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.

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

E

and

H^{4k}(M,\mathbb{Z})

, the 4k-cohomology group of

M

with integer coeficients. Rational Pontryagin class

p_k(E,{\mathbb Q})

is defined to be image of

p_k(E)

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

, the 4k-cohomology group of

M

with rational coeficients 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 Andr 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

\Omega

. In particular the value

\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.

Pontryagin Class

Definition

Properties

Pontryagin classes and curvature

Pontryagin classes of a manifold

Generalizations

See also: