this type are exact.

(b) A map |L_{S}^{c}(G)| ⟶ BG that commutes up to homotopy with the respective maps from BS is defined as follows. Compose the inclusion L̃_{S}^{c}(G) ⟶^{incl} T_{S}^{c}(G) with the functor T_{S}^{c}(G) ⟶ B(G). Here, B(G) is the topological category with one object and the Lie group G as morphisms (and all other categories are discrete), and the functor sends the morphism g ∈ N_{G}(P, Q) to g ∈ G for all objects P, Q of T_{S}^{c}(G). The nerve of B(G) is the topological bar construction BG ≃ BG, and the composite functor induces a map |L̃_{S}^{c}(G)| ⟶ |B(G)| ≃ BG.

Finally, Theorem 9.10 defines the centric linking system of G over S and shows that the p-completed nerve is homotopy equivalent to BG_{p}^{∧}. This combines with the map constructed above, so that Lemma 7.3 implies that the pullback category L_{S}^{c}(G) is isomorphic to the centric linking system of Theorem 9.10, and then, also, that the map |L̃_{S}^{c}(G)| ⟶ BG constructed in step (b) is actually a homotopy equivalence after p-completion. □

10. p-compact groups

A p-compact group is a p-complete version of a finite loop space. As defined by Dwyer and Wilkerson in [DW], a p-compact group is a triple (X, BX, e), where X is a space such that H^{∗}(X; 𝔽_{p}) is finite, BX is a pointed p-complete space, and e: X ⟶ Ω(BX) is a homotopy equivalence. If G is a compact Lie group such that the group of components π_{0}(G) is a finite p-group, then upon setting BĜ = BG_{p}^{∧} and Ĝ = Ω(BĜ), the triple (Ĝ, BĜ, Id) is a p-compact group. For general references on p-compact groups, we refer to the original papers by Dwyer and Wilkerson [DW] and [DW2], and also to the survey article by Jesper Møller [Mø].

When T ≅ (S^{1})^{r} is a torus of rank r, then the p-completion T̂ = Ω(BT _{p}^{∧}) of T is called a p-compact torus of rank r. Both BT̂ ≃ K((ℤ̂_{p})^{r}, 2) and T̂ ≃ K((ℤ̂_{p})^{r}, 1) are Eilenberg-MacLane spaces. A p-compact toral group is a p-compact group (P̂, BP̂, e) such that π_{1}(BP̂) is a p-group, and the identity component of P̂ is a p-compact torus with classifying space the universal cover of BP̂.

If X is either a discrete p-toral group or a p-compact group, and Y is a p-compact group, a homomorphism f: X → Y is by definition a pointed map Bf: BX → BY. Two homomorphisms f, f′: X → Y are conjugate if Bf and Bf′ are freely homotopic; i.e., via a homotopy which need not preserve basepoints. Given a homomorphism f: X → Y, the homotopy fibre of Bf is denoted Y/f(X), or just Y/X if f is understood from the context. With this notation, f is called a monomorphism if H^{∗}(Y/f(X); 𝔽_{p}) is finite. By [DW, Proposition 9.11], a homomorphism f is a monomorphism if and only if H^{∗}(BX; 𝔽_{p}) is a finitely generated H^{∗}(BY; 𝔽_{p})-module via H^{∗}(f; 𝔽_{p}).

If P̂ is an arbitrary p-compact toral group, a discrete approximation to P̂ is a pair (P, f), where P is a discrete p-toral group and Bf: BP → BP̂ induces an isomorphism in mod p cohomology. By [DW, Proposition 6.9], every compact p-toral group has a discrete approximation. Each discrete p-toral group P is a discrete approximation of (P̂, BP̂, Id), where BP̂ = BP _{p}^{∧} and P̂ = Ω(BP̂). Hence every monomorphism f: P →

X from a discrete p-toral group to a p-compact group factors as P → P̂ f̂→ X: a discrete approximation followed by a monomorphism of p-compact groups. Lemma 1.10 says, among other things, that any two discrete approximations of a p-compact toral group are isomorphic.

If f: X → Y is a homomorphism of p-compact groups, the centralizer of f in Y is defined to be the triple (C_{Y} (X, f), BC_{Y} (X, f), Id), where

BC_{Y} (X, f) = Map(BX, BY)_{Bf} and C_{Y} (X, f) = Ω(BC_{Y} (X, f)).

Whenever f is understood, we simply write C_{Y} (X) for C_{Y} (X, f).

A discrete p-toral subgroup of a p-compact group X is a pair (P, f), where P is a discrete p-toral group and P̂ ^{f}→ X is a monomorphism. We write BC_{X}(P, f) = BC_{X}(P̂, f) = Map(BP, BX)_{Bf} and C_{X}(P, f) = C_{X}(P̂, f) for short. By [DW, §§5– 6], C_{X}(P) is a p-compact group, and the homomorphism C_{X}(P) ⟶ X (induced by evaluation at the basepoint of BP) is a monomorphism. The subgroup (P, f) is called central if this monomorphism C_{X}(P) ⟶ X is an equivalence.

Proposition 10.1. Let X be any p-compact group.

(a) X has a maximal discrete p-toral subgroup S ⟶^{f} X. If P ⟶^{u} X is any other discrete p-toral subgroup of X, then Bu ≃ Bf ∘ Bψ for some ψ ∈ Hom(P, S); and (P, u) is maximal if and only if p ∤ χ(X/u(P̂)). Here, Euler characteristics are taken with respect to homology with coefficients in 𝔽_{p}.

(b) The centralizer C_{X}(P, f) of any discrete p-toral subgroup P ⟶^{f} X is again a pcompact group, and a subgroup of X. Also, if P = ⋃_{n=1}^{∞} P_{n}, then BC_{X}(P) ≃ BC_{X}(P_{n}) for n large enough.

(c) A discrete p-toral subgroup P ⟶^{f} X is central if and only if there is a map BP × BX ⟶ BX whose restriction to BP × ∗ is Bf and whose restriction to ∗ × BX is the identity. When this is the case, then P is abelian, and there is a fibration sequence BP _{p}^{∧} ⟶^{f} BX ⟶ B(X/P) where B(X/P) is the classifying space of a p-compact group X/P.

Proof. Point (a) follows mostly from [DW2, Propositions 2.10 & 2.14] together with Lemma 1.10. If (P, u) is not maximal, then since u factors through S, χ(X/u(P̂)) = χ(X/f(Ŝ)) · χ(Ŝ/P̂), and the last factor is a multiple of p.

Point (b) is shown in [DW, Proposition 5.1 & Theorem 6.1]. In point (c), a central subgroup is abelian by [DW2, Theorem 1.2], while the other two claims are shown in [DW, Lemma 8.6 & Proposition 8.3]. □

As in other contexts, the maximal discrete p-toral subgroups of a p-compact group X will be referred to as Sylow p-subgroups of X.

The fusion system of a p-compact group is easily defined: it is just the fusion system of the space BX, as defined in [BLO2, Definition 7.1].

Definition 10.2. For any p-compact group X with Sylow p-subgroup S ⟶^{f} X, let F_{S,f}(X) be the category whose objects are the subgroups of S, and where for P, Q ≤ S,

Mor_{F_{S,f} (X)}(P, Q) = Hom_{X}(P, Q) =^{def} {φ ∈ Hom(P, Q) | Bf|_{BQ} ∘ Bφ ≃ Bf|_{BP} }.

We next want to show that F_{S,f}(X) is saturated. Before doing this, we need to define and study normalizers of discrete p-toral subgroups of p-compact groups. We also need to establish an “adjointness” relation which corresponds to the equivalence (for groups) between homomorphisms Q ⟶ N_{G}(P) and homomorphisms P ⋊ Q ⟶ G.

Fix a p-compact group X and a Sylow p-subgroup f: S ⟶ X. For any subgroup P ≤ S and any discrete p-toral subgroup K ≤ Aut_{X}(P), set

BN_{X}^{K}(P) = (EK ×_{K} BC_{X}(P))_{p}^{∧},

where K acts on BC_{X}(P) = Map(BP, BX)_{Bα} via the action on P. Set N_{X}^{K}(P) = Ω(BN_{X}^{K}(P)). Since the action of K on BP fixes the basepoint, evaluation at the basepoint of BP defines a map

ev : BN_{X}^{K}(P) = (EK ×_{K} Map(BP, BX)_{Bf})_{p}^{∧} ⟶ BX.

If Q is any discrete p-toral group, and ρ ∈ Hom(Q, K), then any homomorphism

EQ ×_{Q} BP ≅ B(P ⋊_{ρ} Q) ⟶ BX

is adjoint to a Q-equivariant map

EQ ⟶ Map(BP, BX)_{f|BP} = BC_{X}(P) ,

where Q acts on BC_{X}(P) via the action on P defined by ρ (and via the trivial action on BX). After taking the Borel construction, this defines a map

BQ ⟶ BN_{X}^{K}(P) = (EK ×_{K} BC_{X}(P))_{p}^{∧} .

In particular, when Q is the group

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

and

B(P ⋊ N_{S}^{K}(P)) ⟶ BS ⟶^{f} BX

is induced by the inclusions and f, then this construction is denoted

Bγ_{P}^{K}: BN_{S}^{K}(P) ⟶ BN_{X}^{K}(P) .

Lemma 10.3. Fix a p-compact group X, a Sylow p-subgroup S ⟶^{f} X, and subgroups P ≤ S and K ≤ Aut_{X}(P) where K is discrete p-toral. Then the induced sequence

BC_{X}(P) ⟶ BN_{X}^{K}(P) ⟶^{τ} BK_{p}^{∧}  (1)

is a fibration sequence. If Q is another discrete p-toral group, then for any homomorphism ρ: Q → N_{X}^{K}(P), there is a fibration sequence

Map(B(P ⋊_{ρ} Q), BX)_{f,ρ} → Map(BQ, BN_{X}^{K}(P))_{Bρ} → Map(BQ, BK_{p}^{∧})_{Bρ} ,

where ρ ∈ Hom(Q, K) is any homomorphism such that Bρ_{p}^{∧} ≃ τ ∘ Bρ, P ⋊_{ρ} Q is the

semidirect product for the action Q ^{ρ}→ K ≤ Aut(P), and the fiber is the space of all maps B(P ⋊_{ρ} Q) → BX which restrict (up to homotopy) to BP ^{Bf|_{P}} → BX and are adjoint to Bρ in the sense described above.

Proof. The action of K on each cohomology group H^{i}(BC_{X}(P); 𝔽_{p}) factors through a finite quotient group of K, thus through the p-group π_{0}(K), and hence is nilpotent. So by [BK, II.5.1], the usual fibration sequence

BC_{X}(P) ⟶ EK ×_{K} BC_{X}(P) ⟶ BK

for the Borel construction over BK is still a fibration sequence after p-completion. Thus (1) is a fibration sequence.

Since [BQ, BK_{p}^{∧}] ≅ [BQ, BK] ≅ Rep(Q, K) (Lemma 1.10), ρ ∈ Hom(Q, K) is uniquely determined up to conjugacy by ρ.

For any fixed homomorphism ρ: Q → K, (1) induces a fibration sequence

BC_{X}(P)^{hQ} → Map(BQ, BN_{X}^{K}(P))ρ̃ → Map(BQ, BK_{p}^{∧})_{Bρ_{p}^{∧}} ,

where ρ̃ denotes the set of connected components of Map(BQ, BN_{X}^{K}(P)) that map into Map(BQ, BK_{p}^{∧})_{Bρ_{p}^{∧}}; and (if ρ̃ ≠ ∅) BC_{X}(P)^{hQ} is the homotopy fixed point set of the

action of Q induced by the pullback of (1) over BQ ⟶_{≃τ ∘Bρ}^{Bρ_{p}^{∧}} BK_{p}^{∧}. We need to identify

this action of Q on BC_{X}(P) with that induced by the action of Q on P via ρ. This follows by comparing the fibrations

\begin{aligned}BC_{X}(P) & \overline{→} & EK ×_{K} BC_{X}(P) & \overline{→} & BK \\ BC_{X}(P) & ‖ & \overline{→} & BN_{X}^{K}(P) & ↓ & \overline{→} & BK_{p}^{∧}, & ↓ \end{aligned}

since the action of Q on BC_{X}(P) induced by BQ ⟶^{Bρ_{p}^{∧}} BK_{p}^{∧} in the bottom fibration coincides with that induced by ρ in the fibration sequence of the top row. By construction, the action of K on BC_{X}(P) induced by the top row is just the action of K on BC_{X}(P) = Map(BP, BX)_{Bf|_{P}} induced by the original action of K on P.

Now set f_{P} = f|_{P} : P ⟶ X for short. We can identify

BC_{X}(P)^{hQ} = (Map(BP, BX)_{Bf_{P}} )^{hQ} = Map_{Q}(EQ, Map(BP, BX)_{Bf_{P}})

≃ Map_{Q}(BP × EQ, BX)f̃ ≃ Map(BP ×_{Q} EQ, BX)f̃ ≃ Map(B(P ⋊_{ρ} Q), BX)f̃,

where f̃ is the set of connected components of maps whose restriction to BP is homotopic to Bf_{P}. Here, BP ×_{Q} EQ ≃ B(P ⋊_{ρ} Q) because the action of Q in BP is induced from the action described above of Aut(P) on BP, and this has a fixed point, providing a section of the fibration

BP → BP ×_{Aut(P)} E Aut(P) → B Aut(P) .

Finally, upon restricting to one component of Map(BQ, BN_{X}^{K}(P))ρ̃, we obtain the fibration in the statement of the proposition. □

Notice that in the particular case where K = 1, Map(BQ, BK) is contractible, and the fibration of Lemma 10.3 reduces to the equivalence

Map(BP × BQ, BX) ≃ Map(BQ, Map(BP, BX)).

Proposition 10.4. Let X be a p-compact group, let S ⟶^{f} X be a Sylow p-subgroup, and set F = F_{S,f}(X) for short. Fix a subgroup P ≤ S, and a discrete p-toral group of automorphisms K ≤ Aut_{F}(P). Then the following hold.

(a) BN_{X}^{K}(P) is the classifying space of a p-compact group which we denote N_{X}^{K}(P),

and

N_{S}^{K}(P) ⟶^{γ_{P}^{K}} N_{X}^{K}(P)

is a discrete p-toral subgroup. Furthermore, the square

\begin{matrix}BN_{S}^{K}(P) & \overline{→^{Bγ_{P}^{K}}} & BN_{X}^{K}(P) \\ BS & ↓^{B incl} & \overline{^{f} →} & BX & ↓^{ev} \end{matrix}  (1)

commutes up to pointed homotopy.

(b) There is φ ∈ Hom_{F}(P, S) such that φ(P) is fully φKφ^{−1}-normalized in F.

(c) P is fully K-normalized in F if and only if N_{S}^{K}(P) is a Sylow p-subgroup of N_{X}^{K}(P).

Proof. (a) By Lemma 10.3, BC_{X}(P) ⟶ BN_{X}^{K}(P) ⟶ BK_{p}^{∧} is a fibration sequence. The loop spaces of the fiber and base of this sequence have finite mod p cohomology, so the same is true of N_{X}^{K}(P) =^{def} Ω(BN_{X}^{K}(P)). Thus N_{X}^{K}(P) is a p-compact group.

The map

BN_{S}^{K}(P) ⟶^{Bγ_{P}^{K}} BN_{X}^{K}(P) = (EK ×_{K} Map(BP, BX)_{Bf|_{P}} )_{p}^{∧}

is defined to be adjoint to the composite

B(P ⋊ N_{S}^{K}(P)) ⟶^{B(incl ⋊ incl)} BS ⟶^{f} BX.  (2)

Hence the composite of Bγ_{P}^{K} with the evaluation map from Map(BP, BX)_{Bf_{P}} to BX (evaluation at the basepoint of BP) is equal to the restriction of (2) to BN_{S}^{K}(P). This proves that (1) is commutative.

If γ_{P}^{K} factored through a quotient group N_{S}^{K}(P)/R for some R ≠ 1, then the restriction of Bf: BS ⟶ BX to BR would be homotopically trivial, but this cannot happen. So if S_{0} ⟶^{f_{0}} N_{S}^{K}(P) is a maximal discrete p-toral subgroup, then γ_{P}^{K} factors through a monomorphism from N_{S}^{K}(P) to S_{0} (Proposition 10.1(a)), and thus γ_{P}^{K} is itself a monomorphism.

(b,c) By Lemma 10.3, any discrete p-toral subgroup BQ ⟶ BN_{X}^{K}(P) lifts to a map B(P ⋊ Q) ⟶ BX which factors through a homomorphism P ⋊ Q ⟶^{β} S. Set P′ = β(P ⋊ 1) ≤ S, φ = β|_{P ⋊1} ∈ Iso_{F}(P, P′), and K′ = φKφ^{−1} ≤ Aut_{X}(P′). Then β(Q) ≤ N_{S}^{K′}(P′), and β|_{1⋊Q} is injective since otherwise BQ ⟶ BN_{X}^{K}(P) would factor through a quotient group of Q and hence wouldn’t be a subgroup. Thus, the largest possible K-normalizer N_{S}^{K′}(P′) occurs when it is a Sylow p-subgroup of N_{X}^{K}(P), so P′ is fully K′-normalized in F, and P is fully K-normalized if and only if N_{S}^{K}(P) is a Sylow p-subgroup of N_{X}^{K}(P). □

We are now ready to show that F_{S,f}(X) is saturated.

Proposition 10.5. Let X be a p-compact group, and let S ⟶^{f} X be a Sylow psubgroup. Then F_{S,f}(X) is a saturated fusion system over S.

Proof. Write F = F_{S,f}(X) for short.

Proof of (I): Fix a subgroup P ≤ S which is fully normalized in F. Let K ≤ Aut_{F}(P) be such that K ≥ Aut_{S}(P) and K/ Inn(P) ∈ Syl_{p}(Out_{F}(P)). Then P is fully Knormalized, since it is fully normalized and N_{S}^{K}(P) = N_{S}(P). So by Proposition 10.4(c), N_{S}^{K}(P) is a Sylow p-subgroup of N_{X}^{K}(P).

Set K′ = Aut_{S}(P) for short, and consider the following commutative diagram of connected spaces:

\begin{aligned}BC_{S}(P)_{p}^{∧} & → & BN_{S}^{K}(P)_{p}^{∧} & → & BK′_{p}^{∧} \\ BC_{X}(P) & ↓^{f_{1}} & \overline{→} & BN_{X}^{K}(P)_{p}^{∧} & ↓^{f_{2}} & \overline{→} & BK_{p}^{∧}. & ↓^{f_{3}} \end{aligned}

Let F_{i} be the homotopy fiber of f_{i} (for i = 1, 2, 3). Each row is a fibration sequence before p-completion; and the actions of K′ on H^{∗}(BC_{S}(P); 𝔽_{p}) and of K on H^{∗}(BC_{X}(P); 𝔽_{p}) factor through finite p-group quotients and hence are nilpotent. So the rows are still fibration sequences after p-completion by [BK, II.5.1].

Each of the maps f_{i} is a monomorphism of p-compact groups, and hence H^{∗}(F_{i}; 𝔽_{p}) is finite for each i. Since BC_{X}(P) is connected, π_{1}(BN_{X}^{K}(P)_{p}^{∧}) surjects onto π_{1}(BK_{p}^{∧}) ≅ π_{0}(K), and hence π_{0}(F_{2}) surjects onto π_{0}(F_{3}). Thus F_{1} is the homotopy fiber of the map F_{2} ⟶ F_{3}, and so χ(F_{2}) = χ(F_{1})·χ(F_{3}).

Since N_{S}^{K}(P) ∈ Syl_{p}(N_{X}^{K}(P)), χ(F_{2}) is prime to p by Proposition 10.1(a). Thus χ(F_{1}) and χ(F_{3}) are both prime to p, and hence C_{S}(P) ∈ Syl_{p}(C_{X}(P)) and (since K is discrete p-toral) K′ = K. Hence Out_{S}(P) = K/ Inn(P) ∈ Syl_{p}(Out_{F}(P)). Also, since C_{S}(P) ∈ Syl_{p}(C_{X}(P)), we can again apply Proposition 10.4(c) (this time with K = 1), to show that P is fully centralized in F. This finishes the proof of (I).

Proof of (II): Fix P ≤ S and φ ∈ Hom_{F}(P, S), and set P′ = φ(P). Assume that P′ is fully centralized in F. Set

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

and set K = Aut_{N_{φ}}(P), K′ = φKφ^{−1} ≤ Aut_{S}(P′), and N′_{φ} = N_{S}^{K′}(P′). Then P′ is fully K′-normalized in F, since it is fully centralized and K′ ≤ Aut_{S}(P′). Consider the following diagram:

BN_{φ}  \overline{^{Bγ_{P}^{K}} →} BN_{X}^{K}(P) \overline{^{proj} →} BK_{p}^{∧}

BN′_{φ}^{^{Bφ} . . . . . . . .}  \overline{^{Bγ_{P′}^{K′}} →} BN_{X}^{K′}(P′) ↓^{≃ Ec_{φ}×(φ^{∗})^{−1}} \overline{^{proj} →} BK′_{p}^{∧}. ↓^{≅ Bc_{φ}}

(1)

Here, the composites in the two rows are induced by the epimorphisms N_{φ} −↠^{ω} K

and N′_{φ} −↠^{ω′} K′ (exactly, not just up to homotopy). By Proposition 10.4(c), N′_{φ} is a Sylow p-subgroup of N_{X}^{K′}(P′) ≃ N_{X}^{K}(P), and hence there is a homomorphism φ ∈ Hom(N_{φ}, N′_{φ}) which makes the left hand square commute up to homotopy.

Since [BN_{φ}, BK′_{p}^{∧}] ≅ Rep(N_{φ}, K′) (Lemma 1.10), the homotopy commutativity of (1) implies that there is g ∈ K′ such that c_{φ} ∘ ω = c_{g} ∘ ω′ ∘ φ. Since ω′ is onto, there is g ∈ N′_{φ} such that ω′(g) = g; and upon replacing φ by c_{g} ∘ φ we can assume that c_{φ} ∘ ω = ω′ ∘ φ.

Fix a homotopy H which makes the left hand square in (1) commute. Then the composite proj ∘H is a loop in Map(BN_{φ}, BK′_{p}^{∧}) based at B(ω′ ∘φ), and this component has the homotopy type of BC_{K′}(ω′ ∘ φ(N_{φ}))_{p}^{∧} by Lemma 1.10 again. So after replacing φ by c_{g′} ∘ φ for some appropriate g′ ∈ N′_{φ}, and after modifying H using the homotopy from Bφ to B(c_{g′} ∘φ) determined by g′, we can arrange that proj ∘H is nullhomotopic in Map(BN_{φ}, BK′_{p}^{∧}). We can now apply Lemma 10.3, to show that the following diagram commutes up to homotopy:

B(P ⋊ N_{φ}) \overline{^{incl ⋊ incl}→} BS  \overline{^{Bf} →} BX

B(P′ ⋊ N′_{φ}) ↓^{B(φ⋊φ)} \overline{^{incl ⋊ incl}→} BS  \overline{^{Bf} →} BX. ‖

In particular, φ ∈ Hom_{F}(N_{φ}, N′_{φ}). Also, the two homomorphisms from P ⋊ N_{φ} to S induced by inclusions and by φ ⋊ φ have the same kernel {(g, g^{−1}) | g ∈ P }, and this implies that φ = φ|_{P}.

Proof of (III): Fix P = ⋃_{n=1}^{∞} P_{n}, where P_{1} ≤ P_{2} ≤ ⋯ is an increasing sequence of subgroups. Let φ ∈ Inj(P, S) be such that φ|_{P_{n}} ∈ Hom_{F}(P_{n}, S) for all n. Thus for each n, (Bf ∘ Bφ)|_{BP_{n}} ≃ Bf|_{BP_{n}}. Also, by Proposition 10.1(b), Map(BP_{n}, BX)_{Bf|BP_{n}} ≃ Map(BP, BX)_{Bf|BP} for n sufficiently large. We can thus choose homotopies H_{n} from (Bf ∘ Bφ)|_{BP_{n}} to Bf|_{BP_{n}} such that H_{n} = H_{n+1}|_{BP_{n}×I}, and set H = ⋃ H_{n}. This shows that Bf ∘ Bφ ≃ Bf|_{BP}, and hence that φ ∈ Hom_{F}(P, S). □

In [CLN], a p-compact toral subgroup P of a p-compact group X is called centric if the inclusion map BP ⟶^{Bf} BX is a centric map; i.e., if

Map(BP, BP)_{Id} ⟶^{Bf ∘−} Map(BP, BX)_{Bf}

is an equivalence. We must check that this is equivalent to the concept of F-centricity (applied to discrete p-toral subgroups) used here.

Lemma 10.6. Let X be a p-compact group, and let S ⟶^{f} X be a Sylow p-subgroup. Set F = F_{S,f}(X). Then for any subgroup P ≤ S, Bf|_{BP} : BP _{p}^{∧} ⟶ BX is a centric map if and only if P is F-centric.

Proof. Assume P is F-centric. In particular, P is fully centralized in F. By Proposition 10.4(a,c) (applied with K = 1), C_{X}(P) is a p-compact group with Sylow p-subgroup C_{S}(P) = Z(P). Also, composition defines a map

Map(BP, BP)_{Id}_{≃BZ(P)} × Map(BP, BX)_{Bf|BP} ⟶ Map(BP, BX)_{Bf|BP} .

So by Proposition 10.1(c), Z(P) is central in C_{X}(P), and there is a p-compact group C_{X}(P)/Z(P) whose Euler characteristic is prime to p and a fibration sequence

BZ(P)_{p}^{∧} ⟶ BC_{X}(P) ⟶ B(C_{X}(P)/Z(P)) .

Then C_{X}(P)/Z(P) must be trivial, so B(C_{X}(P)/Z(P)) ≃ ∗,

BZ(P)_{p}^{∧} ≃ Map(BP, BX)_{Bf|BP} ,

and hence Bf|_{BP} is a centric map.

Conversely, if Bf|_{BP} is a centric map, then BC_{X}(P) ≃ BZ(P) by Lemma 1.10, so C_{S}(P′) = Z(P′) for all P′ ≤ S which is F-conjugate to P, and P is F-centric. □

It remains to construct a linking system associated to F_{S,f}(X) whose p-completed nerve has the homotopy type of BX. This will be done using Proposition 4.6, together with a construction by Castellana, Levi, and Notbohm in [CLN].

Theorem 10.7. Let X be a p-compact group, and let S ⟶^{f} X be a Sylow p-subgroup. Set F = F_{S,f}(X) =^{def} F_{S,Bf}(BX) for short. Then there is a centric linking system L = L_{S,f}^{c}(X) associated to F such that

|L_{S,f}^{c}(X)|_{p}^{∧} ≃ BX .

In other words, (S, F, L) is a p-local compact group whose classifying space is homotopy equivalent to BX.

Proof. By Proposition 10.5, the fusion system F is saturated.

In [CLN], the authors define a category O^{c}(F)_{+} by adding a final object to O^{c}(F); i.e., the category O^{c}(F)_{+} consists of O^{c}(F) together with an additional object ∗, and a unique morphism from each object in O^{c}(F) to ∗. (The actual category they work with contains the same objects as O^{c}(F) by Lemma 10.6.) They then define a homotopy functor

B_{+} : O^{c}(F)_{+} ⟶ hoTop

by setting B_{+}(P) = BP _{p}^{∧} for all F-centric P ≤ S, and B_{+}(∗) = BX (with the obvious maps between them). By Lemmas 1.10 and 10.6, this is a centric diagram in the sense of [DK]. Since O^{c}(F) has a final object, the Dwyer-Kan obstructions to rigidifying B_{+} to a functor to Topall vanish [DK] (see also Corollary A.4), and so this functor can be lifted. In particular, this restricts to a functor B̂ from O^{c}(F) to Top, together with a map from \underrightarrow{hocolim}(B̂) to BX. (See also Corollary A.5.)

By [CLN, Theorem 8.5], this map from \underrightarrow{hocolim}(B̂) to BX induces a homotopy equivalence

(\underrightarrow{hocolim}(B̂))_{p}^{∧} ≃ BX

(the collection of F-centric subgroups of X is “subgroup ample”). Hence by Proposition 5.9, there is a functor B̃: O^{c}(F) ⟶ Top which is a rigidification of the homotopy functor B, and a natural transformation of functors χ: B̃ ⟶ B̂ which is the completion map on each object. Proposition 4.6 now applies to show that there is a centric linking system L = L_{S,f}^{c}(X) associated to F such that

|L|_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O^{c}(F)}(B̃))_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O^{c}(F)}(B̂))_{p}^{∧} ≃ BX_{p}^{∧}.  □

In fact, one can show that there is a unique centric linking system L associated to F_{S,f}(X) such that |L|_{p}^{∧} ≃ BX_{p}^{∧}, but we leave that for a later paper.

Appendix A. Lifting diagrams in the homotopy category

As elsewhere in the paper, we let Top denote the category of spaces, and hoTop the homotopy category. Let ho: Top⟶ hoTop be the forgetful functor. When C is a small category, a functor F from C to Top or hoTop is called centric if for each morphism φ ∈ Mor_{C}(c, d), the natural map

Map(F (c), F (c))_{Id} ⟶_{≃}^{F (φ)∘−} Map(F (c), F (d))_{F (φ)}

is a homotopy equivalence. In [DK], Dwyer and Kan identify the obstructions to rigidifying a centric functor F : C ⟶ hoTop to a functor F̃ : C ⟶ Top; and also describe the space of such rigidifications. We prove here a relative version of their result which is needed in Section 5. This result can, in fact, be derived from the main theorem in [DK], but that argument is so indirect that we find it helpful to give a more direct, and also more elementary, proof.

More precisely, a rigidification of F is a functor F̃: C ⟶ Top, together with a natural transformation of functors F ⟶ ho ∘ F̃ which is a homotopy equivalence on each object. Two rigidifications F̃ and F̃′ are equivalent if there is a third rigidification F̃′′, together with natural transformations of functors F̃ ⟶ F̃′′ ⟵ F̃′ which commute with the natural transformations from F, and hence which define homotopy equivalences F̃(c) ≃ F̃′′(c) ≃ F̃′(c) for each c ∈ Ob(C). This is easily seen to be an equivalence relation by taking pushouts.

The main idea here is to construct a rigidification of F: C ⟶ hoTop by first constructing a space which looks like a “homotopy colimit” of F, and then show that this homotopy colimit automatically induces a rigidification F̃. Recall that the nerve of a small category C is defined by setting

BC = (∐_{n≥0} ∐_{x_{0}→···→x_{n}} Δ^{n})/∼;

and that the homotopy colimit of any functor F : C ⟶ Topis the space

\underrightarrow{hocolim}_{C}(F) = (∐_{n≥0} ∐_{x_{0}→···→x_{n}} F (x_{0}) × Δ^{n})/∼.

Here, in both cases, we divide out by the usual face and degeneracy identifications. Let p_{F} : \underrightarrow{hocolim}(F) ⟶ BC be the projection. It will be convenient to refer to the “skeleta” of the homotopy colimit: let \underrightarrow{hocolim}^{(n)}(F) denote the union of the F (x_{0})×Δ^{i} for all i ≤ n (and all x_{0} → ⋯ → x_{i} in C).

Now assume that F : C ⟶ hoTop is a functor to the homotopy category instead. We assume that for each f : x ⟶ y in C, a concrete map F (f) : F (x) ⟶ F (y) has been chosen. The 1-skeleton \underrightarrow{hocolim}^{(1)}(F) is defined in the same way as before: it is the union of the mapping cylinders of the F (f) taken over all f ∈ Mor(C). It is also straightforward to define the 2-skeletion; but it is convenient at this stage to replace Δ^{2} by a truncated triangle Δ_{t}^{2}. More preciesely, for each sequence x_{0} ⟶^{f} x_{1} ⟶^{g} x_{2}, F (x_{0}) × Δ_{t}^{2} is attached to \underrightarrow{hocolim}^{(1)}(F) via the following picture: where the small segment at the top is mapped using a homotopy between F (g ∘ f) and F (g) ∘ F (f).

[figure: F (g) ∘ F (f); F (g ∘ f); Id; F (f); Id; Id; F (f)]

The first obstructions arise when constructing the 3-skeleton. For each x_{0} → ⋯ → x_{3}, we want to attach F (x_{0}) × Δ_{t}^{3} to \underrightarrow{hocolim}^{(2)}(F), where Δ_{t}^{3} (the “truncated 3simplex”) is the cone over Δ_{t}^{2} with its vertex cut off. The attachment map is easily defined, except on the “top” surface resulting from truncating the cone vertex. Hence, the obstruction to defining the attachment map lies in the group

π_{1}(Map(F (x_{0}), F (x_{3})) , F (x_{0} → x_{3})).

At this point, it becomes necessary to switch from the intuitive picture to formal definitions, by replacing the truncated simplices Δ_{t}^{n} by cubes I^{n}, and regarding simplices as cubes modulo certain identifications. This correspondence will be made explicit later.

Let Δ be the simplicial category, with objects the sets [n] = {0, …, n} for n ≥ 0, and morphisms the order preserving maps between sets. We let ∂_{i} ∈ Mor_{Δ}([n − 1], [n]) denote the i-th face map (with image [n]∖{i}). Define a functor

I^{•}: Δ ⟶ Top

by setting I^{•}([n]) = I^{n+1} (where I is the closed interval I = [0, 1]), and

I^{•}(σ)(t_{0}, …, t_{n}) = ( ∏_{i∈σ^{−1}(0)} t_{i} , ∏_{i∈σ^{−1}(1)} t_{i} , …, ∏_{i∈σ^{−1}(m)} t_{i})

for σ ∈ Mor_{Δ_{j}}([n], [m]). Here, the product over the empty set is always 1. Let Δ_{1} ⊆ Δ_{0} ⊆ Δ be the subcategories with the same objects, where

Mor_{Δ_{0}}([m], [n]) = {σ ∈ Mor_{Δ}([m], [n]) | σ(0) = 0}
Mor_{Δ_{1}}([m], [n]) = {σ ∈ Mor_{Δ}([m], [n]) | σ(0) = 0, σ(m) = n}.

For each n ≥ 0, let I_{1}^{•}([n]) ⊆ I_{0}^{•}([n]) ⊆ I^{•}([n]) be the subspaces

I_{0}^{•}([n]) = {(0, x_{1}, …, x_{n}) ∈ I^{•}([n])} ≅ I^{n}
I_{1}^{•}([n]) = {(0, x_{1}, …, x_{n−1}, 0) ∈ I^{•}([n])} ≅ I^{n−1} .

Then for each j = 0, 1, I^{•}|_{Δ_{j}} restricts to a subfunctor I_{j}^{•}: Δ_{j} ⟶ Top.

Throughout the rest of this section, C denotes a fixed small category. For each n ≥ 0, define Mor^{n} = Mor^{n}(C) to be the set of all sequences c_{0} → c_{1} → ⋯ → c_{n} of composable morphisms in C. In particular, Mor^{0}(C) = Ob(C) and Mor^{1}(C) = Mor(C). For σ ∈ Mor_{Δ}([n], [m]), σ^{∗}: Mor^{m}(C) ⟶ Mor^{n}(C) is defined as usual by taking compositions, inserting identity morphisms, and (if σ ∉ Mor(Δ_{1})) dropping morphisms at one or both ends of the chain. For example, ∂_{i}^{∗} (from Mor^{n}(C) to Mor^{n−1}(C)) is defined by composing two morphisms in the sequence, or by dropping one of them if i = 0 or n. Also, for each ξ = (c_{0} → ⋯ → c_{n}) in Mor^{n}(C) and each 0 ≤ i ≤ j ≤ n, we write

ξ_{ij} = (c_{i} → ⋯ → c_{j}) ∈ Mor^{j−i}(C),

let ξ^{∘}_{ij} ∈ Mor_{C}(c_{i}, c_{j}) denote the composite of this sequence of maps, and set ξ^{∘} = ξ^{∘}_{0n}.

In order to simplify the notation in what follows, whenever F: C ⟶ hoTop is a functor and φ ∈ Mor(C), we let F (φ) denote some chosen representative of the homotopy class of maps defined by F, not the homotopy class itself.

Definition A.1. Fix a functor F: C ⟶ hoTop. An Rg_{∞}-structure F on F consists

of a space F (c) and a homotopy equivalence F (c) ⟶_{≃}^{ν(c)} F (c), defined for each c ∈ Ob(C); together with maps

F (ξ): I^{n−1} = I_{1}^{•}([n]) ⟶ Map(F (c_{0}), F (c_{n})),

defined for each n ≥ 1 and each ξ = (c_{0} → c_{1} → ⋯ → c_{n}) ∈ Mor^{n}(C), which satisfy the following relations.

(a) For all φ ∈ Mor_{C}(c_{0}, c_{1}), F (c_{0} ⟶^{φ} c_{1}) ∘ ν(c_{0}) ≃ ν(c_{1}) ∘ F (φ).

(b) For all m, n ≥ 1, σ ∈ Mor_{Δ_{1}}([m], [n]), ξ ∈ Mor^{n}(C), and t ∈ I^{m−1},

F (σ^{∗}ξ)(t) = F (ξ)(I_{1}^{•}(σ)(t)).

(c) For all n ≥ 2, ξ ∈ Mor^{n}(C), 1 ≤ i ≤ n − 1, t_{1} ∈ I^{i−1}, and t_{2} ∈ I^{n−i−1},

F (ξ)(t_{1}, 0, t_{2}) = F (ξ_{in})(t_{2}) ∘ F (ξ_{0i})(t_{1}).

Schematically, relation (b) can be described via the following commutative diagram:

\xymatrix{I^{m−1} \ar[rr]^{I_{1}^{•}(σ)} \ar[dr]^{F (σ^{∗}(ξ))} &  & I^{n−1} \ar[dl]_{F (ξ)} \\  & Map(F (c_{0}), F (c_{n}))}

while relation (c) can be described via the diagram:

I^{i−1} × I^{n−i−1}  \overline{^{t_{1},t_{2} ↦ (t_{1},0,t_{2})}  →} I^{n−1}

Map(F (c_{0}), F (c_{i})) × Map(F (c_{i}), F (c_{n}))

↓^{F (ξ_{0i})×F (ξ_{in})}

composition→ Map(F (c_{0}), F (c_{n})) .

↓^{F (ξ)}

These relations are more easily understood when one thinks of I_{1}^{•}([n]) ≅ I^{n−1} as the space of all (t_{0}, …, t_{n}) ∈ I^{•}([n]) ≅ I^{n+1} such that t_{0} = 0 = t_{n}. Each coordinate in I^{•}([n]) corresponds to one of the objects in the chain ξ = (c_{0} → ⋯ → c_{n}). When t_{i} = 1 for some 0 < i < n, t_{i} and c_{i} can be removed, giving the face relation

F (ξ)(t_{1}, …, t_{i−1}, 1, t_{i+1}, …, t_{n−1}) = F (∂_{i}ξ)(t_{1}, …, t_{i−1}, t_{i+1}, …, t_{n−1}).

When t_{i} = 0 for 0 < i < n, then F (ξ)(t) can be split as a composite at the object c_{i} (relation (c)). If one of the morphisms in ξ is an identity, then one can remove it and multiply the coordinates corresponding to its two objects.

For instance, when m = 2 and n = 1 (and σ is one of the surjections), condition (b) says that

F (c_{0} ⟶^{Id} c_{0} ⟶^{φ} c_{1}) and F (c_{0} ⟶^{φ} c_{1} ⟶^{Id} c_{1})

are both the constant maps to F (φ). In particular, F (φ) ∘ F (Id_{c_{0}}) = F (φ) = F (Id_{c_{1}}) ∘ F (φ).

When n = 2, condition (c) says that F (c_{0} ⟶^{φ} c_{1} ⟶^{ψ} c_{2}) is a homotopy from F (ψ) ∘ F (φ) to F (ψ ∘ φ). More generally, when ξ ∈ Mor^{n}(C) for n ≥ 2,

F (ξ)(0, …, 0) = F (ξ_{n−1,n}) ∘ ⋯ ∘ F (ξ_{12}) ∘ F (ξ_{01}) and F (ξ)(1, …, 1) = F (ξ^{∘}).

At the other vertices of I^{n−1}, we get all of the other possible composites of the F (ξ^{∘}_{ij}). An Rg_{∞}-structure on F is thus a collection of higher homotopies connecting given homotopies F (ψ) ∘ F (φ) ≃ F (ψ ∘ φ).

From this point of view, one sees that when defining an Rg_{∞}-structure on F, it suffices to define it on all nondegenerate sequences ξ ∈ Mor^{n}(C) (i.e., those containing no identity morphisms), inductively for increasing n, where at each step F (ξ) has already been defined on ∂I^{n−1} and must be extended in some way to I^{n−1}. The starting point can be any choice of maps F (φ), for all φ ∈ Mor(C), in the given homotopy class determined by F (φ), such that

F (φ) ∘ F (Id_{c}) = F (φ) = F (Id_{d}) ∘ F (φ)

for each morphism φ ∈ Mor_{C}(c, d) in C.

If F and F′ are both Rg_{∞}-structures on F, then a morphism Θ: F ⟶ F′ consists of homotopy equivalences θ(c): F (c) ⟶^{≃} F′(c) (for each c ∈ Ob(C)) such that θ(c) ∘ ν(c) ≃ ν′(c), and such that for each ξ = (c_{0} → ⋯ → c_{n}) and each t ∈ I^{n−1},

θ(c_{n}) ∘ F (ξ)(t) = F′(ξ)(t) ∘ θ(c_{0}) ∈ Map(F (c_{0}, F′(c_{n})).

Two Rg_{∞}-structures on F are equivalent if there is a third to which they both have morphisms. One easily sees that (homotopy) pushouts exist for morphisms of Rg_{∞}- structures on F, and hence that this defines an equivalence relation among Rg_{∞}- structures.

For any given Rg_{∞}-structure F on F : C ⟶ hoTop, we define its “homotopy colimit” 𝔖𝔭(F) to be the space

𝔖𝔭(F) = (∐_{n≥0} ∐_{c_{0}→···→c_{n}} F (c_{0}) × I^{n})/∼  (I^{n} = I_{0}^{•}([n]))

where the following identifications are made for each n ≥ 1, each ξ = (c_{0} → ⋯ → c_{n}) ∈ Mor^{n}(C), and each x ∈ F (c_{0}):

(x; I_{0}^{•}(σ)(t))_{[ξ]} ∼ (x; t)_{[σ^{∗}ξ]}  (σ ∈ Mor_{Δ_{0}}([m], [n]), t ∈ I^{m})
(x; (t_{1}, 0, t_{2}))_{[ξ]} ∼ (F (ξ_{0i})(t_{1})(x); t_{2})_{[ξ_{in}]}  (1 ≤ i ≤ n, t_{1} ∈ I^{i−1}, t_{2} ∈ I^{n−i}.)

For example, in the case of a sequence ξ = (c_{0} ⟶^{f} c_{1} ⟶^{g} c_{2}) in Mor^{2}(C), the corresponding square I_{0}^{•}([2]) is attached to the 1-skeleton in the following way:

[figure: F (g) ∘ F (f); F (g ∘ f); Id; F (f); Id; Id; F (f); (1,1); (0,1); (1,0); (0,0); F (ξ); F (c_{0})×I^{2}; ⟶; F (c_{0}); F (c_{0})×I; F (c_{0})×I; F (c_{1}); F (c_{1})×I; F (c_{2})]

The labels in the first picture describe the maps by which a vertex F (c_{0}) or an edge F (c_{0})×I is attached to the space represented by the second picture. Thus the trapezoid in the earlier picture has now been replaced by a square.

One way to understand these relations and their connection with those in Definition A.1 is to think of I_{0}^{•}([n]) ≅ I^{n} as the subspace of all (n + 1)-tuples (0, t_{1}, …, t_{n}) in I^{•}([n]) ≅ I^{n+1}. For ξ ∈ Mor^{n}(C), each coordinate in I^{•}([n]) corresponds to one of the objects in the chain ξ = (c_{0} → ⋯ → c_{n}). When t_{i} = 1 for 0 < i ≤ n, t_{i} and c_{i} can be removed, giving the face relation

(x; (t_{1}, …, t_{i−1}, 1, t_{i+1}, …, t_{n}))_{[ξ]} ∼ (x; (t_{1}, …, t_{i−1}, t_{i+1}, …, t_{n}))_{[∂_{i}ξ]}.

When t_{i} = 0 for 0 < i ≤ n, then ξ splits as a composite at the object c_{i}, we apply F (ξ_{0i})(t_{1}, …, t_{i−1}) to x, and get the second of the above relations. If one of the morphisms in ξ is an identity, then we get a degeneracy relation by removing it and multiplying the two corresponding coordinates.

Consider the maps σ_{n} : I^{n} ⟶ Δ^{n} defined by

σ_{n}(t_{1}, …, t_{n}) = (t_{1}t_{2} ⋯ t_{n}, (1 − t_{1})t_{2} ⋯ t_{n}, (1 − t_{2})t_{3} ⋯ t_{n}, …, (1 − t_{n−1})t_{n}, 1 − t_{n}).

When F is a functor to Topand F is the corresponding locally constant Rg_{∞}-structure (i.e., for each ξ, F (ξ) is the constant map with value F (ξ^{∘})), then the σ_{n} define a homeomorphism from 𝔖𝔭(F) to the usual homotopy colimit \underrightarrow{hocolim}(F). More generally, when F is an arbitrary Rg_{∞}-structure, then there is a map

pr_{F} : 𝔖𝔭(F) ⟶ |C|

defined on each subspace F (c_{0}) × I^{n} by first projecting to the I^{n} and then to Δ^{n} via σ_{n}.

We now define a functor ℜ𝔤(F) : C ⟶ Top by letting ℜ𝔤(F)(c) be the pullback space

\begin{aligned}ℜ𝔤(F)(c) & \overline{→} & 𝔖𝔭(F) \\ |C↓c| & ↓ & \overline{→} & |C| & ↓^{pr_{F}} \end{aligned}

(the ordinary pullback, not the homotopy pullback). A morphism φ ∈ Mor_{C}(c, d) induces a map from |C↓c| to |C↓d| via composition with φ in the usual way, and hence induces a map from ℜ𝔤(F)(c) to ℜ𝔤(F)(d). Equivalently,

ℜ𝔤(F)(c) = (∐_{n≥0} ∐_{c_{0}→···→c_{n}→c} F (c_{0}) × I^{n})/∼  (I^{n} = I_{0}^{•}([n]))

where the identifications are analogous to those used to define 𝔖𝔭(F). This clearly makes ℜ𝔤(F) into a functor from C to Top.

For each c, F (c) can be identified as a subspace of ℜ𝔤(F)(c): the inverse image under

the projection to |C↓c| of the vertex (c ⟶^{Id} c). The composite F (c) ⟶_{≃}^{ν(c)} F (c) ⊆

ℜ𝔤(F)(c) defines a natural transformation F ⟶ ho ∘ℜ𝔤(F) of functors C ⟶ hoTop. The following proposition now shows that this is a natural equivalence, and hence that ℜ𝔤(F) is a rigidification of F.

Proposition A.2. For any Rg_{∞}-structure F on F: C ⟶ hoTop, for each c ∈ Ob(C), F (c) is a deformation retract of ℜ𝔤(F)(c). Thus ℜ𝔤(F) is a rigidification of F.

Proof. Define Φ: ℜ𝔤(F)(c) × I ⟶ ℜ𝔤(F)(c) by setting

Φ((x; t)_{[ξ→c]}, s) = (x; (t, s))_{[ξ→c⟶^{Id} c]}

for all ξ ∈ Mor^{n}(C), t ∈ I^{n}, and s ∈ I. Then Φ(u, 1) = u and Φ(u, 0) ∈ F (c) for all u ∈ ℜ𝔤(F)(c) by definition of ℜ𝔤(F)(c). Furthermore, the homotopy is the identity on F (c), and thus F (c) is a deformation retract. □

For any given F: C ⟶ hoTop, let Rigid(F) be the set of equivalence classes of rigidifications of F, and let Rg_{∞}(F) be the set of equivalence classes of Rg_{∞}-structures on F. A rigidification of F can be regarded as a “locally constant” Rg_{∞}-structure on F; i.e., an Rg_{∞}-structure F where each of the maps F (ξ) (for ξ ∈ Mor^{n}(C)) is constant on I^{n−1}. We thus have maps

Rigid(F) \overrightarrow{⟵}_{ℜ𝔤}^{const} Rg_{∞}(F).

One easily checks that for any rigidification F̃, there is a natural transformation of functors from ℜ𝔤(const(F̃)) to F̃, and hence these are equal in Rigid(F). We do not know whether the other composite is the identity on Rg_{∞}(F), but that will not be needed here.

A natural transformation χ: F ⟶ F′ of functors F, F′: C ⟶ hoTop will be called relatively centric if for each morphism φ ∈ Mor_{C}(c, d) in C, the homotopy commutative square

Map(F (c), F (c))_{Id} \overline{^{F (φ)∘−}→} Map(F (c), F (d))_{F (φ)}

Map(F (c), F′(c))_{χ(c)} ↓^{χ(c)∘−}  \overline{→^{F′(φ)∘−}} Map(F (c), F′(d))_{F′(φ)∘χ(c)} ↓^{χ(d)∘−}

is a homotopy pullback. For example, when F′ is the functor which sends every object to a point, then χ is relatively centric if and only if the functor F defines a centric diagram. Assume we are given a relatively centric natural transformation χ: F ⟶ F′ where F′ is a functor to Top, and assume furthermore that for each c ∈ Ob(C), the homotopy fiber

Γ(c) =^{def} hofiber(Map(F (c), F (c))_{Id} ⟶^{χ(c)∘−} Map(F (c), F′(c))_{χ(c)})

is connected. We claim that this determines functors

β_{i}: C^{op} ⟶ Ab  (all i ≥ 1)

such that β_{i}(c) ≅ π_{i}(Γ(c)) for all c. To show this, we can assume without loss of generality that χ(c) is a fibration for all c, and let Γ(c) be the space of all f ∈ Map(F (c), F (c)) such that χ(c) ∘ f = χ(c). Then Γ(c) is a monoid under composition, and in particular, π_{1}(Γ(c)) is abelian. For each morphism φ ∈ Mor_{C}(c, d) in C, we can choose a representative F (φ) such that the following square commutes:

\begin{matrix}F (c) & \overline{→^{F (φ)}} & F (d) \\ F′(c) & ↓^{χ(c)} & \overline{→^{F′(φ)}} & F′(d) . & ↓^{χ(d)} \end{matrix}

Since χ is relatively centric, the fibers of the map

(χ(d) ∘ −): Map(F (c), F (d))_{F (φ)} ⟶ Map(F (c), F′(d))_{F′(φ)∘χ(c)}

have the homotopy type of Γ(c) and hence are connected. Hence any two choices for F (φ) differ by a path in the fiber over the point F′(φ) ∘ χ(c); i.e., by a homotopy {F_{t}(φ)}_{t∈I} such that χ(d) ∘ F_{t}(φ) = F′(φ) ∘ χ(c) for each t.

For each φ ∈ Mor_{C}(c, d), consider the following diagram:

Map(F (d), F (d))_{Id} \overline{→^{\begin{matrix}−∘F (φ) \\ =w_{1} \end{matrix}}} Map(F (c), F (d))_{F (φ)} ←\begin{matrix}F (φ)∘− \\ =w_{2} \end{matrix} Map(F (c), F (c))_{Id}

Map(F (d), F′(d))_{χ(d)} ↓^{χ(d)∘− =u_{1}}  \overline{→^{−∘F (φ)}} Map(F (c), F′(d))_{χ(d)F (φ)} ↓^{χ(d)∘− =u_{2}}  ←^{F′(φ)∘−} Map(F (c), F′(c))_{χ(c)} ↓^{χ(c)∘− =u_{3}}

where the right hand square commutes by the assumption on F (φ) (and the other since composition is associative). Set Γ(c) = u_{3}^{−1}(χ(c)) and β_{i}(c) = π_{i}(Γ(c), Id_{F (c)}) (and similarly for d). By assumption, w_{2} sends Γ(c) to u_{2}^{−1}(F′(φ) ∘ χ(c)) by a homotopy equivalence, and we let β_{i}(φ) be the composite

π_{i}(Γ(d), Id_{F (d)})_{=β_{i}(d)} ⟶^{w_{1}∘−} π_{i}(u_{2}^{−1}(F′(φ) ∘ χ(c)), F (φ)) ⟶_{≅}^{(w_{2}∘−)^{−1}} π_{i}(Γ(c), Id_{F (c)})_{=β_{i}(c)}.

By the above remarks, this is independent of the choice of map F (φ). Hence this defines a functor on C^{op}: the relations β_{i}(ψ ∘ φ) = β_{i}(ψ) ∘ β_{i}(φ) follow using any choice of homotopy from F (ψ ∘ φ) to F (ψ) ∘ F (φ) which covers F′(ψ ∘ φ). (Recall that we are assuming F′ is a functor to Top, so F′(ψ ∘ φ) = F′(ψ) ∘ F′(φ).)

The following theorem is our main result giving a relative version of the Dwyer-Kan obstruction theory. The special case where F′(c) is a point for all c ∈ Ob(C) is the case shown by Dwyer and Kan in [DK].

Theorem A.3. Fix functors F: C ⟶ hoTop and F′: C ⟶ Top, and let

χ: F ⟶ ho ∘ F′

be a relatively centric natural transformation of functors. For each c ∈ Ob(C), assume that the homotopy fiber

Γ(c) = hofiber(Map(F (c), F (c))_{Id} ⟶^{χ(c)∘−} Map(F (c), F′(c))_{χ(c)});

is connected. Let β_{i}: C^{op} ⟶ Ab (all i ≥ 1) be the functors defined above. Then the obstructions to the existence of a rigidification F̃ \underrightarrow{χ̃} F′ of F ⟶^{χ} F′ lie in the groups \underleftarrow{lim}_{C}^{n+2}(β_{n}) for n ≥ 1; while the obstructions to the uniqueness of (F̃, χ̃) up to

equivalence of rigidifications lie in \underleftarrow{lim}_{C}^{n+1}(β_{n}) for n ≥ 1.

Proof. We use here the description of the higher limits of a functor α: C^{op} ⟶ Ab as the homology groups of the normalized cochain complex

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

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

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

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

Then \underleftarrow{lim}_{C}^{∗}(α) ≅ H^{∗}(C^{∗}(C; α), d) (cf. [GZ, Appendix II, Proposition 3.3] or [Ol, Lemma

2].)

Proof of existence: As above, we replace each χ(c) by a fibration, and replace each F (φ) (for φ ∈ Mor_{C}(c, d) by a map such that χ(d) ∘ F (φ) = F′(φ) ∘ χ(c). We also assume that F (Id_{c}) = Id_{F (c)} for each c. Then

Γ(c) =^{def} {f ∈ Map(F (c), F (c)) | χ(c) ∘ f = χ(c)}

is a topological monoid under composition, and is connected by assumption. So we can ignore basepoints when working in the homotopy groups β_{i}(c) = π_{i}(Γ(c)).

We want to construct an Rg_{∞}-structure F such that F (c) = F (c) for all c ∈ Ob(C), F (φ) = F (φ) for all φ ∈ Mor(C), and such that for each n ≥ 2 and each ξ = (c_{0} → ⋯ → c_{n}) ∈ Mor^{n}(C), the following square commutes (exactly) for each t ∈ I^{n−1}:

\begin{matrix}F (c_{0}) & \overline{→^{F (ξ)(t)}} & F (c_{n}) \\ F′(c_{0}) & ↓^{χ(c_{0})} & \overline{→^{F′(ξ^{∘})}} & F′(c_{n}) . & ↓^{χ(c_{n})} \end{matrix}  (1)

By Proposition A.2, any such structure induces a rigidification F̃ of F, together with a natural transformation of functors χ̃ from F̃ to F′.

Assume, for some n ≥ 2, that F has been defined on Mor^{i}(C) for all i < n. Fix ξ ∈ Mor^{n}(C), a composite of (nonidentity) maps from c_{0} to c_{n}. Consider the following commutative square, which is a homotopy pullback by assumption:

Map(F (c_{0}), F (c_{n}))_{F (ξ^{∘})} ←^{\begin{matrix}F (ξ^{∘})∘− \\ =w \end{matrix}} Map(F (c_{0}), F (c_{0}))_{Id}

Map(F (c_{0}), F′(c_{n}))_{F′(ξ^{∘})∘χ(c_{0})} ↓^{χ(c_{n})∘− =u}  ←^{F′(ξ^{∘})∘−} Map(F (c_{0}), F′(c_{0}))_{χ(c_{0})} . ↓^{χ(c_{0})∘−}

(2)

Conditions (b) and (c) in Definition A.1 determine a map F (ξ)_{0} from ∂I^{n−1} to

Map(F (c_{0}), F (c_{n}))_{F (ξ^{∘})} whose image lies in u^{−1}(F′(ξ^{∘}) ∘ χ(c_{0})). Hence the obstruction to

defining F (ξ) on I^{n−1} is an element

η(ξ) ∈ π_{n−2}(u^{−1}(F′(ξ^{∘}) ∘ χ(c_{0})), ) ⟵_{≅}^{w∘−} π_{n−2}(Γ(c_{0})) = β_{n−2}(c_{0}).

If one of the morphisms in the sequence ξ is an identity morphism, then we define F (ξ) using the appropriate formula in Definition A.1(b), and η(ξ) = 0. Thus η ∈ C^{n}(C; β_{n−2}).

We claim that dη = 0. Fix ω = (c_{0} → ⋯ → c_{n+1}) ∈ Mor^{n+1}(C). Consider the face maps on the n-cube

δ_{i}^{t}: I^{n−1} ⟶ I^{n} where δ_{i}^{t}(t_{1}, …, t_{n−1}) = (t_{1}, …, t_{i−1}, t, t_{i}, …, t_{n−1})

(for all i = 1, …, n and t = 0, 1). The conditions in Definition A.1(b,c) define a map

F_{•}(ω): (I^{n})^{(n−2)} ⟶ Map(F (c_{0}), F (c_{n+1}))_{F (ω^{∘})} ≃ Aut(F (c_{0}))_{1}
= Map(F (c_{0}), F (c_{0}))_{Id},

and hence

∑_{i=1}^{n}(−1)^{i}([F_{•}(ω) ∘ δ_{i}^{1}|_{∂I^{n−1}}] − [F_{•}(ω) ∘ δ_{i}^{0}|_{∂I^{n−1}}]) = 0 ∈ π_{1}(Aut(F (c_{0}))_{1}). (3)

Furthermore, F_{•}(ω) extends to the faces δ_{i}^{0}(I^{n−1}) for 2 ≤ i ≤ n − 1 (again, by the conditions in Definition A.1(c)), and so those terms vanish in (3). So we are left with the equality

0 = [F_{•}(ω) ∘ δ_{1}^{0}|_{∂I^{n−1}}] + ∑_{i=1}^{n}(−1)^{i}[F_{•}(ω) ∘ δ_{i}^{1}|_{∂I^{n−1}}] + (−1)^{n+1}[F_{•}(ω) ∘ δ_{n}^{0}|_{∂I^{n−1}}]

= F (ω_{01})^{∗}(η(∂_{0}ω)) + ∑_{i=1}^{n+1}(−1)^{i}η(∂_{i}ω) = dη(ω).

Thus dη = 0, and so [η] ∈ \underleftarrow{lim}_{C}^{n}(β_{n−2}).

If [η] = 0, then there is ρ ∈ C^{n−1}(C; β_{n−2}) such that η = dρ. Similar (but simpler) arguments to those used above now show that F can be “changed by ρ” on elements of Mor^{n−1}(C), in a way so that the obstruction η vanishes. We can thus arrange that F can be extended to Mor^{n}(C). Upon continuing this procedure, we obtain the Rg_{∞}-structure F.

Proof of uniqueness: Now assume that

F̃_{1} \underrightarrow{χ̃_{1}} F′ \underleftarrow{χ̃_{2}} F̃_{2}

are two rigidifications of χ: F ⟶ F′. In other words, we have a homotopy commutative diagram

\begin{matrix}F & \overline{^{ν_{1}} →} & ho ∘ F̃_{1} \\ [figure] & χ &  \\ ho ∘ F̃_{2} & ↓^{ν_{2}} & \overline{→^{ho (χ̃_{2})}} & ho ∘ F′ & ↓^{ho (χ̃_{1})} \end{matrix}  (4)

of functors C → hoTop and natural transformations between them. We can assume that the maps χ̃(c) and χ̃′(c) are fibrations for each c ∈ Ob(C); otherwise we replace them by fibrations using one of the canonical constructions.

For each c ∈ Ob(C), let θ(c): F̃_{1}(c) ⟶ F̃_{2}(c) be any map such that θ(c) ∘ ν_{1}(c) ≃ ν_{2}(c) as maps from F̃_{1}(c) to F̃_{2}(c). Using the homotopy commutativity of (4), and the homotopy lifting property for χ̃_{2}(c), we can assume that χ̃_{2}(c) ∘ θ(c) = χ̃_{1}(c) (exactly, not just up to homotopy). Let F̂(c) be the mapping cylinder of θ(c), and let χ̂(c): F̂(c) ⟶ F′(c) be the projection induced by χ̃_{1}(c) and χ̃_{2}(c).

Regard F̃_{1}(c) and F̃_{2}(c) as subspaces of F̂(c). We want to extend the locally finite Rg_{∞}-structures F̃ and F̃′ to an Rg_{∞}-structure F̂ covering F′. For each morphism φ ∈ Mor_{C}(c, d) in C, θ(d) ∘ F̃_{1}(φ) ≃ F̃_{2}(φ) ∘ θ(c), and hence F̃_{1}(φ) and F̃_{2}(φ) can be extended to a map F̂(φ) from F̂(c) to F̂(d). Using the homotopy lifting property again, this can be chosen such that χ̂(d) ∘ F̂(φ) = F′(φ) ∘ χ̂(c).

Assume, for some n ≥ 2, that F̂ has been defined on Mor^{i}(C) for i < n in a way so that (1) commutes (with F and F replaced by F̂) for each ξ. Then for each ξ = (c_{0} → ⋯ → c_{n}) in Mor^{n}(C), F̂(ξ) has been defined on

((F̃_{1}(c_{0}) ∪ F̃_{2}(c_{0})) × I^{n−1}) ∪ (F̂(c_{0}) × ∂I^{n−1}),

and must be extended to F̂(c_{0})×I^{n−1} while covering χ̂(c_{n})∘F̂(ξ^{∘}) ∈ Map(F̂(c_{0}), F′(c_{n})). So with the help of diagram (2) again, the obstruction to defining F̂(ξ) is seen to be an element τ(ξ) ∈ π_{n−1}(Γ(c_{0})) = β_{n−1}(c_{0}). Together, these define a cochain τ ∈ C^{n}(C; β_{n−1}). Just as in the proof of existence, one then shows that dτ = 0, and hence that τ represents a class [τ] ∈ \underleftarrow{lim}^{n}(β_{n−1}). If [τ] = 0, then τ = dρ for some ρ ∈ C^{n−1}(C; β_{n−1}), and F̂ can be modified on Mor^{n−1}(C) using ρ in such a way that it can then be extended to Mor^{n}(C). Upon continuing this procedure, we construct an Rg_{∞}-structure F̂ on F, together with a natural transformation to F′ and morphisms of Rg_{∞}-structures

F̃_{1} ⟶ F̂ ⟵ F̃_{2}.

So by Proposition A.2,

F̃_{1} ≃ ℜ𝔤(F̃_{1}) ≃ ℜ𝔤(F̂) ≃ ℜ𝔤(F̃_{2}) ≃ F̃_{2}.  □

We finish the section with two corollaries to Theorem A.3. The first is the main theorem of Dwyer and Kan in [DK]. It is the “absolute case” of Theorem A.3: the case where F′ is the constant functor which sends each object to a point.

A functor F from C to Top or hoTop will be called centric if for each morphism φ ∈ Mor_{C}(c, d) in C, the induced map

Map(F (c), F (c)_{Id} ⟶^{φ∘−} Map(F (c), F (d))_{φ}

is a homotopy equivalence. This is what Dwyer and Kan call a centric diagram.

Corollary A.4. Fix a centric functor F: C ⟶ hoTop. Define α_{i}: C^{op} ⟶ Ab (all i ≥ 1) by setting α_{i}(c) = π_{i}(Map(F (c), F (c))_{Id}) and by letting α_{i}(c ^{φ}→ d) be the composite

π_{i}(Map(F (d), F (d))_{Id}) ^{(−∘F (φ))_{∗}} → π_{i}(Map(F (c), F (d))_{F (φ)}) ← (F (φ)∘−)_{∗} ; ≅ π_{i}(Map(F (c), F (c))_{Id}).

Then the obstructions to the existence of a rigidification F̃ of F lie in the groups \underleftarrow{lim}_{C}^{n+2}(α_{n}) for n ≥ 1; while the obstructions to the uniqueness of F̃ up to equivalence

of rigidifications lie in \underleftarrow{lim}_{C}^{n+1}(α_{n}) for n ≥ 1.

The second corollary is a generalization of [CLN, Proposition B], and follows upon combining Corollary A.4 with an idea taken from the proof of that proposition.

Corollary A.5. Fix a space X, and a centric functor F: C ⟶ hoTop. We also let X denote the constant functor X: C ⟶ Top which sends each object to X and each morphism to Id_{X}.

(a) Assume there is a natural transformation of functors χ: F ⟶ ho ∘ X such that the map χ(c): F (c) ⟶ X is centric for each c ∈ Ob(C). Then there is a rigidification F̃ of F, together with a rigidification χ̃: F̃ ⟶ X of χ.

(b) Assume F̃_{1} and F̃_{2} are two rigidifications of F. Let ψ_{i}: F ⟶ ho ∘ F̃_{i} be natural equivalences, and let χ̃_{i}: F̃_{i} ⟶ X be natural transformations of functors such that for all c ∈ Ob(C), χ̃_{i}(c) ∈ Map(F̃_{i}(c), X) is centric, and the square

\begin{matrix}F (c) & \overline{^{ψ_{1}(c)} →} & F̃_{1}(c) \\ F̃_{2}(c) & ↓^{ψ_{2}(c)} & \overline{^{χ̃_{2}(c)} →} & X & ↓^{χ̃_{1}(c)} \end{matrix}  (1)

commutes up to homotopy. Then F̃_{1} and F̃_{2} are equivalent rigidifications. More precisely, there is a third rigidification F̃_{0} of F, natural transformations of functors

F̃_{1} \underrightarrow{ψ̃_{1}} F̃_{0} \underleftarrow{ψ̃_{2}} F̃_{2}

such that ψ̃_{i}(c) is a homotopy equivalence for each c ∈ Ob(C), a space X_{0} together with a natural transformation χ̃_{0} : F̃_{0} ⟶ X_{0} to the constant functor, and homotopic homotopy equivalences f_{1} ≃ f_{2} : X ⟶ X_{0}, such that the following diagram commutes for each c ∈ Ob(C):

\begin{aligned}F̃_{1}(c) & \overline{\begin{matrix}ψ̃_{1}(c) \\ ≃ \end{matrix} →} & F̃_{0}(c) & ← \begin{matrix}ψ̃_{2}(c) \\ ≃ \end{matrix} & F̃_{2}(c) \\ X & ↓^{χ̃_{1}(c)} & \overline{^{f_{1}} →} & X_{0} & ↓^{χ̃_{0}(c)} & ← ^{f_{2}} & X. & ↓^{χ̃_{2}(c)} \end{aligned}  (2)

Proof. Let C_{+} be the category C with an additional final object ∗ added. For any functor α: C_{+}^{op} ⟶ Ab, \underleftarrow{lim}^{i}(α) = 0 for all i ≥ 1 since C_{+}^{op} has an initial object. A functor F_{+} : C_{+} ⟶ hoTop can be thought of as a triple F_{+} = (F, X, χ), where F = F_{+}|_{C} is a functor from C to hoTop, X = F_{+}(∗) is a space, and χ is a natural transformation of functors from F to the constant functor X. Functors from C_{+} to Top are described in an analogous way.

In the situation of (a), (F, X, χ) is a functor from C_{+} to hoTop. The obstruction groups of Corollary A.4 vanish, and hence it has a rigidification (F̃, X̃, χ̃). Upon composing with an appropriate homotopy equivalence X̃ ⟶^{≃} X, we can arrange that X̃ = X.

In the situation of (b), (F̃_{1}, X, χ̃_{1}) and (F̃_{2}, X, χ̃_{2}) are two functors from C_{+} to Top which are rigidifications of the same functor (F, X, χ̃_{1} ∘ ψ_{1}) by the homotopy commutativity of (1). Since the uniqueness obstructions of Corollary A.4 all vanish, there is a third homotopy lifting (F̃_{0}, X_{0}, χ̃_{0}), together with natural transformations of functors

(F̃_{1}, X, χ̃_{1}) \underleftarrow{ψ̃_{1}} (F̃_{0}, X_{0}, χ̃_{0}) \underrightarrow{ψ̃_{2}} (F̃_{2}, X, χ̃_{2})

which induce homotopy equivalences on all objects. Thus upon setting f_{i} = ψ̃_{i}(∗), we obtain the commutative diagram (2), where all horizontal maps are homotopy equivalences. Finally, ψ̃_{1}(∗) ≃ ψ̃_{2}(∗), since they come from equivalences between liftings of the same homotopy functor, and this finishes the proof of (b). □

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86 Discrete models for the p-local homotopy theory of compact Lie groups and p-compact groups

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

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

Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K.

E-mail address: ran@maths.abdn.ac.uk

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

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