Morse homology

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Morse homology is a way to study the shape of a smooth space using a function and a metric. For a compact smooth manifold M, pick a Morse function f (all critical points are nondegenerate) and a Riemannian metric g. The gradient flow of f with respect to g moves points along paths that locally decrease f.

If the pair (f, g) is Morse–Smale (the stable and unstable manifolds of all critical points intersect nicely), we can build a chain complex. The generators are the critical points of f, grouped by their Morse index (a number that measures how many directions f increases). The differential counts gradient flow lines from a critical point p of index i to a critical point q of index i−1. After a small adjustment (quotienting by reparametrizations of the flow lines), these flow lines form a finite set, and each line contributes a signed count to the differential.

The key fact is that d^2 = 0. This comes from looking at how 1-dimensional families of flow lines can break into two steps via an intermediate critical point; the broken flows form the boundary of a compact 1-dimensional moduli space, and their signed count cancels out.

The homology of this Morse chain complex, called Morse homology, does not depend on the particular Morse function or metric used. Any two choices are connected by a smooth change (a homotopy) that induces an isomorphism of Morse homologies. One way to see this is via continuation maps; another is to relate Morse homology directly to singular or cellular homology.

Because Morse homology is isomorphic to singular homology, the Morse numbers (the counts of critical points by index) satisfy Morse inequalities, linking dynamics to topology.

Morse homology also inspires generalizations. Floer homology extends the ideas to infinite-dimensional settings, while Morse–Bott theory handles functions with critical manifolds. Novikov theory generalizes to closed 1-forms, and there are various further connections in geometry and topology, including ideas related to Witten’s Morse theory.