De Rham's Theorem

De Rham's theorem establishes the fundamental bridge between differential geometry and algebraic topology, proving that de Rham cohomology -- defined analytically via differential forms -- is isomorphic to singular cohomology -- defined purely topologically.

Statement

Theorem5.1De Rham's theorem

For any smooth manifold $M$ , there is a natural isomorphism

H^k_{\mathrm{dR}}(M) \cong H^k_{\mathrm{sing}}(M; \mathbb{R})

for all $k \geq 0$ , where $H^k_{\mathrm{sing}}(M; \mathbb{R})$ denotes singular cohomology with real coefficients. The isomorphism is given by integration: a closed $k$ -form $\omega$ maps to the singular cochain $\sigma \mapsto \int_\sigma \omega$ , where $\sigma$ is a smooth singular $k$ -simplex.

The Integration Map

Definition5.8De Rham homomorphism

Define the de Rham map $I: \Omega^k(M) \to C^k_{\mathrm{sing}}(M; \mathbb{R})$ by

I(\omega)(\sigma) = \int_\sigma \omega

for each smooth singular $k$ -simplex $\sigma: \Delta^k \to M$ . By Stokes' theorem, $I(d\omega)(\sigma) = \int_\sigma d\omega = \int_{\partial \sigma} \omega = I(\omega)(\partial \sigma)$ , so $I$ is a cochain map: $I \circ d = \delta \circ I$ , where $\delta$ is the singular coboundary operator. Hence $I$ induces a map on cohomology.

Proof Outline

Proof

The proof proceeds in several steps.

Step 1: Presheaf setup. Both de Rham cohomology $H^k_{\mathrm{dR}}$ and singular cohomology $H^k_{\mathrm{sing}}(\cdot; \mathbb{R})$ are contravariant functors from smooth manifolds to real vector spaces. The integration map $I$ gives a natural transformation between them.

Step 2: Agreement on convex sets. For any convex open subset $U \subset \mathbb{R}^n$ , the Poincare lemma gives $H^k_{\mathrm{dR}}(U) = 0$ for $k \geq 1$ and $\mathbb{R}$ for $k = 0$ . By the same computation in singular cohomology (using the cone construction), $H^k_{\mathrm{sing}}(U; \mathbb{R})$ has the same values. So $I$ is an isomorphism on convex sets.

Step 3: Mayer-Vietoris comparison. Both theories admit Mayer-Vietoris long exact sequences for open covers $M = U \cup V$ , and the integration map $I$ commutes with all maps in the sequence (restriction, difference, connecting homomorphism). By the five lemma, if $I$ is an isomorphism on $U$ , $V$ , and $U \cap V$ , then it is an isomorphism on $M$ .

Step 4: Induction on a good cover. A good cover of $M$ is an open cover where all finite intersections are either empty or diffeomorphic to $\mathbb{R}^n$ (hence convex in local coordinates). Every smooth manifold admits a good cover. By induction on the number of open sets in the cover, using Mayer-Vietoris at each step, the isomorphism extends from convex sets to all of $M$ . $\blacksquare$

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Significance

RemarkTopological invariance of de Rham cohomology

De Rham's theorem implies that de Rham cohomology, although defined using the smooth structure, is actually a topological invariant: homeomorphic manifolds have isomorphic de Rham cohomology. This is not obvious from the definition, which involves smooth forms.

ExampleConcrete applications

De Rham's theorem allows computation of topological invariants (Betti numbers, Euler characteristic) using calculus. Conversely, it shows that topological constraints (e.g., $H^1(S^2) = 0$ ) have analytical consequences (every closed 1-form on $S^2$ is exact).

RemarkGeneralizations

De Rham's theorem generalizes in many directions: equivariant de Rham theory for group actions, de Rham cohomology with coefficients in flat bundles, the Dolbeault theorem relating $\bar{\partial}$ -cohomology to sheaf cohomology on complex manifolds, and the Hodge theorem relating harmonic forms to cohomology on Riemannian manifolds.