De Rham cohomology

Last updated
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. Irrotationalfield.svg
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 (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.

Contents

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 harmonicform 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

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.

Related Research Articles

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

In vector calculus and differential geometry, the generalized Stokes theorem, also called the Stokes–Cartan theorem, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. It allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.

In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc.

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule.

In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an n-dimensional variety, the theorem says that a cohomology group is the dual space of another one, . Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf.

In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero, and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.

In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic.

In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as standard in commutative algebra and algebraic geometry somewhat later, once the need was felt to adapt methods from calculus and geometry over the complex numbers to contexts where such methods are not available.

Čech cohomology

In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.

In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold M in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes.

In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by Claude Chevalley and Samuel Eilenberg (1948) to coefficients in an arbitrary Lie module.

In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).

In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : EX is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.

In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994), using the Seiberg–Witten theory studied by Nathan Seiberg and Witten during their investigations of Seiberg–Witten gauge theory.

In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support.

In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. The Yang–Mills equations arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. However, the Yang–Mills equations have independently found significant use within mathematics.

In differential geometry, the integration along fibers of a k-form yields a -form where m is the dimension of the fiber, via "integration".

References