Stiefel–Whitney class

Summary

The Stiefel–Whitney classes $w_i(E) \in H^i(B; \Z/2\Z)$ are mod $2$ characteristic classes of real vector bundles $E \to B$. Four axioms determine them: naturality, the Whitney sum formula, a normalization, and vanishing above the rank. The first class detects orientability and the second detects the existence of a spin structure. Stiefel–Whitney numbers of manifolds decide when two closed manifolds are cobordant.