Pontryagin class

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.
Definition.
Given a real vector bundle "E" over "M", its "k"-th Pontryagin class formula_1 is defined as 
where:
The rational Pontryagin class formula_8 is defined to be the image of formula_1 in formula_10, the formula_7-cohomology group of "M" with rational coefficients.
Properties.
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 formula_16 over the 9-sphere. (The clutching function for formula_16 arises from the homotopy group formula_18.) 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 + Sq1("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 
where "e"("E") denotes the Euler class of "E", and formula_20 denotes 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 
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 