the de Rham theorem
/ duh-RAHM /
The de Rham theorem is the bridge between calculus and topology. It says that the holes you detect analytically, with differential forms and the operator d, are exactly the same holes you detect combinatorially, with cycles and triangulations. Two completely different-looking ways of measuring a manifold's shape — one from smooth analysis, one from topology — give the identical answer. This is one of the most satisfying unifications in geometry.
Precisely: for a smooth manifold M, integration of forms over smooth chains defines a natural map from de Rham cohomology to singular cohomology with real coefficients, and the de Rham theorem states this map is an isomorphism: H^k_dR(M) is isomorphic to H^k(M; R) for every k, and moreover the isomorphism respects the ring structure (the wedge product on the left matches the cup product on the right). The map is concrete: a closed form omega is sent to the functional that integrates omega over each smooth k-cycle; Stokes' theorem is exactly what makes this well-defined on cohomology (changing omega by an exact form, or the cycle by a boundary, changes the integral by zero). The pairing 'integrate a closed form over a cycle' is the heart of the statement, an instance of the period pairing.
Its significance is that it lets you compute a topological invariant (singular cohomology) by doing analysis (solving d omega = 0 and dividing by exact forms), and conversely to bring topological tools to bear on differential equations and geometry. The crucial caveat, and the most common error: the isomorphism is with REAL coefficients only. de Rham cohomology cannot see torsion; H^k(M; Z) may have finite cyclic summands (as for RP^2 or the Klein bottle) that vanish entirely after tensoring with R. So the de Rham theorem identifies the real (torsion-free) part precisely and says nothing about the integral torsion. State it as H^k_dR(M) congruent to H^k(M; R), never to H^k(M; Z).
On the torus T^2, the de Rham generators of H^1 are the forms d theta_1 and d theta_2; the singular generators are the two basis loops a, b of pi_1. The de Rham isomorphism is realized by the period matrix: integral over a of d theta_1 = 2 pi, integral over a of d theta_2 = 0, integral over b of d theta_1 = 0, integral over b of d theta_2 = 2 pi. The nonsingular pairing between forms and loops is exactly the isomorphism H^1_dR(T^2) congruent to H^1(T^2; R), both R^2.
The period pairing 'integrate forms over cycles' is the de Rham isomorphism made concrete; on T^2 it is the 2x2 period matrix.
The isomorphism is with H^k(M; R), the real-coefficient cohomology — never with integral cohomology. de Rham theory is structurally incapable of seeing torsion, so it gives the Betti numbers but not the finer integral invariants; for those you must leave this field for singular or simplicial cohomology with Z coefficients.