The homotopy theory of fusion systems

By Carles Broto, Ran Levi, Bob Oliver

Abstract

We define and characterize a class of -complete spaces which have many of the same properties as the -completions of classifying spaces of finite groups. For example, each such has a Sylow subgroup , maps for a -group are described via homomorphisms , and is isomorphic to a certain ring of “stable elements” in . These spaces arise as the “classifying spaces” of certain algebraic objects which we call -local finite groups”. Such an object consists of a system of fusion data in , as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.

The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like -completed classifying spaces of finite groups. These spaces occur as the “classifying spaces” of certain algebraic objects, which we call -local finite groups. A -local finite group consists, roughly speaking, of a finite -group and fusion data on subgroups of , encoded in a way explained below. Our starting point is our earlier paper Reference BLO on -completed classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig Reference Pu, Reference Pu2 of systems of fusion among subgroups of a given -group.

The -completion of a space is a space which isolates the properties of at the prime , and more precisely the properties which determine its mod cohomology. For example, a map of spaces induces a homotopy equivalence if and only if induces an isomorphism in mod cohomology; and in favorable cases (if is -good”). When is a finite group, the -completion of its classifying space encodes many of the properties of at . For example, not only the mod cohomology of , but also the Sylow -subgroup of together with all fusion among its subgroups, are determined up to isomorphism by the homotopy type of .

Our goal here is to give a direct link between -local structures and homotopy types which arise from them. This theory tries to make explicit the essence of what it means to be the -completed classifying space of a finite group, and at the same time yields new spaces which are not of this type, but which still enjoy most of the properties a space of the form would have. We hope that the ideas presented here will have further applications and generalizations in algebraic topology. But this theory also fits well with certain aspects of modular representation theory. In particular, it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic .

A saturated fusion system over a -group consists of a set of monomorphisms, for each pair of subgroups , which form a category under composition, include all monomorphisms induced by conjugation in , and satisfy certain other axioms formulated by Puig (Definitions 1.1 and 1.2 below). In particular, these axioms are satisfied by the conjugacy homomorphisms in a finite group. We refer to Reference Pu and Reference Pu2 for more details of Puig’s work on saturated fusion systems (which he calls “full Frobenius systems” in Reference Pu2). The definitions and results given here, in Section 1 and in Appendix A are only a very brief account of those results of Puig used in our paper.

If is a saturated fusion system over , then two subgroups are called -conjugate if . A subgroup is called -centric if for all that are -conjugate to ; this is equivalent to what Puig calls -selfcentralizing”. Let be the full subcategory of whose objects are the -centric subgroups of . A centric linking system associated to is a category whose objects are the -centric subgroups of , together with a functor which is the identity on objects and surjective on morphisms, and which satisfies other axioms listed below in Definition 1.7. For example, for each object , the kernel of the induced map is isomorphic to , and contains a distinguished subgroup isomorphic to .

The motivating examples for these definitions come from finite groups. If is a finite group and is a prime, then is the fusion system over such that for each , is the set of homomorphisms induced by conjugation in (and inclusion). The -centric subgroups of are the -centric subgroups: those such that for some of order prime to (see Reference BLO, Lemma A.5). In Reference BLO, we defined a category whose objects are the -centric subgroups of which are contained in , and where . Here, is the set of elements of which conjugate into . The category , together with its projection to which sends the morphism corresponding to an element to conjugation by , is the example which motivated our definition of an associated centric linking system.

We define a -local finite group to be a triple , where is a centric linking system associated to a saturated fusion system over a -group . The classifying space of such a triple is the space , where for any small category , the space denotes the geometric realization of the nerve of . This is partly motivated by the result that for any finite Reference BLO, Proposition 1.1. But additional motivation comes from Proposition 2.2 below, which says that if is a centric linking system associated to , then , where is a certain quotient “orbit” category of , and is a lifting of the homotopy functor which sends to . The classifying space of a -local finite group thus comes equipped with a decomposition as the homotopy colimit of a finite diagram of classifying spaces of -groups.

We now state our main results. Our first result is that a -local finite group is determined up to isomorphism by its classifying space. What is meant by an isomorphism of -local finite groups will be explained later.

Theorem A (Theorem 7.4).

A -local finite group is determined by the homotopy type of . In particular, if and are two -local finite groups and , then and are isomorphic.

Next we study the cohomology of -local finite groups. As one might hope, we have the following result, which appears as Theorem 5.8.

Theorem B.

For any -local finite group , is isomorphic to the ring of “stable elements” in ; i.e., the inverse limit of the rings as a functor on the category . Furthermore, this ring is noetherian.

The next theorem gives an explicit description of the mapping space from the classifying space of a finite -group into the classifying space of a -local finite group. It is stated precisely as Corollary 4.5 and Theorem 6.3.

Theorem C.

For any -local finite group , and any -group ,

Furthermore, each component of the mapping space has the homotopy type of the classifying space of a -local finite group which can be thought of as the “centralizer” of the image of the corresponding homomorphism .

The next result describes the space of self equivalences of the classifying space of a -local finite group. It is a generalization of Reference BLO, Theorem C. For a small category , let denote the groupoid whose objects are self equivalences of , and whose morphisms are natural isomorphisms of functors. Let be a centric linking system associated to a saturated fusion system . Self equivalences of which are structure preserving, in a sense to be made precise in section 7 below, are said to be isotypical. We let denote the subgroupoid of whose objects are the isotypical self equivalences of . For a space , let denote the topological monoid of all self homotopy equivalences of . The following theorem is restated below as Theorem 8.1.

Theorem D.

Fix a -local finite group . Then and are equivalent as topological monoids in the sense that their classifying spaces are homotopy equivalent. In particular, their groups of components are isomorphic, and each component of is aspherical.

The statement of Theorem 8.1 also includes a description of the homotopy groups of .

So far, we have not mentioned the question of the existence and uniqueness of centric linking systems associated to a given saturated fusion system. Of course, as pointed out above, any finite group gives rise to an associated -local finite group. However there are saturated fusion systems which do not occur as the fusion system of any finite group. Thus a tool for deciding existence and uniqueness would be useful. The general obstructions to the existence and uniqueness of associated centric linking systems, which lie in certain higher limits taken over the orbit category of the fusion system, are described in Proposition 3.1; and a means of computing these groups is provided by Proposition 3.2. The following result is just one special consequence of this, which settles the question for -groups of small rank. Here, for any finite group , we write for the largest rank of any elementary abelian -subgroup of .

Theorem E (Corollary 3.5).

Fix a saturated fusion system over a -group . If , then there exists a centric linking system associated to , and if , then the associated centric linking system is unique.

In the last section, we present some direct constructions of saturated fusion systems and associated -local finite groups (see Examples 9.3 and 9.4). The idea is to look at the fusion system over a -group (for odd only) generated by groups of automorphisms of and certain of its subgroups, and show that under certain hypotheses the resulting system is saturated. In all of these cases, the -group is nonabelian, and has an index subgroup which is abelian and homocyclic (a product of cyclic groups of the same order). We then give a list of all finite simple groups which have Sylow subgroups of this form, based on the classification theorem, and use that to show that certain of the fusion systems which were constructed are not the fusion systems of any finite groups. In all cases, Theorem E applies to show the existence and uniqueness of centric linking systems, and hence -local finite groups, associated to these fusion systems.

The basic definitions of saturated fusion systems and their associated centric linking systems are given in Section 1. Homotopy decompositions of classifying spaces of -local finite groups are constructed in Section 2. The obstruction theory for the existence and uniqueness of associated centric linking systems, as well as some results about those obstruction groups, are shown in Section 3. Maps from the classifying space of a -group to the classifying space of a -local finite group are studied in Sections 4 and 6, while the cohomology rings of classifying spaces of -local finite groups are dealt with in Section 5. A characterization of classifying spaces of -local finite groups is given in Section 7, and their spaces of self equivalences are described in Section 8. The “exotic” examples of -local finite groups are constructed in Section 9. Finally, some additional results on saturated fusion systems are collected in an appendix.

We would like to thank Dave Benson and Jesper Grodal for their many suggestions throughout the course of this work. In particular, Dave had earlier written and distributed notes which contained some of the ideas of our centric linking systems. We would also like to thank Lluís Puig for giving us a copy of his unpublished notes on saturated fusion systems. Markus Linckelmann, Haynes Miller, Bill Dwyer, and Jon Alperin have all shown interest and made helpful comments and suggestions. Kasper Andersen and Kari Ragnarsson both read earlier versions of this paper in detail, and sent us many suggestions for improvements. Two of the authors would also like to thank Slain’s Castle, a pub in Aberdeen, for their hospitality on New Year’s Day while we worked out the proof that the nerve of a centric linking system is -good.

We would especially like to thank the universities of Aberdeen and Paris-Nord, the CRM and the UAB in Barcelona, and the Max-Planck Institut in Bonn for their hospitality in helping the three authors get together in various combinations; and also the European Homotopy Theory Network for helping to finance these visits.

1. Fusion systems and associated centric linking systems

We begin with the precise definitions of saturated fusion systems and their associated centric linking systems. Additional results about fusion systems due to Puig Reference Pu, Reference Pu2 are in Appendix A.

Given two finite groups , , let denote the set of group homomorphisms from to , and let denote the set of monomorphisms. If and are subgroups of a larger group , then denotes the subset of homomorphisms induced by conjugation by elements of , and the group of automorphisms induced by conjugation in .

Definition 1.1.

A fusion system over a finite -group is a category whose objects are the subgroups of , and whose morphism sets satisfy the following conditions:

(a)

for all .

(b)

Every morphism in factors as an isomorphism in followed by an inclusion.

Note that what we call a fusion system here is what Puig calls a divisible Frobenius system.

If is a fusion system over and , then we write to emphasize that morphisms in the category are all homomorphisms, and for the subset of isomorphisms in . Thus if , and otherwise. Also, and . Two subgroups are called -conjugate if .

The fusion systems we consider here will all satisfy the following additional condition. Here, and throughout the rest of the paper, we write for the set of Sylow -subgroups of . Also, for any and any , denotes the automorphism .

Definition 1.2.

Let be a fusion system over a -group .

A subgroup is fully centralized in if for all that are -conjugate to .

A subgroup is fully normalized in if for all that are -conjugate to .

is a saturated fusion system if the following two conditions hold:

(I)

Any which is fully normalized in is fully centralized in , and .

(II)

If and are such that is fully centralized, and if we set

then there is such that .

The above definition is slightly different from the definition of a “full Frobenius system” as formulated by Lluís Puig Reference Pu2, §2.5, but is equivalent to his definition by the remarks after Proposition A.2. Condition (I) can be thought of as a “Sylow condition”. It says that has order prime to (just as has order prime to if ); and more generally it reflects the fact that for any -subgroup , there is some such that . Another way of interpreting this condition is that if for -conjugate to , then and must also be maximal in the same sense. As for condition (II), it is natural to require that some extension property hold for morphisms in , and is by definition the largest subgroup of to which could possibly extend.

The motivating example for this definition is the fusion system of a finite group . For any , we let be the fusion system over defined by setting for all .

Proposition 1.3.

Let be a finite group, and let be a Sylow -subgroup of . Then the fusion system over is saturated. Also, a subgroup is fully centralized in if and only if , while is fully normalized in if and only if .

Proof.

Fix some , and choose such that contains a Sylow -subgroup of . Then and , and so . This clearly implies that for all that are -conjugate to . Thus is fully normalized in , and is fully normalized in if and only if , if and only if . A similar argument proves that is fully centralized in if and only if .

If is fully normalized in , then since , the obvious counting argument shows that

In particular, is fully centralized in , and this proves condition (I) in Definition 1.2.

To see condition (II), let and be such that and is fully centralized in , and write for short. Set

and similarly

In particular, ( and are conjugate modulo centralizers), and thus and are two -subgroups of . Furthermore,

is prime to (since is fully centralized), so . Since has -power index, all Sylow -subgroups of are conjugate by elements of , and hence there is such that . Thus extends .

Puig’s original motivation for defining fusion systems came from block theory. Let be an algebraically closed field of characteristic . A block in a group ring is an indecomposable 2-sided ideal which is a direct summand. Puig showed Reference Pu that the Brauer pairs associated to a block (the -subpairs”), together with the inclusion and conjugacy relations defined by Alperin and Broué Reference AB, form a saturated fusion system over the defect group of . See, for example, Reference AB or Reference Alp, Chapter IV, for definitions of defect groups and Brauer pairs of blocks.

In practice, when proving that certain fusion systems are saturated, it will be convenient to replace condition (I) by a modified version of the condition, as described in the following lemma.

Lemma 1.4.

Let be a fusion system over a -group which satisfies condition (II) in Definition 1.2, and also satisfies the condition

(I)

Each subgroup is -conjugate to a fully centralized subgroup such that .

Then is a saturated fusion system.

Proof.

We must prove condition (I) in Definition 1.2. Assume that is fully normalized in . By (I), there is which is -conjugate to , and such that is fully centralized in and . In particular,