The homotopy theory of fusion systems

Abstract

The main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p-completed classifying spaces of finite groups. These spaces occur as the “classifying spaces” of certain algebraic objects, which we call p-local finite groups. A p-local finite group consists, roughly speaking, of a finite p-group S and fusion data on subgroups of S, encoded in a way explained below. Our starting point is our earlier paper [BLO] on p-completed classifying spaces of finite groups, together with the axiomatic treatment by Lluis Puig [Pu], [Pu2] of systems of fusion among subgroups of a given p-group. The p-completion of a space X is a space Xp which isolates the properties of X at the prime p, and more precisely the properties which determine its mod p cohomology. For example, a map of spaces X f −−→ Y induces a homotopy equivalence Xp ' −−→ Y p if and only if f induces an isomorphism in mod p cohomology; and H(Xp ;Fp) ∼= H∗(X ;Fp) in favorable cases (if X is “p-good”). When G is a finite group, the p-completion BGp of its classifying space encodes many of the properties of G at p. For example, not only the mod p cohomology of BG, but also the Sylow p-subgroup of G together with all fusion among its subgroups, are determined up to isomorphism by the homotopy type of BGp . Our goal here is to give a direct link between p-local structures and homotopy types which arise from them. This theory tries to make explicit the essence of what it means to be the p-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 BGp 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 p. A saturated fusion system F over a p-group S consists of a set HomF (P,Q) of monomorphisms, for each pair of subgroups P,Q ≤ S, which form a category under composition, include all monomorphisms induced by conjugation in S, 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 [Pu] and [Pu2] for more details of Puig’s work on saturated