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

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

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here c_{2k}(E otimes mathbb{C}) denotes times 2k-th Chern class of the complexification E otimes mathbb{C}=Eoplus i E of E and

Related Topics:
Chern class - Complexification

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H^{4k}(M,mathbb{Z}), the 4k-cohomology group of M with integer coefficients.

Related Topics:
Cohomology - Integer

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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.

Related Topics:
Cohomology - Rational

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

Related Topics:
Differential geometry - Chern class

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