Discrete models for thep–local homotopy theory of compact Lie groups andp–compact groups

Abstract

In our earlier paper [7], we defined and studied a certain class of spaces which in many ways behave like p ‐completed classifying spaces of finite groups. These spaces occur as “classifying spaces” of certain algebraic objects called p ‐local finite groups. The purpose of this paper is to generalize the concept of p ‐local finite groups to what we call p ‐local compact groups. The motivation for introducing this family comes from the observation that p ‐completed classifying spaces of finite and compact Lie groups, as well as classifying spaces of p ‐compact groups (see Dwyer and Wilkerson [11]), share many similar homotopy theoretic properties, but earlier studies of these properties usually required different techniques for each case. Moreover, while p ‐completed classifying spaces of finite and, more generally, compact Lie groups arise from the algebraic and geometric structure of the groups in question, p ‐compact groups are purely homotopy theoretic objects. Unfortunately, many of the techniques used in the study of p ‐compact groups fail for p ‐completed classifying spaces of general compact Lie groups. With the approach presented here, we propose a framework general enough to include p ‐completed classifying spaces of arbitrary compact Lie groups as well as p ‐compact groups. The new idea here is to replace fusion systems over finite p ‐groups, as handled in [7], by fusion systems over discrete p ‐toral groups. A discrete p ‐toral group is a group which contains a discrete p ‐torus (a group of the form .Z=p 1 / r for finite r 0) as a normal subgroup of p ‐power index. A p ‐local compact group consists of a triple