A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

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 BG ∧ p . 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 BG ∧ p 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 Hom F (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

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A p-local finite group consists of a finite p-group S, together with a pair of categories which encode "conjugacy" relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at smaller classes of subgroups of S. We also prove that the homotopy type of the classifying space of a given p-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of F -centric F -radical subgroups (at a minimum) to the set of F -quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to p-constrained finite groups, and prove that they in fact all arise from groups.

Abstract. We define here two new classes of saturated fusion systems, reduced fusion systems and tame fusion systems. These are motivated by our attempts to better understand and search for exotic fusion systems: fusion systems which are not the fusion systems of any finite group. Our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable, and that every reduced fusion system which is not tame is the reduction of some exotic (nonrealizable) fusion system.

A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin 7 (q) (q an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2-completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer-Wilkerson space BDI(4). AMS Classification numbers 918 Ran Levi and Bob OliverAs one step in the classification of finite simple groups, Ron Solomon [22] 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 for any involution x ∈ G, there are subgroups K H C G (x) such that K and C G (x)/H have odd order and H/K ∼ = Spin 7 (q) for some odd prime power q . Solomon showed that such a group G does not exist. The proof of this statement was also interesting, in the sense that the 2-local structure of the group in question appeared to be internally consistent, and 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 [3], 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. He started with the spaces BDI(4) constructed by Dwyer and Wilkerson having the property that(the rank four Dickson algebra at the prime 2). Benson 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. This homotopy fixed point set is denoted here BDI 4 (q).In this paper, we construct a family of 2-local finite groups, in the sense of [6], which have the 2-local structure considered by Solomon, and whose classifying spaces are homotopy equivalent to Benson's spaces BDI 4 (q). The results of [6] combined with those here allow us to make much more precise the statement that these spaces have many of the properties which the 2-completed classifying spaces of the groups studied by Solomon would have had if they existed. To explain what this means, we first recall some definitions.A fusion system over a finite p-group S is a category whose objects are the subgroups of S , and whose morphisms are monomorphisms of groups which include all those induced by conjugation by elements of S . A fusion system is saturated if it satisfies certain axioms formulated by Puig [19], and als...

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We prove a version of the Atiyah-Segal completion theorem for proper actions of an infinite discrete group G. More precisely, for any finite proper G-CW-complex X, K * (EG× G X) is the completion of K * G (X) with respect to a certain ideal. We also show, for such G and X, that K G (X) can be defined as the Grothendieck group of the monoid of G-vector bundles over X.