Discrete models for the $p$–local homotopy theory of compact Lie groups and $p$–compact groups

Abstract

We define and study a certain class of spaces which includes $p$–completed classifying spaces of compact Lie groups, classifying spaces of $p$–compact groups, and $p$–completed classifying spaces of certain locally finite discrete groups. These spaces are determined by fusion and linking systems over “discrete $p$–toral groups”—extensions of ${\left(\mathbb{Z}∕p^{\infty }\right)}^r$ by finite $p$–groups—in the same way that classifying spaces of $p$–local finite groups as defined in our paper [The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003) 779–856] are determined by fusion and linking systems over finite $p$–groups. We call these structures “$p$–local compact groups”.