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| The Poincar\'e lemma states that every closed differential form |
The Poincar\'e lemma states that every closed differential form |
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is locally \PMlinkname{exact}{ExactDifferentialForm}.
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is locally exact.
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| \begin{theorem} (Poincar\'e Lemma) |
\begin{theorem} (Poincar\'e Lemma) |
| \cite{conlon} Suppose $X$ is a smooth |
\cite{conlon} Suppose $X$ is a smooth |
| manifold, $\Omega^k(X)$ is the set of smooth differential |
manifold, $\Omega^k(X)$ is the set of smooth differential |
| $k$-forms on $X$, and suppose $\omega$ is a closed form |
$k$-forms on $X$, and suppose $\omega$ is a closed form |
| in $\Omega^k(X)$ for some $k>0$. |
in $\Omega^k(X)$ for some $k>0$. |
| \begin{itemize} |
\begin{itemize} |
| \item |
\item |
| Then for every $x\in X$ there is a neighbourhood $U\subset X$, and a |
Then for every $x\in X$ there is a neighbourhood $U\subset X$, and a |
| $(k-1)$-form $\eta\in \Omega^{k-1}(U)$, such that |
$(k-1)$-form $\eta\in \Omega^{k-1}(U)$, such that |
| $$ d\eta = \iota^\ast \omega,$$ |
$$ d\eta = \iota^\ast \omega,$$ |
| where $\iota$ is the inclusion $\iota:U\hookrightarrow X$. |
where $\iota$ is the inclusion $\iota:U\hookrightarrow X$. |
| \item If $X$ is contractible, this $\eta$ exists globally; there exists a |
\item If $X$ is contractible, this $\eta$ exists globally; there exists a |
| $(k-1)$-form $\eta\in \Omega^{k-1}(X)$ such that |
$(k-1)$-form $\eta\in \Omega^{k-1}(X)$ such that |
| $$ d\eta = \omega.$$ |
$$ d\eta = \omega.$$ |
| \end{itemize} |
\end{itemize} |
| \end{theorem} |
\end{theorem} |
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| \subsubsection*{Notes} |
\subsubsection*{Notes} |
| Despite the name, the Poincar\'e lemma is an |
Despite the name, the Poincar\'e lemma is an |
| extremely important result. For instance, in algebraic topology, |
extremely important result. For instance, in algebraic topology, |
| the definition of the $k$th de Rham cohomology group |
the definition of the $k$th de Rham cohomology group |
| $$ |
$$ H^k(X) = \frac{ \operatorname{Ker}\{ d:\Omega^k(X)\to \Omega^{k+1}(X)\}}{ \operatorname{Im}\{ d:\Omega^{k-1}(X)\to \Omega^{k}(X)\}}$$ |
| H^k(X) = \frac{ \operatorname{Ker}\{ d\colon \Omega^k(X)\to \Omega^{k+1}(X)\}}{ \operatorname{Im}\{ d\colon \Omega^{k-1}(X)\to \Omega^{k}(X)\}} |
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| $$ |
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| can be seen as a measure of the degree in which the Poincar\'e lemma fails. |
can be seen as a measure of the degree in which the Poincar\'e lemma fails. |
| If $H^k(X)=0$, then every $k$ form is exact, but if $H^k(X)$ is non-zero, then |
If $H^k(X)=0$, then every $k$ form is exact, but if $H^k(X)$ is non-zero, then |
| $X$ has a non-trivial topology (or ``holes'') such that $k$-forms are not |
$X$ has a non-trivial topology (or ``holes'') such that $k$-forms are not |
| globally exact. For instance, in $X=\sR^2\setminus\{0\}$ with polar coordinates $(r,\phi)$, |
globally exact. For instance, in $X=\sR^2\setminus\{0\}$ with polar coordinates $(r,\phi)$, |
| the $1$-form $\omega=d\phi$ is not globally exact. |
the $1$-form $\omega=d\phi$ is not globally exact. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {conlon} L. Conlon, \emph{Differentiable Manifolds: A first course}, |
\bibitem {conlon} L. Conlon, \emph{Differentiable Manifolds: A first course}, |
| Birkh\"auser, 1993. |
Birkh\"auser, 1993. |
| \end{thebibliography} |
\end{thebibliography} |
