example of de Rham cohomology

If ω is a differential formMathworldPlanetmath on a smooth manifoldMathworldPlanetmath X, then it is always true that if ω is exact (ω=dη for some other differential form η), then ω is closed (dω=0). On some manifolds, the opposite is also the case: all closed formsPlanetmathPlanetmath of degree at least 1 are exact. However, in general this is not true. The idea of de Rham cohomologyMathworldPlanetmath is to measure the extent to which closed differential forms are not exact in terms of real vector spaces.

The simplest example of a differential manifold (apart from the empty manifold) is the zero-dimensional manifold consisting of a single point. Here the only differential forms are those of degree 0; actually, ΩX=Ω0X if X is a single point. Applying the definition of the de Rham cohomology gives HdRX=HdR0X.

Next, we use the fact that the de Rham cohomology is a homotopy invariant functor to show that for any n0 the de Rham cohomology groups of n are



HdRi(n)=0for i>0.

The reason for this is that n is contractible (homotopy equivalent to a point), and so has the same de Rham cohomology. More generally, any contractible manifold has the de Rham cohomology of a point; this is essentially the statement of the Poincaré lemma.

The first example of a non-trivial HdRi for i>0 is the circle S1. In fact, we have




where ω is any 1-form on S1 with S1ω0. The standard volume formMathworldPlanetmath dϕ on S1, which it inherits from 2 if we view S1 as the unit circlePlanetmathPlanetmath, is such a form. The notation dϕ is somewhat misleading since it is not the differential of a global function ϕ; this is exactly the reason it appears in HdR1(S1). (However, by the Poincaré lemma, it can locally be viewed as the differential of a function.)

For arbitrary n>0, the dimensions of the de Rham cohomology groups of Sn are given by dimHdRi(Sn)=1 for i=0 or i=n, and dimHdRi(Sn)=0 otherwise. A couple of methods exist for calculating the de Rham cohomology groups for Sn and other, more complicated, manifolds. The Mayer-Vietoris sequence is an example of such a tool.

Title example of de Rham cohomology
Canonical name ExampleOfDeRhamCohomology
Date of creation 2013-03-22 14:25:01
Last modified on 2013-03-22 14:25:01
Owner pbruin (1001)
Last modified by pbruin (1001)
Numerical id 5
Author pbruin (1001)
Entry type Example
Classification msc 55N05
Classification msc 58A12