De Rham cohomology

In mathematics, de Rham cohomology (named 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 differential forms on some smooth manifold , with the exterior derivative as the differential:
where is the space of smooth functions on , is the space of -forms, and so forth. Forms that are the image of other forms under the exterior derivative, plus the constant function in , are called exact and forms whose exterior derivative is are called closed (see "Closed and exact differential forms"); the relationship then says that exact forms are closed.
In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and the -form corresponding to the derivative of angle from a reference point at its centre, typically written as (described at "Closed and exact differential forms"). There is no function defined on the whole circle such that is its derivative; the increase of in going once around the circle in the positive direction implies a multivalued function . Removing one point of the circle obviates this, at the same time changing the topology of the manifold.
The idea behind de Rham cohomology is to define equivalence classes of closed forms on a manifold. One classifies two closed forms as cohomologous if they differ by an exact form, that is, if is exact. This classification induces an equivalence relation on the space of closed forms in . One then defines the -th de Rham cohomology group formula_2 to be the set of equivalence classes, that is, the set of closed forms in modulo the exact forms.
Note that, for any manifold composed of disconnected components, each of which is connected, we have that
This follows from the fact that any smooth function on with zero derivative everywhere is separately constant on each of the connected components of .
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 -sphere.
For the -sphere, formula_4, and also when taken together with a product of open intervals, we have the following. Let , and be an open real interval. Then
The -torus.
The formula_6-torus is the Cartesian product: formula_7. Similarly, allowing formula_8 here, we obtain
We can also find explicit generators for the de Rham cohomology of the torus directly using differential forms. Given a quotient manifold formula_10 and a differential form formula_11 we can say that formula_12 is formula_13-invariant if given any diffeomorphism induced by formula_13, formula_15 we have formula_16. In particular, the pullback of any form on formula_17 is formula_13-invariant. Also, the pullback is an injective morphism. In our case of formula_19 the differential forms formula_20 are formula_21-invariant since formula_22. But, notice that formula_23 for formula_24 is not an invariant formula_25-form. This with injectivity implies that
Since the cohomology ring of a torus is generated by formula_27, taking the exterior products of these forms gives all of the explicit representatives for the de Rham cohomology of a torus.
Punctured Euclidean space.
Punctured Euclidean space is simply formula_28 with the origin removed.
The Möbius strip.
We may deduce from the fact that the Möbius strip, , can be deformation retracted to the -sphere (i.e. the real unit circle), that:
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 formula_2 to singular cohomology groups formula_32 De Rham's theorem, proved by Georges de Rham in 1931, states that for a smooth manifold , this map is in fact an isomorphism.
More precisely, consider the map
defined as follows: for any formula_34, let be the element of formula_35 that acts as follows:
The theorem of de Rham asserts that this is an isomorphism between de Rham cohomology and singular cohomology.
The exterior 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 formula_37, where formula_38 is the sheaf of abelian groups determined by formula_39 for all connected open sets formula_40, and for open sets formula_41 such that formula_42, the group morphism formula_43 is given by the identity map on formula_44 and where formula_45 is a good open cover of formula_46 (i.e. all the open sets in the open cover formula_45 are contractible to a point, and all finite intersections of sets in formula_45 are either empty or contractible to a point). In other words formula_38 is the constant sheaf given by the sheafification of the constant presheaf assigning formula_39.
Stated another way, if formula_46 is a compact manifold of dimension formula_52, then for each formula_53, there is an isomorphism
where the left-hand side is the formula_55-th de Rham cohomology group and the right-hand side is the Čech cohomology for the constant sheaf with fibre formula_56
Proof.
Let formula_57 denote the sheaf of germs of formula_55-forms on formula_46 (with formula_60 the sheaf of formula_61 functions on formula_46). By the Poincaré lemma, the following sequence of sheaves is exact (in the category of sheaves):
This sequence now breaks up into short exact sequences
Each of these induces a long exact sequence in cohomology. Since the sheaf of formula_61 functions on a manifold admits partitions of unity, the sheaf-cohomology formula_66 vanishes for formula_67. 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.
If is a compact Riemannian manifold, then each equivalence class in formula_68 contains exactly one harmonic form. That is, every member formula_12 of a given equivalence class of closed forms can be written as
where formula_71 is exact and formula_72 is harmonic: formula_73.
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 -torus, one may envision a constant -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 -torus is two. More generally, on an formula_6-dimensional torus formula_75, one can consider the various combings of formula_76-forms on the torus. There are formula_6 choose formula_76 such combings that can be used to form the basis vectors for formula_79; the formula_76-th Betti number for the de Rham cohomology group for the formula_6-torus is thus formula_6 choose formula_76.
More precisely, for a differential manifold , one may equip it with some auxiliary Riemannian metric. Then the Laplacian formula_84 is defined by
with formula_86 the exterior derivative and formula_87 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 formula_76 separately.
If formula_89 is compact and oriented, the dimension of the kernel of the Laplacian acting upon the space of -forms is then equal (by Hodge theory) to that of the de Rham cohomology group in degree formula_76: the Laplacian picks out a unique harmonic form in each cohomology class of closed forms. In particular, the space of all harmonic formula_76-forms on formula_89 is isomorphic to formula_32 The dimension of each such space is finite, and is given by the formula_76-th Betti number.
Hodge decomposition.
Let formula_89 be a compact oriented Riemannian manifold. The "Hodge decomposition" states that any formula_76-form on formula_89 uniquely splits into the sum of three components:
where formula_71 is exact, formula_100 is co-exact, and formula_72 is harmonic.
One says that a form formula_100 is co-closed if formula_103 and co-exact if formula_104 for some form formula_105, and that formula_72 is harmonic if the Laplacian is zero, formula_73. 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 inner product on formula_108:
By use of Sobolev spaces or distributions, the decomposition can be extended for example to a complete (oriented or not) Riemannian manifold.