De Rham theorem
A theorem expressing the real cohomology groups of a differentiable manifold in terms of the complex of differential forms (cf. Differential form) on
. If
is the de Rham complex of
, where
is the space of all infinitely-differentiable
-forms on
equipped with the exterior differentiation, then de Rham's theorem establishes an isomorphism between the graded cohomology algebra
of the complex
and the cohomology algebra
of
with values in
. An explicit interpretation of this isomorphism is that to each closed
-form
there is associated a linear form
on the space of
-dimensional singular cycles
in
.
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 of the complex
of forms with compact supports is isomorphic to the real cohomology algebra
of the manifold
with compact supports. The cohomology of
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
with values in any field
of characteristic 0 is isomorphic to the cohomology of the corresponding de Rham polynomial complex over
. In the case when
is the singular complex of an arbitrary topological space
one obtains in this way a graded-commutative differential graded
-algebra
with cohomology algebra
isomorphic to the singular cohomology algebra
(see [4]). If
is a smooth affine algebraic variety over
, then the cohomology algebra
is isomorphic to the cohomology algebra of the complex of regular differential forms on
(see de Rham cohomology).
References
[1] | G. de Rham, "Sur l'analysis situs des variétés à ![]() |
[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) |
De Rham theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=De_Rham_theorem&oldid=46588