# proof of Poincaré lemma

Let $X$ be a smooth manifold^{}, and let $\omega $ be a closed differential form of degree $k>0$ on $X$. For any $x\in X$, there exists a contractible neighbourhood $U\subset X$ of $x$ (i.e. $U$ is homotopy equivalent to a single point), with inclusion map

$$\iota :U\hookrightarrow X.$$ |

To construct such a neighbourhood, take for example an open ball in a coordinate chart around $x$. Because of the homotopy invariance of de Rham cohomology^{}, the $k$th de Rham cohomology group ${\mathrm{H}}^{k}(U)$ is isomorphic to that of a point; in particular,

$${\mathrm{H}}^{k}(U)=0\mathit{\hspace{1em}}\text{for all}k0.$$ |

Since $d({\iota}^{*}\omega )={\iota}^{*}(d\omega )=0$, this implies that there exists a $(k-1)$-form $\eta $ on $U$ such that $d\eta ={\iota}^{*}\omega $. In the case where $X$ is a contractible manifold, such an $\eta $ exists globally since we can choose $U=X$ above.

Title | proof of Poincaré lemma |
---|---|

Canonical name | ProofOfPoincareLemma |

Date of creation | 2013-03-22 14:24:36 |

Last modified on | 2013-03-22 14:24:36 |

Owner | pbruin (1001) |

Last modified by | pbruin (1001) |

Numerical id | 4 |

Author | pbruin (1001) |

Entry type | Proof |

Classification | msc 53-00 |

Classification | msc 55N05 |