# Pontryagin class

In mathematics, the Pontryagin classes are certain characteristic classes. The Pontryagin class lies in cohomology groups with index a multiple of four. It applies to real vector bundles.

 Contents

## Definition

Given a vector bundle [itex]E[itex] over [itex]M[itex] its k-th Pontryagin class [itex]p_k(E)[itex] can be defined as

[itex]p_k(E)=p_k(E,\mathbb{Z})=(-1)^kc_{2k}(E \otimes \mathbb{C})\in H^{4k}(M,\mathbb{Z}),[itex]

here [itex]c_{2k}(E \otimes \mathbb{C})[itex] denotes times 2k-th Chern class of the complexification [itex]E \otimes \mathbb{C}=E\oplus i E[itex] of [itex]E[itex] and [itex]H^{4k}(M,\mathbb{Z})[itex], the 4k-cohomology group of [itex]M[itex] with integer coefficients.

Rational Pontryagin class [itex]p_k(E,{\mathbb Q})[itex] is defined to be image of [itex]p_k(E)[itex] in [itex]H^{4k}(M,\mathbb{Q})[itex], the 4k-cohomology group of [itex]M[itex] with rational coefficients

Pontryagin classes have a meaning in real differential geometry — unlike the Chern class, which assumes a complex vector bundle at the outset.

## Properties

If all Pontryagin classes and Stiefel-Whitney classes of [itex]E[itex] vanish then the bundle is stably trivial, i.e. its Whitney sum with a trivial bundle is trivial. The total Pontryagin class [itex]p(E)=1+p_1(E)+p_2(E)+...\in H^{*}(M,\mathbb{Z}),[itex] is multiplicative with respect to Whitney sum of vector bundles, i.e [itex]p(E\oplus F)=p(E)\cup p(F)[itex] for two vector bundles [itex]E[itex] and [itex]F[itex] over [itex]M[itex], i.e.

[itex]p_1(E\oplus F)=p_1(E)+p_1(F),[itex]
[itex]p_2(E\oplus F)=p_2(E)+p_1(E)\cup p_1(F)+p_2(F) [itex]

and so on. Given a 2k-dimensional vector bundle E we have

[itex]p_k(E)=e(E)\cup e(E),[itex]

where [itex]e(E)[itex] denotes Euler class of E, and the notation is 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

[itex]p_n(E,\mathbb{Q})\in H^{4k}(M,\mathbb{Q})[itex]

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, its k-th Pontryagin class can be realized by the 4k-form

[itex] Tr(\Omega\wedge...\wedge\Omega)[itex]

constructed with 2k copies of the curvature form [itex]\Omega[itex]. In particular the value

[itex] p_n(E,\mathbb{Q})=[Tr(\Omega\wedge...\wedge\Omega)]\in H^{4k}_{dR}(M)[itex]

does not depend on the choice of connection. Here

[itex] H^{*}_{dR}(M)[itex]

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's theorem states that if manifolds are homeomorphic then their rational Pontryagin classes

[itex]p_k(M,\mathbb{Q}) \in H^{4k}(M,\mathbb{Q})[itex]

are the same.

If the dimension is at least five, there at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.

## Generalizations

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

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy