Pontryagin class
Encyclopedia
In mathematics
, the Pontryagin classes are certain characteristic class
es. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundle
s.
is defined as
Here
denotes the 2k-th Chern class
of the complexification
of E and
, the 4k-cohomology
group of
with integer
coefficients.
The rational Pontryagin class
is defined to be the image of 
in 
, the 4k-cohomology
group of
with rational
coefficients.
is (modulo 2-torsion) multiplicative with respect to
Whitney sum of vector bundles, i.e.,
for two vector bundles E and F over M. In terms of the individual Pontryagin classes
,
and so on.
The vanishing of the Pontryagin classes and Stiefel-Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle
over the 9-sphere. (The clutching function for 
arises from the stable homotopy group 
.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel-Whitney class 
of 
vanishes by the Wu formula 
. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of 
with any trivial bundle remains nontrivial.
Given a 2k-dimensional vector bundle E we have
where
denotes the Euler class
of E, and
denotes the cup product
of cohomology classes; under the splitting principle
, this corresponds to the square of the Vandermonde polynomial being the discriminant
.
and André Weil
around 1948, the rational Pontryagin classes
can be presented as differential forms which depend polynomially on the curvature form
of a vector bundle. This Chern-Weil theory
revealed a major connection between algebraic topology and global differential geometry.
For a vector bundle
E over a n-dimensional differentiable manifold
M equipped with a connection
, the total Pontryagin class is expressed as
where
denotes the curvature form
, and
denotes the de Rham cohomology
groups.
.
Novikov proved in 1966 that if manifolds are homeomorphic
then their rational Pontryagin classes:
are the same.
If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
. The Pontryagin number vanishes if the dimension of manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of a manifold
as follows:
Given a smooth 4n-dimensional manifold M and a collection of natural numbers
such that 
the Pontryagin number
is defined by
where
denotes the k-th Pontryagin class and [M] the fundamental class
of M.
structure.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Pontryagin classes are certain characteristic class
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s.
Definition
Given a real vector bundle E over M, its k-th Pontryagin class is defined asHere
denotes the 2k-th Chern classChern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
of the complexification
Complexification
In mathematics, the complexification of a real vector space V is a vector space VC over the complex number field obtained by formally extending scalar multiplication to include multiplication by complex numbers. Any basis for V over the real numbers serves as a basis for VC over the complex...
of E and, the 4k-cohomologyCohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
group of
with integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
coefficients.
The rational Pontryagin class
is defined to be the image of in , the 4k-cohomologyCohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
group of
with rationalRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
coefficients.
Properties
The total Pontryagin classis (modulo 2-torsion) multiplicative with respect to
Whitney sum of vector bundles, i.e.,
for two vector bundles E and F over M. In terms of the individual Pontryagin classes
,and so on.
The vanishing of the Pontryagin classes and Stiefel-Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle
over the 9-sphere. (The clutching function for arises from the stable homotopy group .) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel-Whitney class of vanishes by the Wu formula . Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of with any trivial bundle remains nontrivial. Given a 2k-dimensional vector bundle E we have
where
denotes the Euler classEuler class
In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...
of E, and
denotes the cup productCup 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 graded commutative product operation in cohomology, turning the cohomology of a space X into a...
of cohomology classes; under the splitting principle
Splitting principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles.In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computations are well understood for line bundles and for direct...
, this corresponds to the square of the Vandermonde polynomial being the discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
.
Pontryagin classes and curvature
As was shown by Shiing-Shen ChernShiing-Shen Chern
Shiing-Shen Chern was a Chinese American mathematician, one of the leaders in differential geometry of the twentieth century.-Early years in China:...
and André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
around 1948, the rational Pontryagin classes
can be presented as differential forms which depend polynomially on the curvature form
Curvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
of a vector bundle. This Chern-Weil theory
Chern-Weil theory
In mathematics, Chern–Weil theory computes topological invariants of vector bundles and principal bundles in terms of connections and curvature. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry...
revealed a major connection between algebraic topology and global differential geometry.
For a vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
E over a n-dimensional differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
M equipped with a connection
Connection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms....
, the total Pontryagin class is expressed as
where
denotes the curvature formCurvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
, and
denotes the de Rham cohomology
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...
groups.
Pontryagin classes of a manifold
The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundleTangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
.
Novikov proved in 1966 that if manifolds are homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
then their rational Pontryagin classes:
are the same.If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
Pontryagin numbers
Pontryagin numbers are certain topological invariants of a smooth manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. The Pontryagin number vanishes if the dimension of manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
as follows:
Given a smooth 4n-dimensional manifold M and a collection of natural numbers
such that the Pontryagin number
is defined bywhere
denotes the k-th Pontryagin class and [M] the fundamental class
Fundamental class
In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
of M.
Properties
- Pontryagin numbers are oriented cobordismCobordismIn mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class. - Pontryagin numbers of closed Riemannian manifold (as well as Pontryagin classes) can be calculated as integrals of certain polynomial from curvature tensor of Riemannian manifold.
- Such invariants as signatureSignature (topology)In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d=4k divisible by four ....
and
-genus can be expressed through Pontryagin numbers.
Generalizations
There is also a quaternionic Pontryagin class, for vector bundles with quaternionQuaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
structure.