Heegaard–Floer homology

This chapter reviews the Heegaard-Floer homology, due to Peter Ozsvath and Zoltan Szabo. The Heegaard-Floer homology is a homological invariant of three-manifolds and knots in three-manifolds. A crucial observation is that the Euler characteristic of the Heegaard-Floer homology for knots in S³ coincides with the Alexander polynomial. This can be seen by considering some standard Heegaard decomposition of S³ related to a particular diagram of a knot with a fixed system of meridians. The chapter explains the Alexander polynomial invariant and describes how it relates to Heegaard-Floer homology. Heegaard-Floer homology is well suited to problems in knot theory and three-manifold topology which can be formulated in terms of the existence of four-dimensional cobordisms; one of the most striking applications is that the Heegaard-Floer homology is an unknot detector.