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A theorem expressing the real cohomology groups of a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303301.png" /> in terms of the complex of differential forms (cf. [[Differential form|Differential form]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303302.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303303.png" /> is the de Rham complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303305.png" /> is the space of all infinitely-differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303306.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303307.png" /> equipped with the exterior differentiation, then de Rham's theorem establishes an isomorphism between the graded cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303308.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d0303309.png" /> and the cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033011.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033012.png" />. An explicit interpretation of this isomorphism is that to each closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033013.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033014.png" /> there is associated a linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033015.png" /> on the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033016.png" />-dimensional singular cycles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033017.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033018.png" />.
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A theorem expressing the real cohomology groups of a differentiable manifold  $  M $
 +
in terms of the complex of differential forms (cf. [[Differential form|Differential form]]) on $  M $.  
 +
If $  E  ^ {*} ( M) = \sum _ {p = 0 }  ^ {n} E  ^ {p} ( M) $
 +
is the de Rham complex of $  M $,  
 +
where $  E  ^ {p} ( M) $
 +
is the space of all infinitely-differentiable $  p $-
 +
forms on $  M $
 +
equipped with the exterior differentiation, then de Rham's theorem establishes an isomorphism between the graded cohomology algebra $  H  ^ {*} ( E  ^ {*} ( M)) $
 +
of the complex $  E  ^ {*} ( M) $
 +
and the cohomology algebra $  H  ^ {*} ( M, \mathbf R ) $
 +
of $  M $
 +
with values in $  \mathbf R $.  
 +
An explicit interpretation of this isomorphism is that to each closed $  p $-
 +
form $  \omega $
 +
there is associated a linear form $  \gamma \rightarrow \int _  \gamma  \omega $
 +
on the space of $  p $-
 +
dimensional singular cycles $  \gamma $
 +
in $  M $.
  
 
The theorem was first established by G. de Rham [[#References|[1]]], although the idea of a connection between cohomology and differential forms goes back to H. Poincaré.
 
The theorem was first established by G. de Rham [[#References|[1]]], although the idea of a connection between cohomology and differential forms goes back to H. Poincaré.
  
There are various versions of de Rham's theorem. For example, the cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033019.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033020.png" /> of forms with compact supports is isomorphic to the real cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033021.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033022.png" /> with compact supports. The cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033023.png" /> with values in a locally constant sheaf of vector spaces is isomorphic to the cohomology of the complex of differential forms with values in the corresponding flat vector bundle [[#References|[3]]]. The cohomology of a simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033024.png" /> with values in any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033025.png" /> of characteristic 0 is isomorphic to the cohomology of the corresponding de Rham polynomial complex over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033026.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033027.png" /> is the singular complex of an arbitrary topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033028.png" /> one obtains in this way a graded-commutative differential graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033029.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033030.png" /> with cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033031.png" /> isomorphic to the singular cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033032.png" /> (see [[#References|[4]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033033.png" /> is a smooth affine algebraic variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033034.png" />, then the cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033035.png" /> is isomorphic to the cohomology algebra of the complex of regular differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033036.png" /> (see [[De Rham cohomology|de Rham cohomology]]).
+
There are various versions of de Rham's theorem. For example, the cohomology algebra $  H  ^ {*} ( E _ {c}  ^ {*} ( M)) $
 +
of the complex $  E _ {c}  ^ {*} ( M) $
 +
of forms with compact supports is isomorphic to the real cohomology algebra $  H _ {c}  ^ {*} ( M, \mathbf R ) $
 +
of the manifold $  M $
 +
with compact supports. The cohomology of $  M $
 +
with values in a locally constant sheaf of vector spaces is isomorphic to the cohomology of the complex of differential forms with values in the corresponding flat vector bundle [[#References|[3]]]. The cohomology of a simplicial set $  S $
 +
with values in any field $  k $
 +
of characteristic 0 is isomorphic to the cohomology of the corresponding de Rham polynomial complex over $  k $.  
 +
In the case when $  S $
 +
is the singular complex of an arbitrary topological space $  X $
 +
one obtains in this way a graded-commutative differential graded $  k $-
 +
algebra $  A _ { \mathop{\rm dR}  } ( X) $
 +
with cohomology algebra $  H  ^ {*} ( A _ { \mathop{\rm dR}  } ( X)) $
 +
isomorphic to the singular cohomology algebra $  H  ^ {*} ( X, k) $(
 +
see [[#References|[4]]]). If $  X $
 +
is a smooth affine algebraic variety over $  \mathbf C $,  
 +
then the cohomology algebra $  H  ^ {*} ( X, \mathbf C ) $
 +
is isomorphic to the cohomology algebra of the complex of regular differential forms on $  M $(
 +
see [[De Rham cohomology|de Rham cohomology]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. de Rham,  "Sur l'analysis situs des variétés à <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030330/d03033037.png" /> dimensions"  ''J. Math. Pures Appl. Sér. 9'' , '''10'''  (1931)  pp. 115–200</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. de Rham,  "Differentiable manifolds" , Springer  (1984)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.S. Raghunathan,  "Discrete subgroups of Lie groups" , Springer  (1972)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  D. Lehmann,  "Théorie homotopique des forms différentiélles (d'après D. Sullivan)"  ''Astérisque'' , '''45'''  (1977)</TD></TR></table>
+
<table><tr><td valign="top">[1]</td> <td valign="top">  G. de Rham,  "Sur l'analysis situs des variétés à $n$ dimensions"  ''J. Math. Pures Appl. Sér. 9'' , '''10'''  (1931)  pp. 115–200</td></tr><tr><td valign="top">[2]</td> <td valign="top">  G. de Rham,  "Differentiable manifolds" , Springer  (1984)  (Translated from French)</td></tr><tr><td valign="top">[3]</td> <td valign="top">  M.S. Raghunathan,  "Discrete subgroups of Lie groups" , Springer  (1972)</td></tr><tr><td valign="top">[4]</td> <td valign="top">  D. Lehmann,  "Théorie homotopique des forms différentiélles (d'après D. Sullivan)"  ''Astérisque'' , '''45'''  (1977)</td></tr></table>

Latest revision as of 17:01, 1 July 2020


A theorem expressing the real cohomology groups of a differentiable manifold $ M $ in terms of the complex of differential forms (cf. Differential form) on $ M $. If $ E ^ {*} ( M) = \sum _ {p = 0 } ^ {n} E ^ {p} ( M) $ is the de Rham complex of $ M $, where $ E ^ {p} ( M) $ is the space of all infinitely-differentiable $ p $- forms on $ M $ equipped with the exterior differentiation, then de Rham's theorem establishes an isomorphism between the graded cohomology algebra $ H ^ {*} ( E ^ {*} ( M)) $ of the complex $ E ^ {*} ( M) $ and the cohomology algebra $ H ^ {*} ( M, \mathbf R ) $ of $ M $ with values in $ \mathbf R $. An explicit interpretation of this isomorphism is that to each closed $ p $- form $ \omega $ there is associated a linear form $ \gamma \rightarrow \int _ \gamma \omega $ on the space of $ p $- dimensional singular cycles $ \gamma $ in $ M $.

The theorem was first established by G. de Rham [1], although the idea of a connection between cohomology and differential forms goes back to H. Poincaré.

There are various versions of de Rham's theorem. For example, the cohomology algebra $ H ^ {*} ( E _ {c} ^ {*} ( M)) $ of the complex $ E _ {c} ^ {*} ( M) $ of forms with compact supports is isomorphic to the real cohomology algebra $ H _ {c} ^ {*} ( M, \mathbf R ) $ of the manifold $ M $ with compact supports. The cohomology of $ M $ with values in a locally constant sheaf of vector spaces is isomorphic to the cohomology of the complex of differential forms with values in the corresponding flat vector bundle [3]. The cohomology of a simplicial set $ S $ with values in any field $ k $ of characteristic 0 is isomorphic to the cohomology of the corresponding de Rham polynomial complex over $ k $. In the case when $ S $ is the singular complex of an arbitrary topological space $ X $ one obtains in this way a graded-commutative differential graded $ k $- algebra $ A _ { \mathop{\rm dR} } ( X) $ with cohomology algebra $ H ^ {*} ( A _ { \mathop{\rm dR} } ( X)) $ isomorphic to the singular cohomology algebra $ H ^ {*} ( X, k) $( see [4]). If $ X $ is a smooth affine algebraic variety over $ \mathbf C $, then the cohomology algebra $ H ^ {*} ( X, \mathbf C ) $ is isomorphic to the cohomology algebra of the complex of regular differential forms on $ M $( see de Rham cohomology).

References

[1] G. de Rham, "Sur l'analysis situs des variétés à $n$ dimensions" J. Math. Pures Appl. Sér. 9 , 10 (1931) pp. 115–200
[2] G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French)
[3] M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972)
[4] D. Lehmann, "Théorie homotopique des forms différentiélles (d'après D. Sullivan)" Astérisque , 45 (1977)
How to Cite This Entry:
De Rham theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_Rham_theorem&oldid=13456
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article