Splitting principle for real vector bundles

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.