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Wikipedia / Pontryagin class

In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.

Definition

is defined as

where:

with rational coefficients.

Properties

The total Pontryagin class

is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e.,

,

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 over the 9-sphere. (The clutching function arises from the homotopy group vanishes by the Wu formula. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum with any trivial bundle remains nontrivial.

we have

denotes the Euler class denotes the cup product of cohomology classes.

Pontryagin classes and curvature

As was shown by Shiing-Shen Chern 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-dimensional differentiable manifold equipped with a connection, the total Pontryagin class is expressed as

denotes the curvature form denotes the de Rham cohomology groups.

Pontryagin classes of a manifold

The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle.

Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic 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 classes from Chern classes

. Then, given this relation, we can see.For example, we can apply this formula to find the Pontryagin classes of a complex vector bundle on a curve and a surface. For a curve, we haveso all of the Pontryagin classes of complex vector bundles are trivial.

In general, looking at first two terms of the product by dimension reasons.

Pontryagin classes on a quartic K3 surface

is a smooth subvariety is a K3 surface. If we use the normal sequence

we can find

. This number can be used to compute the third stable homotopy group of spheres.

Pontryagin numbers

Pontryagin numbers are certain topological invariants of a smooth manifold is not divisible by 4. It is defined in terms of the Pontryagin classes of the manifold as follows:

and a collection of natural numbers

  1. ,

is defined by

the fundamental class.

Properties

  1. Pontryagin numbers are oriented cobordism invariant; and together with Stiefel-Whitney numbers they determine an oriented manifold's oriented cobordism class.

  2. Pontryagin numbers of closed Riemannian manifolds (as well as Pontryagin classes) can be calculated as integrals of certain polynomials from the curvature tensor of a Riemannian manifold.

  3. Invariants such as signature and \hat A-genus can be expressed through Pontryagin numbers. For the theorem describing the linear combination of Pontryagin numbers giving the signature see Hirzebruch signature theorem.

Generalizations

There is also a quaternionic Pontryagin class, for vector bundles with quaternion structure.

See also

References