Equivalences of classifying spaces completed at the prime two

Bob Oliver

We prove here the Martino-Priddy conjecture for the prime $2$:  the 
$2$-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 $2$-subgroups which preserves fusion.  This is a consequence of a 
technical algebraic result, which says that for a finite group $G$, the 
second higher derived functor of the inverse limit vanishes for a certain 
functor $\calz_G$ on the $2$-subgroup orbit category of $G$.  The proof of 
this result uses the classification theorem for finite simple groups.