Ken Brown asked whether there is any group G such that the coset lattice of G is contractible.  I’ll start by telling you what this question means.  I’ll then describe some recent progress of myself and John Shareshian towards a negative answer.
An essential tool of our work is Smith Theory, which examines the action of a p-group on a topological space, and examines its fixed point set.  As time allows, I will survey some results from this theory.