Equivalences of Classifying Spaces Completed at the Prime Two (Memoirs of the American Mathematical Society) by Robert Oliver (PDF)

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We prove here the Martino-Priddy conjecture at the prime $2$: the $2$-completions of the classifying spaces of two finite 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 $mathcal{Z}_G$ on the $2$-subgroup orbit category of $G$. The proof of this result uses the classification theorem for finite simple groups.

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