TheInfoList

OR: In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, de Rham cohomology (named after Georges de Rham) is a tool belonging both to
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
and to
differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, capable of expressing basic topological information about
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a
cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
based on the existence of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship.

# Definition

The de Rham complex is the
cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s on some
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, with the
exterior derivative 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. The r ...
as the differential: :$0 \to \Omega^0\left(M\right)\ \stackrel\ \Omega^1\left(M\right)\ \stackrel\ \Omega^2\left(M\right)\ \stackrel\ \Omega^3\left(M\right) \to \cdots ,$ where is the space of
smooth functions In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
on , is the space of -forms, and so forth. Forms that are the image of other forms under the
exterior derivative 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. The r ...
, plus the constant function in , are called exact and forms whose exterior derivative is are called closed (see ''
Closed and exact differential forms 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 diff ...
''); 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 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 diff ...
''). 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 In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
. 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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
(no-holes condition). In this case the exterior derivative $d$ restricted to closed forms has a local inverse called a homotopy operator. Since it is also
nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...
, it forms a dual
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
with the arrows reversed 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 class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es 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 $H^_\left(M\right)$ 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 :$H^_\left(M\right) \cong \R ^m .$ 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 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 ...
. Another useful fact is that the de Rham cohomology is a
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
invariant. While the computation is not given, the following are the computed de Rham cohomologies for some common
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
objects:

## The -sphere

For the -sphere, $S^n$, and also when taken together with a product of open intervals, we have the following. Let , and be an open real interval. Then :$H_^\left(S^n \times I^m\right) \simeq \begin \R & k = 0\textk = n, \\ 0 & k \ne 0\textk \ne n. \end$

## The -torus

The $n$-torus is the Cartesian product: $T^n = \underbrace_$. Similarly, allowing $n \geq 1$ here, we obtain :$H_^\left(T^n\right) \simeq \R ^.$ We can also find explicit generators for the de Rham cohomology of the torus directly using differential forms. Given a quotient manifold $\pi: X \to X/G$ and a differential form $\omega \in \Omega^k\left(X\right)$ we can say that $\omega$ is $G$-invariant if given any diffeomorphism induced by $G$, $\cdot g:X \to X$ we have $\left(\cdot g\right)^*\left(\omega\right) = \omega$. In particular, the pullback of any form on $X/G$ is $G$-invariant. Also, the pullback is an injective morphism. In our case of $\R^n/\Z^n$ the differential forms $dx_i$ are $\Z^n$-invariant since $d \left(x_i + k\right) = dx_i$. But, notice that $x_i + \alpha$ for $\alpha \in \R$ is not an invariant $0$-form. This with injectivity implies that : Since the cohomology ring of a torus is generated by $H^1$, 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 $\mathbb^n$ with the origin removed. :$H^k_\left(\mathbb^n\setminus\\right) \cong \begin \mathbb^2 & n = 1, k = 0\\ \mathbb & n > 1, k = 0, n - 1\\ 0 & \text\end.$

## The Möbius strip

We may deduce from the fact that the
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Au ...
, , can be
deformation retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...
ed to the -sphere (i.e. the real unit circle), that: :$H_^\left(M\right) \simeq H_^\left(S^1\right).$

# De Rham's theorem

Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
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 In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
from de Rham cohomology $H^_\left(M\right)$ to singular cohomology groups $H^k\left(M;\R\right).$ De Rham's theorem, proved by Georges de Rham in 1931, states that for a smooth manifold , this map is in fact an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. More precisely, consider the map :$I: H_^p\left(M\right) \to H^p\left(M; \R\right),$ defined as follows: for any 
omega Omega (; capital: Ω, lowercase: ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and final letter in the Greek alphabet. In the Greek numeric system/ isopsephy ( gematria), it has a value of 800. Th ...
\in H_^p(M), let be the element of $\text\left(H_p\left(M\right), \R \right) \simeq H^p\left(M; \R \right)$ 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 In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of ...
endows the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of these groups with a ring structure. A further result of the theorem is that the two
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually un ...
s are isomorphic (as graded rings), where the analogous product on singular cohomology is the
cup product 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''. This defines an associative (and distributive) graded commuta ...
.

# Sheaf-theoretic de Rham isomorphism

For any smooth manifold ''M'', let $\underline$ be the constant sheaf on ''M'' associated to the abelian group $\mathbb$; in other words, $\underline$ is the sheaf of locally constant real-valued functions on ''M.'' Then we have a natural isomorphism :$H^*_\left(M\right) \cong H^*\left(M, \underline\right)$ between the de Rham cohomology and the sheaf cohomology of $\underline$. (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 $\check^*(\mathcal, \underline)$ for any good cover $\mathcal$ of ''M''.)

## Proof

The standard proof proceeds by showing that the de Rham complex, when viewed as a complex of sheaves, is an acyclic resolution of $\underline$. In more detail, let ''m'' be the dimension of ''M'' and let $\Omega^k$ denote the sheaf of germs of $k$-forms on ''M'' (with $\Omega^0$ the sheaf of $C^$ functions on ''M''). By the Poincaré lemma, the following sequence of sheaves is exact (in the
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abel ...
of sheaves): :$0 \to \underline \to \Omega^0 \,\xrightarrow\, \Omega^1 \,\xrightarrow\, \Omega^2\,\xrightarrow \dots \xrightarrow\, \Omega^m \to 0.$ This long exact sequence now breaks up into short exact sequences of sheaves :$0 \to \mathrm \, d_ \,\xrightarrow\, \Omega^k \,\xrightarrow\, \mathrm \, d_ \to 0,$ where by exactness we have isomorphisms $\mathrm \, d_ \cong \mathrm \, d_k$ for all ''k''. Each of these induces a long exact sequence in cohomology. Since the sheaf $\Omega^0$ of $C^$ functions on ''M'' admits partitions of unity, any $\Omega^0$-module is a
fine sheaf In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied to sheaves: flabby ( ...
; in particular, the sheaves $\Omega^k$ are all fine. Therefore, the sheaf cohomology groups $H^i(M,\Omega^k)$ vanish for $i > 0$ since all fine sheaves on paracompact spaces are acyclic. So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. At one end of the chain is the sheaf cohomology of $\underline$ 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 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 cohom ...
, and the Atiyah–Singer index theorem. However, even in more classical contexts, the theorem has inspired a number of developments. Firstly, the
Hodge theory 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 cohom ...
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, 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 cohom ...
.

## Harmonic forms

If is a compact
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ' ...
, then each equivalence class in $H^k_\left(M\right)$ contains exactly one
harmonic form 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 cohom ...
. That is, every member $\omega$ of a given equivalence class of closed forms can be written as :$\omega = \alpha + \gamma$ where $\alpha$ is exact and $\gamma$ is harmonic: $\Delta\gamma = 0$. Any
harmonic function In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \ ...
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 In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not t ...
, 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 In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
of a -torus is two. More generally, on an $n$-dimensional torus $T^n$, 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 $H^k_\left(T^n\right)$; the $k$-th Betti number for the de Rham cohomology group for the $n$-torus is thus $n$ choose $k$. More precisely, for a differential manifold , one may equip it with some auxiliary Riemannian metric. Then the Laplacian $\Delta$ is defined by :$\Delta=d\delta+\delta d$ with $d$ the
exterior derivative 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. The r ...
and $\delta$ the codifferential. The Laplacian is a homogeneous (in grading)
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear re ...
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
acting upon the
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s: we can look at its action on each component of degree $k$ separately. If $M$ is compact and
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
of the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the Laplacian acting upon the space of -forms is then equal (by
Hodge theory 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 cohom ...
) 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 $H^k\left(M;\R\right).$ The dimension of each such space is finite, and is given by the $k$-th
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
.

## Hodge decomposition

Let $M$ be a compact
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ' ...
. The ''Hodge decomposition'' states that any $k$-form on $M$ uniquely splits into the sum of three components: :$\omega = \alpha + \beta + \gamma ,$ where $\alpha$ is exact, $\beta$ is co-exact, and $\gamma$ is harmonic. One says that a form $\beta$ is co-closed if $\delta \beta = 0$ and co-exact if $\beta = \delta \eta$ for some form $\eta$, and that $\gamma$ is harmonic if the Laplacian is zero, $\Delta\gamma = 0$. 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 inner product on $\Omega^k\left(M\right)$: :$\left(\alpha,\beta\right)=\int_M \alpha \wedge .$ By use of
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s or distributions, the decomposition can be extended for example to a complete (oriented or not) Riemannian manifold.Jean-Pierre Demailly
Complex Analytic and Differential Geometry
Ch VIII, § 3.

*
Hodge theory 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 cohom ...
* Integration along fibers (for de Rham cohomology, the pushforward is given by integration) *
Sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
* $\partial \bar \partial$-lemma for a refinement of exact differential forms in the case of compact Kähler manifolds.

* * * *