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Concept
Classe de Pontriaguine
Résumé
In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
Given a real vector bundle E over M, its k-th Pontryagin class is defined as
where:
denotes the -th Chern class of the complexification of E,
is the -cohomology group of M with integer coefficients.
The rational Pontryagin class is defined to be the image of in , the -cohomology group of M with rational coefficients.
The total Pontryagin class
is (modulo 2-torsion) multiplicative with respect to
Whitney sum of vector bundles, i.e.,
for two vector bundles E and F over M. In terms of the individual Pontryagin classes pk,
and so on.
The vanishing of the Pontryagin classes and Stiefel–Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle over the 9-sphere. (The clutching function for arises from the homotopy group .) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class w9 of E10 vanishes by the Wu formula w9 = w1w8 + Sq1(w8). Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of E10 with any trivial bundle remains nontrivial.
Given a 2k-dimensional vector bundle E we have
where e(E) denotes the Euler class of E, and denotes the cup product of cohomology classes.
As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes
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, the total Pontryagin class is expressed as
where Ω denotes the curvature form, and H*dR(M) denotes the de Rham cohomology groups.
Source officielle
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