De Rham cohomology

In mathematics, de Rham cohomology 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. It is in different, definite senses dual both to singular homology, and to Alexander-Spanier cohomology.

The differential k-forms on any smooth manifold M form an abelian group (in fact a real vector space) called

Ω^k(M)

under addition.
The exterior derivative d gives mappings

d:Ω^k(M) → Ω^k+1(M).
There is a fundamental relationship

d² = 0;

this follows essentially from symmetry of second derivatives. Therefore vector spaces of k-forms along with the exterior derivative are a cochain complex, the de Rham complex:

In differential geometry terminology, forms which are exterior derivatives are called exact and forms
whose exterior derivatives are 0 are called closed (see closed and exact differential forms); the relationship d² = 0 then says that
:exact forms are closed.
The cohomology groups of the de Rham complex, which are the vector spaces of closed forms modulo exact forms, are called the de Rham cohomology groups

H^k_DR(M).

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

De Rham's theorem, proved by Georges de Rham in 1931, states that for a compact oriented smooth manifold M, these groups are isomorphic as real vector spaces with the singular cohomology groupss

H^p(M;R).

Further, the two cohomology rings are isomorphic (as graded rings).

The general Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains.

Harmonic forms

For a differential manifold M, we can equip it with some auxiliary Riemannian metric. Then the Laplacian Δ, defined by

*d*d + d*d*

using the exterior derivative and Hodge dual defines a homogeneous (in grading) linear differential operator acting upon the exterior algebra of differential forms: we can look at its action on each component of degree p separately.

If M is compact and oriented, the dimension of its kernel acting upon the space of p-forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree p: the Laplacian picks out a unique harmonic form in each cohomology class of closed formss, in particular the space of
all harmonic p-forms on M is isomorphic to H^p(M;R).

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