Cyclic Group Actions on Contractible 4-Manifolds

TL;DR

This paper investigates the extendability of free cyclic group actions on certain 3-manifolds as boundaries of contractible 4-manifolds, revealing smooth obstructions and new examples of locally linear but non-smooth extensions.

Contribution

It demonstrates that free cyclic actions on specific Brieskorn spheres do not extend smoothly over contractible 4-manifolds and introduces new examples of locally linear but non-smooth extendability.

Findings

Free cyclic actions on Brieskorn spheres do not extend smoothly over contractible 4-manifolds.

Existence of new examples where actions extend locally linearly but not smoothly.

Obstructions to smooth extension of group actions on 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 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.