Title: A homotopy characterization of p-completed classifying spaces of finite groups
Speaker: Kári Ragnarsson
Speaker Info: Google, Inc.
The homotopy type of classifying spaces of finite groups is easy to characterize: they are the Eilenberg-MacLane spaces of type K(G,1). When working p-locally (as one often must) things are more complicated as the homotopy type of a p-completed classifying space does not depend on the group itself, but just on its p-local fusion system, which is the category of p-subgroups and homomorphisms induced by conjugation (this is the Martino--Priddy conjecture). Seeing fusion systems arise in different contexts, Puig axiomatized them and introduced abstract fusion systems. Thus, to characterize the homotopy type of p-completed classifying spaces, one should really study classifying spaces for fusion systems. To this end, Broto--Levi--Oliver introduced p-local finite groups as a rather complicated algebro-categorical model for the classifying space of a fusion system. Around the same time, Haynes Miller proposed a purely homotopy-theoretic characterization for a p-completed classifying spaces as a space that admits a transfer retract to the classifying space of a finite p-group, and tentatively suggested that they should be equivalent to the Broto--Levi--Oliver model. In this talk I will explain these two models and then go through an outline proof of Miller's conjecture. This is work in progress, joint with Matthew Gelvin.Date: Monday, April 23, 2012