A theorem identifying the de Rham cohomology of a smooth manifold, defined via differential forms, with its singular (or simplicial) cohomology with real coefficients, via the integration pairing.

Principle

Integration of closed differential forms over singular chains induces an isomorphism between the cohomology of the differential complex of forms and the topological cohomology groups with real coefficients.

Demonstration

For the circle S^1 the space of closed 1-forms modulo exact forms is one-dimensional and integration around the loop gives an isomorphism with H^1(S^1; R) ≅ R; explicit primitives and period computations exhibit the correspondence.

Misapplication

Using the statement for non-smooth spaces, for coefficients other than the reals without adjustment, or conflating de Rham cohomology with cohomology with arbitrary coefficients leads to error; neglecting orientation or regularity assumptions invalidates the integration pairing.

Consequence

Enables computation of topological invariants using differential tools and elliptic theory; it shows that purely analytic objects (differential forms up to exactness) capture topological cohomology over R.

Reversal

Singular cohomology need not present differential representatives or analytic structure; the theorem does not endow arbitrary topological cohomology classes with canonical smooth forms unless the manifold is smooth.

Boundary

Requires a smooth manifold (or a suitable smooth structure); for singular spaces, orbifolds, or schemes one must use adapted theories (e.g., sheaf cohomology, distributional forms) and the classical de Rham isomorphism may fail.

Semantic Tension

Sits between analytic/differential descriptions and algebraic/topological invariants; tension arises when one attempts to generalize the isomorphism beyond smooth, real-coefficient contexts.

Synthesis

The de Rham theorem states that on a smooth manifold the cohomology computed from differential forms equals the topological real cohomology, realized concretely by integrating forms over cycles.