= Gysin proof of mod-two projective-space cohomology
{c}
{title2=$H^*(\mathbb{RP}^n;\mathbb F_2)=\mathbb F_2[a]/(a^{n+1})$}
The unit sphere bundle of the tautological line is the double cover $S^n\to\mathbb{RP}^n$. In its <Gysin sequence>, multiplication by the mod-two Euler class $a$ is an isomorphism in successive interior degrees, since the sphere has no interior cohomology. Connectedness in degree zero makes multiplication into degree one injective. At the top, vanishing above dimension $n$ makes the transfer from $H^n(S^n;\mathbb F_2)$ surjective; together with that injection it makes the top group one-dimensional. Thus every $a^j$, $0\leq j\leq n$, is the nonzero generator and the next power vanishes.
Back to article page
