In
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, 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 forms with prescribed properties.
On any smooth manifold, every
exact form 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 ...
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 th ...
of
differential forms 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 as the differential:
:
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, 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 (no-holes condition). In this case the exterior derivative
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 t ...
with the arrows reversed compared to the de Rham complex. This is the situation described in the
Poincaré lemma 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 ...
.
The idea behind de Rham cohomology is to define
equivalence classes 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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
, 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
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 ...
. 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 defor ...
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,
, 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
representative
Representative may refer to:
Politics
* Representative democracy, type of democracy in which elected officials represent a group of people
* House of Representatives, legislative body in various countries or sub-national entities
* Legislator, som ...
s 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, , 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 deformat ...
ed to the -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
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 "same" ...
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
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 th ...
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 ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
s), 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 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
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
:
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
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.
Motivation
Let ''X'' be a topol ...
; 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 In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category), which is used to defi ...
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 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 ...
, 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 ...
of sheaves):
:
This
long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
now breaks up into
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
s 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
In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X:
* there is a neighbourhood of where all but a finite number of the functions of are 0, ...
, any
-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
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 ...
,
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 is a
compact Riemannian manifold, then each equivalence class in
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 coh ...
. 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
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,
: \f ...
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 tou ...
, 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 simplici ...
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
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 ...
, one may equip it with some auxiliary
Riemannian metric
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 the
Laplacian is defined by
:
with
the
exterior derivative and
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 ...
differential operator 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 forms: we can look at its action on each component of degree
separately.
If
is
compact and
oriented, 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 coor ...
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 learn ...
of the Laplacian acting upon the space of
-forms is then equal (by
Hodge theory) 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
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 simplici ...
.
Hodge decomposition
Let
be a
compact oriented Riemannian manifold. 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
*
Integration along fibers (for de Rham cohomology, the
pushforward
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things.
* Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...
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 ...
*
-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