= de Rham cohomology of a finite quotient
{c}
{title2=$H^p_{\mathrm{dR}}(M/G)\cong H^p_{\mathrm{dR}}(M)^G$}
For a free smooth action of a <finite group> $G$ on a <smooth manifold> $M$, the quotient projection induces
$$
H^p_{\mathrm{dR}}(M/G)\cong H^p_{\mathrm{dR}}(M)^G.
$$
Invariant <differential forms> descend uniquely through local inverse branches of the <covering map>. Averaging by $|G|^{-1}\sum_{g\in G}g^*$ commutes with the <exterior derivative>, produces invariant representatives of invariant classes, and produces an invariant primitive for every invariant exact form. Hence taking invariant forms and taking invariant <cohomology> give the same result over the real numbers.
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