= Top-degree differential forms on even-dimensional real projective space are exact
{title2=$H^{2n}_{\mathrm{dR}}(\mathbb{RP}^{2n})=0$}
Let $n\geq1$. For the double <covering map> $\pi:S^{2n}\to\mathbb{RP}^{2n}$ and the <antipodal map> $a$, every pulled-back top <differential form> satisfies $a^*\pi^*\omega=\pi^*\omega$. Since $\deg a=-1$, its integral over the <sphere> is its own negative and hence vanishes. Top-dimensional <de Rham cohomology> of the oriented <sphere> therefore makes it an <exact differential form>.
Average a primitive under $a$ to obtain an invariant primitive. It descends through the double <covering map>, proving that $\omega$ is exact on <Real projective space>. This is the <invariant primitive under a finite group action> argument. The condition $n\geq1$ is essential: $\mathbb{RP}^0$ is a <point>, whose degree-zero <de Rham cohomology> is $\mathbb R$.
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