# example of de Rham cohomology

If $\omega $ is a differential form^{} on a smooth manifold^{} $X$, then it is always true that if $\omega $ is exact ($\omega =d\eta $ for some other differential form $\eta $), then $\omega $ is closed ($d\omega =0$). On some manifolds, the opposite is also the case: all closed forms^{} of degree at least 1 are exact. However, in general this is not true. The idea of de Rham cohomology^{} 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, $\mathrm{\Omega}X={\mathrm{\Omega}}^{0}X\cong \mathbb{R}$ if $X$ is a single point. Applying the definition of the de Rham cohomology gives ${\mathrm{H}}_{\mathrm{dR}}X={\mathrm{H}}_{\mathrm{dR}}^{0}X\cong \mathbb{R}$.

Next, we use the fact that the de Rham cohomology is a homotopy invariant functor to show that for any $n\ge 0$ the de Rham cohomology groups of ${\mathbb{R}}^{n}$ are

$${\mathrm{H}}_{\mathrm{dR}}^{0}({\mathbb{R}}^{n})\cong \mathbb{R}$$ |

and

$${\mathrm{H}}_{\mathrm{dR}}^{i}({\mathbb{R}}^{n})=0\mathit{\hspace{1em}}\text{for}i0\text{.}$$ |

The reason for this is that ${\mathbb{R}}^{n}$ 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 ${\mathrm{H}}_{\mathrm{dR}}^{i}$ for $i>0$ is the circle ${S}^{1}$. In fact, we have

$${\mathrm{H}}_{\mathrm{dR}}^{0}({S}^{1})\cong \mathbb{R}$$ |

and

$${\mathrm{H}}_{\mathrm{dR}}^{1}({S}^{1})\cong \mathbb{R}\cdot [\omega ],$$ |

where $\omega $ is any 1-form on ${S}^{1}$ with ${\int}_{{S}^{1}}\omega \ne 0$. The standard volume form^{} $d\varphi $ on ${S}^{1}$, which it inherits from ${\mathbb{R}}^{2}$ if we view ${S}^{1}$ as the unit circle^{}, is such a form. The notation $d\varphi $ is somewhat misleading since it is not the differential of a global function $\varphi $; this is exactly the reason it appears in ${\mathrm{H}}_{\mathrm{dR}}^{1}({S}^{1})$. (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 ${S}^{n}$ are given by $dim{\mathrm{H}}_{\mathrm{dR}}^{i}({S}^{n})=1$ for $i=0$ or $i=n$, and $dim{\mathrm{H}}_{\mathrm{dR}}^{i}({S}^{n})=0$ otherwise. A couple of methods exist for calculating the de Rham cohomology groups for ${S}^{n}$ 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 |