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 ...
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 manifolds in a form particularly adapted to computation and the concrete representation of
cohomology class
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 ...
es. It is a
cohomology theory 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 application ...
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 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 application ...
s on some
smooth manifold , 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 re ...
as the differential:
:
where is the space of
smooth functions 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 re ...
, plus the constant function in , are called exact and forms whose exterior derivative is are called closed (see ''
Closed and exact differential forms''); 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''). 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 . 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 In homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain ...
.
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 cl ...
,
it forms a dual
chain complex 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
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
:
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. 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 deform ...
invariant. While the computation is not given, the following are the computed de Rham cohomologies for some common
topological objects:
The -sphere
For the
-sphere,
, and also when taken together with a product of open intervals, we have the following. Let , and be an open real interval. Then
:
The -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
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 A ...
, , can be
deformation retracted to the -sphere (i.e. the real unit circle), that:
:
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
A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
. It says that the pairing of differential forms and chains, via integration, gives a
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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 , 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
:
defined as follows: for any
, let 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 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 commutati ...
.
Sheaf-theoretic de Rham isomorphism
For any smooth manifold ''M'', let
be the
constant sheaf
Constant or The Constant may refer to:
Mathematics
* Constant (mathematics), a non-varying value
* Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
* Control variable or scientific const ...
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
natural isomorphism
:
between the de Rham cohomology and the
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally whe ...
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
acyclic resolution of
. In more detail, let ''m'' be the dimension of ''M'' and let
denote the
sheaf of germs of
-forms on ''M'' (with
the sheaf of
functions on ''M''). By the
Poincaré lemma, the following sequence of sheaves is exact (in the
abelian category of sheaves):
:
This
long exact sequence now breaks up into
short exact sequences of sheaves
:
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
fine sheaf; in particular, the sheaves
are all fine. Therefore, the sheaf cohomology groups
vanish for
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
and at the other lies the de Rham cohomology.
Related ideas
The de Rham cohomology has inspired many mathematical ideas, including
Dolbeault cohomology In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault co ...
,
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 co ...
, and the
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the sp ...
. 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 co ...
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 co ...
.
Harmonic forms
If is a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
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
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 -
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 n ...
, 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 of a -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 , one may equip it with some auxiliary
Riemannian metric. Then the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
is defined by
:
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 re ...
and
the
codifferential
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 ...
. 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 ...
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 retur ...
acting upon the
exterior algebra 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 application ...
s: we can look at its action on each component of degree
separately.
If
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
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 coord ...
of the
kernel 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 co ...
) 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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
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
-form on
uniquely splits into the sum of three 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 inner product on
:
:
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.
See also
*
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 co ...
*
Integration along fibers In differential geometry, the integration along fibers of a ''k''-form yields a (k-m)-form where ''m'' is the dimension of the fiber, via "integration". It is also called the fiber integration.
Definition
Let \pi: E \to B be a fiber bundle over ...
(for de Rham cohomology, the
pushforward is given by
integration)
*
Sheaf theory
*
-lemma for a refinement of exact differential forms in the case of compact
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
s.
Citations
References
*
*
*
*
External links
*
Idea of the De Rham Cohomology' i
Mathifold Project*
{{DEFAULTSORT:De Rham Cohomology
Cohomology theories
Differential forms