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# De Rham cohomology

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 Title: De Rham cohomology Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### De Rham cohomology

Vector field corresponding to a differential form on the punctured plane that is closed but not exact, showing that the de Rham cohomology of this space is non-trivial.

In 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.

## Contents

• Definition 1
• De Rham cohomology computed 2
• The n-sphere 2.1
• The n-torus 2.2
• Punctured Euclidean space 2.3
• The Möbius strip 2.4
• De Rham's theorem 3
• Sheaf-theoretic de Rham isomorphism 4
• Proof 4.1
• Related ideas 5
• Harmonic forms 5.1
• Hodge decomposition 5.2
• References 7

## Definition

The de Rham complex is the cochain complex of exterior differential forms on some smooth manifold M, with the exterior derivative as the differential.

0 \to \Omega^0(M)\ \stackrel{d}{\to}\ \Omega^1(M)\ \stackrel{d}{\to}\ \Omega^2(M)\ \stackrel{d}{\to}\ \Omega^3(M) \to \cdots

where Ω0(M) is the space of smooth functions on M, Ω1(M) is the space of 1-forms, and so forth. Forms which are the image of other forms under the exterior derivative, plus the constant 0 function in Ω0(M) are called exact and forms whose exterior derivative is 0 are called closed (see closed and exact differential forms); the relationship d 2 = 0 then says that exact forms are closed.

The converse, however, is not in general true; closed forms need not be exact. A simple but significant case is the 1-form of angle measure on the unit circle, written conventionally as (described at closed and exact differential forms). There is no actual function θ defined on the whole circle of which is the derivative; the increment of 2π in going once round the circle in the positive direction means that we can't take a single-valued θ. We can, however, change the topology by removing just one point.

The idea of de Rham cohomology is to classify the different types of closed forms on a manifold. One performs this classification by saying that two closed forms α, β ∈ Ωk(M) are cohomologous if they differ by an exact form, that is, if αβ is exact. 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 H^{k}_{\mathrm{dR}}(M) to be the set of equivalence classes, that is, the set of closed forms in Ωk(M) modulo the exact forms.

Note that, for any manifold M with n connected components

H^{0}_{\mathrm{dR}}(M) \cong \mathbf{R}^n.

This follows from the fact that any smooth function on M with zero derivative (i.e. locally constant) is constant on each of the connected components of M.

## De Rham cohomology computed

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. Another useful fact is that the de Rham cohomology is a homotopy invariant. While the computation is not given, the following are the computed de Rham cohomologies for some common topological objects:

### The n-sphere

For the n-sphere, Sn, and also when taken together with a product of open intervals, we have the following. Let n > 0, m ≥ 0, and I an open real interval. Then

H_{\mathrm{dR}}^{k}(S^n \times I^m) \simeq \begin{cases} \mathbf{R} & \mbox{if } k = 0,n, \\ 0 & \mbox{if } k \ne 0,n. \end{cases}

### The n-torus

Similarly, allowing n > 0 here, we obtain

H_{\mathrm{dR}}^{k}(T^n) \simeq \mathbf{R}^{n \choose k}.

### Punctured Euclidean space

Punctured Euclidean space is simply Euclidean space with the origin removed.

\begin{align} \forall n \in \mathbb{N}, H_{\mathrm{dR}}^{k}(\mathbf{R}^n \setminus \{\vec{0}\}) &\simeq \begin{cases} \mathbf{R} & \mbox{if } k = 0,n-1 \\ 0 & \mbox{if } k \ne 0,n-1 \end{cases} \\ &\simeq H_{\mathrm{dR}}^{k}(S^{n-1}) \end{align}

### The Möbius strip

This follows from the fact that the Möbius strip, M, can be deformation retracted to the 1-sphere:

H_{\mathrm{dR}}^{k}(M) \simeq H_{\mathrm{dR}}^{k}(S^1).

## De Rham's theorem

• Hazewinkel, Michiel, ed. (2001), "De Rham cohomology",