Heegaard Floer homology

Summary

Heegaard Floer homology is a family of invariants of closed oriented 3-manifolds, introduced by Peter Ozsváth and Zoltán Szabó in the early 2000s. It is a kind of Floer homology built from a Heegaard splitting: one counts holomorphic disks in a symmetric product of the Heegaard surface. Its versions are written $\widehat{HF}$, $HF^+$, $HF^-$ and $HF^\infty$. It has a relative version for knots, knot Floer homology, and it gives invariants of smooth $4$-dimensional cobordisms.