Brieskorn spheres and rational homology ball symplectic fillings

Antonio Alfieri; Alberto Cavallo; Irena Matkovič

paper

Brieskorn spheres and rational homology ball symplectic fillings

arXiv:2605.13812

Abstract

Given a canonically oriented Brieskorn sphere $Y = Σ (a_{1}, ..., a_{n})$ , 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 $Σ (3, 4, 5),$ $Σ (2, 5, 7)$ and $Σ (2, 3, 6 k + 1)$ for $k \geq 1$ . Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.