On symplectic fillings of virtually overtwisted torus bundles

TL;DR

This paper classifies symplectic fillings of virtually overtwisted torus bundles by reducing the problem to tight lens spaces, providing a complete classification for elliptic and parabolic cases and describing the hyperbolic case.

Contribution

It extends the classification of symplectic fillings to virtually overtwisted torus bundles using Menke's decomposition theorem, linking them to tight lens space fillings.

Findings

Complete classification for elliptic and parabolic torus bundles.

All hyperbolic bundle fillings derive from tight lens space fillings.

Conditions identified for when different lens space fillings produce the same bundle filling.

Abstract

We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1-handle attachment, and we give a condition under which distinct tight lens space fillings yield the same torus bundle filling.