---
tag: 94BWTKKGJM
kind: concept
name: Morse homology
msc: [57R70, 57R58, "58E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Morse_homology
same_as:
  wikidata: Q3139850
  wikipedia: Morse homology
---

## Summary

Morse homology is the homology of a chain complex built from a [[Morse function|KPQ96640ZJ]] on a closed manifold.
The chains are generated by the critical points, graded by their index.
The boundary map counts gradient flow lines between critical points whose indices differ by one.
Its homology is isomorphic to the [[singular homology|M0Q5CFJ7ZH]] of the manifold, whatever Morse function and metric are used.
This point of view is the model for [[Floer homology|NCKFPAGHXF]] in infinite-dimensional settings.

## Explanations

### L1 human

Pour water on a landscape and watch it flow downhill from passes to pits, or from peaks to passes.
Morse homology records which critical points are joined by such flow lines and in how many ways.
From this purely local bookkeeping one recovers the global holes of the space.
For example, on an upright torus there are two flow lines from the top down to each saddle, and they cancel in pairs.
The resulting homology has one generator in degree $0$, two in degree $1$ and one in degree $2$, as it should.