De Rham cohomology

The de Rham cohomology is a cohomology theory for smooth manifolds. It is based on the Stokes' theorem, in its generalized form. An analog of the de Rham cohomology of complex manifolds is the Dolbeault cohomology

De Rham complex

Definition

Is a smooth manifold and the amount of p- forms. The De Rham complex is the Kokettenkomplex

The images are given by the Cartan derivative.

De Rham complex in three-dimensional space

If you choose to as the underlying manifold as the de Rham complex has a particular shape. In this case, the Cartan derivations correspond to known from vector analysis, differential operators gradient, divergence and rotation. Specifically, it means that the diagram

Commutes, so you the same result no matter what arrows you follow. The illustrations and are diffeomorphisms. Thus, the Sharp - isomorphism and the Hodge star operator.

Definition of de Rham cohomology

Let be a smooth manifold. The -th De Rham cohomology group is defined as the - th cohomology group of the De Rham complex. In particular, for

History

In his Paris thesis (1931 ) Georges de Rham proved his theorem a conjecture of Élie Cartan, which in turn went back to considerations of Henri Poincaré. Since the cohomology of a topological space has been an issue until several years later, he actually worked with the homology and (due to the set of Stokes ) dual complex of n- chains.

Homotopy

Let and be two homotopy equivalent smooth manifolds, then for each

Thus, since two homotopic smooth manifolds up to isomorphism have the same de Rham cohomology, this cohomology is a topological invariant of a smooth manifold. This is remarkable, because in the definition of the de Rham group the differentiable structure of the manifold plays an important role. It has so first no reason to assume that a topological manifold with different differentiable structures has the same de Rham groups.

Set of de Rham

The central message in the theory of de Rham cohomology is called the set of de Rham. He states that the de Rham cohomology of smooth manifolds is naturally isomorphic to the singular cohomology with coefficients in the real numbers. With the singular homology is called. It is therefore

Be an element of the p-th singular homology group. Then the isomorphism is the mapping

Described. It was identified ( see also Universal Koeffiziententheorem ). This mapping is called De Rham homomorphism or De Rham isomorphism.

Examples of some de Rham groups

Computing the de Rham groups is often difficult, so now follow a few examples. It is always assumed that the considered manifolds are smooth.

• Let be a connected manifold, then is equal to the amount of constant functions and has dimension one.
• Let be a zero -dimensional manifold, then the dimension of is equal to the thickness of and all other cohomology groups is disappearing.
• Be an open star-shaped domain, then for all. This is the Poincaré lemma, which states that d Ⓜ = 0, is even exactly on a star field every closed differential form, ( ie there is a " potential form " χ, so that ω = dχ applies ).
• In particular, since the Euclidean space is a star-shaped domain.
• Let be a simply - connected manifold, then applies.