Rational Pontryagin classes of Euclidean fiber bundles

Abstract

The Pontryagin classes of an $m$–dimensional real vector bundle satisfy well-known vanishing relations due to their close relationship with the Chern classes of the complexified vector bundle. It is also known that the rational Pontryagin classes qualify as characteristic classes for Euclidean fiber bundles, ie bundles whose fibers are homeomorphic to $m$–dimensional Euclidean space. In this more general setting of fiber bundles, the vanishing relations which one has for vector bundles fail to hold.