Brieskorn spheres and rational homology ball symplectic fillings

TL;DR

This paper investigates the symplectic fillability of Brieskorn spheres, confirming conjectures about obstructions to rational homology ball fillings under certain conditions and classifying fillable structures.

Contribution

It proves new obstructions to symplectic fillings of Brieskorn spheres and classifies fillable structures for specific cases, advancing understanding of contact topology.

Findings

Obstructed rational homology ball fillings for certain Brieskorn spheres.

Classified Brieskorn spheres with up to two fillable structures.

Identified exceptions in Milnor fillable structures for specific spheres.

Abstract

Given a canonically oriented Brieskorn sphere $Y=\Sigma(a_1,...,a_n)$, we confirm some statements conjectured by Gompf. More specifically, we obstruct the existence of rational homology ball symplectic fillings for any contact structure on $-Y$ if $n=3$, and when there is no half convex Giroux torsion for $n>3$. Furthermore, we show that the same result holds for the Milnor fillable structure on $Y$ with the possible exception of $\Sigma(3,4,5),$ $\Sigma(2,5,7)$ and $\Sigma(2,3,6k+1)$ for $k\geq1$. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.