 Stiefel–Whitney class

The Stiefel–Whitney classes of a real rank-$r$ vector bundle $E\to X$ are characteristic cohomology classesThey are natural under pullback vector bundles and obey the Whitney product formula for Stiefel–Whitney classes, $w(E\oplus F)=w(E)w(F)$. For a real line bundle $L$, the normalization is $w(L)=1+w_1(L)$, with $w_1$ the class classifying its orientation double cover. These properties determine the classes through the splitting principle for real vector bundles.

The first Stiefel–Whitney class vanishes exactly when the vector bundle is orientable. The top Stiefel–Whitney class is its Euler class with mod-two coefficients and obstructs a nowhere-zero section. A trivial vector bundle has total class one; the Möbius line bundle has nonzero $w_1$.

Vanishing of every positive-degree Stiefel–Whitney class need not imply triviality. The tangent bundle of the two-dimensional sphere has $w_1=0$ and $w_2=\chi (S^2)\mathrm{mod}2=0$, although its integral Euler class is nonzero. The classes detect useful obstructions while forgetting information lost by reduction modulo two.