Symplectic fillings and cobordisms of lens spaces

TL;DR

This paper completes the classification of symplectic fillings of tight contact structures on lens spaces, showing that any such filling relates to a universal tight filling and describing the maximal Stein filling.

Contribution

It provides a full classification of symplectic fillings for tight contact structures on lens spaces and introduces methods to construct symplectic cobordisms between them.

Findings

Any symplectic filling of a virtually overtwisted contact structure on L(p,q) has a related filling of the universally tight structure.

The Stein filling with maximal second homology is obtained via plumbing disk bundles.

Partial results on constructing symplectic cobordisms between lens spaces are reported.

Abstract

We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling $X$ of a virtually overtwisted contact structure on $L(p,q)$ has another symplectic structure that fills the universally tight contact structure on $L(p,q)$. Moreover, we show that the Stein filling of $L(p,q)$ with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results.