StiefelWhitney

Stiefel-Whitney classes, named for Heinz Stiefel and Hassler Whitney, are characteristic classes of real vector bundles with coefficients in Z/2. For a real vector bundle E of rank n over a paracompact space X, there are classes w_i(E) in H^i(X; Z/2) for i ≥ 0, vanishing for i > n. The classes combine into the total Stiefel-Whitney class w(E) = 1 + w_1(E) + w_2(E) + … + w_n(E). They arise from the classifying map of E to the classifying space BO(n) and pull back the universal Stiefel-Whitney classes on BO(n).

Key properties of the Stiefel-Whitney classes include naturality (functoriality under bundle maps) and the Whitney sum formula: w(E ⊕ F) = w(E) ∪ w(F). The i-th class w_i(E) encodes obstructions to constructing certain structures on E. In particular, w_1(E) is the obstruction to orientability of E, so w_1(TM) vanishes iff a manifold M is orientable. If E is oriented, w_2(E) is the primary obstruction to lifting the structure group to Spin(n); thus w_2(E) vanishes precisely when a spin structure exists on E under appropriate conditions. The top class w_n(E) reduces modulo 2 to the Euler class e(E) in H^n(X; Z/2) for oriented bundles.

In geometric applications, the Stiefel-Whitney classes of the tangent bundle TM of a manifold M yield global invariants: w_1(M) detects nonorientability, and the vanishing of w_2(M) is a necessary condition for a spin structure on M. The concept plays a central role in obstruction theory, cobordism, and the study of bundle existences.