Homotopy equivalences of p-completed classifying spaces of finite groups

Abstract

We study homotopy equivalences of p-completions of classifying spaces of finite groups. To each finite group G and each prime p, we associate a finite category ℒ p c (G) with the following properties. Two p-completed classifying spaces BG p ∧ and BG’ p ∧ have the same homotopy type if and only if the associated categories ℒ p c (G) and ℒ p c (G’) are equivalent. Furthermore, the topological monoid Aut(BG p ∧) of self equivalences is determined by the self equivalences of the associated category ℒ p c (G).