Stiefel-Whitney Class

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

In particular, the th 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.