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\cl{\scriptsize (For the last version of this text, type {\em hodgegaillard} on Google. Date of this version: June 16, 2010.)}\vskip2em%\vfill 

\cl{\Large A Hodge Theorem for Noncompact Manifolds} 

\n\textbf{Theorem}\ \  \textit{If $M$ is a riemannian manifold, then the inclusion of the complex of coclosed harmonic forms into the de Rham complex induces a linear isomorphism in cohomology. If $M$ has at most countably many connected components, this linear isomorphism is a Fr\'echet isomorphism.} 

The simplest example is that of the real line with its standard metric. In degree zero the complex of coclosed harmonic forms is $\mathbb C\oplus\mathbb Cx$, and in degree one it is $\mathbb Cdx$, which gives the right cohomology. 

[Manifolds are assumed to be $C^\infty$ and Hausdorff.] 

\n\textbf{Proof.} Theorem 5 in Section I.9.10 of Bourbaki \cite{2} implying that $M$ is paracompact, we can assume that it is connected, and also that it is non-compact (the result being classical in the compact case). Then the claim follows easily (using the Open Mapping Theorem and the fact that the de Rham cohomology is a Fr\'echet space) from the surjectivity of the laplacian on the de Rham complex (see \emph{Algebra Background} below). Let us check this surjectivity. In \cite[p.~158]{4} de Rham proves (using results of Aronszajn, Krzywicki and Szarski \cite{1}) that a harmonic form which has a zero of infinite order vanishes identically; this implies in particular that the laplacian satisfies Property (A) in Definition~5 of Malgrange \cite[p.~333]{3}; it is well known that the laplacian satisfies also Condition (P) --- called \textbf{ellipticity} nowadays --- in Definition 6 of \cite[p.~338]{3};  in view of Theorem 5 in \cite[p.~341]{3} this implies the desired surjectivity. QED

\n\emph{Algebra Background}. Let $A$ be a module over some unnamed ring, and let $d,\delta$ be two endomorphisms of $A$ satisfying $d^2=0=\delta^2$. Put $\Delta:=d\delta+\delta d$. Assume $A=\Delta A+A_{d,\delta}$ where $A_{d,\delta}$ stands for $\ker d\cap\ker\delta$. Write $A_{\delta,\Delta}$ for $\ker\Delta\cap\ker\delta$. Note $dA_{\delta,\Delta}\subset A_{\delta,\Delta}$. 

We claim that the natural map $$H(A_{\delta,\Delta},d)\to H(A,d)$$ between homology modules is bijective.  

Injectivity. Assume $\delta da=0$ form some $a$ in $A$. We must find an $x$ in $A_{\delta,\Delta}$ such that $dx=da$. We have $a=\Delta b+c$ for some $b\in A$ and some $c\in A_{d,\delta}$. One easily checks that $x:=\delta db+c$ does the trick. 

Surjectivity. Let $a$ be in $\ker d$. We must find $x\in A$, $y\in A_{d,\delta}$ such that $a=dx+y$. We have $a=\Delta b+c$ for some $b\in A$ and some $c\in A_{d,\delta}$. One easily checks that $x:=\delta b$, $y:=\delta db+c$ works. 
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\bibitem{1} Aronszajn N., Krzywicki A., Szarski J., A unique continuation theorem for exterior differential forms on Riemannian manifolds, \textit{Ark. Mat.}, Volume~4, Number~5 (1962) 417-453. Available at \href{http://www.springerlink.com/content/c853518221310827/?p=5beb9fd076f6487fb274a999415eb402&pi=5}{springerlink.com}. 
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\bibitem{2} Bourbaki, N., \textbf{Topologie g\'en\'erale}, Vol. 1, Chapitres 1 \`a 4, Hermann, 1971. 
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\bibitem{3} Malgrange B., Existence et approximation des solutions des \'equations aux d\'eriv\'ees partielles et des \'equations de convolution, \textit{Ann. Inst. Fourier}, Grenoble \textbf{6} (1955-56) 271-354. Available at \href{http://www.numdam.org/numdam-bin/fitem?id=AIF_1956__6__271_0}{numdam.org}.  
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\bibitem{4} de Rham G., \textbf{Differentiable manifolds}, Springer-Verlag, 1984. 
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\n{\tiny hodge.theorem.100616}\hfill Pierre-Yves Gaillard 
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