De Rham cohomology

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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 Template:Mvar, with the exterior derivative as the differential.

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

where Ω0(M) is the space of smooth functions on Template:Mvar, 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 Template:Mvar 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 Template:Mvar. 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 Template:Mvar-th de Rham cohomology group ${\displaystyle H_{\mathrm {dR} }^{k}(M)}$ 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 Template:Mvar with Template:Mvar connected components

${\displaystyle H_{\mathrm {dR} }^{0}(M)\cong \mathbf {R} ^{n}.}$

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

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 Template:Mvar-sphere

For the [[n-sphere|Template:Mvar-sphere]], and also when taken together with a product of open intervals, we have the following. Let n > 0, m ≥ 0, and Template:Mvar an open real interval. Then

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

The Template:Mvar-torus

Similarly, allowing n > 0 here, we obtain

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

Punctured Euclidean space

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

{\displaystyle {\begin{aligned}\forall n\in \mathbb {Z} ^{+},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\neq 0,n-1\end{cases}}\\&\simeq H_{\mathrm {dR} }^{k}(S^{n-1})\end{aligned}}}

The Möbius strip

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

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

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 ${\displaystyle H_{\mathrm {dR} }^{k}(M)}$ 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 Template:Mvar, this map is in fact an isomorphism.

More precisely, consider the map

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

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

${\displaystyle H_{p}(M)\ni [c]\longmapsto \int _{c}\omega .}$

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 Template:Mvar is the sheaf of abelian groups determined by F(U) = R for all connected open sets UM, and for open sets U, V such that UV, the group morphism resV,U : F(V) → F(U) is given by the identity map on R, and where U is a good open cover of Template:Mvar (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 Template:Mvar is a compact Cm+1 manifold of dimension Template:Mvar, then for each km, there is an isomorphism

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

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

Proof

Let Ωk denote the sheaf of germs of Template:Mvar-forms on Template:Mvar (with Ω0 the sheaf of Cm+1 functions on Template:Mvar). By the Poincaré lemma, the following sequence of sheaves is exact (in the category of sheaves):

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

This sequence now breaks up into short exact sequences

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

Each of these induces a long exact sequence in cohomology. Since the sheaf of Cm+1 functions on a manifold admits partitions of unity, the sheaf-cohomology Hik) 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

{{#invoke:see also|seealso}} If Template:Mvar is a compact Riemannian manifold, then each equivalence class in ${\displaystyle H_{\mathrm {dR} }^{k}(M)}$ contains exactly one harmonic form. That is, every member Template:Mvar of a given equivalence class of closed forms can be written as

${\displaystyle \omega =d\alpha +\gamma }$

where Template:Mvar is some form, and Template:Mvar is harmonic: Δγ = 0.

Any harmonic function on a compact connected 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 2-torus is two. More generally, on an Template:Mvar-dimensional torus Tn, one can consider the various combings of Template:Mvar-forms on the torus. There are Template:Mvar choose Template:Mvar such combings that can be used to form the basis vectors for ${\displaystyle H_{\text{dR}}^{k}(T^{n})}$; the Template:Mvar-th Betti number for the de Rham cohomology group for the Template:Mvar-torus is thus Template:Mvar choose Template:Mvar.

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

${\displaystyle \Delta =d\delta +\delta d}$

with Template:Mvar the exterior derivative and Template:Mvar 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 Template:Mvar separately.

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

Hodge decomposition

Letting Template:Mvar be the codifferential, one says that a form Template:Mvar is co-closed if δω = 0 and co-exact if ω = δα for some form Template:Mvar. The Hodge decomposition states that any Template:Mvar-form can be split into three L2 components:

${\displaystyle \omega =d\alpha +\delta \beta +\gamma }$

where Template:Mvar is harmonic: Δγ = 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 Ωk(M):

${\displaystyle (\alpha ,\beta )=\int _{M}\alpha \wedge *\beta .}$

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

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