A Survey of Heegaard Floer Homology

Abstract

This work has two goals. The first is to provide a conceptual introduction to Heegaard Floer homology, the second is to survey the current state of the field, without aiming for completeness. After reviewing the structure of Heegaard Floer homology, we list some of its most important applications. Many of these are purely topological results, not referring to Heegaard Floer homology itself. Then, we briefly outline the construction of Lagrangian intersection Floer homology. We construct the Heegaard Floer chain complex as a special case of the above, and try to motivate the role of the various seemingly ad hoc features such as admissibility, the choice of basepoint, and Spin^c-structures. We also discuss the proof of invariance of the homology up to isomorphism under all the choices made, and how to define Heegaard Floer homology using this in a functorial way (naturality). Next, we explain why Heegaard Floer homology is computable, and how it lends itself to the various combinatorial descriptions. The last chapter gives an overview of the definition and applications of sutured Floer homology, which includes sketches of some of the key proofs. Throughout, we have tried to collect some of the important open conjectures in the area. For example, a positive answer to two of these would give a new proof of the Poincar\'e conjecture.