Complementary legs and symplectic rational balls

TL;DR

This paper investigates when small Seifert fibered spaces with complementary legs can symplectically bound rational homology balls, revealing distinctions from the smooth category and classifying fillable contact structures on spherical 3-manifolds.

Contribution

It characterizes symplectic fillability of small Seifert fibered spaces with complementary legs based on orientation and contact structure, completing the classification of fillable contact structures on spherical 3-manifolds.

Findings

Certain small Seifert fibered spaces with complementary legs do not bound rational homology balls.

Characterization of when these spaces bound rational homology balls depending on $e_0$ and contact structure.

Limited number of fillable contact structures on spherical 3-manifolds with finite fundamental group.

Abstract

We show that a small Seifert fibered space with complementary legs does not symplectically bound a rational homology ball for at least one choice of orientation. In the case $e_0\leq -1$, we characterize when a small Seifert fibered space with uniquely complementary legs symplectically bounds a rational homology ball. In the case $e_0\geq 0$, we characterize when a small Seifert fibered space with complementary legs, equipped with a balanced contact structure, symplectically bounds a rational homology ball. Our results highlight a sharp contrast with the smooth category, where many more such Seifert fibered spaces are known to bound smooth rational homology balls. As a consequence of the results above, we also complete the classification of contact structures on oriented spherical $3$-manifolds that admit symplectic rational homology ball fillings. In particular, we show that a closed, oriented $3$-manifold with finite fundamental group admits at most six contact structures, up to isotopy, which are symplectically fillable by rational homology balls.