Pontryagin class

In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in with degrees a multiple of four.

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Definition Given a real vector bundle $E$ over $M$, its $k$-th Pontryagin class $p\_k(E)$ is defined asLawson & Michelson 90, Equation (B.12)

$p\_k(E)\; =\; p\_k(E,\; \backslash Z)\; =\; (-1)^k\; c\_\{2k\}(E\backslash otimes\; \backslash Complex)\; \backslash in\; H^\{4k\}(M,\; \backslash Z),$

where:

• $c\_\{2k\}(E\backslash otimes\; \backslash Complex)$ denotes the $2k$-th Chern class of the complexification $E\backslash otimes\; \backslash Complex\; =\; E\backslash oplus\; iE$ of $E$,
• $H^\{4k\}(M,\; \backslash Z)$ is the $4k$-cohomology group of $M$ with integer coefficients.

The rational Pontryagin class $p\_k(E,\; \backslash Q)$ is defined to be the image of $p\_k(E)$ in $H^\{4k\}(M,\; \backslash Q)$, the $4k$-cohomology group of $M$ with Rational number coefficients.

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Properties The total Pontryagin class

$p(E)=1+p\_1(E)+p\_2(E)+\backslash cdots\backslash in\; H^*(M,\backslash Z),$

is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e.,Lawson & Michelson 90, Equation (B.10)

$2p(E\backslash oplus\; F)=2p(E)\backslash smile\; p(F)$

for two vector bundles $E$ and $F$ over $M$. In terms of the individual Pontryagin classes $p\_k$,

$2p\_1(E\backslash oplus\; F)=2p\_1(E)+2p\_1(F),$

$2p\_2(E\backslash oplus\; F)=2p\_2(E)+2p\_1(E)\backslash smile\; p\_1(F)+2p\_2(F)$

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 $E\_\{10\}$ over the N-sphere. (The clutching function for $E\_\{10\}$ arises from the homotopy group $\backslash pi\_8(\backslash mathrm\{O\}(10))\; =\; \backslash Z/2\backslash Z$.) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class $w\_9$ of $E\_\{10\}$ vanishes by the Wu formula $w\_9\; =\; w\_1\; w\_8\; +\; Sq^1(w\_8)$. Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of $E\_\{10\}$ with any trivial bundle remains nontrivial.

Given a $2\; k$-dimensional vector bundle $E$ we have

$p\_k(E)=e(E)\backslash smile\; e(E),$

where $e(E)$ denotes the Euler class of $E$, and $\backslash smile$ denotes the cup product of cohomology classes.

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

$p\_k(E,\backslash mathbf\{Q\})\backslash in\; H^\{4k\}(M,\backslash mathbf\{Q\})$

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, the total Pontryagin class is expressed as

$p=\backslash left[1-\backslash frac$

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