An application of group theory to topology

 Let p be a prime.  To every finite group is associated a topological
space known as the p-completion of its classifying space.  The
Martino-Priddy conjecture states that for two groups G and H, these
spaces are homotopically equivalent if and only if there is a special
type of isomorphism between the Sylow p-subgroups of G and H
(an isomorphism of fusion systems, e.g., elements conjugate in G
are mapped to elements conjugate in H).

  The combined work of several authors has proved this conjecture
and some extensions, partly by assuming the classification of
finite simple groups.  Recently, J. Lynd and I removed this assumption.
I plan to discuss the main ideas of these results.