The homotopy theory of fusion systems

Source abstract

We define and characterize a class of p p -complete spaces X X which have many of the same properties as the p p -completions of classifying spaces of finite groups. For example, each such X X has a Sylow subgroup B S ⟶ X BS\longrightarrow X , maps B Q ⟶ X BQ\longrightarrow X for a p p -group Q Q are described via homomorphisms Q ⟶ S Q\longrightarrow S , and H ∗ ( X ; F p ) H^*(X;\mathbb {F}_p) is isomorphic to a certain ring of “stable elements” in H ∗ ( B S ; F p ) H^*(BS;\mathbb {F}_p) . These spaces arise as the “classifying spaces” of certain algebraic objects which we call “ p p -local finite groups”. Such an object consists of a system of fusion data in S S , as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.