Editing Thom space (section)

==Consequences for differentiable manifolds==

If we take the bundle in the above to be the [[tangent bundle]] of a smooth manifold, the conclusion of the above is called the [[Wu formula]], and has the following strong consequence: since the Steenrod operations are invariant under homotopy equivalence, we conclude that the Stiefel–Whitney classes of a manifold are as well. This is an extraordinary result that does not generalize to other characteristic classes. There exists a similar famous and difficult result establishing topological invariance for rational [[Pontryagin classes]], due to [[Sergei Novikov (mathematician)|Sergei Novikov]].