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 $M$ .