De Rham's Theorem

De Rham's theorem establishes a fundamental bridge between differential geometry and algebraic topology, showing that singular cohomology of smooth manifolds can be computed using differential forms.

De Rham Cohomology

Definition

Let $M$ be a smooth manifold and $\Omega^k(M)$ the space of smooth $k$ -forms on $M$ . The de Rham complex is $0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \Omega^2(M) \xrightarrow{d} \cdots$ where $d$ is the exterior derivative. The de Rham cohomology is $H^k_{dR}(M) = \ker(d : \Omega^k \to \Omega^{k+1}) / \operatorname{im}(d : \Omega^{k-1} \to \Omega^k) = \frac{\text{closed $k$-forms}}{\text{exact $k$-forms}}$

The Theorem

Theorem7.9De Rham's Theorem

For any smooth manifold $M$ , the integration map $I : H^k_{dR}(M) \to H^k(M; \mathbb{R}), \quad [\omega] \mapsto \left(\sigma \mapsto \int_\sigma \omega\right)$ is an isomorphism of $\mathbb{R}$ -vector spaces for all $k \geq 0$ . Moreover, this isomorphism is compatible with the ring structures: the wedge product on de Rham cohomology corresponds to the cup product on singular cohomology.

The proof proceeds by showing both cohomology theories satisfy the Eilenberg-Steenrod axioms on smooth manifolds and agree on a point, then invoking a Mayer-Vietoris argument for good covers.

ExampleDe Rham cohomology of $S^n$

The $n$ -sphere $S^n$ for $n \geq 1$ has $H^0_{dR}(S^n) \cong \mathbb{R}$ (constant functions), $H^n_{dR}(S^n) \cong \mathbb{R}$ (generated by the volume form), and $H^k_{dR}(S^n) = 0$ for $0 < k < n$ . By de Rham's theorem, this matches the singular cohomology $H^k(S^n; \mathbb{R})$ .

Consequences

Theorem7.10Poincare Lemma

If $M$ is a contractible smooth manifold (e.g., an open star-shaped subset of $\mathbb{R}^n$ ), then every closed $k$ -form with $k \geq 1$ is exact. That is, $H^k_{dR}(M) = 0$ for $k \geq 1$ .

RemarkComputational power

De Rham's theorem converts cohomological computations into calculus: one can compute $H^k(M; \mathbb{R})$ by finding closed forms that are not exact. Conversely, topological invariance of singular cohomology implies that de Rham cohomology depends only on the diffeomorphism type of $M$ , which is far from obvious analytically.

The interplay between topology and analysis through de Rham theory extends to Hodge theory on Riemannian manifolds, where harmonic forms provide canonical representatives for cohomology classes, connecting to spectral geometry and mathematical physics.