The homotopy theory of fusion systems

Abstract

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

An algebraic model for finite loop spaces

“…Hence by [9,Proposition 5.4], H 3 .O.F 1 /Iˆ/ D 0. By an argument similar to that used by the authors to prove [8,Proposition 3.1], the obstruction to extending L 0 to L 1 lies in H 3 .O.F 1 /Iˆ/, and hence L 1 does exist. This is now generalized as follows.…”

Stable splitting and cohomology of p–local finite groups over the extraspecial p–group of order p³ and exponent p

“…Recall that A is F ec -radical if and only if SL 2 (F p ) ⊂ W G (A) (see Ruiz-Viruel [9, Lemma 4.1]). In [1] and [11], proofs of the above theorem are given only for H * (BG; Z (p) ). A proof for H * (BG) is explained in Section 11.…”

“…(5) 2 F 4 (2) , J 4 , for p=3, (6) Th for p=5, For case (1), we know that H * (E : W) ∼ = H * (E) W . Except for these extensions and exotic cases, all H even (G; Z) (p) are studied by Tezuka and Yagita [11].…”

Reduced fusion systems over 2-groups of sectional rank at most 4

“…Background on fusion systems A saturated fusion system over a p-group S is a category F whose objects are the subgroups of S, and where for each P, Q ≤ S, Mor F (P, Q) is a set of injective homomorphisms from P to Q which includes all morphisms induced by conjugation in S, and which satisfies a set of axioms which are described, for example, in [AKO,§ I.2], [BLO2,Definition 1.2], or [Cr,Definition 4.11]. We write Hom F (P, Q) = Mor F (P, Q) to emphasize that the morphisms are all homomorphisms.…”

“…BACKGROUND ON FUSION SYSTEMS proposition. (These conditions are used to define saturation in [BLO2] and other papers.) Proposition 1.2 ( [AKO,Proposition I.2.5]).…”