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  "body": "\n## Summary\n\nHeegaard Floer homology is a family of invariants of closed oriented [[3-manifolds|R20TB2KEQQ]], introduced by Peter Ozsváth and Zoltán Szabó in the early 2000s.\nIt 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.\nIts versions are written $\\widehat{HF}$, $HF^+$, $HF^-$ and $HF^\\infty$.\nIt has a relative version for knots, [[knot Floer homology|5P608WRETM]], and it gives invariants of smooth $4$-dimensional cobordisms.\n\n## Explanations\n\n### L1 human\n\nA 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.\nHeegaard 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.\nThe homology of this complex does not depend on the drawing, only on the space.\nFor the $3$-sphere, the simplest version $\\widehat{HF}$ is a single copy of the coefficient field.\n",
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