= Top de Rham cohomology of a compact connected oriented manifold
{title2=$H^n_{\mathrm{dR}}(M)\cong\mathbb R$}
For a compact connected oriented boundaryless $n$-dimensional <smooth manifold>, choose a <Riemannian metric>. Harmonic functions are constant because $\langle f,\Delta f\rangle=\|df\|^2$, and <Hodge star commutes with the Hodge Laplacian>. Thus the <Hodge star operator> identifies the constant functions with the <harmonic differential forms> of top degree, exactly $\mathbb R\omega_g$. The <Hodge decomposition theorem> identifies these with the top <de Rham cohomology>. The <metric volume form> has nonzero class by <Stokes theorem>. The harmonic-representative theorem is stated in https://www.ocw.mit.edu/courses/18-966-geometry-of-manifolds-spring-2007/d6848bb391c032ef27993e984fef4558_lect15.pdf[Denis Auroux's Hodge theory lecture].
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