Fillable structures on negative-definite Seifert fibred spaces

TL;DR

This paper classifies fillable contact structures on negative-definite Seifert fibred spaces, computes their maximal twisting number explicitly, and relates tight structures to Stein structures on surface singularity resolutions.

Contribution

It provides a complete classification of fillable contact structures on negative-definite star-shaped plumbings and links tight structures to Stein structures via lattice cohomology.

Findings

Negative maximal twisting number is unique for these spaces.

Explicit computation of the twisting number using lattice cohomology.

Negative-twisting tight structures are induced by Stein structures.

Abstract

We classify fillable contact structures on all negative-definite star-shaped plumbings. Along the way, we show that such Seifert fibred spaces admit a unique negative maximal twisting number, and compute it explicitly using the Alexander filtration in lattice cohomology. In particular, we show that the negative-twisting tight structures on these manifolds are induced by the Stein structures on the minimal resolution of the underlying complex surface singularity. As an application, we provide a necessary condition for a negative-definite Seifert fibred space to admit a separating contact-type embedding in a strong symplectic filling of a generalised $L$-space.