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 2 k -th Chern class of the complexification E otimes mathbb{C}=Eoplus i E of E and H^{4k}(M,mathbb{Z}), the 4 k -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 4 k -Cohomology group of M with Rational number 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 Glossary_of_differential_geometry_and_topology#W 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 Glossary_of_differential_geometry_and_topology#W of vector bundles, i.e p(Eoplus F)=p(E)cup p(F) for two vector bundles E and F over M, i.e. :p_1(Eoplus F)=p_1(E)+p_1(F), :p_2(Eoplus F)=p_2(E)+p_1(E)cup p_1(F)+p_2(F) and so on. Given a 2 k -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 form, its k -th Pontryagin class can be realized by the 4 k -Differential form

: Tr(Omegawedge...wedgeOmega)

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

: p_n(E,mathbb{Q})=[Tr(Omegawedge...wedgeOmega)]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 homeomorphism 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#Homotopy_equivalence_of_spaces and Pontryagin classes.

=Generalizations=

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

==See also==

*Chern-Simons form *Pontryagin number