Heegaard Floer homologies and contact structures

TL;DR

This paper introduces an invariant in Heegaard Floer homology that distinguishes tight contact structures from overtwisted ones on three-manifolds, linking contact topology with Floer homology.

Contribution

It constructs a new invariant in Floer homology that detects Stein fillability and relates contact structures to open book decompositions and knot Floer homology.

Findings

Invariant vanishes for overtwisted contact structures

Invariant is non-zero for Stein fillable contact structures

Uses Giroux's open book decomposition and knot Floer homology

Abstract

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant which takes values in the three-manifold's Floer homology $\HFa$. This invariant vanishes for overtwisted contact structures and is non-zero for Stein fillable ones. The construction uses of Giroux's interpretation of contact structures in terms of open book decompositions, and the knot Floer homologies introduced in math.GT/0209056.