First de Rham cohomology of the two-sphere
= First de Rham cohomology of the two-sphere
{title2=$H^1_{\mathrm{dR}}(S^2)=0$}
A <closed differential form> of degree one on the <two-sphere> has primitives on its north-punctured and south-punctured charts by the <Poincare lemma>. The overlap is connected, so their difference is constant. Adjust one constant and glue the primitives. Thus every closed one-form is an <exact differential form>, proving the first <de Rham cohomology> vanishes.