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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.

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 a real rank-$r$ vector bundle $E\to B$, the Thom isomorphism theorem with ${\mathbb{F}}_2$ coefficients identifies the relative cohomology sequence of its disk bundle and sphere bundle withThe multiplier is the top Stiefel–Whitney class, also called the mod-two Euler class. No orientation of $E$ is required. For the real tautological line bundle, the sphere bundle is the antipodal cover of Real projective space; this sequence proves that all powers of the degree-one generator through the dimension of the base are nonzero.