# The homotopy theory of fusion systems

## Abstract

We define and characterize a class of spaces -complete which have many of the same properties as the of classifying spaces of finite groups. For example, each such -completions 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 .“ finite groups”. Such an object consists of a system of fusion data in -local 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 classifying spaces of finite groups. These spaces occur as the “classifying spaces” of certain algebraic objects, which we call -completed* finite groups -local*. A finite group consists, roughly speaking, of a finite -local -group and fusion data on subgroups of encoded in a way explained below. Our starting point is our earlier paper ,Reference BLO on classifying spaces of finite groups, together with the axiomatic treatment by Lluís Puig -completedReference 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

Our goal here is to give a direct link between

A *saturated fusion system*

If if

*if* -centric

*centric linking system*associated to

The motivating examples for these definitions come from finite groups. If

We define a to be a triple

*classifying space*of such a triple is the space

We now state our main results. Our first result is that a

Next we study the cohomology of

The next theorem gives an explicit description of the mapping space from the classifying space of a finite

The next result describes the space of self equivalences of the classifying space of a

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

In the last section, we present some direct constructions of saturated fusion systems and associated

The basic definitions of saturated fusion systems and their associated centric linking systems are given in Section 1. Homotopy decompositions of classifying spaces of

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

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

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

If if

The fusion systems we consider here will all satisfy the following additional condition. Here, and throughout the rest of the paper, we write

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

The motivating example for this definition is the fusion system of a finite group

Puig’s original motivation for defining fusion systems came from block theory. Let

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.