The ith Stiefel-Whitney class of a real vector bundle (or tangent bundle or a real manifold) is in the ith cohomology group of the base space involved. It is an obstruction to the existence of (n - i + 1) real linearly independent vector fields on that vector bundle, where n is the dimension of the fiber. Here, obstruction means that the ith Stiefel-Whitney class being nonzero implies that there do not exist (n - i + 1) everywhere linearly independent vector fields (although the Stiefel-Whitney classes are not always the obstruction). In particular, the nth Stiefel-Whitney class is the obstruction to the existence of an everywhere nonzero vector field, and the first Stiefel-Whitney class of a manifold is the obstruction to orientability.

Chern class | obstruction | Pontryagin class | Stiefel-Whitney number