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\begin{document}

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\section*{de Rham theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{on_cohomology}{On cohomology}\dotfill \pageref*{on_cohomology} \linebreak
\noindent\hyperlink{OnCochains}{On cochains}\dotfill \pageref*{OnCochains} \linebreak
\noindent\hyperlink{synthetic_version}{Synthetic version}\dotfill \pageref*{synthetic_version} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{on_cohomology_2}{On cohomology}\dotfill \pageref*{on_cohomology_2} \linebreak
\noindent\hyperlink{ReferencesdeRhamTheoremOnCocycles}{On cocycles}\dotfill \pageref*{ReferencesdeRhamTheoremOnCocycles} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{on_cohomology}{}\subsubsection*{{On cohomology}}\label{on_cohomology}

The \textbf{de Rham theorem} (named after [[Georges de Rham]]) asserts that the [[de Rham cohomology]] $H^n_{dR}(X)$ of a smooth [[manifold]] $X$ (without boundary) is [[isomorphism|isomorphic]] to the [[real cohomology]] $H^n(X, \mathbb{R})$, hence its [[ordinary cohomology]] with [[real number]] [[coefficients]], as computed for instance by the [[singular cohomology|singular]] or [[Čech cohomology]] with real coefficients.

The theorem has several dozens of different proofs. For example in the ech approach one can make a double complex whose first row is the ech complex of a covering and first column is the de Rham complex and other entries are mixed and use spectral sequence argument (see the textbook of Bott and Tu, or the geometry lectures book by Postnikov, semester III).

This is maybe best formulated, understood and proven in the context of [[abelian sheaf cohomology]]:

Write $\mathbb{R}_c$ for

\begin{itemize}%
\item the abelian group $\mathbb{R}$


\item regarded not as a [[Lie group]] with the standard [[manifold]] structure on $\mathbb{R}$ but as a topologically discrete group on the underlying set of $\mathbb{R}$


\item and then regarded as a [[sheaf]] on $X$: the [[constant sheaf]] that sends connected $U \subset X$ to the set of \emph{constant} maps $U \to \mathbb{R}$.



\end{itemize}
Write $\mathbf{B}^n \mathbb{R}_c$ for the corresponding [[Eilenberg-MacLane object]] in [[chain complex]]es of sheaves of abelian groups: this is the complex of sheaves with $\mathbb{R}_c$ in degree $n$:

\begin{displaymath}
\mathbf{B}^n \mathbb{R}_c
  = 
  (\cdots \to 0 \to \mathbb{R}_c \to 0 \to \cdots \to 0) 
  \,.
\end{displaymath}
Next, write $\bar \mathbf{B}^n \mathbb{R}$ (without the subscript $c$!) for the [[Deligne cohomology|Deligne complex]] for $\mathbb{R}$

\begin{displaymath}
\bar \mathbf{B}^n \mathbb{R}
  = 
  (C^\infty(-,\mathbb{R}) \stackrel{d_{dR}}{\to}
    \Omega^1(-) \stackrel{d_{dR}}{\to}
    \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{closed}(-)) 
  \,.
\end{displaymath}
(The notation here is borrowed from that used at [[motivation for sheaves, cohomology and higher stacks]]: we can think of $\bar \mathbf{B}^n \mathbb{R}$ as a differential refinement of the object $\mathbf{B}^n \mathbb{R}_c$).

Then we have:

\begin{itemize}%
\item ``ordinary'' $\mathbb{R}$-valued cohomology of $X$ is the [[abelian sheaf cohomology]] with coefficients in $\mathbf{B}^n \mathbb{R}_c$.


\item [[de Rham cohomology]] of $X$ is the [[abelian sheaf cohomology]] with coefficients in $\bar \mathbf{B}^n \mathbb{R}$ (this is semi-obvious, requires a bit more discussion).


\item the [[Poincare lemma]] says that every closed [[differential form]] is locally exact, and hence there is a [[quasi-isomorphism]] of chain complexes of sheaves

\begin{displaymath}
\mathbf{B}^n \mathbb{R}_c \stackrel{\simeq}{\to}
  \bar \mathbf{B}^n \mathbb{R}
\end{displaymath}
given by injecting for each $U \subset X$ the set $\mathbb{R}$ as the constant functions into $C^\infty(U,\mathbb{R})$.



\end{itemize}
It is this quasi-isomorphism of coefficient objects that induces the de Rham isomorphism of [[abelian sheaf cohomology|abelian sheaf]] [[cohomology group]]s, which is ordinarily written as

\begin{displaymath}
H^n(X,\mathbb{R}) \simeq 
  H^n_{dR}(X)
  \,.
\end{displaymath}
\ldots{}

\hypertarget{OnCochains}{}\subsubsection*{{On cochains}}\label{OnCochains}

The equivalence on cohomology asserted by the de Rham theorem is but a [[decategorification]] of a more refined statement: a [[quasi-isomorphism]] of [[cochain complex]]es. This even respects the product structure:

for $X$ a [[smooth manifold]] there is an [[equivalence in an (infinity,1)-category|equivalence]] of [[A-infinity algebra]]s

\begin{displaymath}
(\Omega^\bullet(X), d_{dR})
  \stackrel{\simeq}{\to}
  (C(X), \cup)
\end{displaymath}
between the [[de Rham complex]] and the collection of [[singular cohomology|singular cochain]]s equipped with the [[cup product]].

This is due to (\hyperlink{Gugenheim}{Gugenheim, 1977}).

Furthermore, the [[E-infinity algebra]] structure on [[differential forms]] (trivially induced by the [[commutative dga]] structure) and [[singular cochains]] (as witnessed by the action of the [[sequence operad]] of McClure and Smith on [[singular cochains]]) is also preserved.

\hypertarget{synthetic_version}{}\subsubsection*{{Synthetic version}}\label{synthetic_version}

The de Rham theorem also holds internally in the context of suitable [[smooth topos]]es $\mathcal{T}$ modelling the axioms of [[synthetic differential geometry]].

Specifically

\begin{itemize}%
\item the internal [[singular cohomology|singular chain complex]] in $\mathcal{T}$ is given as the $R$-linear dual of the free internal $R$-module on the [[internal hom]] objects $[\Delta^n,X]$, where $R$ is the internal incarnation of the real numbers;


\item the de Rham complex is given by [[differential forms in synthetic differential geometry]].



\end{itemize}
The de Rham theorem in $\mathcal{T}$ then asserts that for $X$ a manifold regarded as an object in the well-adapted smooth topos $\mathcal{T}$ the morphism

\begin{displaymath}
\int : H^p(X) \to H_p(X,R)^*
\end{displaymath}
in $\mathcal{T}$ is an [[isomorphism]] for all $p \in \mathbb{N}$. This implies the standard (external) de Rham theorem.

This is discussed in chapter IV of

\begin{itemize}%
\item [[Ieke Moerdijk]], [[Gonzalo Reyes]], \emph{[[Models for Smooth Infinitesimal Analysis]]} for the toposes $\mathcal{T} = \mathcal{G}, \mathcal{F}$ (see \href{http://ncatlab.org/nlab/show/Models+for+Smooth+Infinitesimal+Analysis#Models}{models}).

\end{itemize}
A little bit a long these lines for [[diffeological space]]s is also in

\begin{itemize}%
\item Patrick Iglesias-Zemmour, \emph{De Rham calculus} (\href{http://math.huji.ac.il/~piz/documents/D6.pdf}{pdf})

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[equivariant de Rham theorem]]


\item [[Poincaré lemma]]


\item [[Stokes theorem]]


\item \textbf{de Rham theorem}

\begin{itemize}%
\item [[kernel of integration is the exact differential forms]]

\end{itemize}

\item [[Dolbeault theorem]]


\item [[Hodge theorem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{on_cohomology_2}{}\subsubsection*{{On cohomology}}\label{on_cohomology_2}

Standard textbook references include

\begin{itemize}%
\item [[Raoul Bott]], Loring Tu, \emph{Algebraic topology and differential forms},


\item [[M M Postnikov]], Lectures on geometry, vol. III, Differentiable manifolds


\item Arne Lorenz, \emph{Abstract de Rham theorem}, \href{http://wwwb.math.rwth-aachen.de/~arne/Talks/deRham.pdf}{pdf slides} (exposition of the standard de Rham theorem)



\end{itemize}
In analytic geometry also

\begin{itemize}%
\item M. E. Herrera, \emph{De Rham theorems on semianalytic sets}, Bull. Amer. Math. Soc. \textbf{7} 3 (1967) 414--418, \href{http://dx.doi.org/10.1090/S0002-9904-1967-11772-5}{doi}, \href{http://www.ams.org/mathscinet-getitem?mr=214094}{MR214094}

\end{itemize}
\hypertarget{ReferencesdeRhamTheoremOnCocycles}{}\subsubsection*{{On cocycles}}\label{ReferencesdeRhamTheoremOnCocycles}

The refinement of the de Rham theorem from an [[isomorphism]] of [[cohomology group]]s to an [[equivalence]] of [[A-∞ algebras]] of cochains and forms was first stated in

\begin{itemize}%
\item [[Victor Gugenheim]], \emph{On Chen's iterated integrals} , Illinois J. Math. Volume 21, Issue 3 (1977), 703-715.

\end{itemize}
proven using Chen's [[iterated integral]]s.

A review is in section 3 of

\begin{itemize}%
\item [[Camilo Arias Abad]], [[Florian Schätz]], \emph{The $A_\infty$ de Rham theorem and integration of representations up to homotopy} (\href{http://arxiv.org/abs/1011.4693}{arXiv:1011.4693})

\end{itemize}
[[!redirects deRham theorem]]

[[!redirects de Rham isomorphism]] [[!redirects de Rham isomorphisms]]



\end{document}