Stiefel-Whitney Class

The $i$th Stiefel-Whitney class of a Real Vector Bundle (or Tangent Bundle or a Real Manifold) is in the $i$th 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 $i$th Stiefel-Whitney class being Nonzero implies that there do not exist $(n-i+1)$ everywhere linearly dependent Vector Fields (although the Stiefel-Whitney classes are not always the Obstruction).

In particular, the $n$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.

See also Chern Class, Obstruction, Pontryagin Class, Stiefel-Whitney Number