de Rham cohomology
/ duh-RAHM /
de Rham cohomology is a way to count the holes of a smooth manifold using only calculus — no triangulations, no combinatorics, just differential forms and the operator d. Its slogan is that closed forms that are not exact detect holes: a form with zero derivative that nevertheless is not itself a derivative is registering some global feature the manifold has. The k-th de Rham cohomology measures, in a precise sense, the independent k-dimensional holes.
Precisely, on a smooth manifold M the de Rham complex is the chain of spaces Omega^0(M) -> Omega^1(M) -> Omega^2(M) -> ... linked by the exterior derivative d, with d^2 = 0. The k-th de Rham cohomology is the quotient H^k_dR(M) = (closed k-forms) / (exact k-forms) = ker(d on Omega^k) / image(d on Omega^{k-1}). Because exact forms are closed (d^2 = 0) the quotient makes sense, and two closed forms are cohomologous if they differ by an exact form. These are real vector spaces; their dimensions are the Betti numbers b_k. The wedge product descends to a product on cohomology, making H*_dR(M) = direct sum of H^k_dR(M) a graded-commutative ring, the de Rham cohomology ring.
It is a powerful invariant: it is homotopy invariant (homotopy-equivalent manifolds have isomorphic de Rham cohomology), computable by the Mayer-Vietoris sequence, and pairs with cycles via integration thanks to Stokes. Its values encode connectivity (b_0 counts components), loops, and higher holes. The essential honesty caveat: de Rham cohomology computes cohomology with REAL coefficients only. It is completely blind to torsion — the finite-order phenomena that integral cohomology sees (for example RP^2 has 2-torsion in H_1 with Z coefficients, but its de Rham cohomology in degree 1 is just 0). Never say de Rham 'is' the cohomology; it is the real-coefficient shadow, made precise by the de Rham theorem.
For the circle S^1: H^0_dR = R (one component, the constants) and H^1_dR = R, generated by the angle form d theta which is closed but not exact. So b_0 = b_1 = 1 — the single 1-dimensional hole of the circle. For the 2-sphere S^2: H^0 = R, H^1 = 0 (no 1-holes, simply connected), H^2 = R (the area form, the 2-dimensional cavity). For the 2-torus T^2: H^0 = R, H^1 = R^2 (the two independent loops d theta_1, d theta_2), H^2 = R.
de Rham cohomology of S^1, S^2, T^2 reads off their holes; the dimensions are the Betti numbers.
de Rham cohomology has REAL coefficients and cannot see torsion — it agrees with singular cohomology tensored with R, not the full integral theory. So it never detects a Klein bottle's or RP^2's torsion classes; for those you need integral singular or simplicial cohomology (a different field).