Categories: Homology theory | Algebraic topology | Differential geometry | Differential topology

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

1 Definition

2 Properties

2.1 Pontryagin classes and curvature

3 Pontryagin classes of a manifold

4 Generalizations

5 See also

Definition

Given a vector bundle $E$ over $M$ its k-th Pontryagin class $p k (E)$ can be defined as

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

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

Rational Pontryagin class $p_k(E,{\mathbb Q})$ is defined to be image of $p k (E)$ in $H^{4k}(M,\mathbb{Q})$ , the 4k-cohomology group of $M$ 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 $E$ vanish then the bundle is stably trivial, i.e. its Whitney sum with a trivial bundle is trivial. The total Pontryagin class $p(E)=1+p_1(E)+p_2(E)+...\in H^{*}(M,\mathbb{Z}),$ is multiplicative with respect to Whitney sum of vector bundles, i.e $p(E\oplus F)=p(E)\cup p(F)$ for two vector bundles $E$ and $F$ over $M$ , i.e.

$p_1(E\oplus F)=p_1(E)+p_1(F),$

$p_2(E\oplus F)=p_2(E)+p_1(E)\cup p_1(F)+p_2(F)$

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

$p_k(E)=e(E)\cup e(E),$

where $e (E)$ 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

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

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

$Tr(\Omega\wedge...\wedge\Omega)$

constructed with 2k copies of the curvature form $Ω$ . In particular the value

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

does not depend on the choice of connection. Here

$H^{*}_{dR}(M)$

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

$p_k(M,\mathbb{Q}) \in H^{4k}(M,\mathbb{Q})$

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.