University of Chicago
Department of Mathematics

Fusion systems are an abstraction of the p-local structure of finite groups: They model the inclusion of a Sylow p-subgroup in an ambient finite group, together with ambient conjugation (or "fusion") data. The work of Broto-Levi-Oliver has shown how these algebraic, almost combinatorial objects are closely related to the homotopy type of the p-completion of classifying spaces of finite groups. In this talk, I will examine how the fundamental notion of group actions on finite sets can be understood in the context of fusion systems. I will try to motivate the notion of a "fusion action system," as well as the basic properties they should satisfy. There is also a strong parallel with the work of BLO: Fusion action systems have a notion of "classifying space," which, in the presence of an actual ambient group G acting on a finite set X, reconstructs the Borel construction EG\times_G X up to p-completion. I shall finish with an overview of this classifying space story, and suggest some applications to fusion system theory.