NationMaster - Encyclopedia: De Rham cohomology

In mathematics, de Rham cohomology 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. It is in different, definite senses dual both to singular homology, and to Alexander-Spanier cohomology. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ... In mathematics, particularly in algebraic topology Alexander_Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. ...

1 Definition
2 de Rham cohomology computed
3 de Rham's theorem
4 Sheaf-theoretic de Rham isomorphism
- 4.1 Proof
5 Related ideas
- 5.1 Harmonic forms
- 5.2 Hodge decomposition
6 See also

Definition

The set of smooth, differentiable differential k-forms on any smooth manifold M form an abelian group (in fact a real vector space) called In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

Ω^k(M)

under addition. The exterior derivative d gives mappings 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

d:Ω^k(M) → Ω^k+1(M).

There is a fundamental relationship

d ² = 0;

this follows essentially from symmetry of second derivatives. Therefore vector spaces of k-forms along with the exterior derivative are a cochain complex, the de Rham complex: In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a function f(x1, x2, ..., xn) of n variables. ... In homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ...

$C^infty(M) = Omega^0(M)to Omega^1(M) to Omega^2(M) to Omega^3(M) to ldots.$

In differential geometry terminology, forms which are exterior derivatives are called exact and forms whose exterior derivatives are 0 are called closed (see closed and exact differential forms); the relationship d ² = 0 then says that exact forms are closed. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β...

The converse, however, is not in general true; closed forms need not be exact. 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 and β in $Ω k (M)$ are 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 $Ω k (M)$ . One then defines the k-th de Rham cohomology group

H^k_dR(M)

to be the set of equivalence classes, that is, the set of closed forms in $Ω k (M)$ modulo the exact forms.

Note that, for any manifold M with n connected components, Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ...

H⁰_dR(M) = Rⁿ

where the equals actually denotes that the two are homomorphic. This follows from the fact that any $C^infty$ function on M with zero derivative is locally constant on each of the connected components. In abstract algebra, a homomorphism is a structure-preserving map. ...

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: In algebraic topology and related branches of mathematics, the Mayer-Vietoris sequence (named after Walther Mayer and Leopold Vietoris) is an exact sequence that often helps one to compute homology groups. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... Topology (Greek topos = place and logos = word) is a branch of mathematics concerned with the study of topological spaces. ...

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 an open real interval. Then:

$H_{dR}^{k}(S^n times I^m) simeq begin{cases} mathbb{R} & mbox{if } k = 0,n 0 & mbox{if } k ne 0,n end{cases}$

The n-torus:

Similarly, allowing n > 0 here, we obtain:

$H_{dR}^{k}(T^n) simeq mathbb{R}^{n choose k}$

Punctured Euclidean space:

Punctured Euclidean space is simply Euclidean space with the origin removed. For n > 0, we have: In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

$H_{dR}^{k}(mathbb{R}^n - {0})$	$simeq begin{cases} mathbb{R} & mbox{if } k = 0,n-1 0 & mbox{if } k ne 0,n-1 end{cases}$
	$simeq H_{dR}^{k}(S^{n-1})$

The Möbius strip, M:

This more or less follows from the fact that the Möbius strip may be, loosely speaking, "contracted" to the 1-sphere: A MÃ¶bius strip made with a piece of paper and tape. ...

$H_{dR}^{k}(M) simeq H_{dR}^{k}(S^1)$

de Rham's theorem

De Rham's theorem, proved by Georges de Rham in 1931, states that for a compact oriented smooth manifold M, the groups H^k_dR(M) are isomorphic as real vector spaces with the singular cohomology groups Georges de Rham (10 September 1903-9 October 1990) was a Swiss mathematician, known for his contributions to differential topology. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... This article or section should be merged with Orientable manifold. ... In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...

H^k(M;R).

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. In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. ...

The general Stokes' theorem is an expression of duality between de Rham cohomology and the homology of chains. The Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... The word duality has a variety of different meanings in different contexts: In mathematics, see duality (mathematics). ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In set theory, a chain is a total order subset of a poset. ...

Sheaf-theoretic de Rham isomorphism

The de Rham cohomology is isomorphic to the Čech cohomology H^*(U,F), where F is the sheaf of abelian groups determined by F(U) = R for all open sets U in M, and for open sets U and V such that U ⊂ V, the group morphism res_V,U : F(V) → F(U) is given by the identity map on R, and where U is a good open cover of M (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). In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... ÄŒech cohomology is a particular type of cohomology in mathematics. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a cover of a set X is a collection of subsets C of X whose union is X. In symbols, if C = {Uα : α ∈ A} is an indexed family of subsets of X, then C is a cover if More generally, if Y is a subset of X... In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i. ...

Stated another way, if M is a compact C^m+1 manifold of dimension m, then for each k≤m, there is an isomorphism A differentiability class in mathematics is a class of functions which share differentiability features. ...

$H^k_{dR}(M,mathbb{R})cong check{H}^k(M,mathbb{R})$

where the left-hand side is the k-th de Rham cohomology group and the right-hand side is the sheaf cohomology for the constant sheaf with fibre R. In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric... In mathematics, a function f from a topological space A to a set B is called locally constant, iff for every a in A there exists a neighborhood U of a, such that f is constant on U. Every constant function is locally constant. ...

Proof

Let Ω^k denote the sheaf of germs of k-forms on M (with Ω⁰ the sheaf of C^m+1 functions on M). By the Poincaré lemma, the following sequence of sheaves is exact (in the category of sheaves): In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dÎ± = 0 for a given form Î± to be a closed form, and Î± = dÎ² for an exact form, with Î± given and Î² unknown. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

$0rightarrowmathbb{R}rightarrowOmega^0rightarrow^{!!!!!!d} Omega^1rightarrow^{!!!!!!d} Omega^2rightarrow^{!!!!!!d} dotsrightarrow^{!!!!!!d} Omega^mrightarrow 0.$

This sequence now breaks up into short exact sequences In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

$0rightarrow Omega^{k-1}rightarrow^{!!!!!!d} Omega^krightarrow^{!!!!!!d} dOmega^krightarrow 0.$

Each of these induces a long exact sequence in cohomology. Since the sheaf of C^m+1 functions on a manifold admits partitions of unity, the sheaf-cohomology Hⁱ(Ω^k) 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. In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, a partition of unity of a topological space X is a set of continuous functions {Ïi} from X to the unit interval [0,1] such that every point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all...

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 theorem 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. In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology is a special generalization of de Rham cohomology to complex manifolds. ... In mathematics, Hodge theory is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric... In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is an important unifying result that connects topology and analysis. ... For the Hodge theorem in mathematics, see Hodge theory Hodge index theorem This is a disambiguation pageâ€”a list of articles associated with the same title. ... In mathematics, Hodge theory is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric...

Harmonic forms

If M is a compact Riemannian manifold, then each equivalence class in H^k_dR(M) contains exactly one harmonic form. That is, every member ω of a given equivalence class of closed forms can be written as Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M. It was developed by W. V. D. Hodge in the 1930s as an extension...

ω = d α + γ

where α is some form, and γ is harmonic: Δγ=0.

Recall that any harmonic function on a compact 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 two-torus is two. More generally, on an n-dimensional torus Tⁿ, one can consider the various combings of k-forms on the torus. There are n choose k such combings that can be used to form the basis vectors for H^k_dR(Tⁿ); the k-th Betti number for the de Rham cohomology group for the n-torus is thus n choose k. In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ... A torus. ... In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ...

More precisely, for a differential manifold M, one may equip it with some auxiliary Riemannian metric. Then the Laplacian Δ is defined by In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

Δ = d δ + δ d

with d the exterior derivative and δ 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 k separately. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... The word linear comes from the Latin word linearis, which means created by lines. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

If M 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 k: the Laplacian picks out a unique harmonic form in each cohomology class of closed forms. In particular, the space of all harmonic k-forms on M is isomorphic to H^k(M;R). The dimension of each such space is finite, and is given by the k-th Betti number. Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... This article or section should be merged with Orientable manifold. ... 2-dimensional renderings (ie. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, Hodge theory is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations dα = 0 for a given form α to be a closed form, and α = dβ for an exact form, with α given and β... In algebraic topology, the Betti numbers of a topological space X are a sequence b0, b1, ... of topological invariants. ...

Hodge decomposition

Letting δ be the codifferential, one says that a form ω is co-closed if δω=0 and co-exact if ω=δα for some form α. The Hodge decomposition states that any k-form ω can be split into three L² components: In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

ω = d α + δβ + γ

where γ is harmonic: Δ γ = 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 $Ω k (M)$ : In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

$(alpha,beta)=int_M alpha wedge *beta$

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 for the discussion of the convergence of a Cauchy sequence of forms to a limiting form. This language helps overcome some of the limitations of requiring compact support, such as in Alexander-Spanier cohomology. In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its derivatives up to some order k the condition of finite Lp norm, for given p ≥ 1. ... In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematics, particularly in algebraic topology Alexander_Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. ...

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