---
tag: 2WJJBN2T5P
kind: concept
name: Heegaard Floer homology
msc: [57R58, 57K31, 57K18]
specializes: [NCKFPAGHXF]
sources:
  - url: https://en.wikipedia.org/wiki/Floer_homology
---

## Summary

Heegaard Floer homology is a family of invariants of closed oriented [[3-manifolds|R20TB2KEQQ]], introduced by Peter Ozsváth and Zoltán Szabó in the early 2000s.
It is a kind of [[Floer homology|NCKFPAGHXF]] built from a [[Heegaard splitting|EDE9919B5R]]: 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|5P608WRETM]], and it gives invariants of smooth $4$-dimensional cobordisms.

## Explanations

### L1 human

A three-dimensional space can be cut along a surface into two simple pieces, and the way the pieces are glued is drawn as two families of curves on the surface.
Heegaard Floer homology turns this drawing into a chain complex: its generators come from the crossings of the two families, and its differential counts certain maps of disks.
The homology of this complex does not depend on the drawing, only on the space.
For the $3$-sphere, the simplest version $\widehat{HF}$ is a single copy of the coefficient field.