Equivalences of classifying spaces completed at odd primes
Bob Oliver
Universite Paris-Nord

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.  

(just submitted to be made into preprint)