# Morse complex

Let $M$ be a smooth manifold, and $u:M\to\mathbb{R}$ be a Morse function. Let $C_{n}^{u}(M)$ be a vector space of formal $\mathbb{C}$-linear combinations of critical points of $u$ with index $n$. Then there exists a differential $\partial_{n}:C_{n}\to C_{n-1}$ based on the Morse flow making $C_{*}$ into a chain complex called the Morse complex such that the homology 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 MorseComplex 2013-03-22 13:53:18 2013-03-22 13:53:18 bwebste (988) bwebste (988) 4 bwebste (988) Definition msc 58E05