De Rham cohomology

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The De Rham cohomology (after Georges de Rham ) is a mathematical construction from the algebraic topology , which develops the cohomology for smooth manifolds , i.e. for curves, surfaces and other geometric objects that look like a locally from the perspective of analysis Euclidean space. This cohomology uses Stokes' theorem in its generalized form, which extends the fundamental theorem of analysis and opens a connection line from differential geometry to algebraic topology. The analog of the De Rham cohomology for complex manifolds is the Dolbeault cohomology .

De Rham Complex

definition

Be a smooth manifold and the amount of p-forms on . The De Rham complex is the coquette complex

.

The illustrations are given by the Cartan derivation .

De Rham complex in three-dimensional space

If you choose that as the underlying manifold, the De-Rham complex has a special shape. In this case the Cartan derivatives correspond to the differential operators gradient , divergence and rotation known from vector analysis . Specifically, it says that the diagram

commutes , so you get the same result regardless of which arrows you follow. The figures and are diffeomorphisms . Such is the Sharp isomorphism and the Hodge star operator .

Definition of De Rham cohomology

Be a smooth manifold. The -th De-Rham cohomology group is defined as the -th cohomology group of the De-Rham complex. In particular applies to

history

In his Paris dissertation (1931), Georges de Rham proved with his sentence a conjecture by Élie Cartan , which in turn was based on considerations of Henri Poincaré . Since the cohomology of a topological space was only discussed a few years later, he actually worked with homology and the (based on Stokes theorem ) dual complex of n -chains.

Homotopy invariance

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

.

Since two homotopic, smooth manifolds have the same De Rham cohomology except for isomorphism, this cohomology is a topological invariant of a smooth manifold. This is remarkable, since the differentiable structure of the manifold plays an important role in the definition of the De Rham group. So there is no reason to assume that a topological manifold with different differentiable structures has the same De Rham groups.

De Rham's theorem

The central statement in the theory of De Rham cohomology is called de Rham's theorem. It says that the De Rham cohomology of smooth manifolds is of course isomorphic to the singular cohomology with coefficients in the real numbers. With that is singular homology referred. So it applies

Let be an element of the pth singular homology group. Then the isomorphism becomes through the mapping

described. It was identified with (see also universal coefficient theorem ). This mapping is called the De Rham homomorphism or De Rham isomorphism.

Examples of some De Rham groups

Calculating the De Rham groups is often difficult, so few examples are given below. It is always assumed that the considered manifolds are smooth.

  • Let be a connected manifold, then is equal to the set 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 vanish.
  • Be an open star region , then applies to all . This is Poincaré's lemma , which says that in a star region every closed differential form , dω = 0, is even exact (that is, there is a “potential form” χ such that ω = dχ).
  • In particular, it is true that Euclidean space is a star region.
  • Let be a simply-connected manifold then .

literature

  • Raoul Bott , Loring W. Tu: Differential forms in algebraic topology. Springer, New York NY et al. 1982, ISBN 0-387-90613-4 ( Graduate Texts in Mathematics 82).
  • Klaus Jänich : Vector analysis. 5th edition. Springer Verlag, Berlin et al. 2005, ISBN 3-540-23741-0 ( Springer textbook ).
  • Georges de Rham : Sur l'analysis situs des variétés à n dimensions. In: Journal de Mathématiques pures et appliquées. 10, 1931, ISSN  0021-7824 , pp. 115-200, online .
  • André Weil : Sur les théorèmes de de Rham. In: Commentarii mathematici Helvetici. 26, 1952, pp. 119–145, online , (reprinted in: André Weil: Œuvres Scientifiques. Volume 2: 1951–1964. Reprinted edition. Springer, Berlin et al. 2009, ISBN 978-3-540-87735-6 , p . 17-43).