# Poincaré formula

Let $K$ be finite oriented simplicial complex^{} of dimension^{} $n$. Then

$$\chi (K)=\sum _{p=0}^{n}{(-1)}^{p}{R}_{p}(K),$$ |

where $\chi (K)$ is the Euler characteristic^{} of $K$, and ${R}_{p}(K)$ is the $p$-th Betti number of $K$.

This formula also works when $K$ is any finite CW complex. The Poincaré formula is also known as the Euler-Poincaré formula, for it is a generalization of the Euler formula^{} for polyhedra.

If $K$ is a compact^{} connected^{} orientable surface with no boundary and with genus h, then $\chi (K)=2-2h$. If $K$ is non-orientable instead, then $\chi (K)=2-h$.

Title | Poincaré formula |
---|---|

Canonical name | PoincareFormula |

Date of creation | 2013-03-22 13:40:15 |

Last modified on | 2013-03-22 13:40:15 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 11 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 05C99 |

Synonym | Euler-Poincaré formula |

Synonym | Euler-Poincare formula |

Related topic | EulersPolyhedronTheorem |

Related topic | Polytope |