A fundamental result asserting an isomorphism between the cohomology of the complex of smooth differential forms on a smooth manifold (de Rham cohomology) and the singular (or sheaf) cohomology with real coefficients; the isomorphism respects algebraic structure such as cup products.

Principle

Integration of differential forms over singular chains (or the map from the de Rham complex to a resolution of the constant sheaf) produces a chain map inducing an isomorphism on cohomology: closed forms represent topological cohomology classes and exact forms map to boundaries.

Demonstration

On the circle S1, the 1-form dθ is closed but not exact, producing a one-dimensional H1 de Rham cohomology; the de Rham classes correspond to the singular cohomology H1(S1; R) ≅ R, exhibiting the isomorphism concretely.

Misapplication

Applying the theorem indiscriminately to non-smooth topological spaces, singular algebraic varieties without passing to algebraic de Rham theory, or expecting an isomorphism with integer coefficients; also misusing it by ignoring necessary finiteness or orientability hypotheses when they matter for integrals.

Consequence

Topological invariants of smooth manifolds can be computed and studied using differential forms and analytic methods; the theorem underlies Hodge theory on Riemannian manifolds and supplies bridges between analysis, geometry, and topology.

Reversal

Viewed from the opposite perspective, singular cohomology provides a purely topological construction whose algebraic structure is realized analytically by differential forms; the reversal highlights two equivalent viewpoints—analytic versus combinatorial/topological.

Boundary

Applies to smooth manifolds (or smooth manifolds with boundary, with appropriate relative versions) and more generally to smooth paracompact manifolds or smooth manifolds of finite type; it does not hold as stated for arbitrary topological spaces or for coefficients in Z without extra structure.

Semantic Tension

Tension arises between the differential-analytic nature of de Rham cohomology and algebraic or étale versions (algebraic de Rham, étale cohomology): the statements are analogous but require different hypotheses and coefficient fields.

Synthesis

De Rham Theorem identifies the analytic cohomology of differential forms with topological cohomology for smooth spaces, enabling computation of topological invariants via calculus and anchoring deeper links such as Hodge theory.