De Rham cohomology

Discussion of the sequence is most clear in de Rham cohomology.

Of course one does not see any of this in de Rham cohomology.

The first de Rham cohomology group is 0 if and only if all closed 1-forms are exact.

Flux is the integral of a differential form and, consequently, a de Rham cohomology class.

Non-contractible spaces need not have trivial de Rham cohomology.

It is defined in the same way as de Rham cohomology except that one uses square-integrable differential forms.

In other words, for each de Rham cohomology class on M, there is a unique harmonic representative.

One major consequence of this is that the de Rham cohomology groups on a compact manifold are finite-dimensional.

Another useful fact is that the de Rham cohomology is a homotopy invariant.

Then the de Rham cohomology groups with compact support are the homology of the chain complex :

In de Rham cohomology, the cup product of differential forms is induced by the wedge product.

The idea of de Rham cohomology is to classify the different types of closed forms on a manifold.

This gives a homomorphism from de Rham cohomology to singular cohomology.

From the differential forms and the exterior derivative, one can define the de Rham cohomology of the manifold.

This product can be understood as induced by the exterior product of differential forms in de Rham cohomology.

Its cohomology is the de Rham cohomology of M.

This definition of algebraic de Rham cohomology is available for algebraic varieties over any field k.

Another version of the theorem with real coefficients features the de Rham cohomology with values in the orientation bundle.

The noncommutative or quantum de Rham cohomology is defined as the cohomology of this complex.

This theorem also underlies the duality between de Rham cohomology and the homology of chains.

These are homological in nature, in the way that differential forms give rise to De Rham cohomology.

It was developed by Hodge in the 1930s as an extension of de Rham cohomology, and has major applications on three levels:

That is, Chern classes are cohomology classes in the sense of de Rham cohomology.

A corollary of the Poincaré lemma is that de Rham cohomology is homotopy-invariant.

On the other hand, the differential forms, with exterior derivative, d, as the connecting map, form a cochain complex, which defines de Rham cohomology.