= Top-degree de Rham cohomology of a compact Lie group
{title2=$H^n_{\mathrm{dR}}(G;\mathbb R)\cong\mathbb R^{\pi_0(G)}$}
A compact <Lie group> has finitely many <connected components>, each an oriented compact manifold of dimension $n$. Componentwise integration identifies its <top-degree de Rham cohomology> with one real coordinate per component. For a connected group, if top-degree classes are represented by invariant forms, the <invariant volume form on a Lie group> spans the cohomology: invariant top forms are one-dimensional, and its positive integral proves that its class is nonzero by the <Generalized Stokes theorem>. A disconnected group cannot have an invariant representative of every top-degree class, since left translations permute its components. For example, $S^1\times\mathbb Z/2$ has two independent degree-one classes but only one independent invariant top form.
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