Top-degree de Rham cohomology of a compact Lie group ($H_{\mathrm{dR}}^n(G;\mathbb{R})\cong {\mathbb{R}}^{{\pi }_0(G)}$)

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.