Heegaard Floer homology and contact structures

Abstract

Given a contact structure on a closed, oriented three-manifold $Y$, we describe an invariant that takes values in the three-manifold's Floer homology $\widehat{\mathrm{HF}}$. This invariant vanishes for overtwisted contact structures and is nonzero for Stein-fillable ones. The construction uses Giroux's interpretation of contact structures in terms of open-book decompositions.