A saturated fusion system associated to a finite group G encodes the p-structure of the group as the Sylow p-subgroup enriched with additional conjugation. The fusion system contains just the right amount of algebraic information to for instance reconstruct the p-completion of BG, but not BG itself. Abstract saturated fusion systems F without ambient groups exist, and these have (p-completed) classifying spaces BF as well.

In a joint project with Tomer Schlank and Nat Stapleton, we combine the theory of abstract fusion systems with the work by Hopkins-Kuhn-Ravenel and Stapleton on transchromatic character maps, and we generalize several results from finite groups to fusion systems.

A main ingredient of this project is studying the free loop spaces L(BG) and L(BF) for groups and fusion systems, and constructing transfer maps from L(BG) to L(BH) when H is a subgroup of G.

The pretalk will be an introduction to fusion systems and how their classifying spaces are stable retracts of classifying spaces for the underlying Sylow subgroups, which turns out to just involve learning to calculate with bisets.