An evolution of the Topological Spherical Space Form Problem
Dennis Dreesen, Paul Igodt, Nansen Petrosyan · Lirias · 2009
Until the late 70's, a central question regarding finite group actions was the topological spherical space form problem. That is, when does a finite group act freely on a sphere Sn? In 1978, using Swan's criteria, Madsen, Thomas and Wall settled the question by giving a complete algebraic characterization. One of the problems that has evolved is to classify groups that can at least act freely and properly discontinuously on Sn×Rk. In 2001, Adem and Smith showed that a countable group G acts freely and properly discontinuously on some Sn × Rk if and only if G has periodic cohomology. In this talk, we will explore this cohomological periodicity, its implied conditions, and their relations to the Euclidean space form problem. This is a joint work with Dennis Dreesen and Nansen Potrosyan.