Cyclic group actions on contractible $4$–manifolds

Abstract

There are known infinite families of Brieskorn homology $3$–spheres which can be realized as boundaries of smooth contractible $4$–manifolds. In this paper we show that smooth free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible $4$–manifold. We give a new infinite family of examples in which the actions extend locally linearly but not smoothly.