Complementary legs and symplectic rational balls

TL;DR

The paper addresses which rational homology 3-spheres bound symplectic rational homology balls, focusing on small Seifert fibered spaces with complementary legs and on spherical 3-manifolds. It develops theta-invariant obstructions and leverages Lisca's lens-space criteria and Lecuona's smooth classifications to obtain a sharp dichotomy: for $e_0\u2264{-}2$ there are no symplectic fillings, while for $e_0\u2265{-}1$ fillings occur only in explicit families $Y(e_0;r,s,1-r)$ with a uniquely determined balanced contact structure in many cases. The work completes the spherical-manifold classification across both orientations, identifying lens spaces $L(m^2,mh-1)$ and dihedral-type manifolds (including $-T_3$) as the complete set of oriented spherical manifolds admitting symplectic rational ball fillings. By providing concrete theta-invariant calculations and a cobordism-based framework, the authors clarify the contrast between smooth and symplectic fillings and supply practical obstructions for ruling out fillings in broad families of Seifert fibered spaces.

Abstract

We completely characterize when a small Seifert fibered space with complementary legs symplectically bounds a rational homology ball in the case $e_0\leq -1$, and we establish strong obstructions for other values of $e_0$. 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. We also complete the classification of contact structures on spherical $3$-manifolds with either orientations that admit symplectic rational homology ball fillings.

Figures (12)

• Figure 1: A surgery diagram for the small Seifert fibered space $Y(e_{0};r_{1},r_{2},r_{3})$, with normalized Seifert invariants.
• Figure 2: Surgery diagram for a small Seifert fibered space with complementary legs. Here $r/s>1$, $r/s=[a^1_1,\ldots, a^1_{n_1}]$, $p/q=[a^2_1,\ldots, a^2_{n_2}]$, and $r/(r-s)=[a^3_1,\ldots, a^3_{n_3}]$.
• Figure 3: A surgery diagram for $S^1\times S^2$ with a Seifert fibered structure where $F$ is a regular fiber.
• Figure 4: Surgery diagram for a small Seifert fibered space with complementary legs from Lecuona's paper Lecuona2019. Here $r/s>1$, $r/s=[a^1_1,\ldots, a^1_{n_1}]$, and $r/(s-r)=[a^3_1,\ldots, a^3_{n_3}]$, where $0\leq t\leq n_2$, $a_t^2>2$, and $[a_t^2-1, a_{t+1}^2,\ldots, a_{n_2}^2]$ is in $\mathcal{R}$.
• Figure 5: Equating Figures \ref{['fig:balanced']} and \ref{['fig:lecuona']} when $e_0=-2$.

Theorems & Definitions (61)

• Theorem 1.1: Lecuona 2019, Lecuona2019
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Proposition 1.9
• Remark 1.10