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 $M$ .

单词	MorseComplex
释义	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 $M$ .
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