Definition
The De Rham theorem states that for any smooth manifold $M$, the de Rham cohomology groups $H_{\mathrm{dR}}^{k}(M)$ are naturally isomorphic to the singular cohomology groups of $M$ with real coefficients, $H^{k}(M;\mathbb{R})$. The isomorphism is realized by the integration map, which sends a differential $k$-form $\omega$ to the cohomology class that evaluates on a singular $k$-chain $\sigma$ as $\int_{\sigma}\omega$.
Overview
The theorem provides a bridge between the analytic world of differential forms and the topological world of singular cochains. It implies that the topological invariants known as Betti numbers of a smooth manifold can be computed analytically as the dimensions of the corresponding spaces of closed differential forms modulo exact forms. The result holds for all smooth manifolds, irrespective of dimension, orientability, or compactness, although certain technicalities (e.g., partitions of unity) are required in the non‑compact case.
Etymology / Origin
The theorem is named after the Swiss mathematician Georges de Rham (1903–1990), who introduced de Rham cohomology in a 1931 paper and proved the isomorphism with singular cohomology in 1931–1932. His work laid the foundations for modern differential topology and algebraic topology.
Characteristics
- Integration map: The natural transformation $\mathcal{I}\colon \Omega^{k}(M) \rightarrow C^{k}(M;\mathbb{R})^{*}$ defined by $\mathcal{I}(\omega)(\sigma)=\int_{\sigma}\omega$ induces the isomorphism on cohomology.
- Naturality: For a smooth map $f\colon M\rightarrow N$, the diagram
$$ \begin{aligned} H_{\mathrm{dR}}^{k}(N) &\xrightarrow{f^{}} H_{\mathrm{dR}}^{k}(M)\ \downarrow\cong & \quad\downarrow\cong\ H^{k}(N;\mathbb{R}) &\xrightarrow{f^{}} H^{k}(M;\mathbb{R}) \end{aligned} $$ commutes, reflecting functoriality. - Proof techniques: Standard proofs employ the Mayer–Vietoris sequence, partitions of unity, and the Poincaré lemma. Modern approaches use sheaf theory, interpreting de Rham cohomology as the sheaf cohomology of the sheaf of smooth differential forms.
- Consequences:
- Equality of Betti numbers and de Rham dimensions: $b_{k}= \dim_{\mathbb{R}} H_{\mathrm{dR}}^{k}(M)$.
- Compatibility with cup product: the wedge product of forms corresponds to the cup product in singular cohomology under the isomorphism.
- Foundations for Hodge theory on compact Riemannian manifolds, where additional structure (harmonic forms) refines the isomorphism.
Related Topics
- Differential forms and the exterior derivative.
- Singular cohomology and simplicial complexes.
- Poincaré lemma (local exactness of closed forms).
- Sheaf cohomology and the de Rham complex as a fine resolution.
- Mayer–Vietoris sequence in both de Rham and singular contexts.
- Hodge theory (harmonic representatives of cohomology classes).
- Betti numbers and other topological invariants of manifolds.
- Smooth manifolds and their differentiable structures.