More Brieskorn spheres bounding rational balls

TL;DR

This paper introduces new families of Brieskorn spheres that non-trivially bound rational homology balls, expanding the known examples and providing simpler proofs of classical results in the field.

Contribution

It presents novel infinite families of Brieskorn spheres bounding rational homology balls and simplifies existing proofs of classical results.

Findings

New families: xt(2,4n+3,12n+7) and xt(3,3n+2,12n+7) for even n

Revisits and simplifies classical results on Brieskorn spheres bounding integral homology balls

Extends the understanding of which Brieskorn spheres bound rational homology balls

Abstract

We call an integral homology sphere $\textit{non-trivially}$ bounds a rational homology ball if it is obstructed from bounding an integral homology ball. After Fintushel and Stern's well-known example $\Sigma(2,3,7)$, Akbulut and Larson recently provided the first infinite families of Brieskorn spheres non-trivially bounding rational homology balls: $\Sigma(2,4n+1,12n+5)$ and $\Sigma(3,3n+1,12n+5)$ for odd~$n$. Using their technique, we present new such families: $\Sigma(2,4n+3,12n+7)$ and $\Sigma(3,3n+2,12n+7)$ for even $n$. Also manipulating their main argument, we simply recover some classical results of Akbulut and Kirby, Fickle, Casson and Harer, and Stern about Brieskorn spheres bounding integral homology balls.