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 ,
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 stable 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 of vanishes by the Wu formula . Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of with any trivial bundle remains nontrivial. (Hatcher 2009, p. 76)
Given a 2k-dimensional vector bundle E we have
where denotes the Euler class of E, and denotes the cup product of cohomology classes; under the splitting principle, this corresponds to the square of the Vandermonde polynomial being the discriminant.
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