De Rham cohomology
In
On any smooth manifold, every
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
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 dθ (described at Closed and exact differential forms). There is no function θ defined on the whole circle such that dθ 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.
One prominent example when all closed forms are exact is when the underlying space is contractible to a point, i.e., it is simply connected (no-holes condition). In this case the exterior derivative restricted to closed forms has a local inverse called a homotopy operator.[3][4] Since it is also nilpotent,[3] it forms a dual chain complex with the arrows reversed[5] compared to the de Rham complex. This is the situation described in the Poincaré lemma.
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 α − β 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
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
Punctured Euclidean space
Punctured Euclidean space is simply with the origin removed.
The Möbius strip
We may deduce from the fact that the
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
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
Sheaf-theoretic de Rham isomorphism
For any smooth manifold M, let be the constant sheaf on M associated to the abelian group ; in other words, is the sheaf of locally constant real-valued functions on M. Then we have a
between the de Rham cohomology and the sheaf cohomology of . (Note that this shows that de Rham cohomology may also be computed in terms of Čech cohomology; indeed, since every smooth manifold is paracompact Hausdorff we have that sheaf cohomology is isomorphic to the Čech cohomology for any good cover of M.)
Proof
The standard proof proceeds by showing that the de Rham complex, when viewed as a complex of sheaves, is an
This
where by exactness we have isomorphisms for all k. Each of these induces a long exact sequence in cohomology. Since the sheaf of functions on M admits partitions of unity, any -module is a
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 M is a compact Riemannian manifold, then each equivalence class in contains exactly one
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
with the exterior derivative and the
: we can look at its action on each component of degree separately.If is
Hodge decomposition
Let be a
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.[6]
See also
- Hodge theory
- integration)
- Sheaf theory
- -lemma for a refinement of exact differential forms in the case of compact Kähler manifolds.
Citations
- ^ Lee 2013, p. 440.
- ^ Tao, Terence (2007) "Differential Forms and Integration" Princeton Companion to Mathematics 2008. Timothy Gowers, ed.
- ^ OCLC 56347718.
- OCLC 9683855.
- S2CID 199472766.
- ^ Jean-Pierre Demailly, Complex Analytic and Differential Geometry Ch VIII, § 3.
References
- Lee, John M. (2013). Introduction to Smooth Manifolds. ISBN 978-1-4419-9981-8.
- ISBN 978-0-387-90613-3
- MR 1288523
- Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: ISBN 978-0-387-90894-6