= Unoriented Gysin sequence
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> with
$$
\cdots\to H^{j-r}(B;\mathbb F_2)\xrightarrow{\cup w_r(E)}H^j(B;\mathbb F_2)\to H^j(S(E);\mathbb F_2)\to H^{j-r+1}(B;\mathbb F_2)\to\cdots.
$$
The 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.
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