= Suspension isomorphism for top de Rham cohomology of spheres
{title2=$H^{n-1}_{\mathrm{dR}}(S^{n-1})\cong H^n_{\mathrm{dR}}(S^n),\quad n>1$}
Cover the <sphere> by complements of its two poles. Each open set is contractible, and their intersection deformation retracts to the equatorial <sphere>. The <Poincaré lemma> and <homotopy invariance of de Rham cohomology> make the two neighboring groups in the <Mayer--Vietoris sequence for de Rham cohomology> zero, so its connecting map is an <isomorphism>. Its inverse takes a top-degree form to the difference of its local primitives on the equator, up to the connecting-map sign convention.
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