Tautological degree-one class on a real projective bundle

Assume AC, and let $E\to B$ be a numerable real vector bundle of rank $n\ge 1$ over an paracompact Hausdorff CGWH base of CW homotopy type. Put $P(E)$ and ${\gamma }_E$ as in Real projective bundle and tautological line; both are numerable.

The projective total space is paracompact Hausdorff CGWH of CW type and the tautological line has the refined numeration of the projective-bundle definition. The stable-Grassmannian classification theorem therefore gives a classifying map $c:P(E)⟶{\mathrm{Gr}}_1({\mathbb{R}}^{\mathrm{\infty }})={\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ with $c^{\ast }{\gamma }_1\cong {\gamma }_E$, where ${\gamma }_1$ is the tautological line. Here a classifying map means precisely a map with this bundle-pullback isomorphism.

Let $a\in H^1({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)$ be the fixed generator of $H^{\ast }({\mathbb{R}\mathbb{P}}^{\mathrm{\infty }};{\mathbb{F}}_2)\cong {\mathbb{F}}_2[a]$ supplied by Mod-two cohomology ring of infinite real projective space. The tautological degree-one class of $E$ is $x_E:=c^{\ast }a\in H^1(P(E);{\mathbb{F}}_2),$ computed for a chosen classifying map $c$ of ${\gamma }_E$. The definition uses neither $w_1$ nor any Stiefel–Whitney class, and no Thom class. Independence of $x_E$ from the choice of $c$ is proved in The tautological degree-one class is well defined and fiber generating; all later statements about $x_E$ are read modulo that lemma. For $n=1$ the identification $P(E)\cong B$ of the projective-bundle definition presents $x_E$ as a class in $H^1(B;{\mathbb{F}}_2)$.