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.

Properties

If all Pontryagin classes and Stiefel-Whitney classes of E vanish then the bundle is stably trivial,

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i.e. its Whitney sum with a trivial bundle is trivial.

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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(Eoplus F)=p(E)cup p(F) for two vector bundles E and F over M, i.e.

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:p_1(Eoplus F)=p_1(E)+p_1(F),

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:p_2(Eoplus F)=p_2(E)+p_1(E)cup p_1(F)+p_2(F)

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and so on.

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Given a 2k-dimensional vector bundle E we have

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:p_k(E)=e(E)cup e(E),

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where e(E) denotes Euler class of E, and the notation is the cup product of cohomology classes.

Related Topics:
Euler class - Cup product

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Pontryagin classes and curvature

As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes

Related Topics:
Shiing-Shen Chern - André Weil

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

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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.

Related Topics:
Curvature form - Chern-Weil theory

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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

Related Topics:
Vector bundle - Differentiable manifold - Connection - Form

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: Tr(Omegawedge...wedgeOmega)

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

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: p_n(E,mathbb{Q})=in H^{4k}_{dR}(M)

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does not depend on the choice of connection. Here

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

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denotes the de Rham cohomology groups.

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