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.

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.

Every exact form is closed, but the reverse is not necessarily true. On the other hand, there is a relation between failure of exactness and existence of "holes". De Rham cohomology groups are a set of invariants of smooth manifolds which make aforementioned relation quantitative,[1] and will be discussed in this article.

The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds.
Terence Tao, Differential Forms and Integration[2]

Definition

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

where Ω0(M) is the space of smooth functions on M, Ω1(M) is the space of 1-forms, and so forth. Forms that 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 d2 = 0 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 1-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 θ defined on the whole circle such that is its derivative; the increase of 2π in going once around the circle in the positive direction implies a multivalued function θ. 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 α, β ∈ Ωk(M) as 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 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 composed of m disconnected components, each of which is connected, we have that

This follows from the fact that any smooth function on M with zero derivative everywhere is separately 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, , and also when taken together with a product of open intervals, we have the following. Let n > 0, m ≥ 0, and I be an open real interval. Then

The n-torus

The -torus is the Cartesian product: . Similarly, allowing 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 and a differential form we can say that is -invariant if given any diffeomorphism induced by , we have . In particular, the pullback of any form on is -invariant. Also, the pullback is an injective morphism. In our case of the differential forms are -invariant since . But, notice that for is not an invariant -form. This with injectivity implies that

Since the cohomology ring of a torus is generated by , 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 with the origin removed.

The Möbius strip

We may deduce from the fact that the Möbius strip, M, can be deformation retracted to the 1-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 to singular cohomology groups 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

defined as follows: for any , let I(ω) be the element of 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 , where is the sheaf of abelian groups determined by for all connected open sets , and for open sets such that , the group morphism is given by the identity map on and where is a good open cover of (i.e. all the open sets in the open cover are contractible to a point, and all finite intersections of sets in are either empty or contractible to a point). In other words is the constant sheaf given by the sheafification of the constant presheaf assigning .

Stated another way, if is a compact Cm+1 manifold of dimension , then for each , there is an isomorphism

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

Proof

Let denote the sheaf of germs of -forms on (with the sheaf of functions on ). 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 functions on a manifold admits partitions of unity, the sheaf-cohomology vanishes for . 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.

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

where is exact and is harmonic: .

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 -dimensional torus , one can consider the various combings of -forms on the torus. There are choose such combings that can be used to form the basis vectors for ; the -th Betti number for the de Rham cohomology group for the -torus is thus choose .

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

with the exterior derivative and 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 separately.

If 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 : the Laplacian picks out a unique harmonic form in each cohomology class of closed forms. In particular, the space of all harmonic -forms on is isomorphic to The dimension of each such space is finite, and is given by the -th Betti number.

Hodge decomposition

Let be a compact oriented Riemannian manifold. The Hodge decomposition states that any -form on uniquely splits into the sum of three L2 components:

where is exact, is co-exact, and is harmonic.

One says that a form is co-closed if and co-exact if for some form , and that is harmonic if the Laplacian is zero, . 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 :

By use of Sobolev spaces or distributions, the decomposition can be extended for example to a complete (oriented or not) Riemannian manifold.[3]

See also

  • Hodge theory
  • Integration along fibers (for de Rham cohomology, the pushforward is given by integration)

Citations

  1. Lee 2013, p. 440.
  2. Terence, Tao. "Differential Forms and Integration" (PDF). Cite journal requires |journal= (help)
  3. Jean-Pierre Demailly, Complex Analytic and Differential Geometry Ch VIII, § 3.

References

  • Lee, John M. (2013). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-1-4419-9981-8.
  • 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
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