= Top-degree differential forms on even-dimensional real projective space are exact
{title2=$H^{2n}_{\mathrm{dR}}(\mathbb{RP}^{2n})=0$}
For the double covering $\pi:S^{2n}\to\mathbb{RP}^{2n}$ and the <antipodal map> $a$, every pulled-back top form satisfies $a^*\pi^*\omega=\pi^*\omega$. Since $\deg a=-1$, its integral over the sphere is its own negative and hence vanishes. It is therefore exact on the sphere. Averaging a primitive under $a$ and descending it proves that $\omega$ is exact on real projective space.
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