De Rham cohomology

In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.

It is a cohomology theory based on the existence of differential forms with prescribed properties.

On any smooth manifold, every exact form is closed, but the converse may fail to hold.

Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship.

[1] The de Rham complex is the cochain complex of differential forms on some smooth manifold M, with the exterior derivative as the differential: where Ω0(M) is the space of smooth functions on M, Ω1(M) is the space of 1-forms, and so forth.

An illustrative case is a circle as a manifold, and the 1-form corresponding to the derivative of angle from a reference point at its centre, typically written as dθ (described at Closed and exact differential forms).

There is no function θ defined on the whole circle such that dθ is its derivative; the increase of 2π in going once around the circle in the positive direction implies a multivalued function θ.

Removing one point of the circle obviates this, at the same time changing the topology of the manifold.

One prominent example when all closed forms are exact is when the underlying space is contractible to a point or, more generally, if it is simply connected (no-holes condition).

restricted to closed forms has a local inverse called a homotopy operator.

The idea behind de Rham cohomology is to define equivalence classes of closed forms on a manifold.

This classification induces an equivalence relation on the space of closed forms in Ωk(M).

One then defines the k-th de Rham cohomology group

Note that, for any manifold M composed of m disconnected components, each of which is connected, we have that This follows from the fact that any smooth function on M with zero derivative everywhere is separately constant on each of the connected components of M. One may often find the general de Rham cohomologies of a manifold using the above fact about the zero cohomology and a Mayer–Vietoris sequence.

here, we obtain We can also find explicit generators for the de Rham cohomology of the torus directly using differential forms.

This with injectivity implies that Since the cohomology ring of a torus is generated by

, taking the exterior products of these forms gives all of the explicit representatives for the de Rham cohomology of a torus.

We may deduce from the fact that the Möbius strip, M, can be deformation retracted to the 1-sphere (i.e. the real unit circle), that: Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains.

It says that the pairing of differential forms and chains, via integration, gives a homomorphism from de Rham cohomology

The exterior product endows the direct sum of these groups with a ring structure.

So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms.

However, even in more classical contexts, the theorem has inspired a number of developments.

This relies on an appropriate definition of harmonic forms and of the Hodge theorem.

Any harmonic function on a compact connected Riemannian manifold is a constant.

Thus, this particular representative element can be understood to be an extremum (a minimum) of all cohomologously equivalent forms on the manifold.

In this case, there are two cohomologically distinct combings; all of the others are linear combinations.

-th Betti number for the de Rham cohomology group for the

More precisely, for a differential manifold M, one may equip it with some auxiliary Riemannian metric.

is compact and oriented, the dimension of the kernel of the Laplacian acting upon the space of k-forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree

: By use of Sobolev spaces or distributions, the decomposition can be extended for example to a complete (oriented or not) Riemannian manifold.