Morse complex

Let M be a smooth manifoldMathworldPlanetmath, and u:M be a Morse function. Let Cnu(M) be a vector spaceMathworldPlanetmath of formal -linear combinationsMathworldPlanetmath of critical points of u with index n. Then there exists a differential n:CnCn-1 based on the Morse flow making C* into a chain complex called the Morse complex such that the homologyPlanetmathPlanetmath of the complex is the singular homology of M. In particular, the number of critical points of u of index n on M is at least the n-th Betti number, and the alternating sum of the number of critical points of u is the Euler characteristic of M.

Title Morse complex
Canonical name MorseComplex
Date of creation 2013-03-22 13:53:18
Last modified on 2013-03-22 13:53:18
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 4
Author bwebste (988)
Entry type Definition
Classification msc 58E05