Symplectic rational homology ball fillings of Seifert fibered spaces

TL;DR

This paper investigates which small Seifert fibered spaces can serve as boundaries of symplectic rational homology balls, providing classifications, restrictions, and evidence related to Stein fillings and Lagrangian disk fillings.

Contribution

It characterizes when small Seifert fibered spaces bound symplectic rational homology balls and offers restrictions for others, including classifications of spherical 3-manifolds and evidence for Gompf's conjecture.

Findings

Only specific lens spaces admit such fillings, as previously identified by Lisca.

Brieskorn spheres likely do not bound Stein domains in ^2, supporting Gompf's conjecture.

Restrictions are established on Lagrangian disk fillings of Legendrian knots in small Seifert fibered spaces.

Abstract

We characterize when some small Seifert fibered spaces can be the convex boundaries of symplectic rational homology balls and give strong restrictions for others to bound such manifolds. In particular, we show that the only spherical $3$-manifolds, oriented as links of the corresponding quotient singularities, which admit symplectic rational homology ball fillings, are the lens spaces $L(p^2,pq-1)$ previously identified by Lisca. In a different direction, we provide evidence for Gompf's conjecture that Brieskorn spheres do not bound Stein domains in $\mathbb{C}^2$. Finally, we establish restrictions on Lagrangian disk fillings of certain Legendrian knots in small Seifert fibered spaces.