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).