= de Rham cohomology of a product with a circle
{c}
{title2=$H^k_{\mathrm{dR}}(M\times S^1)\cong H^k_{\mathrm{dR}}(M)\oplus H^{k-1}_{\mathrm{dR}}(M)$}
Choose a closed one-form $\nu$ on $S^1$ with integral one and let $p:M\times S^1\to M$ be projection. Every rotation-invariant $k$-form is uniquely
$$
p^*\alpha+p^*\beta\wedge\nu,
\qquad
\alpha\in\Omega^k(M),\quad\beta\in\Omega^{k-1}(M).
$$
The <exterior derivative> acts componentwise. Averaging therefore proves that
$$
([\alpha],[\beta])\longmapsto[p^*\alpha+p^*\beta\wedge\nu]
$$
is the displayed isomorphism.
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