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  "body": "\n## Summary\n\nMorse homology is the homology of a chain complex built from a [[Morse function|KPQ96640ZJ]] on a closed manifold.\nThe chains are generated by the critical points, graded by their index.\nThe boundary map counts gradient flow lines between critical points whose indices differ by one.\nIts homology is isomorphic to the [[singular homology|M0Q5CFJ7ZH]] of the manifold, whatever Morse function and metric are used.\nThis point of view is the model for [[Floer homology|NCKFPAGHXF]] in infinite-dimensional settings.\n\n## Explanations\n\n### L1 human\n\nPour water on a landscape and watch it flow downhill from passes to pits, or from peaks to passes.\nMorse homology records which critical points are joined by such flow lines and in how many ways.\nFrom this purely local bookkeeping one recovers the global holes of the space.\nFor example, on an upright torus there are two flow lines from the top down to each saddle, and they cancel in pairs.\nThe resulting homology has one generator in degree $0$, two in degree $1$ and one in degree $2$, as it should.\n",
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