# 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 $\u2102$-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 |
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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 |