Group Actions on Products of Spheres
In this talk, we will explore free group actions using the techniques of group cohomology. In particular, we are interested in free actions of finite abelian groups on products of spheres. It is known that the only finite abelian groups which act freely on the circle are cyclic. As a generalization, it has been conjectured that if a product of cyclic groups acts freely on a product of k spheres, then the rank of the group is at most k. This conjecture remains unproven, but partial results are known. For example, we use the long exact sequence for Tate cohomology to prove the case where the spheres are equidimensional and the action is trivial on homology. We will start by introducing the basic ideas of homology and group cohomology, and time permitting, we will build up to this result and see how group cohomology arises in the solution to the problem.