De Rham cohomology

For Grothendieck's algebraic de Rham cohomology, see Crystalline 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 mathematics, de Rham cohomology (after Georges de Rham) 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.

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 $Ω 0 (M)$ is the space of smooth functions on $M$ , $Ω 1 (M)$ 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 $Ω 0 (M)$ 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 $dθ$ (described at closed and exact differential forms). There is no actual function $θ$ defined on the whole circle of which $dθ$ is the derivative; the increment of $2 π$ 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 $α, β \in Ω k (M)$ are cohomologous if they differ by an exact form, that is, if $α - β$ is exact. This classification induces an equivalence relation on the space of closed forms in $Ω k (M)$ . 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 $Ω k (M)$ 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 $n > 0, m \geq 0$ , 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

Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains. It says that the pairing of differential forms and chains, via integration, gives a homomorphism from de Rham cohomology $H^{k}_{\mathrm{dR}}(M)$ to singular cohomology groups $H k (M; R)$ . De Rham's theorem, proved by Georges de Rham in 1931, states that for a smooth manifold $M$ , this map is in fact an isomorphism.

More precisely, consider the map

I: H_{dR}^p(M) \to H^p(M; \mathbf{R}),

defined as follows: for any $[\omega] \in H_{dR}^p(M)$ , let $I (ω)$ be the element of $\text{Hom}(H_p(M; \mathbf{R}), \mathbf{R}) \simeq H^p(M; \mathbf{R})$ that acts as follows:

H_p(M) \ni [c] \longmapsto \int_c \omega.

The theorem of de Rham asserts that this is an isomorphism between de Rham cohomology and singular cohomology.

The wedge product endows the direct sum of these groups with a ring structure. A further result of the theorem is that the two cohomology rings are isomorphic (as graded rings), where the analogous product on singular cohomology is the cup product.

Sheaf-theoretic de Rham isomorphism

The de Rham cohomology is isomorphic to the Čech cohomology $H * (U, F)$ , where $F$ is the sheaf of abelian groups determined by $F (U) = R$ for all connected open sets $U \subset M$ , and for open sets $U, V$ such that $U \subset V$ , the group morphism $res V, U : F (V) \to F (U)$ is given by the identity map on $R$ , and where $U$ is a good open cover of $M$ (i.e. all the open sets in the open cover $U$ are contractible to a point, and all finite intersections of sets in $U$ are either empty or contractible to a point).

Stated another way, if $M$ is a compact $C m +1$ manifold of dimension $m$ , then for each $k \leq m$ , there is an isomorphism

H^k_{\mathrm{dR}}(M)\cong \check{H}^k(M,\mathbf{R})

where the left-hand side is the $k$ -th de Rham cohomology group and the right-hand side is the Čech cohomology for the constant sheaf with fibre $R$ .

Proof

Let $Ω k$ denote the sheaf of germs of $k$ -forms on $M$ (with $Ω 0$ the sheaf of $C m +1$ functions on $M$ ). By the Poincaré lemma, the following sequence of sheaves is exact (in the category of sheaves):

0 \to \mathbf{R} \to \Omega^0 \,\xrightarrow{d}\, \Omega^1 \,\xrightarrow{d}\, \Omega^2\,\xrightarrow{d} \dots \xrightarrow{d}\, \Omega^m \to 0.

This sequence now breaks up into short exact sequences

0 \to d\Omega^{k-1} \,\xrightarrow{\subset}\, \Omega^k \,\xrightarrow{d}\, d\Omega^k \to 0.

Each of these induces a long exact sequence in cohomology. Since the sheaf of $C m +1$ functions on a manifold admits partitions of unity, the sheaf-cohomology $H i (Ω k)$ vanishes for $i > 0$ . So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. At one end of the chain is the Čech cohomology and at the other lies the de Rham cohomology.

Related ideas

The de Rham cohomology has inspired many mathematical ideas, including Dolbeault cohomology, Hodge theory, and the Atiyah-Singer index theorem. However, even in more classical contexts, the theorem has inspired a number of developments. Firstly, the Hodge theory proves that there is an isomorphism between the cohomology consisting of harmonic forms and the de Rham cohomology consisting of closed forms modulo exact forms. This relies on an appropriate definition of harmonic forms and of the Hodge theorem. For further details see Hodge theory.

Harmonic forms

Hodge decomposition

Letting $δ$ be the codifferential, one says that a form $ω$ is co-closed if $δω = 0$ and co-exact if $ω = δα$ for some form $α$ . The Hodge decomposition states that any $k$ -form can be split into three $L 2$ components:

\omega = d\alpha +\delta \beta + \gamma

where $γ$ is harmonic: $Δ γ = 0$ . This follows by noting that exact and co-exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms. Here, orthogonality is defined with respect to the $L 2$ inner product on $Ω k (M)$ :

(\alpha,\beta)=\int_M \alpha \wedge *\beta.

A precise definition and proof of the decomposition requires the problem to be formulated on Sobolev spaces. The idea here is that a Sobolev space provides the natural setting for both the idea of square-integrability and the idea of differentiation. This language helps overcome some of the limitations of requiring compact support.

References

Bott, Raoul; Tu, Loring W. (1982), Differential Forms in Algebraic Topology, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90613-3
Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523
Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90894-6

External links

Hazewinkel, Michiel, ed. (2001), "De Rham cohomology", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4