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[parent] proof of Poincaré lemma (Proof)

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

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



"proof of Poincaré lemma" is owned by pbruin.
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Cross-references: implies, isomorphic, de Rham cohomology group, de Rham cohomology, homotopy invariance, coordinate chart, open ball, inclusion map, point, homotopy equivalent, neighbourhood, contractible, degree, closed differential form, smooth manifold

This is version 1 of proof of Poincaré lemma, born on 2004-06-12.
Object id is 5912, canonical name is ProofOfPoincareLemma.
Accessed 2217 times total.

Classification:
AMS MSC53-00 (Differential geometry :: General reference works )
 55N05 (Algebraic topology :: Homology and cohomology theories :: Cech types)

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