Morse homology

Summary

Morse homology is the homology of a chain complex built from a Morse function 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 of the manifold, whatever Morse function and metric are used. This point of view is the model for Floer homology in infinite-dimensional settings.