Stiefel–Whitney class ($w_i(E)$)

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

For a real line bundle on a closed manifold, a transverse section has a hypersurface as its zero set. Along a closed loop transverse to this hypersurface, trivialize the pullback over the cut interval. Each simple zero reverses the sign of the component function. The total sign change is exactly the bundle's transition sign around the loop, detected by its first Stiefel–Whitney class. Thus zero-set intersection parity with every loop equals the evaluation of $w_1(L)$, proving the displayed mod-two dual class. No orientability of the manifold or bundle is required.

An immersion of a smooth $d$-manifold in ${\mathbb{R}}^{d+1}$ supplies a rank-one normal bundle $\nu$ with $TM\oplus \nu \cong {\epsilon }^{d+1}$. The Whitney product formula for Stiefel–Whitney classes therefore requires $w(\nu )=w(TM{)}^{-1}$. A real line bundle has no positive-degree classes beyond $w_1$, so every higher-degree part of the inverse total class must vanish. For ${\mathbb{RP}}^2\times {\mathbb{RP}}^2$, the stabilized tangent bundle of real projective space gives $w(TM{)}^{-1}=(1+a)(1+b)$ in ${\mathbb{F}}_2[a,b]/(a^3,b^3)$. Its nonzero degree-two part $ab$ excludes an immersion in ${\mathbb{R}}^5$ even though the normal line need not be orientable.

A complex line bundle has a canonically oriented underlying real vector bundle of rank two. Its first Stiefel–Whitney class vanishes, and its second Stiefel–Whitney class is the mod-two reduction of its First Chern class.

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.

For a map $f:{\mathbb{RP}}^{2k+1}\to {\mathbb{CP}}^k$ nontrivial on degree-two mod-two cohomology, an immersion in the homotopy class of $(f,f)$ has a rank-$(2k-1)$ normal bundle with total Stiefel–Whitney class $(1+x{)}^{2k+2}$. Its forbidden degree-$2k$ Stiefel–Whitney class is $(k+1)x^{2k}$, so $k$ must be odd.

For the Bockstein associated with $0\to \mathbb{Z}/2\to \mathbb{Z}/4\to \mathbb{Z}/2\to 0$,For a real line bundle $L$, one first has $\beta (w_1(L))=w_1(L{)}^2$. The general identity follows from the splitting principle for real vector bundles, the Bockstein derivation rule, and the elementary-symmetric-polynomial identity obtained by separating whether an index lies in a chosen $j$-element subset.

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