THE THEORY OF p-LOCAL GROUPS: A SURVEY

CARLES BROTO, RAN LEVI, AND BOB OLIVER

The motivation for this project comes from the study of the p-local homotopy theory of classifying spaces of finite groups, or more generally of compact Lie groups. By “plocal homotopy theory” of a space we mean the homotopy theory of its p-completion. It turns out that there is a close connection between the p-local homotopy theory of BG and the “p-local structure” of the group G, by which we mean the fusion (conjugacy relations) in a Sylow p-subgroup of G. This connection then suggested to us the construction of certain spaces (classifying spaces of “p-local finite groups” and “p-local compact groups”) which have many of the same properties as have p-completed classifying spaces of finite and compact Lie groups.

A brief survey of the Bousfield-Kan p-completion functor will be given in Section 1. For the purpose of this introduction it suffices to say that this is a functor from spaces to spaces, which focuses on the properties of a space which are visible through its mod p homology.

The p-fusion data of a finite group G consists of a Sylow p-subgroup S ≤ G, together with information on how subgroups of S are related to each other via conjugation in G. This data includes an abundance of information about BG. For example, the well known theorem of Cartan and Eilenberg [CE, Theorem XII.10.1], stating that H^{∗}(G; 𝔽_{p}) is given by the subring of ”stable elements” in H^{∗}(S; 𝔽_{p}), can be interpreted as saying that mod p cohomology of finite groups is determined by p-fusion.

A much stronger version of the Cartan-Eilenberg theorem is given by the Martino-Priddy conjecture [MP]. The conjecture, recently proved by the third author, states roughly that the classifying spaces of two groups have the same p-local homotopy type if and only if the groups share the same p-fusion data. A more precise formulation of this result, as well as a discussion of some of the ideas which go into its proof, are presented in Section 3.

Unfortunately, p-fusion is not quite sufficient to give all homotopy theoretic information associated to the classifying space of a finite group G. For instance, the topological monoid of homotopy equivalences from BG_{p}^{∧} to itself cannot easily be described using the fusion alone, and the work outlined below in Sections 2–3 was initially motivated by the need to obtain an algebraic description of this space. A partial answer to this problem, phrased in terms of p-fusion, appears in [BL]; but the complete description in [BLO1] requires the use of an additional structure, the “centric linking system” of the finite group.

The introduction of this new concept led to several new results, as described in detail in Section 2. For instance, it gave a different condition for determining when two finite groups have equivalent p-completed classifying spaces: a condition which is less useful than that given by the Martino-Priddy conjecture, but which is much easier to prove (Theorem 2.2). It also made it possible to give a complete algebraic description of the topological monoid of self homotopy equivalences of BG_{p}^{∧} for G finite. In addition, the introduction of centric linking systems for groups paved the way to the more general concept of p-local finite groups.

1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 55Q52, 20D20.

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

C. Broto is partially supported by DGICYT grant PB97–0203.

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

The p-fusion in a finite group G with a Sylow p-subgroup S can be made into a category, whose objects are the subgroups of S, and whose morphisms are those monomorphisms induced by conjugation in G. This category is called the “fusion category of G at the prime p”. Fusion systems can be defined more abstractly; they are categories of subgroups of a given p-group whose morphisms are certain monomorphisms between them, and which satisfy certain axioms. The fusion systems we will be interested in satisfy certain extra conditions first formulated by Puig [Pu3],[Pu4], and are known as “saturated fusion systems”.

Our first goal is to construct a classifying space for each saturated fusion system. This is motivated in part by representation theory: we refer to Section 5.3 for a brief discussion of the fusion system of a block as defined by Alperin & Broué. But it also provides a new class of spaces, which have many of the same properties of p-completed classifying spaces of finite groups, and which can be characterized in homotopy theoretic terms. To construct these classifying spaces, we define what it means to be a centric linking system associated to a saturated fusion system. Just like the examples arising from finite groups, centric linking systems associated to abstract saturated fusion systems, if they exist, are categories which can be thought of as lifts of the respective fusion systems. The p-completed nerve of a centric linking system is the desired classifying space of the corresponding fusion system. This does, however, lead to the unsatisfactory situation that we do not yet know whether or not there exists a linking system associated to a given fusion system, and whether or not it is unique (see Section 4.8).

A “p-local finite group” is now defined to be a triple consisting of a p-group, a saturated fusion system over it and an associated centric linking system. The classifying space of a p-local finite group is defined to be the p-completed nerve of its linking system. When the triple consists of a Sylow p-subgroup of a finite group G, together with its fusion and linking systems, then its classifying space has the homotopy type of BG_{p}^{∧}.

A closely related topic, which provides extra motivation, is the study of p-compact groups. These objects were introduced in the 1980’s by Dwyer and Wilkerson [DW3] and extensively studied by them and several other authors. A p-compact group is a loop space X (i.e., X ≃ Ω(BX) for some pointed “classifying space” BX), such that H^{∗}(X; 𝔽_{p}) is finite and BX is p-complete. The concept of a p-compact group was designed to be a homotopy theoretic analogue of a classifying space of a compact Lie group. For instance if G is a compact Lie group whose group of components is a pgroup, then BG_{p}^{∧} is the classifying space of a p-compact group. On the other hand if π_{0}(G) is not a p-group then the loop space Ω(BG_{p}^{∧}) is not generally 𝔽_{p}-finite. In spite of the fact that there are many common aspects to the homotopy theory of classifying spaces of finite, compact Lie and p-compact groups, the techniques used in the study of p-compact groups fail or become hard to use if the 𝔽_{p}-finiteness condition is dropped.

Our approach suggests a way to fix this problem. The report on “p-local compact groups” given here is incomplete, as the research is still in progress at the time of writing this survey. However, the definition of p-local compact groups is a natural extension of that of p-local finite groups, the main change being the replacement of the finite p-group S in the definition of a p-local finite group by what we call a “discrete p-toral group” (Section 6.1). Once we have defined saturated fusion systems and associated centric

THE THEORY OF p-LOCAL GROUPS: A SURVEY 3

linking systems over such groups, the definition of a p-local compact group becomes the obvious generalization of that of p-local finite groups. We are then able to show that the family of spaces obtained as classifying spaces of p-local compact groups contains all p-completed classifying spaces of compact Lie groups and p-compact groups.

The theory presented here suggests a vast ground for further exploration. As one example, we outline some results concerning extensions of p-local finite groups (Section 5.4). Some of the other sections also discuss open problems and topics for further investigation.

This survey is written in order to make it easier for interested readers to learn about the subject. As the theory of p-local groups is still relatively new, the main references are contained in fairly large and quite technical papers. It is our hope that this survey will be a valuable reference for readers to get the general picture before they plunge into more comprehensive articles. The paper is designed respectively, namely, we attempted to give good motivation to all statements, but most of them appear either with a very brief sketch of proof or with no proof at all. For all statements however, a precise reference for where a proof can be found is listed.

Contents

1. The p-completion functor 4 2. Fusion and linking systems of finite groups 5

2.1. Fusion systems of groups 5 2.2. Centric linking systems of groups 5 2.3. Equivalences of p-completed classifying spaces 6 2.4. Maps from classifying spaces of p-groups 6 2.5. Fusion and linking systems for spaces 7 2.6. Isotypical equivalences of fusion and linking categories 8 2.7. Self homotopy equivalences of BG_{p}^{∧} 9

3. The Martino-Priddy conjecture 9 3.1. Fusion preserving isomorphisms 9 3.2. Obstruction theory 10 3.3. Reduction to simple groups 11 3.4. The odd primary case 11 3.5. Obstruction groups at the prime 2 12

4. Abstract finite fusion and linking systems 13 4.1. Fusion systems 14 4.2. Centric linking systems 14 4.3. p-local finite groups 15 4.4. Centralizers and mapping spaces 16 4.5. Homology decompositions 17 4.6. Cohomology of the classifying space 18 4.7. Fusion and linking systems determined by their classifying space 20

4 CARLES BROTO, RAN LEVI, AND BOB OLIVER

4.8. Existence and uniqueness of linking systems or classifying spaces 21 5. Examples and methods of construction 22

5.1. Construction of exotic p-local finite groups 22 5.2. Fusion systems of a type studied by Solomon 24 5.3. The fusion system of a block 25 5.4. Extensions of p-local finite groups 25

6. p-local compact groups 27 6.1. Discrete p-toral groups 27 6.2. Fusion and linking systems over discrete p-toral groups 28 6.3. Reduction to finite subcategories 28 6.4. A homology decomposition of the classifying space 29 6.5. Reduced linking systems 29 6.6. Compact Lie groups 30 6.7. p-compact groups 31

References 31

1. The p-completion functor

We start with a brief description of the p-completion functor of Bousfield and Kan [BK], which we denote by (−)_{p}^{∧}. It is a functor from the category of spaces to itself, and comes equipped with a natural transformation λ: Id ⟶ (−)_{p}^{∧}. A space X is p-complete if λ_{X}: X ⟶ X_{p}^{∧} is a homotopy equivalence. One property of pcompletion (one of the few) which applies to all spaces is that a map f: X ⟶ Y induces a homotopy equivalence f_{p}^{∧}: X_{p}^{∧} ⟶ Y _{p}^{∧} if and only if f induces an isomorphism H^{∗}(X; 𝔽_{p}) ≅ H^{∗}(Y; 𝔽_{p}).

A space X is called p-good if λ_{X}: X ⟶ X_{p}^{∧} induces an isomorphism H^{∗}(X_{p}^{∧}; 𝔽_{p}) ≅ H^{∗}(X; 𝔽_{p}), or equivalently if X_{p}^{∧} is p-complete. Spaces which are p-bad (i.e., not p-good) remain so permanently: repeated application of the completion functor will never produce a p-complete space. Simply connected spaces, or more generally nilpotent spaces, are p-good. Spaces whose fundamental group is finite are p-good. More generally, any space X for which π_{1}(X) contains a p-perfect subgroup of finite index (i.e., a subgroup of finite index generated by its commutators and p-th powers) is also p-good. All spaces discussed in this article (at least, all spaces for which we want to take the p-completion) are p-good.

For any p-good space X, the map λ_{X}: X ⟶ X_{p}^{∧} is a final object among homotopy classes of maps out of X which induce a mod-p homology isomorphism. If X and Y are any two p-good spaces, then their p-completions are homotopy equivalent if and only if there exists some space Z, and maps X ⟶^{f} Z ⟵^{g} Y, such that f and g are both mod-p homology equivalences. We say that X and Y have the same p-local homotopy type, or that they are mod p equivalent, if X_{p}^{∧} ≃ Y _{p}^{∧}.

Among our main objects of study here are the p-completed classifying spaces of compact Lie groups. For any compact Lie group G, BG is p-good since its fundamental group is finite. Also, π_{1}(BG_{p}^{∧}) ≅ π_{0}(G)/O^{p}(π_{0}(G)), where O^{p}(Γ) is the maximal pperfect subgroup of a finite group Γ (equivalently, the smallest normal subgroup of p-power index).

While p-completion does not change the mod-p homology of BG when G (or π_{0}(G)) is finite, the space BG_{p}^{∧} is not generally aspherical. In fact, for finite G, BG_{p}^{∧} is aspherical only if G contains a normal subgroup of p-power index and order prime to p; in all other cases BG_{p}^{∧} has infinitely many non-trivial homotopy groups. The homotopy theory of spaces of this form has some fascinating aspects. For a survey on some of the classical homotopy theory associated with BG_{p}^{∧}, the reader is referred to [CL].

2. Fusion and linking systems of finite groups

The main idea in this section is to describe how the homotopy theory of BG_{p}^{∧} is related to certain categories, which we call the fusion and centric linking systems of G at the prime p (we emphasize “centric” since there is a notion, presently less useful, of a more general linking system, which will not be mentioned here). Throughout the section, we fix a finite group G and a prime p.

2.1. Fusion systems of groups. The fusion system F_{p}(G) of G is the category whose objects are the p-subgroups of G, and where for any pair of p-subgroups P, Q ≤ G,

Mor_{F_{p}(G)}(P, Q) = Hom_{G}(P, Q) =^{def} {α ∈ Hom(P, Q) | α = c_{x}, some x ∈ G}.

Thus, if we define

N_{G}(P, Q) = {x ∈ G | xP x^{−1} ≤ Q}

(the transporter), then

Mor_{F_{p}(G)}(P, Q) ≅ N_{G}(P, Q)/C_{G}(P).

If S is any Sylow p-subgroup of G, then F_{S}(G) ⊆ F_{p}(G) will denote the full subcategory whose objects are the subgroups of S. Since every p-subgroup of G is conjugate to a subgroup of any given S ∈ Syl_{p}(G), the inclusion of F_{S}(G) in F_{p}(G) is an equivalence of categories.

2.2. Centric linking systems of groups. For any finite group H, let O^{p}(H) denote the minimal normal subgroup of p power index, or equivalently the maximal normal p-perfect subgroup of H. A p-subgroup P ≤ G is p-centric if Z(P) ∈ Syl_{p}(C_{G}(P)), or equivalently if C_{G}(P) = Z(P) × O^{p}(C_{G}(P)) and O^{p}(C_{G}(P)) has order prime to p. For such subgroups we write O^{p}(C_{G}(P)) = C′_{G}(P) for short.

The centric linking system L_{p}^{c}(G) of G is the category whose objects are the p-centric subgroups of G, and where

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

for any pair of objects.

The category L_{p}^{c}(G) is the same as what Puig [Pu1, §VI.1] called the “O-localité” of G (when restricted to p-centric subgroups).

If S is any Sylow p-subgroup of G, then L_{S}^{c}(G) ⊆ L_{p}^{c}(G) will denote the full subcategory whose objects are the subgroups of S which are p-centric in G. As was the case for the corresponding inclusion of fusion systems, this inclusion is always an equivalence of categories.

The following theorem helps to explain the usefulness of centric linking systems when studying the homotopy theory of BG_{p}^{∧}.

Theorem 2.1. For any finite group G and any prime p, |L_{p}^{c}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.

Proof. See [BLO1, Proposition 1.1]. The idea of the proof is to construct a larger category L̃_{p}^{c}(G), with the same objects as L_{p}^{c}(G), but where Mor_{L̃_{p}^{c}(G)}(P, Q) = N_{G}(P, Q) for all P, Q. Let B(G) be the category with one object o_{G}, and where End_{B(G)}(o_{G}) = G. One then shows that the maps

|L_{p}^{c}(G)| ⟵^{|π|} |L̃_{p}^{c}(G)| \underrightarrow{α̃_{G}} |B(G)| ≅ BG,

are 𝔽_{p}-homology equivalences, where π is the obvious surjective functor, and where α̃_{G} is induced by the functor which is the inclusion on all morphism sets N_{G}(P, Q) ⊆ G. □

2.3. Equivalences of p-completed classifying spaces. The Martino-Priddy conjecture states roughly that the homotopy type of BG_{p}^{∧} depends only on the fusion of G in S ∈ Syl_{p}(G), or equivalently on the fusion system F_{p}(G). This will be discussed in more detail in Section 3, and stated precisely in Theorem 3.1.

In this section, we state a weaker result, which says that the homotopy type of BG_{p}^{∧} depends only on its linking system L_{p}^{c}(G). This does provide a (finite) combinatorial condition for two p-completed classifying spaces to be homotopy equivalent. The condition is, however, more complicated to check than the one stated in the Martino-Priddy conjecture, and hence less satisfactory. The proof of this statement is however much easier than the proof of the Martino-Priddy conjecture.

Theorem 2.2. For any pair G, G′ of finite groups and any prime p, BG_{p}^{∧} ≃ BG′_{p}^{∧} if and only if L_{p}^{c}(G) ≃ L_{p}^{c}(G′).

Proof. See [BLO1, Theorem A]. The “if” part of this theorem follows immediately from Theorem 2.1. For the “only if part” a construction of a centric linking system for a space is required. See Section 2.5 for more details. □

2.4. Maps from classifying spaces of p-groups. One of our goals is to describe in many cases maps between p-completed classifying spaces of finite groups. The simplest case to consider is that where the source is the classifying space of a p-group.

If H and K are any two discrete groups then the space of (unpointed) maps from BH to BK is very simple to describe, and the result is classical. Set

Rep(H, K) = Hom(H, K)/ Inn(K),

the set of conjugacy classes of homomorphisms (“representations”) from H to K. Then the natural map

B: Rep(H, K) ⟶ [BH, BK]

is a bijection. Also, for each ρ: H ⟶ K, the homomorphism

C_{K}(ρ) × H ⟶^{incl ·ρ} K,

where C_{K}(ρ) =^{def} C_{K}(Im(ρ)), induces a map of spaces BC_{K}(ρ) × BH ⟶ BK, whose adjoint

BC_{K}(ρ) ⟶^{≅} Map(BH, BK)_{Bρ}

is a homotopy equivalence. The simplest way to see this result is to first show that [BH, BK]_{∗} ≅ Hom(H, K) (i.e., pointed homotopy classes of pointed maps) and that each component of the pointed mapping space Map_{∗}(BH, BK) is contractible; and then examine the fibration

Map_{∗}(BH, BK) ⟶ Map(BH, BK) ⟶ BK.

The following “folk theorem” describes one situation in which this result can be generalized from classifying spaces to p-completed classifying spaces.

Theorem 2.3. For any finite p-group P, and any finite group G, the p-completion map BG → BG_{p}^{∧} induces a (weak) homotopy equivalence

Map(BP, BG)_{p}^{∧} ⟶^{≃} Map(BP, BG_{p}^{∧}).

In particular, the map (ρ ↦ Bρ_{p}^{∧}) defines a bijection

Rep(P, G) ⟶_{≅}^{B} [BP, BG_{p}^{∧}].

For each ρ: P → G, the induced product map C_{G}(ρ) × P → G induces (after taking adjoints) a homotopy equivalence

BC_{G}(ρ)_{p}^{∧} ⟶ Map(BP, BG_{p}^{∧})_{Bρ} .

In particular, the mapping space Map(BP, BG_{p}^{∧}) is p-complete.

Proof. See [BL, Proposition 2.1], but this theorem was known to the experts well before we wrote down a proof. □

This result will be generalized in Section 3 (Theorem 4.2).

2.5. Fusion and linking systems for spaces. The “only if” part of Theorem 2.2 follows from another construction: a functor from spaces to categories which in particular sends BG_{p}^{∧} to L_{S}^{c}(G). This is described as follows.

Fix a space X, a p-group S, and a map f: BS ⟶ X. We define F_{S,f}(X) to be the category whose objects are the subgroups of S, and where for P, Q ≤ S,

Mor_{F_{S,f} (X)}(P, Q) = {α ∈ Hom(P, Q) | f|_{BP} ≃ f|_{BQ} ∘ Bα}.

This is clearly a fusion system over S (though not necessarily saturated!), and can be thought of as the fusion system of the space X with respect to the pair (S, f).

We next define the analogous linking system for X with respect to (S, f). Let L_{S,f}(X) be the category whose objects are the subgroups of S, and where for P, Q ≤ S,

Mor_{L_{S,f} (X)}(P, Q) = {(α, [ϕ]) | α ∈ Hom(P, Q), [ϕ] a homotopy class of paths

in Map(BP, X) from f|_{BP} to f|_{BQ} ∘ Bα}.

A morphism of L_{S,f}(X) from P to Q thus consists of a morphism α ∈ Hom_{F_{S,f} (X)}(P, Q), together with a homotopy class of homotopies from f|_{BP} to f|_{BQ} ∘ Bα.

There is an obvious “forgetful” functor

π_{S,f}(X): L_{S,f}(X) ⟶ F_{S,f}(X)

which is the identity on objects and sends a morphism (α, [ϕ]) to α. Note that the set of morphisms in L_{S,f}(X) sitting over any given α ∈ Hom_{F_{S,f} (X)}(P, Q) is in bijective correspondence with π_{1}(Map(BP, X)_{f|BP}).

We also want a centric linking system in this situation. Let L_{S,f}^{c}(X) be the full subcategory of L_{S,f}(X) whose objects are the subgroups P ≤ S such that C_{S}(P′) = Z(P′) for all P′ isomorphic to P in F_{S,f}(X).

The “only if” part of Theorem 2.2 now follows easily from the following:

Theorem 2.4. For any finite group G and any prime p, fix S ∈ Syl_{p}(G), and let θ: BS ⟶ BG_{p}^{∧} be the inclusion. Then there are equivalences of categories

F_{S,θ}(BG_{p}^{∧}) ≅ F_{S}(G)  and  L_{S,θ}^{c}(BG_{p}^{∧}) ≅ L_{S}^{c}(G).

Proof. See [BLO1, Proposition 2.7]. The first isomorphism says that for any P, Q ≤ S and any α ∈ Hom(P, Q), α ∈ Hom_{G}(P, Q) (i.e., α is induced by conjugation in G) if and only if the composite BP ⟶^{Bα} BQ ⟶^{incl} BG is homotopic to the inclusion BP ⊆ BG, and this follows from Theorem 2.3.

The isomorphism between the two linking categories is slightly more delicate. Note first that by definition (and the isomorphism of fusion systems), both categories have as objects the subgroups of S which are p-centric in G. For any α ∈ Hom_{G}(P, Q) (any P, Q ≤ S), the number of morphisms in F_{S}(G) which cover α is equal to |Z(P)|, and the number of morphisms in F_{S,θ}(BG_{p}^{∧}) which cover α is equal to the order of π_{1}(Map(BP, BG_{p}^{∧})_{incl}) ≅ Z(P) (see Theorem 2.3). This suggests that there should be a natural bijection between these morphism sets, and such a bijection is constructed by mapping the transporter N_{G}(P, Q) surjectively to both sets and showing that one has the same identifications in the two cases. □

Theorem 2.4 suggests a way to extend the centric linking system of a finite group G to all p-subgroups. Let L_{p}(G) be the category whose objects are the p-subgroups of G, and where Mor_{L_{p}(G)}(P, Q) = N_{G}(P, Q)/O^{p}(C_{G}(P)). This is easily seen to be well defined on morphisms. As usual, for S ∈ Syl_{p}(G), L_{S}(G) ⊆ L_{p}(G) denotes the full subcategory whose objects are the subgroups of S. Then the argument just sketched also shows that L_{S}(G) ≅ L_{S,θ}(BG_{p}^{∧}).

2.6. Isotypical equivalences of fusion and linking categories. Let C be any of the categories F_{p}(G), L_{p}^{c}(G), F_{S,f}(X), or L_{S,f}(X), or any subcategory of one of these categories. Let C ⟶^{ϵ} Gr be the forgetful functor to the category of groups. A self equivalence φ: C ⟶ C is called isotypical if there is a natural isomorphism of functors from ϵ ∘ φ to ϵ.

When C is one of the above categories, we let Aut(C) and Aut_{typ}(C) denote the monoids of self equivalences and isotypical self equivalences, respectively of C. More generally, we let Aut_{typ}(C) be the strict monoidal category whose objects are the isotypical self equivalences of C, and whose morphisms are the natural isomorphisms of functors. We can then define Out_{typ}(C) to be the group of automorphisms modulo natural isomorphisms; i.e., the group of connected components of the nerve |Aut_{typ}(C)|.

2.7. Self homotopy equivalences of BG_{p}^{∧}. Using Theorems 2.1 and 2.4, one can also describe the monoid of self homotopy equivalences of BG_{p}^{∧} in terms of automorphisms of the category L_{p}^{c}(G). For any space X, we let Aut(X) denote the topological monoid of all self homotopy equivalences of X; and by analogy with the notation for self equivalences of categories, let Out(X) = π_{0}(Aut(X)) denote the group of homotopy classes of self equivalences.

For any finite group G, O_{p′}(G) ◁ G denotes the largest normal subgroup of G of order prime to p.

Theorem 2.5. For any finite group G and any prime p, the connected components of the space Aut(BG_{p}^{∧}) are all aspherical, and there are isomorphisms

Out(BG_{p}^{∧}) ≅ Out_{typ}(L_{p}^{c}(G)) and π_{1}(Aut(BG_{p}^{∧})) ≅ Z(G/O_{p′}(G))_{p}^{∧}.

Moreover, there is a homotopy equivalence

B Aut(BG_{p}^{∧}) ≅ B|Aut_{typ}(L_{p}^{c}(G))|.

Proof. See [BLO1, Theorems B & C] and [BL, Theorem 1.1]. □

3. The Martino-Priddy conjecture

Originally, the results in Section 1 were motivated partly as a means of studying Aut(BG_{p}^{∧}) (Theorem 2.5), but also partly as a means of finding algebraic or combinatorial conditions for two p-completed classifying spaces to be homotopy equivalent (Theorem 2.2). What we really want is to describe both of these in terms of the fusion systems, or equivalently in terms of fusion among subgroups of a Sylow subgroup.

3.1. Fusion preserving isomorphisms. Fix a pair of finite groups G and G′, a prime p, and Sylow subgroups S ∈ Syl_{p}(G) and S′ ∈ Syl_{p}(G′). An isomorphism S ⟶^{φ} S′ is called fusion preserving if for all P, Q ≤ S and all P ⟶_{≅}^{α} Q, α is induced by

conjugation in G if and only if φ(P) ⟶_{≅}^{φαφ^{−1}} φ(Q) is induced by conjugation in G′.

Clearly, a fusion preserving isomorphism S ⟶ S′ in the above situation induces an isomorphism of categories F_{S}(G) ≅ F_{S′}(G′), and hence an equivalence of the larger categories F_{p}(G) ≃ F_{p}(G′). Conversely, given an isotypical equivalence between the fusion systems of G and G′, it is not hard to construct a fusion preserving isomorphism between their Sylow p-subgroups [BLO1, Lemma 5.1].

A classical result in group cohomology is a theorem by Cartan and Eilenberg (see [CE, Theorem XII.10.1]), which states roughly that for any finite group G, the cohomology ring H^{∗}(BG; 𝔽_{p}) is determined by S ∈ Syl_{p}(G) and fusion in S (see Section 4.6). The key point of the Martino-Priddy conjecture is the much stronger statement that the homotopy type of the p-completed classifying space BG_{p}^{∧} is determined by p-fusion.

Theorem 3.1 (Martino-Priddy conjecture). For any pair G, G′ of finite groups and any prime p, the following three conditions are equivalent:

(a) BG_{p}^{∧} ≃ BG′_{p}^{∧}.

(b) There is a fusion preserving isomorphism S ⟶_{≅}^{φ} S′ between Sylow p-subgroups of G and G′.

(c) There is an isotypical equivalence F_{p}(G) ≃ F_{p}(G′).

The implications (a) ⟹ (b) ⟺ (c) were proved by Martino and Priddy [MP]. In particular, (b) and (c) are equivalent by the above remarks. The remaining implication was recently proven by the third author ([O1] and [O2]), using the classification theorem for finite simple groups.

3.2. Obstruction theory. The first step in analyzing the Martino-Priddy conjecture is to reduce it to a problem of higher derived functors of inverse limits over the orbit category of G. For any finite group G and any prime p, the orbit category O_{p}(G) of G is the category whose objects are the p-subgroups of G, and where for all p-subgroups P, Q ≤ G,

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

Let Z_{G}: O_{p}(G)^{op} ⟶ Ab be the functor

Z_{G}(P) = \begin{cases}Z(P) & if P is p-centric \\ 0 & otherwise. \end{cases}

By [BLO1, Proposition 6.1], the obstruction (for S ∈ Syl_{p}(G) and S′ ∈ Syl_{p}(G′)) to lifting a fusion preserving isomorphism

S ⟶_{≅}^{φ} S′

to an equivalence

L_{p}^{c}(G) ⟶^{≃} L_{p}^{c}(G′)

lies in the group

\underleftarrow{lim}^{2}_{O_{p}(G)}(Z_{G}).

This can be seen directly by choosing maps

Mor_{L_{p}^{c}(G)}(P, Q) ⟶ Mor_{L_{p}^{c}(G′)}(φ(P), φ(Q))

(for all P, Q ≤ S), and then examining the 2-cocycle in

C^{2}(O_{p}(G); Z_{G}) =^{def} ∏_{P →Q→R} Z(P)

which describes the failure of the images of commutative triangles in L_{p}^{c}(G) to commute in L_{p}^{c}(G′).

The Martino-Priddy conjecture was proven by showing:

Theorem 3.2. For any finite group G and any prime p, \underleftarrow{lim}^{i}_{O_{p}(G)}(Z_{G}) = 0 if i ≥ 2, or if

p is odd and i ≥ 1.

It is the proof of this result which depends on the classification of finite simple groups.

3.3. Reduction to simple groups. To reduce the proof of Theorem 3.2 to a problem about simple groups, consider a maximal normal series of G:

1 = H_{0} ◁ H_{1} ◁ H_{2} ◁ ⋯ ◁ H_{n} = G.

Then for each i, H_{i}/H_{i−1} is a product of copies of some fixed simple group L_{i} (see [Go, Theorem 2.1.5]). Define subfunctors Z_{G}^{i} ⊆ Z_{G} by setting Z_{G}^{i}(P) = Z(P) ∩ H_{i} if P is p-centric, and Z_{G}^{i}(P) = 0 otherwise. From the long exact sequences

⋯ ⟶ \underleftarrow{lim}^{n}_{O_{p}(G)}(Z_{G}^{i−1}) ⟶ \underleftarrow{lim}^{n}_{O_{p}(G)}(Z_{G}^{i}) ⟶ \underleftarrow{lim}^{n}_{O_{p}(G)}(Z_{G}^{i}/Z_{G}^{i−1})

⟶ \underleftarrow{lim}^{n−1}_{O_{p}(G)}(Z_{G}^{i−1}) ⟶ ⋯,

we see that \underleftarrow{lim}^{n}(Z_{G}) = 0 (for any given n) if \underleftarrow{lim}^{n}_{O_{p}(G)}(Z_{G}^{i}/Z_{G}^{i−1}) = 0 for all i.

Higher limits of the functors Z_{G}^{i}/Z_{G}^{i−1} are described in terms of higher limits of certain other functors Y_{L}^{Γ}, defined when L is quasisimple (i.e., L is perfect and L/Z(L) is simple) and Inn(L) ≤ Γ ≤ Aut(L). For any such L and Γ, let c: L ⟶ Γ denote the homomorphism which sends g ∈ L to conjugation by g, and define

Y_{L}^{Γ} : O_{p}(Γ)^{op} ⟶ Ab

by setting

Y_{Γ}^{L}(P) = \begin{cases}(c^{−1}P)^{P} /Z(L)^{P} & if P ∩ Inn(L) is p-centric in Inn(L) \\ 0 & otherwise . \end{cases}

For example, when L is simple and Γ = Inn(L), then Y_{L}^{Γ} = Z_{L}. In terms of these functors, we get

Proposition 3.3 ([O2, Theorem 3.3]). Assume H_{i−1} ◁ H_{i} ◁ G are normal subgroups, where H_{i}/H_{i−1} is a minimal normal subgroup of G/H_{i−1}, and define Z_{G}^{i−1} ⊆ Z_{G}^{i} as above. If H_{i}/H_{i−1} is abelian, or if there is a p-subgroup Q ≤ G such that [Q, H_{i}] ≤ H_{i−1} and C_{H_{i}}(Q) ≤ H_{i−1}, then the quotient functor Z_{G}^{i}/Z_{G}^{i−1} is acyclic (its higher limits vanish in degrees ≥ 1). Otherwise, there are a quasisimple group L such that H_{i}/H_{i−1} is a product of copies of the simple group L/Z(L) and a subgroup Γ ≤ Aut(L) containing Inn(L), together with homomorphisms

\underleftarrow{lim}^{n}_{O_{p}(Γ)}(Y_{L}^{Γ}) ⟶ \underleftarrow{lim}^{n}_{O_{p}(G)}(Z_{G}^{i}/Z_{G}^{i−1})

which are onto when n = 1 and isomorphisms when n ≥ 2.

Proposition 3.3 describes how the vanishing of higher limits of Z_{G} for arbitrary finite groups G is reduced to a question involving finite simple groups. More precisely, it involves the vanishing of higher limits of the functors YΓ ; L̃, when L̃ is a central extension of a simple group L and Γ is an extension of L by outer automorphisms. In fact, when p is odd, a more direct approach was found which does not involve these functors YΓ ; L̃, but they do still have to be studied when p = 2.

3.4. The odd primary case. When p is odd, then for any p-group P, we define 𝔛(P) ≤ P to be the largest subgroup for which there is a normal series

1 = Q_{0} ◁ Q_{1} ◁ ⋯ ◁ Q_{n} = 𝔛(P)

such that Q_{i} ◁ P for all i, and such that

[Ω_{1}(C_{P} (Q_{i−1})), Q_{i}; p−1] = 1 ∀ i = 1, …, n.

Here, for any p-group P, Ω_{1}(P) = ⟨g ∈ P | g^{p} = 1⟩. Also, for subgroups H, K ≤ G, [H, K; n] denotes the n-fold commutator [⋯ [[H, K], K] ⋯, K].

In the following proposition, when G is a group and S ∈ Syl_{p}(G), we say that a subgroup P ≤ S is weakly Aut(G)-closed in S if there is no other subgroup P ≠ P′ ≤ S which lies in the Aut(G)-orbit of P. Also, J_{e}(P) denotes the Thompson subgroup: the subgroup generated by all (maximal) elementary abelian p-subgroups of P of maximal rank.

Proposition 3.4 ([O1, Propositions 4.1 & 3.7]). For any odd prime p and any finite group G, Z_{G} is acyclic if for each nonabelian simple group L which occurs in the decomposition series of G, and any S ∈ Syl_{p}(L), there is a subgroup Q ≤ 𝔛(S) which is centric and weakly Aut(L)-closed in S. In particular, this always holds if J_{e}(S) ≤ 𝔛(S).

In fact, [O1, Proposition 4.1] is slightly stronger, in that 𝔛(S) in the above statement is replaced by a larger subgroup (but depending also on L). When p is odd, L is simple, and S ∈ Syl_{p}(L), then in almost all cases, either 𝔛(S) = S, or S contains a unique elementary abelian subgroup of maximal rank (and clearly J_{e}(S) ≤ 𝔛(S) when either of these happens). The only exceptions to this (i.e., the only cases where 𝔛(S) ≨ S and there is more than one elementary abelian subgroup of maximal rank) occur when p = 3 and L ≅ P SU_{n}(3^{k}); in which case the more restrictive hypothesis of Proposition 3.4 holds. In fact, no examples are known of a p-group P (for p odd) for which 𝔛(P) does not contain J_{e}(P).

3.5. Obstruction groups at the prime 2. Since \underleftarrow{lim}^{1}(Z_{G}) can be nonzero when p = 2, it is not surprising that this case is harder. We describe here some of the techniques used to prove, for L simple, that the higher limits of YΓ ; L̃ vanish in degrees ≥ 2 whenever L̃ is perfect, L̃/Z(L̃) ≅ L, and Γ ≤ Aut(L̃) contains Inn(L̃). To simplify the following discussion, we restrict attention to the case L̃ = L and Γ = Inn(L); i.e., the case where YΓ ; L̃ = Z_{L}.

In most cases, one first filters the functor Z_{L} by subfunctors

0 = F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{k−1} ⊆ F_{k} = Z_{L},

such that for each i there is a p-centric subgroup P_{i} ≤ L such that

(F_{i}/F_{i−1})(P) ≅ \begin{cases}Z(P) & if P conjugate P_{i} \\ 0 & otherwise. \end{cases}

We say that “P_{i} contributes to \underleftarrow{lim}^{k}(Z_{L})” if

\underleftarrow{lim}^{k}(F_{i}/F_{i−1}) ≠ 0.

Of course, via the long exact sequences which connect the higher limit of F_{i}, F_{i−1}, and F_{i}/F_{i−1}, the contribution of one subgroup to \underleftarrow{lim}^{k}(Z_{L}) can “cancel” the contribution of another subgroup to \underleftarrow{lim}^{k±1}(Z_{L}).

For simple groups L (and p = 2), there are some cases where a 2-subgroup contributes to \underleftarrow{lim}^{2}(Z_{L}). For example, this occurs when L is a projective special linear group P SL_{n}(2) for n ≥ 4, an alternating group A_{n} for n ≡ 0, 1 (mod 8), the Mathieu group M_{24}, or the Held group He. In all such cases, this contribution to \underleftarrow{lim}^{2}(Z_{L}) is cancelled by another subgroup contributing to \underleftarrow{lim}^{1}(Z_{L}). There do not seem to be any cases in which a 2-subgroup contributes to \underleftarrow{lim}^{i}(Z_{L}) for i ≥ 3.

In general, there are graded abelian groups Λ^{∗}(Γ; M), defined for each finite group Γ and each ℤ_{(p)}[Γ]-module M, such that

\underleftarrow{lim}^{∗}(F_{i}/F_{i−1}) ≅ Λ^{∗}(N_{L}(P)/P; Z(P))

[JMO, Lemma 5.4]. Thus P contributes to \underleftarrow{lim}^{i}(Z_{L}) if and only if Λ^{i}(N_{L}(P)/P; Z(P)) is nonvanishing. Some of the very nice properties of these functors include:

Proposition 3.5. (a) If p ∤ |Γ|, then Λ^{∗}(Γ; M) = \begin{cases}M^{Γ} & if ∗ = 0 \\ 0 & if ∗ > 0. \end{cases}

(b) Λ^{∗}(Γ; M) = 0 if p|| Ker[Γ ⟶ Aut(M)]|

(c) Λ^{∗}(Γ; M) = 0 if O_{p}(Γ) ≠ 1

(d) Λ^{∗}(Γ; M) ≅ H̃_{Γ}^{∗−1}(St(Γ); M) (where the Steinberg complex St(Γ) is the nerve of the poset of nontrivial p-subgroups of Γ)

(e) If |M| < ∞ and Λ^{n}(Γ; M) ≠ 0, then there are p-subgroups

1 = P_{0} ≨ P_{1} ≨ ⋯ ≨ P_{n} ≤ Γ

such that P_{i} ◁ P_{n} and P_{i} = O_{p}(N(P_{1}) ∩ ⋯ ∩ N(P_{i})) for all i, such that P_{n} ∈ Syl_{p}(N(P_{1}) ∩ ⋯ ∩ N(P_{n−1})); and such that 𝔽_{p}[P_{n}] ⊆ M (as a P_{n}-module). In particular, if M is finite, then rk_{p}(M) ≥ p^{n}.

Proof. See [JMO, Proposition 6.1(i,ii)] for points (a)–(c), [Gr] for point (d), and [O2, Proposition 4.4] for point (e). □

As one simple example, let V ≅ 𝔽_{2}^{2} be the faithful 2-dimensional representation of Σ_{3}. One can show that

Λ^{2}(Σ_{3} × Σ_{3}; V^{⊗2}) ≅ ℤ/2,

and hence that P ≤ L contributes to \underleftarrow{lim}^{2}(Z_{L}) if N(P)/P ≅ Σ_{3} × Σ_{3} and Z(P) ≅ V^{⊗2} as a module over N(P)/P. This situation does occur in some simple groups, such as P SL_{n}(2) for n ≥ 4, and the sporadic groups M_{24} and He. On the other hand, the simplest example where Λ^{3} ≠ 0 is the case

Λ^{3}(Σ_{3} × Σ_{3} × Σ_{3}; V^{⊗3}) ≠ 0,

and this does not seem to occur as a pair (N(P)/P, Z(P)) in any simple group.

4. Abstract finite fusion and linking systems

The results of the preceding section suggest that it may be possible to formulate a more general context in which to study the homotopy theory of classifying spaces. In other words, we would like to define an algebraic object for which the notion of a classifying space makes sense, and in which the p-local homotopy theory of that classifying space is intrinsic to the algebraic structure and vice versa.

The existence of these algebraic objects was predicted by Dave Benson in the mid-1990’s. In the introduction to his paper [Be1], where he relates the Dwyer-Wilkerson space BDI(4) to what he calls the “classifying spaces” of certain “nonexistent finite simple groups” (see Section 5.2), he writes: “This prompts the speculation that there should exist a theory of ‘p-local groups’ in which one only gives a Sylow p-subgroup and a fusion pattern. The fusion pattern should obey a set of axioms which are strong enough to be able to build a p-completed classifying space.”. Later, in some unpublished and privately distributed notes, Benson made this more precise by formulating some of the axioms of what we now call centric linking systems, and predicting that their p-completed realizations should function as classifying spaces of the fusion patterns.

The concept of a fusion system over a p-group S was in fact defined in the 1990’s by Lluis Puig [Pu3] with an entirely different purpose in mind. Fusion systems of finite groups are particular cases of these more general objects. The version presented here is a modification of his definition, but is completely equivalent to it.

In this section, we first present the definition of abstract (saturated) fusion systems, and then define what it means to be a centric linking system associated to an abstract fusion system. These concepts then lead to the definition of a p-local finite group and its classifying space.

4.1. Fusion systems. A fusion system F over a finite p-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.

The first requirement in the definition is intuitively obvious. It is less clear that the second requirement has to be stated as an axiom. However, it clearly holds when the fusion system is that of a finite group, and it is essential for the theory. Puig calls fusion systems satisfying (b) “divisible” fusion systems.

The next definitions may appear quite mysterious and are quite hard to motivate. The fusion system of a finite group turns out to satisfy an extra set of conditions, which basically makes the theory work. Puig has managed to distil precisely the necessary features in his definition of a saturated fusion system. Before we can explain what this means, we must distinguish certain collection of objects in F. By analogy with groups, two objects P and Q which are isomorphic in F are said to be F-conjugate.

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

A fusion system F is saturated if the following two conditions hold:

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

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

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

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

4.2. Centric linking systems. Before we can define a centric linking system associated to a given saturated fusion system, we need to explain what it means to be centric in this context.

Let F be any fusion system over a p-group S. A subgroup P ≤ S is F-centric if C_{S}(P′) = Z(P′) whenever P′ is F-conjugate to P.

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

π: L ⟶ F^{c},

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

(A) π is the identity on objects 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}

Condition (A) means that F^{c} is a quotient category of L, which is obtained by dividing out a free action of the centers of source objects. Conditions (B) and (C) ensure compatibility of the different ingredients with each other.

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

The first thing one wants to make sure of is that genuine finite groups give rise to plocal finite groups, which is indeed the case. For any finite group G and S ∈ Syl_{p}(G) the fusion system F_{S}(G) is saturated and L_{S}^{c}(G) is an associated centric linking system. Thus (S, F_{S}(G), L_{S}^{c}(G)) is a p-local finite group whose classifying space |L_{S}^{c}(G)|_{p}^{∧} is homotopy equivalent to BG_{p}^{∧}. It is interesting to point out that another consequence of the proof of the Martino-Priddy conjecture is that the natural centric linking system of a finite group G is (up to equivalence) the only one associated to the fusion system of G (see Section 4.8).

In the examples coming from finite groups it is clear that the nerve of the centric linking system is p-good, so that its p-completion is p-complete. It is one crucial ingredient among what makes it possible to reconstruct both the fusion system and the centric linking system from the classifying space of the corresponding p-local finite group. This is also the case for p-local finite groups; and in fact, we can also describe explicitly the fundamental group of the classifying space.

For any saturated fusion system F over a p-group S, define

O_{F}^{p}(S) = ⟨g^{−1}α(g) | g ∈ P ≤ S, α ∈ O^{p}(Aut_{F}(P))⟩ ◁ S.

This subgroup O_{F}^{p}(S) is the hyperfocal subgroup of F as defined by Puig [Pu4]. If F = F_{S}(G) is the fusion system of a finite group G with respect to S ∈ Syl_{p}(G), then the hyperfocal subgroup theorem [Pu2, §1.1] says that O_{F}^{p}(S) = S ∩ O^{p}(G) (where O^{p}(G) is the smallest normal subgroup of p-power index).

Theorem 4.1. Let (S, F, L) be any p-local finite group at the prime p. Then |L| is p-good. Also, the composite

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

induced by the inclusion B(S) ⟶^{θ_{S}} L, is surjective, and induces an isomorphism

π_{1}(|L|_{p}^{∧}) ≅ S/O_{F}^{p}(S).

Proof. See [BLO2, Proposition 1.11] and [BCGLO, §1]. □

4.4. Centralizers and mapping spaces. Our next task is to study mapping spaces from the classifying space of a finite p-group to that of a p-local finite group. The result is a generalization of Theorem 2.3, which we now describe. If G is a finite group and Q a finite p-group, then the set of components [BQ, BG_{p}^{∧}] is described in terms of homomorphisms from Q to G modulo conjugation in G. This motivates us to define, for any p-local finite group (S, F, L) and any p-group Q,

Rep(Q, F) = Hom(Q, S)/∼,

where ∼ is the equivalence relation defined by setting ρ ∼ ρ′ if there is some χ ∈ Iso_{F}(ρ(Q), ρ′(Q)) such that ρ′ = χ ∘ ρ.

Components of mapping spaces Map(BQ, BG_{p}^{∧}) are described in terms of p-com-pleted classifying spaces of centralizers. Thus before we can state the analogous result for p-local finite groups, we need to define what is meant by the centralizer fusion and linking systems. The original definition of the centralizer fusion system is due to Puig.

Fix a p-local finite group (S, F, L) and a subgroup Q ≤ S which is fully centralized in F. Define C_{F}(Q) to be the category whose objects are the subgroups of C_{S}(Q), and where

Hom_{C_{F} (Q)}(P, P′) = {φ ∈ Hom_{F}(P, P′)|∃φ ∈ Hom_{F}(P Q, P′Q), φ|_{P} = φ, φ|_{Q} = 1_{Q}}.

Define C_{L}(Q) to be the category whose objects are the C_{F}(Q)-centric subgroups of C_{S}(Q), and where Mor_{C_{L}(Q)}(P, P′) is the set of those φ ∈ Mor_{L}(P Q, P′Q) whose underlying homomorphisms are the identity on Q and send P into P′. By [BLO2, Proposition 2.5], the triple (C_{S}(Q), C_{F}(Q), C_{L}(Q)) is always itself a p-local finite group in this situation. When Q is fully normalized, there is an analogous construction of a “normalizer” p-local finite group (N_{S}(Q), N_{F}(Q), N_{L}(Q)) [BLO2, Lemma 6.2].

If G is a finite group, S ∈ Syl_{p}(G) and Q ≤ S is fully centralized in F = F_{S}(G) (i.e., C_{S}(Q) ∈ Syl_{p}(C_{G}(Q))), then C_{F}(Q) and C_{L}(Q) are isomorphic to F_{C_{S}(Q)}(C_{G}(Q)) and L_{C_{S}(Q)}^{c}(C_{G}(Q)) respectively. Also Rep(Q, F) ≅ Rep(Q, G). With this in mind all that is left is to translate the statement of Theorem 2.3 to the new terminology.

Theorem 4.2. Let (S, F, L) be a p-local finite group, and let f: BS ⟶ |L|_{p}^{∧} be the natural inclusion followed by completion. Then the following hold, for any p-group Q.

(a) The map

Rep(Q, F) ⟶^{≅} [BQ, |L|_{p}^{∧}],

defined by sending the class of ρ: Q ⟶ S to f ∘ Bρ, is a bijection.

(b) For any homomorphism ρ: Q ⟶ S such that ρQ is fully centralized in F,

Γ′_{L,ρQ}: |C_{L}(ρQ)|_{p}^{∧} ⟶^{≃} Map(BQ, |L|_{p}^{∧})_{Bρ}

is a homotopy equivalence.

In particular, Map(BQ, |L|_{p}^{∧}) is p-complete.

Proof. See [BLO2, Corollary 4.5 & Theorem 6.3]. □

Our original motivation for considering the centric linking system of a finite group G was as a tool for describing the monoid of self homotopy equivalences of BG_{p}^{∧}. The description in Section 2.7 can be extended directly to this more general situation. We refer to Sections 2.6 and 2.7 for definitions of isotypical equivalences, and the category Aut(C) of self equivalences of a category C.

Theorem 4.3. For any p-local finite group (S, F, L), the topological monoids Aut(|L|_{p}^{∧}) and |Aut_{typ}(L)| are equivalent in the sense that their classifying spaces are homotopy equivalent. In particular,

Out(|L|_{p}^{∧}) ≅ Out_{typ}(L)  and  π_{i}(Aut(|L|_{p}^{∧})) ≅ \begin{cases}\underleftarrow{lim}_{F}^{0}(Z) & if i = 1 \\ 0 & if i ≥ 2. \end{cases}

Proof. See [BLO2, Theorem 8.1]. □

4.5. Homology decompositions. One of the standard techniques when studying maps between p-completed classifying spaces of finite groups is to replace them by (the p-completion of) a homotopy colimit of simpler spaces. There are many ways of decomposing BG_{p}^{∧} [Dw], of which the two most frequently used are the following:

• The subgroup decomposition: BG is mod p equivalent to the homotopy direct limit, over the orbit category of G (Section 3.2), of the classifying spaces of its p-radical subgroups [JMO]. The same type of decomposition holds for the collections of the p-centric subgroups, or the p-centric and p-radical subgroups of G.

• The centralizer decomposition: BG is mod p equivalent to the homotopy direct limit, over the fusion category of nontrivial elementary abelian p-subgroups E ≤ G, of the classifying spaces of the centralizers C_{G}(E) [JM].

Both of these decompositions have analogs for classifying spaces of p-local finite groups.

For any p-local finite group (S, F, L), the subgroup decomposition of |L|_{p}^{∧} is taken over the orbit category of F. This is the category O(F) whose objects are the subgroups of S, and whose morphisms are defined by

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

Also, O^{c}(F) denotes the full subcategory of O(F) whose objects are the F-centric subgroups of S. If L is a centric linking system associated to F, then π̃ denotes the composite functor

π̃: L −↠^{π} F^{c} −↠ O^{c}(F).

There is a difference between the orbit category of a fusion system and the orbit category of a group. If G is a group and S ∈ Syl_{p}(G), then

Mor_{O_{S}(G)}(P, Q) ≅ Q\N_{G}(P, Q), while Mor_{O(F_{S}(G))}(P, Q) ≅ Q\N_{G}(P, Q)/C_{G}(P).

In general, of course, these can be very different; but if P is p-centric, they differ only by the action of the group C′_{G}(P) which is of order prime to p.

Let Top denote the category of spaces. Let C and D be small categories and let C ^{ϕ}→ D and C ^{F}→ Topbe functors. The left homotopy Kan extension of F along ϕ is a functor L_{ϕ}F : D → Topwith the property that

\underrightarrow{hocolim}_{C} F ≃ \underrightarrow{hocolim}_{D}(L_{ϕ}F).

Details on the construction and properties of L_{ϕ}F are given in [HV].

Proposition 4.4. Fix a saturated fusion system F and an associated centric linking system L, and let π̃: L ⟶ O^{c}(F) be the projection functor. Let

B̃: O^{c}(F) ⟶ Top

be the left homotopy Kan extension along π̃ of the constant functor L ⟶^{∗} Top. Then B̃ is a homotopy lifting of the homotopy functor P ↦ BP, and

|L| ≃ \underrightarrow{hocolim}_{O^{c}(F)}(B̃).  (1)

Proof. See [BLO2, Proposition 2.2]. The proof is mostly a formality, following from elementary properties and the construction of homotopy Kan extensions. □

The centralizer decomposition of |L| is also mostly a formality in this context. Recall the definition of centralizer fusion and linking systems in Section 4.4: for any p-local finite group (S, F, L) and any fully centralized subgroup Q ≤ S, (C_{S}(Q), C_{F}(Q), C_{L}(Q)) is again a p-local finite group. Also, for any fusion system F over a p-group S, we let F^{e} denote the full subcategory of F whose objects are the nontrivial elementary abelian p-subgroups of S which are fully centralized in F.

Theorem 4.5. Fix a p-local finite group (S, F, L). Then there is a functor

C̃: F^{e} ⟶ Top,

which is a homotopy lifting of the homotopy functor E ↦ |C_{L}(E)|, such that

\underrightarrow{hocolim}_{E∈F^{e}}(C̃) ≃ |L|.

Proof. See [BLO2, Theorem 2.6]. The functor C̃ is defined explicitly there as a left Kan extension. □

4.6. Cohomology of the classifying space. Arguably one of the most fundamental theorems in group cohomology is the statement that if G is a finite group and S ∈ Syl_{p}(G), then for every p-local G-module M, H^{∗}(G, M) is the module of “stable elements” in H^{∗}(S, M); i.e., the module of elements stable with respect to fusion in G. In our context, if we restrict attention to trivial p-local coefficients, then the same statement holds for classifying space |L|_{p}^{∧} of a p-local finite group. As one consequence of this result, H^{∗}(|L|_{p}^{∧}; 𝔽_{p}) is noetherian for any p-local finite group (S, F, L).

For any fusion system F over a p-group S, let H^{∗}(F; 𝔽_{p}) be the subring of H^{∗}(BS; 𝔽_{p}) consisting of those elements which are stable under all fusion in F; i.e.,

H^{∗}(F; 𝔽_{p}) = {x ∈ H^{∗}(BS; 𝔽_{p}) | α^{∗}(x) = x|_{BP} , all α ∈ Hom_{F}(P, S)} .

The proof that H^{∗}(|L|; 𝔽_{p}) ≅ H^{∗}(F; 𝔽_{p}), when (S, F, L) is a p-local finite group, is based on the construction of a certain (S, S)-biset: a set which has left and right actions of S which commute with each other. If P ≤ S and φ ∈ Hom(P, S), let S ×_{(P,φ)} S denote the biset

S ×_{(P,φ)} S = (S × S)/∼, where (x, gy) ∼ (xφ(g), y) for x, y ∈ S, g ∈ P ,

and set

[S ×_{(P,φ)} S] = (H^{∗}(BS) ⟶^{φ^{∗}} H^{∗}(BP) ⟶^{trf_{P}} H^{∗}(BS)) ∈ End(H^{∗}(BS; 𝔽_{p})).

Here, trf_{P} denotes the transfer map. If B is a disjoint union of bisets B_{i} of this form, we let [B] be the sum of the endomorphisms [B_{i}]. If B is an (S, S)-biset, then for P ≤ S and φ ∈ Inj(P, S), we let B|_{(P,S)} denote the restriction of B to a (P, S)-biset, and let B|_{(φ,S)} denote the (P, S)-biset where the left P -action is induced by φ.

Proposition 4.6 ([BLO2, Proposition 5.5]). For any saturated fusion system F over a p-group S, there is an (S, S)-biset Ω with the following properties:

(a) Ω is a disjoint union of bisets of the form S×_{(P,φ)}S for P ≤ S and φ ∈ Hom_{F}(P, S).

(b) For each P ≤ S and each φ ∈ Hom_{F}(P, S), Ω|_{(P,S)} and Ω|_{(φ,S)} are isomorphic as (P, S)-bisets.

(c) |Ω|/|S| ≡ 1 (mod p).

Furthermore, for any biset Ω which satisfies these properties, [Ω] is an idempotent in End(H^{∗}(BS; 𝔽_{p})), and

Im[H^{∗}(BS; 𝔽_{p}) ⟶^{[Ω]} H^{∗}(BS; 𝔽_{p})] = H^{∗}(F; 𝔽_{p}).

It was Markus Linckelmann and Peter Webb who first formulated conditions (a), (b), and (c) in the above proposition, and who saw the significance of finding a biset with these properties.

Theorem 4.7. For any p-local finite group (S, F, L), the natural homomorphism

H^{∗}(|L|_{p}^{∧}; 𝔽_{p}) ⟶^{≅} H^{∗}(F; 𝔽_{p}),

induced by the inclusion of BS in |L|, is an isomorphism. Furthermore, the ring H^{∗}(|L|_{p}^{∧}; 𝔽_{p}) is noetherian.

Proof. See [BLO2, Theorem 5.8]. The idea is to use a certain decomposition theorem for unstable algebras over the Steenrod algebra, due to Dwyer and Wilkerson [DW1, Theorem 1.2]. Their theorem implies both that H^{∗}(F; 𝔽_{p}) is the inverse limit of the cohomology rings H^{∗}(|C_{L}(E)|; 𝔽_{p}) as E runs over the elementary abelian p-subgroups 1 ≠ E ≤ S, and also that higher derived functors of this inverse system vanish. Since |L|_{p}^{∧} is the homotopy direct limit of spaces homotopy equivalent to |C_{L}(E)|_{p}^{∧} (Theorem 4.5), we can then conclude from the spectral sequence of a homotopy colimit that H^{∗}(F; 𝔽_{p}) ≅ H^{∗}(|L|; 𝔽_{p}). One of the requirements when applying the Dwyer-Wilkerson theorem is that the homomorphism

H^{∗}(F; 𝔽_{p}) ⟶ H^{∗}(BS; 𝔽_{p})

be a split monomorphism, and the splitting is provided by Proposition 4.6.

The argument just sketched is carried out inductively, since we need to assume the theorem holds for the centralizers. This means that a separate argument is needed for fusion systems F with nontrivial center; i.e., those for which F = C_{F}(E) (and hence L = C_{L}(E)) for some 1 ≠ E ≤ S. □

The (S, S)-biset Ω of Proposition 4.6, associated to a given saturated fusion system F over S, can also be used to construct a spectrum associated to F which is equivalent to the suspension spectrum of any classifying space for F (if any exists). Any finite (S, S)-biset induces, via inclusions and transfer maps, a stable map from the suspension spectrum Σ^{∞}BS to itself. The properties of Ω listed in Proposition 4.6 imply that the induced map [Ω] is an idempotent in the ring of homotopy classes of maps from Σ^{∞}BS to itself, and that the image of H^{∗}([Ω]; 𝔽_{p}) is equal to H^{∗}(F; 𝔽_{p}). The infinite mapping telescope of [Ω] is thus a stable summand of BS whose mod p cohomology is isomorphic to H^{∗}(F; 𝔽_{p}). Hence by Theorem 4.7, if |L|_{p}^{∧} is any classifying space for F, then this stable summand of Σ^{∞}(BS) is homotopy equivalent as a spectrum to Σ^{∞}(|L|_{p}^{∧}).

4.7. Fusion and linking systems determined by their classifying space. A priori, one might think that the classifying space of a p-local finite group (S, F, L) should contain only part of the information given by the fusion and linking systems F and L. But in fact, one can recover both categories from the homotopy type of |L|_{p}^{∧}. This is done with the help of the functors from spaces to categories described in Section 2.5.

Theorem 4.8. Fix a p-local finite group (S, F, L), and let f: BS ⟶ |L|_{p}^{∧} be the natural inclusion. Then there are equivalences of categories

F = F_{S,f}(|L|_{p}^{∧}) and L ≅ L_{S,f}^{c}(|L|_{p}^{∧}).

Proof. See [BLO2, Proposition 7.3]. The proof is very similar to that of Theorem 2.4. □

In other words, if (S, F, L) and (S′, F′, L′) are two p-local finite groups and |L|_{p}^{∧} ≃ |L′|_{p}^{∧}, then (S, F, L) and (S′, F′, L′) are isomorphic as triples, via isomorphisms of groups and categories which commute with all of the structures which link them. To see how this follows from Theorem 4.8, note that by Theorem 4.2, for any homotopy equivalence |L|_{p}^{∧} \begin{matrix}ψ \\ ≃ \end{matrix}→ |L′|_{p}^{∧}, there is an isomorphism S α ; ≅→ S′ such that the following square commutes up to homotopy:

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

Here, f and f′ are the natural inclusions.

We also note here that for any given fusion system F over a p-group S, there are bijective correspondences

\{ \begin{matrix}linking \\ systems \\ assoc. to F \end{matrix} \} ≅ \{ \begin{matrix}classifying \\ spaces \\ for F \end{matrix} \} ≅ \{ \begin{matrix}liftings of the \\ homotopy functor \\ P ↦ BP \end{matrix} \}

which are given as follows:

\xymatrix{L \ar[dr]^{|−|_{p}^{∧}} \ar[rr]^{Lπ̃(∗)} &  & B̃ \ar[dl]^{hocolim} \\  & X \ar[ul]^{L_{S,θ}^{c}(−)}}

More precisely, the first set (from the left) consists of all linking systems associated to F up to isomorphism (isomorphisms of categories which commute with the projections to F and the distinguished monomorphisms). The second set contains all classifying spaces (p-completed nerves of linking systems associated to F), modulo the relation that |L|_{p}^{∧} and |L′|_{p}^{∧} are equivalent if there is a homotopy equivalence between them which commutes (up to homotopy) with the natural inclusions of BS. The third set consists of all functors from the orbit category O^{c}(F) to spaces which lift the homotopy functor P ↦ BP, modulo natural homotopy equivalences of functors to Top. The bijection between the first two sets follows from Theorem 4.8, and the commutativity of the triangle involving Kan extension, homotopy colimit, and | − |_{p}^{∧} was shown in Proposition 4.4. It remains to check that each lifting of (P ↦ BP) is the left Kan extension of some linking system, and this is shown in the proof of [BLO2, Proposition 2.3], where an explicit procedure is given for constructing a linking system associated to any given homotopy lifting.

It is also worth noting that the obstructions to lifting a given saturated fusion system F to centric linking system L described in Section 4.8 below coincide with those for lifting homotopy functors on the orbit category associated to F described by Dwyer and Kan [DK]. It was this observation which first suggested to us the above bijections, and in fact which first suggested to us the idea that two groups have equivalent p-completed classifying spaces if and only if their linking systems are equivalent.

4.8. Existence and uniqueness of linking systems or classifying spaces. One of the main open questions in this subject is that of the existence and uniqueness of linking systems associated to a given saturated fusion system. The obstructions for these problems are well understood, and are closely related to those for the Martino-Priddy conjecture as discussed in Section 2.

Fix a saturated fusion system F over a p-group S. Define a functor

Z_{F}: O^{c}(F)^{op} ⟶ Ab

by setting Z_{F}(P) = Z(P) = C_{S}(P) for each F-centric subgroup P ≤ S. The obstruction to the existence of a centric linking system associated to F lies in \underleftarrow{lim}^{3}(Z_{F}), and the obstruction to its existence lies in \underleftarrow{lim}^{2}(Z_{F}) [BLO2, Proposition 3.1]. This is completely analogous to the obstructions to the existence and uniqueness of an extension of a group G by another group K which acts on G via outer automorphisms: obstructions which lie in H^{3}(K; Z(G)) and H^{2}(K; Z(G)), respectively.

One way to see the obstruction to the existence of associated linking systems is to construct directly a “category” which satisfies all of the conditions for a linking system, except that the composition of morphisms need not be associative. This can be done in such a way that the failure of associativity assigns to each triple of composable morphisms P_{0} → P_{1} → P_{2} → P_{3} in O^{c}(F) an element of Z(P_{0}), and these elements combine to form a 3-cocycle with coefficients in Z_{F}.

We do have some results about the existence and uniqueness of associated linking systems in various special cases. For example,

Proposition 4.9. Let F be a saturated fusion system over a p-group S. Then there exists a linking system associated to F if rk(S) < p^{3}, and the linking system is unique if rk(S) < p^{2}.

Proof. [BLO2, Corollary 3.4]. The point of this proof is that the corresponding higher limit obstruction groups for existence and uniquness vanish under the hypotheses of the proposition. □

Note that the Martino-Priddy conjecture is a special case of the problem of uniqueness of centric linking systems. The proof of this conjecture, as sketched in Section 3, shows in fact that there is exactly one linking system associated to the fusion system of a finite group. If we still hope to find a proof of this conjecture which is independant of the classification of finite simple groups, then this is not only for esthetic reasons, but also because it seems likely that any such proof could extend to a proof of the existence and uniqueness of linking systems (hence classifying spaces) associated to any given saturated fusion system.

5. Examples and methods of construction

One of the weak points in our investigations into the theory of p-local finite groups is the problem of constructing “exotic” examples: examples which do not come from finite groups. The key difficulty seems to be that of showing that newly constructed fusion systems are saturated. Some attempts to develop general techniques for constructing saturated fusion systems are described in Section 5.1. We then describe some which arise more “naturally”, notably those of the type studied by Solomon and Benson, and those which come from block theory. We finish this section with a discussion of “extensions” of p-local finite groups: a subject which is still to a large extent under development.

We first note the following general result which is very useful when constructing saturated fusion systems. Proposition 5.1 ([BCGLO, Theorem 2.3]). Let F be a fusion system over a p-group S. Assume the following hold:

(I) If P ≤ S is F-centric and fully normalized in F, then Aut_{S}(P) ∈ Syl_{p}(Aut_{F}(P)).

(II) For each F-centric subgroup P ≤ S and each φ ∈ Hom_{F}(P, S), if we set

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

then φ extends to some φ ∈ Hom_{F}(N_{φ}, S).

Let F′ ⊆ F be the subcategory with the same objects, and whose morphisms are the composites of restrictions of morphisms in F between F-centric subgroups. Then F′ is saturated.

In fact, [BCGLO, Theorem 2.3] is formulated more generally, and deals with fusion systems which satisfy the axioms of saturation only on subgroups which are both Fcentric and F-radical (though an extra axiom is then needed). But the case formulated above seems to be the most important one.

5.1. Construction of exotic p-local finite groups. The following theorem is the only result we know which gives a geometric criterion for showing that a fusion system is saturated. If S is a finite p-group, then we say that a map BS ⟶^{f} X is Sylow if every map BP ⟶ X, for a p-group P, factors through f up to homotopy.

Theorem 5.2. Fix a space X, a p-group S, and a map f: BS ⟶ X. Assume that

(a) f is Sylow, and

(b) f|_{BP} is a centric map for each F_{S,f}(X)-centric subgroup P ≤ S.

Let F′ ⊆ F_{S,f}(X) be the subcategory with the same objects, and whose morphisms are the composites of restrictions of morphisms in F_{S,f}(X) between F-centric subgroups. Then F′ is saturated, and the triple (S, F′, L_{S,f}^{c}(X)) is a p-local finite group.

Proof. See [BLO4]. One uses direct geometric arguments to show that F_{S,f}(X) satisfies the axioms of saturation on centric subgroups, and then applies Proposition 5.1. □

One might expect that it is difficult to find interesting examples of maps BS ⟶ X which satisfy the centricity condition (b) above. But in fact, with the help of [BLO2, Proposition 4.2], which says that homotopy colimits commute with mapping spaces Map(BP, −) under certain conditions, spaces X can be constructed which satisfy the above conditions without being completed classifying spaces of finite groups.

The following is one example of how Theorem 5.2 can be applied. If F_{0} is a fusion system over a p-group S, and for each i = 1, …, m we are given subgroups Q_{i} ≤ S and groups of automorphisms Δ_{i} ≤ Out(Q_{i}), then we let

F =^{def} ⟨F_{0}; F_{Q_{1}}(Δ_{1}), …, F_{Q_{m}}(Δ_{m})⟩

be the fusion system over S defined as follows. For each pair of subgroups P, P′ ≤ S, Hom_{F}(P, P′) is the set of composites

P = P_{0} ⟶^{φ_{1}} P_{1} ⟶^{φ_{2}} P_{2} ⟶ ⋯ ⟶ P_{k−2} ⟶^{φ_{k−1}} P_{k−1} ⟶^{φ_{k}} P_{k} = P′,

where for each i, either φ_{i} ∈ Hom_{F_{0}}(P_{i−1}, P_{i}); or P_{i−1}, P_{i} ≤ Q_{j} for some j, and φ_{i} = α|_{P_{i−1}} for some α ∈ Aut(Q_{j}) with [α] ∈ Δ_{j}.

Proposition 5.3. Fix a finite group G, a Sylow p-subgroup S ≤ G, and subgroups Q_{1}, …, Q_{m} ≤ S such that no Q_{i} is conjugate to a subgroup of Q_{j} for i ≠ j. Set K_{i} = Out_{G}(Q_{i}), and fix subgroups Δ_{i} ≤ Out(Q_{i}) which contain K_{i}. Assume for each i that

(1) p ∤ [Δ_{i}:K_{i}];

(2) Q_{i} is p-centric in G, but for each P ≨ Q_{i} there is α ∈ Δ_{i} such that α(P) is not p-centric in G; and

(3) for all α ∈ Δ_{i}∖K_{i}, K_{i} ∩ αK_{i}α^{−1} has order prime to p.

Then the fusion system F =^{def} ⟨F_{S}(G); F_{Q_{1}}(Δ_{1}), …, F_{Q_{m}}(Δ_{m})⟩ is saturated, and has an associated centric linking system.

Proof. See [BLO4]. One first checks that for each i, there is a unique extension G_{i} of Q_{i} by Δ_{i} which contains N_{G}(Q_{i}) (up to isomorphism). Let X be the union of BG and the BG_{i}, where each BG_{i} is attached to BG along their common subspace BN_{G}(Q_{i}). The proposition then follows from Theorem 5.2, applied to the space X_{p}^{∧}. Condition (2) guarantees that the fusion system of X_{p}^{∧} is generated by its restriction to centric subgroups; i.e., that F′ = F_{S,f}(X_{p}^{∧}) in the notation of Theorem 5.2. □

Proposition 5.3 is a generalization of [BLO2, Proposition 9.1]. Some more concrete applications of that result, for primes p ≥ 5, are given in [BLO2, §9]: constructions which can be thought of (very roughly) as mixing features of the fusion systems of two different groups.

The above proposition can also be applied when p = 3, to construct exotic 3-local finite groups with Sylow subgroup the groups

S_{±} = ⟨a, b, x | [a, b] = 1, xax^{−1} = ab, xbx^{−1} = ba^{∓3}⟩

of order 81. We do not yet know whether it can be used to construct any exotic 2-local finite groups.

In all of our examples, except those described in the next section, the proof that a saturated fusion system is not the fusion system of a finite group always involves using the classification of finite simple groups.

5.2. Fusion systems of a type studied by Solomon. Well before Puig formulated the axioms defining a saturated fusion system, Ron Solomon had essentially found one which does not arise from the p-fusion system of any finite group. This was a bi-product of one step in the classification of finite simple groups [Sol]. Solomon considered the problem of classifying all finite simple groups whose Sylow 2-subgroups are isomorphic to those of the Conway group Co_{3}. The end result of his paper was that Co_{3} is the only such group. In the process of proving this, he needed to consider groups G in which all involutions are conjugate, and such that the centralizer of each involution contains a normal subgroup isomorphic to Spin_{7}(q) with odd index, where q is an odd prime power. Solomon showed that such a group G does not exist. However, the 2-local structure that he found turned out to be perfectly consistent. It was only by analyzing its interaction with the p-local structure (where p is the prime of which q is a power) that he found a contradiction.

In a later paper [Be1], Dave Benson, inspired by Solomon’s work, constructed certain spaces which can be thought of as the 2-completed classifying spaces which the groups studied by Solomon would have if they existed. Benson’s construction was arguably the first indication of the existence of spaces which “behave” like p-completed classifying spaces of finite groups, but which are themselves not of this form. To construct these spaces he started with the spaces BDI(4) constructed by Dwyer and Wilkerson having the property that

H^{∗}(BDI(4); 𝔽_{2}) ≅ 𝔽_{2}[x_{1}, x_{2}, x_{3}, x_{4}]^{GL_{4}(2)}

(the rank four Dickson algebra at the prime 2). He then considered, for each odd prime power q, the homotopy fixed point set of the ℤ-action on BDI(4) generated by an “Adams operation” ψ^{q} constructed by Dwyer and Wilkerson. Denote this homotopy fixed point set by BDI_{4}(q).

In [LO], the second and third authors have shown that for each odd prime power q, there is a saturated fusion system F = F_{Sol}(q) over a Sylow 2-subgroup S = S(q) ≤ Spin_{7}(q), with the properties that all involutions in S are F-conjugate, and that when z ∈ Z(S) is the generator, then C_{F}(z) is the fusion system of Spin_{7}(q). The obstruction theory described in Section 4.8 applies to show that there is a unique centric linking system L_{Sol}^{c}(q) associated to F_{Sol}(q). We thus get a 2-local finite group (S(q), F_{Sol}(q), L_{Sol}^{c}(q)), which by Solomon’s theorem (together with some other group theoretic results) cannot be associated to any finite group.

The classifying spaces BSol(q) =^{def} |L_{Sol}^{c}(q)|_{2}^{∧} turn out to be equivalent to the spaces constructed by Benson in [Be1]. For fixed q, one can take the union of the spaces BSol(q^{n}) for all n (strictly speaking this is not true as stated, but it can be done up to homotopy), and the 2-completion of this union is homotopy equivalent to BDI(4).

Conversely, if ψ^{q} ∈ Aut(BDI(4)) is an “Adams map” (its restriction to a maximal torus is induced by the Frobenius automorphism (x ↦ x^{q}) on the algebraic closure 𝔽_{q}), then BSol(q) is homotopy equivalent to BDI_{4}(q) — the homotopy fixed point set of ψ^{q} when regarded as an action of the monoid ℕ. For more details, see [LO, §4].

Other “exotic” p-local finite groups have been constructed by the first author together with Møller [BM]. Their method resembles Benson’s construction of BDI_{4}(q). Namely, they first take homotopy fixed point sets of actions of Adams maps on certain p-compact groups, and then show that these are the classifying spaces of p-local finite groups.

5.3. The fusion system of a block. Part of the motivation for constructing classifying spaces for fusion systems, and part of the reason in general for looking at abstract fusion and linking systems, comes from representation theory. Fix a finite group G, a prime p, and an algebraically closed field k of characteristic p. A block in k[G] is a factor in the maximal decomposition of k[G] as a product of rings, or equivalently a minimal central idempotent. Alperin and Broué [AB] defined, for any block b, inclusion and conjugacy relations among the Brauer pairs associated to b (the “b-subpairs”); and Puig [Pu3] showed that these satisfy the axioms of a saturated fusion system over the defect group of the block. We refer to [AB] or [Alp] for more details (including the definitions of defect groups and Brauer pairs). The existence of a (unique or canonical) linking system associated to the fusion system of a block would thus imply the existence of a classifying space for the block, which might in turn have implications in representation theory (see [Li]).

5.4. Extensions of p-local finite groups. We describe here some work in progress which will appear in [BCGLO]; work which in certain specialized situations allows us to study extensions of p-local finite groups. It is still unclear how to define such extensions in general.

One of the reasons for this difficulty is that an extension of (finite) groups need not induce a fibration sequence of their p-completed classifying spaces. For example, if 1 → H → G → K → 1 is an extension where K has order prime to p, then BK_{p}^{∧} is contractible, while BH_{p}^{∧} and BG_{p}^{∧} need not be homotopy equivalent (since H and G need not have the same cohomology mod p). Two cases where an extension does induce a fibration sequence of p-completed classifying spaces are those where the quotient group is a p-group, and the case of central extensions. It is this last case which is the simplest to consider.

If F is a fusion system over a p-group S, then a subgroup A ≤ S is called central if C_{F}(A) = F; i.e., if each φ ∈ Hom_{F}(P, Q) extends to a morphism φ ∈ Hom_{F}(P A, QA) which is the identity on A. Clearly, if A is central in F, then A ≤ Z(S). In such a situation, there is an obvious way to define a quotient fusion system F/A over S/A, by letting Hom_{F/A}(P/A, Q/A) be the image in Hom(P/A, Q/A) of Hom_{F}(P, Q), and this is saturated by [BLO2, Lemma 5.6]. If, furthermore, L is a centric linking system associated to F, then there is a canonical way to construct a centric linking system L/A associated to F/A ([BLO2, Lemma 5.6] again). A central extension of a p-local finite group (S, F, L) by an abelian group A can now be defined as a p-local finite group (S̃, F̃, L̃), together with an isomorphism A ≅ A′ ≤ S̃ where A′ is central in F̃, such that (S̃/A, F̃/A, L̃/A) ≅ (S, F, L).

Theorem 5.4. Central extensions of a p-local finite group (S, F, L) by a finite abelian p-group A are in one-to-one correspondence with principal fibrations

BA → X → |L|_{p}^{∧},

and also in natural one-to-one correspondence with H^{2}(|L|_{p}^{∧}; A).

Proof. See [BCGLO, §8]. □

We next look at extensions where the quotient group is a p-group.

Theorem 5.5. Let (S, F, L) be a p-local finite group. Let λ: π_{1}(|L|_{p}^{∧}) −↠ π be any surjection of groups, and set S_{0} = Ker[S −↠ π_{1}(|L|_{p}^{∧}) −↠ π]. Let f: |L|_{p}^{∧} ⟶ Bπ be the classifying map for λ, and let X be its homotopy fiber. Then there is a p-local finite group (S_{0}, F_{0}, L_{0}), where F_{0} is a subcategory of F, such that |L_{0}|_{p}^{∧} ≃ X.

Conversely, assume that |L_{0}|_{p}^{∧} ⟶ X ⟶ Bπ is a fibration sequence, where the fiber is the classifying space of a p-local finite group (S_{0}, F_{0}, L_{0}). Then X ≃ |L|_{p}^{∧} for some p-local finite group (S, F, L), where S_{0} ◁ S, F_{0} ⊆ F, and S/S_{0} ≅ π.

Proof. See [BCGLO, §5]. □

Note that in the above proposition, we do not say that the centric linking system L_{0} is a subcategory of L. In fact, it is not in general a subcategory, since F_{0}-centric subgroups of S_{0} need not be F-centric. This helps to illustrate another of the key problems in this subject: the lack of a good concept of subobjects (or morphisms), since even group inclusions and group homomorphisms do not in general send p-centric subgroups to p-centric subgroups.

We can also describe “extensions” where the quotient is a group of order prime to p. For the purpose of the following theorem, for any saturated fusion system F over a p-group S, we say that a saturated fusion subsystem F′ ⊆ F over S has index prime to p if

Aut_{F′}(P) ≥ O^{p′}(Aut_{F}(P))

for each P ≤ S. Here, for any group G, O^{p′}(G) is the largest normal subgroup of index prime to p; i.e., the subgroup generated by the Sylow p-subgroups of G.

Theorem 5.6. Fix a saturated fusion system F over a p-group S. Then there is a normal subgroup Out_{F}^{0}(S) ◁ Out_{F}(S) of index prime to p, and a map

θ̂: Mor(F^{c}) ⟶ Out_{F}(S)/ Out_{F}^{0}(S)

with the following properties:

(a) θ̂(β ∘ α) = θ̂(β)·θ̂(α) for each composable pair of morphisms β, α in F^{c}.

(b) The restriction of θ̂ to Aut_{F}(S) is the natural surjection.

(c) θ̂ sends inclusions to the identity.

(d) For each subgroup T ≤ Out_{F}(S)/ Out_{F}^{0}(S), the subcategory F_{T} ⊆ F with the same objects and whose morphisms are generated by restrictions of morphisms in θ̂^{−1}(T), is a saturated fusion subsystem of F of index prime to p.

(e) Each saturated fusion subsystem F′ ⊆ F of index prime to p is equal to F_{T} for some subgroup T ≤ Out_{F}(S)/ Out_{F}^{0}(S).

(f) If L is a centric linking system associated to F with projection functor π: L ⟶ F, and if T ≤ Out_{F}(S)/ Out_{F}^{0}(S) is a subgroup, then L_{T} =^{def} π^{−1}(F_{T}) ⊆ L is a centric linking system associated to F_{T}.

Proof. See [BCGLO, §6], where the subgroup Out_{F}^{0}(S) is defined explicitly. (In fact, it is the smallest subgroup of Out_{F}(S) for which a map θ̂ exists satisfying conditions (a–c).) Note, in point (f), that if T ◁ Out_{F}(S)/ Out_{F}^{0}(S) is a normal subgroup with quotient group σ, then |L_{T} |_{p}^{∧} ⟶ |L|_{p}^{∧} ⟶ Bσ is not in general a fibration sequence.

□

The map θ̂ of Theorem 5.6 cannot be extended to arbitrary morphisms in F, and still satisfy conditions (a–c). For example, any such extension would have to send the inclusion 1 ⟶ S to the identity in Out_{F}(S)/ Out_{F}^{0}(S) — and Condition (a) would then imply that all automorphisms of S also get sent to the identity.

6. p-local compact groups

In very recent work still in progress, we have begun to extend the results on p-local finite groups to a more general situation, one which includes p-completed classifying spaces of compact Lie groups and classifying spaces of p-compact groups. One of our hopes is that this will provide a new tool to study certain maps between these spaces, in particular self equivalences of these spaces.

6.1. Discrete p-toral groups. A p-toral group is a compact Lie group P whose identity component is a torus T, such that P/T ≅ π_{0}(P) is a finite p-group. By contrast, a discrete p-toral group is a discrete group P, with normal subgroup P_{0} ◁ P, such that P_{0} is isomorphic to a finite product of copies of ℤ/p^{∞} (= ℤ[\frac{1}{p}]/ℤ) and P/P_{0} is a finite p-group. Any discrete p-toral group P contains a unique minimal subgroup P_{0} of finite index, which we call its connected component. Also, P_{0} ≅ (ℤ/p^{∞})^{r} for some r =^{def} rk(P) = rk(P_{0}). We define |P | = (rk(P), |P/P_{0}|) ∈ ℕ × ℕ, and order the elements of ℕ × ℕ lexicographically. This ordering of the “orders” of discrete p-toral groups is what allows us to adapt with very little change the definition of a saturated fusion system over a finite p-group to this new situation.

The motivation for studying fusion and linking systems over discrete p-toral groups comes from regarding them as “discrete approximations” of p-toral groups. The p-toral subgroups of compact Lie groups play the same role as p-subgroups of finite groups (more on this below).

Proposition 6.1. Each p-toral group P contains a dense subgroup P_{δ} ≤ P which is a discrete p-toral group of the same rank, and any two such subgroups are conjugate in P. Furthermore, the inclusion induces an isomorphism H^{∗}(BP; 𝔽_{p}) ≅ H^{∗}(BP_{δ}; 𝔽_{p}), and hence a homotopy equivalence (BP_{δ})_{p}^{∧} ≃ BP _{p}^{∧}.

Proof. See [DW3, Proposition 6.9]. Let T ≤ P be the identity connected component, and let T_{δ} ≤ T be the subgroup of elements of p-power order. Clearly, T_{δ} is the unique discrete p-toral subgroup of T of the same rank, and there is a bijective correspondence between dense discrete p-toral subgroups of P which contain T_{δ} and splittings of the group extension 1 → T/T_{δ} → P/T_{δ} → P/T → 1. Since P/T is a p-group and T/T_{δ} is uniquely p-divisible, H^{i}(P/T; T/T_{δ}) = 0 for all i > 0, and hence the above group extension has a splitting which is unique up to conjugation.

The proof that H^{∗}(BP; 𝔽_{p}) ≅ H^{∗}(BP_{δ}; 𝔽_{p}) is easily reduced to the case where P ≅ S^{1} and P_{δ} ≅ ℤ/p^{∞}, and this result is classical. □

One advantage in working with discrete p-toral groups rather than p-toral groups is that all subgroups of discrete p-toral groups are again discrete p-toral groups, unlike the case for compact p-toral groups. But it’s also simpler to work with discrete groups, and not have to worry about their topology.

6.2. Fusion and linking systems over discrete p-toral groups. 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.

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

The fusion system F is saturated if the following three conditions hold.

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

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

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

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

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

We note here that Out_{S}(P) is a finite p-group for any pair P ≤ S of discrete p-toral groups. So condition (I) includes the condition that Out_{F}(P) is finite for all P ≤ S. The only real difference between this new definition and the definition of a saturated fusion system over a finite p-group is thus condition (III), which can be thought of as a “continuity” condition.

When S is a discrete p-toral group and F is a saturated fusion system over S, then the definitions of F-centric subgroups of S, and of centric linking systems associated to F, are identical to the definitions given in Section 4 above when S is a finite pgroup. A p-local compact group is now defined to be a triple (S, F, L), where S is a discrete p-toral group, F is a saturated fusion system over S, and L is a linking system associated to F. The classifying space of such a triple (S, F, L) is the p-completed nerve |L|_{p}^{∧}. Thus, as in the case for fusion and linking systems over finite p-groups, a linking system for a fusion system F over a discrete p-toral group S can be thought of as a means of associating a classifying space to F.

6.3. Reduction to finite subcategories. The main difficulty when generalizing results about p-local finite groups to the discrete p-toral case is that the categories are no longer finite. It is, however, possible to replace any linking system L associated to a saturated fusion system F over a discrete p-toral group S, by a finite full subcategory L_{0} ⊆ L such that |L_{0}|_{p}^{∧} ≃ |L|_{p}^{∧}. This is done by restricting L to those F-centric subgroups which are also F-radical; i.e., those P ≤ S for which Out_{F}(P) contains no nontrivial normal p-subgroup.

Proposition 6.2. Let F be a saturated fusion system over a discrete p-toral group S, and let L be an associated linking system to F. Then S contains only finitely many Fconjugacy classes of subgroups which are both F-centric and F-radical. Let L_{0} ⊆ L be any finite full subcategory whose objects include F-conjugacy class representatives for all F-radical F-centric subgroups of S. Then the inclusion |L_{0}|_{p}^{∧} ⊆ |L|_{p}^{∧} is a homotopy equivalence.

Proof. See [BLO3, Corollary 3.5 & §5]. □

6.4. A homology decomposition of the classifying space. The orbit category of a fusion system F over a discrete p-toral group S is defined exactly as before: it has the same objects as F, and

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

Also, we write O^{c}(F) to denote the full subcategory of O(F) whose objects are the F-centric subgroups of S.

When F is a saturated fusion system over a discrete p-toral group S, then we have the same correspondence between linking systems associated to F, classifying spaces associated to F, and liftings over O^{c}(F) of the homotopy functor (P ↦ BP) as we do when S is finite (Section 4.7). In particular, any classifying space for F has a decomposition as the homotopy direct limit of the corresponding homotopy lifting, as described by the next proposition.

Proposition 6.3. Fix a saturated fusion system F over a discrete p-toral group S, and let F_{0} ⊆ F^{c} be any finite full subcategory whose objects include F-conjugacy class representatives for all F-radical F-centric subgroups of S. Let L_{0} be any linking system associated to F_{0}, and let π̃_{0} : L_{0} ⟶ O(F_{0}) be the projection functor. Let

B̃: O(F_{0}) ⟶ Top

be the left homotopy Kan extension over π̃_{0} of the constant functor L_{0} ⟶^{∗} Top. Then B̃ is a homotopy lifting of the homotopy functor P ↦ BP, and

|L|_{p}^{∧} ≃ |L_{0}|_{p}^{∧} ≃ (\underrightarrow{hocolim}_{O(F_{0})}(B̃))_{p}^{∧} .  (1)

Proof. See [BLO3, Proposition 5.7]. □

6.5. Reduced linking systems. Constructing centric linking systems over discrete p-toral groups is not as straightforward as it sometimes is in the case of finite p-groups. For this reason, we define another type of category which is intermediate between fusion systems and linking systems.

Let F be a fusion system over a discrete p-toral group S, and let F_{0} ⊆ F^{c} be any full subcategory. A reduced linking system associated to F_{0} consists of a category L_{0}, together with a functor π: L_{0} ⟶ F_{0} and distinguished homomorphisms

P ⟶^{δ_{P}} Aut_{L_{0}}(P), which satisfy axioms (A), (B), and (C) in Section 4.2, except that Ker(δ_{P}) = Z(P)_{0}, and that the group Z(P)/Z(P)_{0} ≅ π_{0}(Z(P)) (not Z(P) itself) acts freely on morphism sets Mor_{L}(P, Q) and induces bijections

Mor_{L}(P, Q)/π_{0}(Z(P)) ⟶^{≅} Hom_{F}(P, Q).

Fortunately, we do know that each reduced linking system lifts to a unique linking system.

Proposition 6.4. Let F be a fusion system over the discrete p-toral group S. Then any reduced linking system L associated to F lifts to a centric linking system L associated to F which is unique up to isomorphism.

Proof. See [BLO3, Proposition 5.10]. □

6.6. Compact Lie groups. It was immediately obvious that every finite group can be regarded as a p-local finite group. In fact, we were already studying the fusion system and linking system for finite groups long before we considered the more abstract definitions. The definitions of fusion systems, and of reduced linking systems, for compact Lie groups are also straightforward, but the definition of their linking systems is less obvious.

Fix a compact Lie group G and a prime p. Let T ≤ G be a maximal torus, and fix a Sylow p-subgroup S/T ∈ Syl_{p}(N_{G}(T)/T). Then S is a maximal p-toral subgroup of G, and any other maximal p-toral subgroup of G is conjugate to S. These subgroups play the role of “Sylow p-subgroups” when working with compact Lie groups. For example, an arbitrary p-toral subgroup P ≤ G is maximal p-toral if and only if χ(G/P) is prime to p, where χ denotes as usual the Euler characteristic.

Now let S ≤ S be any discrete p-toral subgroup which is dense in S and has the same rank. By Proposition 6.1, there is such a subgroup, and any two such subgroups are conjugate in S. Since the closure of any discrete p-toral subgroup of G is p-toral, we now see that S is a maximal discrete p-toral subgroup of G, and that any other maximal discrete p-toral subgroup is G-conjugate to S.

Define the fusion category F_{S}(G) exactly as was done for finite G: its objects consist of all subgroups of S, and Mor_{F_{S}(G)}(P, Q) is the set of all group homomorphisms which are induced by conjugation in G. A subgroup P ≤ S is F_{S}(G)-centric if and only if Z(P) is a maximal discrete p-toral subgroup of C_{G}(P), or equivalently if and only if C_{G}(P) = Z(P) × C′_{G}(P) for some (unique) subgroup C′_{G}(P) which is finite of order prime to p.

We now define the reduced linking system L_{S}^{c}(G) to be the category whose objects are the F_{S}(G)-centric subgroups of S, and where

Mor_{L_{S}^{c}(G)}(P, Q) = N_{G}(P, Q)/(Z(P)_{0} × C′_{G}(P)).

By Proposition 6.4, there is a unique centric linking system L_{S}^{c}(G) associated to L_{S}^{c}(G).

Theorem 6.5. Fix a compact Lie group G and a maximal discrete p-toral subgroup S ≤ G. Then (S, F_{S}(G), L_{S}(G)) is a p-local compact group, with classifying space |L_{S}(G)|_{p}^{∧} ≃ BG_{p}^{∧}.

Proof. See [BLO3, §6]. The proofs that F_{S}(G) is a fusion system and L_{S}^{c}(G) an associated centric linking system are straightforward. The proof that |L_{S}(G)|_{p}^{∧} ≃ BG_{p}^{∧} is based on the homology decomposition of BG_{p}^{∧} in [JMO]. □

6.7. p-compact groups. A p-compact group is a triple (X, BX, e), where BX is a pointed, connected, p-complete space, e: X ⟶ Ω(BX) is a homotopy equivalence, and H^{∗}(X; 𝔽_{p}) is finite. These objects were introduced in the 1980’s by Dwyer and Wilkerson [DW3], and extensively studied by them and many other authors. The concept of a p-compact group was designed to be a homotopy theoretic analog of a classifying space of a compact Lie group. For instance, if G is a compact Lie group whose group of components is a p-group, then BG_{p}^{∧} is the classifying space of a pcompact group. In contrast, if π_{0}(G) is not a p-group, then the mod p cohomology of the loop space Ω(BG_{p}^{∧}) is not in general finite.

One of our original motivations for defining p-local compact groups was to provide a wider framework for studying p-compact groups. This requires first checking that every p-compact group X can be considered as a p-local compact group; in particular, that BX ≃ |L|_{p}^{∧} for some linking system L associated to a saturated fusion system F over a discrete p-toral group.

By [DW3, §8–9], for any p-compact group X, there is a unique maximal p-toral subgroup, hence a unique maximal discrete p-toral subgroup f: BS ⟶ BX (Proposition 6.1); and any other map from the classifying space of a discrete p-toral group to BX factors through f. We set

F_{S,f}(X) =^{def} F_{S,f}(BX) and L_{S,f}^{c}(X) =^{def} L_{S,f}^{c}(BX),

where the fusion and linking systems of the space BX are defined as in Section 2.5. Using the properties of the mapping spaces Map(BP, BX) described in [DW3, §5–6] (for a discrete p-toral group P), we check that F_{S,f}(X) is a saturated fusion system over S, and that L_{S,f}^{c}(X) is a reduced centric linking system associated to F_{S,f}(X). Let L_{S,f}^{c}(X) be the centric linking system associated to L_{S,f}^{c}(X) (and hence to F_{S,f}(X)) by Proposition 6.4. We then show:

Theorem 6.6. For any p-compact group X and any maximal discrete p-toral subgroup f: BS ⟶ BX, (S, F_{S,f}(X), L_{S,f}(X)) is a p-local compact group with classifying space

|L_{S,f}(X)|_{p}^{∧} ≃ BX .

Proof. See [BLO3, §7]. The proof that |L_{S,f}(X)|_{p}^{∧} ≃ BX is based on the decomposition, shown in [CLN], of BX as a homotopy direct limit of BP ’s for p-toral (or discrete ptoral) subgroups P ≤ X. □

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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: bob@math.univ-paris13.fr