We define and characterize a class of p-complete spaces X which have many of the same properties as the p-completions of classifying spaces of finite groups. For example, each such X has a Sylow subgroup BS −−→ X , maps BQ −−→ X for a p-group Q are described via homomorphisms Q −−→ S, and H ∗(X ;Fp) is isomorphic to a certain ring of “stable elements” in H (BS;Fp). These spaces arise as the “classifying spaces” of certain algebraic objects which we call “p-local finite groups”. Such an object… CONTINUE READING