Morse complex

Let $M$ be a smooth manifold, and $u:M\to ℝ$ be a Morse function. Let $C_n^u(M)$ be a vector space of formal $ℂ$-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 $M$.