Let be a smooth manifold, and be a Morse function. Let be a vector space of formal -linear combinations of critical points of with index . Then there exists a differential based on the Morse flow making into a chain complex called the Morse complex such that the homology of the complex is the singular homology of . In particular, the number of critical points of of index on is at least the -th Betti number, and the alternating sum of the number of critical points of is the Euler characteristic of .