= Stiefel–Whitney class
{c}
{title2=$w_i(E)$}
{wiki}
The <Stiefel–Whitney classes> of a real rank-$r$ <vector bundle> $E\to X$ are characteristic <cohomology classes>
$$
w_i(E)\in H^i(X;\mathbb F_2),\qquad w_0(E)=1,\qquad w_i(E)=0\ (i>r).
$$
They 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 $w_2=\chi(S^2)\bmod2=0$, although its integral <Euler class> is nonzero. The classes detect useful obstructions while forgetting information lost by reduction modulo two.
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