Pontryagin class

Summary

The Pontryagin classes $p_i(E) \in H^{4i}(B; \Z)$ are characteristic classes of real vector bundles $E \to B$ with integer coefficients. They are defined from the Chern classes of the complexification $E \otimes \C$. Pontryagin classes of the tangent bundle of a manifold give the Pontryagin numbers, which are invariants of oriented cobordism. They enter the Hirzebruch signature theorem and Milnor's construction of exotic spheres.