Cyclic Branched Coverings of Brieskorn Spheres Bounding Acyclic 4-Manifolds

TL;DR

This paper demonstrates that certain cyclic symmetries on Brieskorn homology spheres cannot extend smoothly over any bounding acyclic 4-manifold, using equivariant Yang-Mills theory to obstruct such extensions.

Contribution

It introduces a novel application of equivariant Yang-Mills moduli spaces to obstruct smooth extensions of cyclic actions on Brieskorn spheres.

Findings

Cyclic actions with fixed points do not extend smoothly to bounding acyclic 4-manifolds.

Obstructions are identified via equivariant Yang-Mills invariants.

Standard invariants of homology spheres are insufficient to detect these obstructions.

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.