Filtering the Heegaard Floer contact invariant

TL;DR

This paper introduces a new invariant of contact structures in three dimensions derived from Heegaard Floer homology, which helps distinguish between overtwisted and Stein fillable contact structures and has applications in obstructing Stein fillability.

Contribution

It defines a novel contact invariant in Heegaard Floer homology with specific properties and computability from open book decompositions, advancing contact topology tools.

Findings

Invariant is zero for overtwisted structures

Invariant is infinite for Stein fillable structures

Obstructs Stein fillability in certain contact 3-manifolds

Abstract

We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsv\'ath-Szab\'o contact class.