About:

This technical lecture was part of the "Classifying spaces in homotopy theory: in honour of Ran Levi's 60th Birthday" workshop at ICMS.

We were delighted to welcome speaker Dave Benson (he/him) to give this talk.

The talk started by giving a review of the cohomology of finite groups, and a glimpse of their role in group actions on topological spaces, before introducing the p-completion of Bousfield and Kan. The p-completed classifying space only reflects behaviour of the group at the prime p, and this is formalised in the notions of fusion system and linking system. The loop space on the classifying space gives you back the group, but the loop space on the p-completion is an interesting object which is in Koszul duality to the classifying space. The singuarity category of each is equivalent to the cosingularity category of the other. This was illustrated with some examples coming from some small(ish) finite groups.