Stiefel–Whitney class ($w_i(E)$)

The Stiefel–Whitney classes of a real rank-$d$ vector bundle are classesTheir total class is $w(E)=1+w_1(E)+\cdots +w_d(E)$.

Let $p:\mathbb{P}(E)\to X$ and put $u=w_1(L_E)$. The mod-two projective bundle formula makes $H^{\ast }(\mathbb{P}(E);{\mathbb{F}}_2)$ free over $H^{\ast }(X;{\mathbb{F}}_2)$ on $1,u,\dots ,u^{d-1}$. The unique relationdefines the Stiefel–Whitney classes.

After pulling a real vector bundle back along a suitable iterated projective or flag bundle, it splits into real line bundles, and the pullback on mod-two cohomology is injective. Identities among Stiefel–Whitney classes can therefore be checked after splitting.

For real vector bundles $E$ and $F$,After applying the splitting principle for real vector bundles, both sides are the product of $1+w_1(L)$ over all line summands.

Real line bundles are classified by $H^1(X;{\mathbb{F}}_2)$, and tensor product corresponds to addition:

Underlet $x$ and $v$ be the degree-one generators. Then