Contact Invariants in Floer Homology

Abstract

In a pair of seminal papers Peter Ozsvath and Zoltan Szabo defined a collection of homology groups associated to a 3-manifold they named Heegaard-Floer homologies. Soon after, they associated to a contact structure ξ on a 3-manifold, an element of its Heegaard-Floer homology, the contact invariant c(ξ). This invariant has been used to prove a plethora of results in contact topology of 3-manifolds. In this series of lectures we introduce and review some basic facts about Heegaard Floer Homology and its generalization to manifolds with boundary due to Andras Juhasz, the Sutured Floer Homology. We use the open book decompositions in the case of closed manifolds, and partial open book decompositions in the case of contact manifolds with convex boundary to define contact invariants in both settings, and show some applications to fillability questions.