Filtering the Heegaard Floer contact invariant

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 Ozsváth-Szabó contact class.

36 pages, 17 figures. This article supersedes arXiv:1503.01685v2. v5: Proposition 3.3 is replaced due to an error in its proof. Section 4 is modified accordingly. The contents of Section 5 are removed for future investigation. New application obstructing Stein fillability of contact 3-manifolds with non-vanishing Ozsvath-Szabo contact class is added as Section 5