Author: Bob Oliver
Title: Equivalences of classifying spaces completed at odd primes
We prove here the Martino-Priddy conjecture for an odd prime p: the
p-completions of the classifying spaces of two groups G and G' are
homotopy equivalent if and only if there is an isomorphism between their
Sylow p-subgroups which preserves fusion. A second theorem is a
description for odd p of the group of homotopy classes of self homotopy
equivalences of the p-completion of BG, in terms of automorphisms of a
Sylow p-subgroup of G which preserve fusion in G. These are both
consequences of a technical algebraic result, which says that for an odd
prime p and a finite group G, all higher derived functors of the inverse
limit vanish for a certain functor on the p-subgroup orbit category of G.