De Rham cohomology

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. It is in different, definite senses dual both to singular homology, and to Alexander-Spanier cohomology.

Definition

The set of smooth, differentiable differential k-forms on any smooth manifold M form an abelian group (in fact a real vector space) called

Ω^k(M)

under addition. The exterior derivative d gives mappings

d:Ω^k(M) → Ω^k+1(M).

There is a fundamental relationship

d² = 0;

this follows essentially from symmetry of second derivatives. Therefore vector spaces of k-forms along with the exterior derivative are a cochain complex, the de Rham complex:

C^\infty(M) = \Omega^0(M)\to \Omega^1(M) \to \Omega^2(M) \to \Omega^3(M) \to \cdots.

In differential geometry terminology, forms which are exterior derivatives are called exact and forms whose exterior derivatives are 0 are called closed (see closed and exact differential forms); the relationship d² = 0 then says that exact forms are closed.

The converse, however, is not in general true; closed forms need not be exact. 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 and β in $\Omega^k(M)$ are cohomologous if they differ by an exact form, that is, if $\alpha-\beta$ is exact. This classification induces an equivalence relation on the space of closed forms in $\Omega^k(M)$ . One then defines the k-th de Rham cohomology group

H^k_dR(M)

to be the set of equivalence classes, that is, the set of closed forms in $\Omega^k(M)$ modulo the exact forms.

Note that, for any manifold M with n connected components,

H⁰_dR(M) = Rⁿ

where the equals actually denotes that the two are homomorphic. This follows from the fact that any $C^\infty$ function on M with zero derivative is locally constant on each of the connected components.

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, 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_{dR}^{k}(S^n \times I^m) \simeq \begin{cases} \mathbb{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_{dR}^{k}(T^n) \simeq \mathbb{R}^{n \choose k}

Punctured Euclidean space:

Punctured Euclidean space is simply Euclidean space with the origin removed. For n > 0, we have:

H_{dR}^{k}(\mathbb{R}^n - \{0\})

\simeq \begin{cases} \mathbb{R} & \mbox{if } k = 0,n-1 \\ 0 & \mbox{if } k \ne 0,n-1 \end{cases}

\simeq H_{dR}^{k}(S^{n-1})

The Möbius strip, M:

This more or less follows from the fact that the Möbius strip may be, loosely speaking, "contracted" to the 1-sphere:

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

de Rham's theorem

De Rham's theorem, proved by Georges de Rham in 1931, states that for a compact oriented smooth manifold M, the groups H^k_dR(M) are isomorphic as real vector spaces with the singular cohomology groups

H^k(M;R).

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.

The general Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains.

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 in M, and for open sets U and V such that U ⊂ V, the group morphism res_V,U : F(V) → 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≤m, there is an isomorphism

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

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

Proof

Let Ω^k denote the sheaf of germs of k-forms on M (with Ω⁰ 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\rightarrow\mathbb{R}\rightarrow\Omega^0\rightarrow^{\!\!\!\!\!\!d}\ \Omega^1\rightarrow^{\!\!\!\!\!\!d}\  \Omega^2\rightarrow^{\!\!\!\!\!\!d}\ \dots\rightarrow^{\!\!\!\!\!\!d}\ \Omega^m\rightarrow 0.

This sequence now breaks up into short exact sequences

0\rightarrow \Omega^{k-1}\rightarrow^{\!\!\!\!\!\!d}\ \Omega^k\rightarrow^{\!\!\!\!\!\!d}\ d\Omega^k\rightarrow 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 Hⁱ(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 theorem 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

If M is a compact Riemannian manifold, then each equivalence class in H^k_dR(M) contains exactly one harmonic form. That is, every member ω of a given equivalence class of closed forms can be written as

\omega = d\alpha+\gamma

where α is some form, and γ is harmonic: Δγ=0.

Recall that any harmonic function on a compact Riemannian manifold is a constant. Thus, this particular representative element can be understood to be an extremum (a minimum) of all cohomologously equivalent forms on the manifold. For example, on a 2-torus, one may envision a constant 1-form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length). In this case, there are two cohomologically distinct combings; all of the others are linear combinations. In particular, this implies that the 1st Betti number of a two-torus is two. More generally, on an n-dimensional torus Tⁿ, one can consider the various combings of k-forms on the torus. There are n choose k such combings that can be used to form the basis vectors for H^k_dR(Tⁿ); the k-th Betti number for the de Rham cohomology group for the n-torus is thus n choose k.

More precisely, for a differential manifold M, one may equip it with some auxiliary Riemannian metric. Then the Laplacian Δ is defined by

\Delta=d\delta+\delta d

with d the exterior derivative and δ the codifferential. The Laplacian is a homogeneous (in grading) linear differential operator acting upon the exterior algebra of differential forms: we can look at its action on each component of degree k separately.

If M is compact and oriented, the dimension of the kernel of the Laplacian acting upon the space of k-forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree k: the Laplacian picks out a unique harmonic form in each cohomology class of closed forms. In particular, the space of all harmonic k-forms on M is isomorphic to H^k(M;R). The dimension of each such space is finite, and is given by the k-th Betti number.

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 ω can be split into three L² 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 L² inner product on $\Omega^k(M)$ :

(\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 for the discussion of the convergence of a Cauchy sequence of forms to a limiting form. This language helps overcome some of the limitations of requiring compact support, such as in Alexander-Spanier cohomology.

References

Raoul Bott and Loring Tu, Differential Forms in Algebraic Topology. Springer-Verlag, 1981.
Philip Griffiths and Joe Harris, Principles of Algebraic Geometry. Wiley-Interscience, 1978.

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