Cyclic Branched Coverings of Brieskorn Spheres Bounding Acyclic 4-Manifolds

Abstract

We show that standard cyclic actions on Brieskorn homology 3-spheres with non-empty fixed set do not extend smoothly to any contractible smooth 4-manifold it may bound. The quotient of any such extension would be an acyclic $4$-manifold with boundary a related Brieskorn homology sphere. We briefly discuss well known invariants of homology spheres that obstruct acyclic bounding 4-manifolds, and then use a method based on equivariant Yang-Mills moduli spaces to rule out extensions of the actions.