Group actions on exotic spheres

It has long been known that the standard $n$-sphere is the most symmetric omotopy $n$-sphere. It is also known that the homotopy spheres which bound parallelizable manifolds are more symmetric than those which are detected by stable homotopy theory. Many fundamental questions about the symmetries of homotopy spheres remain open. For instance, it is not known whether every homotopy sphere admits a smooth nontrivial circle action. Moreover, it is not known whether every homotopy sphere admits smooth nontrivial actions of every finite cyclic group.

In this talk, I will discuss recent progress on using stable homotopy theory to produce actions of finite cyclic groups on homotopy spheres. This is joint work with Nick Kuhn and J.D. Quigley.