PlanetMath: proof of Poincaré lemma

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proof of Poincaré lemma

(Proof)

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

$\displaystyle \iota\colon U\hookrightarrow X.$

To construct such a neighbourhood, take for example an open ball in a coordinate chart around

th de Rham cohomology group ${\rm H}^k(U)$ is isomorphic to that of a point; in particular,

$\displaystyle {\rm H}^k(U)=0\quad\hbox{for all $k>0$}.$

Since $d(\iota^*\omega)=\iota^*(d\omega)=0$ , this implies that there exists a

-form $\eta$ on

such that $d\eta=\iota^*\omega$ . In the case where

is a contractible manifold, such an $\eta$ exists globally since we can choose

above.

"proof of Poincaré lemma" is owned by pbruin.

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AMS MSC:	53-00 (Differential geometry :: General reference works )
	55N05 (Algebraic topology :: Homology and cohomology theories :: Cech types)

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