De Rham cohomology and the integration pairing

Let M be a Hausdorff second-countable smooth manifold without boundary. Its smooth real differential forms form a complex with exterior derivative d and $d^2=0$. Define $H^k_{\mathrm{dR}}(M)$ as closed k-forms modulo exact k-forms. A closed form has derivative zero; an exact form is the derivative of a globally defined smooth form one degree lower. Local primitives supplied by the Poincaré lemma do not automatically patch into a global primitive, and this difference is precisely what the quotient records.

The construction does not require M to be compact or oriented. Orientation is needed for integrating top forms over an entire oriented manifold, but each smooth singular simplex already carries its own standard orientation for integration. Pullback along smooth maps commutes with exterior derivative and therefore induces maps backward on de Rham cohomology. Wedge products descend to a graded ring with the expected degree signs, linking this differential complex to the algebraic cohomology operations studied earlier.

For a smooth singular k-simplex $\sigma$, assign a k-form $\omega$ the number $\int_{\Delta^k}\sigma^*\omega$. Stokes' theorem says that evaluating dω on a higher simplex equals evaluating ω on its oriented boundary, matching the cochain differential convention used earlier. Hence a closed form gives a cocycle, and an exact form gives a coboundary. Changing a cycle by a boundary does not change the period of a closed form. Changing a form by an exact one does not change its period on a cycle.

The inclusion of smooth singular chains into ordinary continuous singular chains induces the same homology for smooth manifolds, via approximation results. De Rham's theorem then identifies the resulting real cohomology map as an isomorphism; on cohomology it also respects wedge and cup products. Multiplicativity is a theorem on classes, not a claim that the elementary simplex-integration cochain map strictly multiplies all cochains. A proof uses local Poincaré lemmas and a global gluing argument, rather than only checking Stokes on one simplex.

Over the reals, cohomology evaluates as a linear functional on real homology, so all periods determine a de Rham class. A torsion integral cycle has zero period for a closed real form: a nonzero integer multiple bounds, and Stokes makes that multiple of its period zero in the real numbers. Likewise finite-order integral cohomology classes vanish after passage to real coefficients. A real calculation may therefore be complete for de Rham cohomology while missing genuine integral information detected by Ext or other coefficient systems.

The period test applies only after closedness has been checked. A nonclosed form can have a nonzero integral around a contractible loop because its exterior derivative contributes over the filling. Conversely a local expression that looks like a differential may fail to come from a single-valued global function. The angular coordinate on a punctured plane is the standard warning: its differential is globally defined as a one-form, while the angle itself requires branches. A primitive written on one branch does not establish global exactness.