Pontryagin class

In mathematics, the Pontryagin classes, named for Lev Pontryagin, are certain characteristic classes. The Pontryagin class lies in cohomology groups with degree a multiple of four. It applies to real vector bundles.

Given a real vector bundle E over M, its k-th Pontryagin class p_k(E) is defined as

where:

The rational Pontryagin class p_k(E, Q) is defined to be the image of p_k(E) in H^4k(M, Q), the 4k-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 p_k,

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 E₁₀ over the 9-sphere. (The clutching function for E₁₀ arises from the stable homotopy group π₈(O(10)) = Z/2Z.) 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. (Hatcher 2009, p. 76)

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