the Hodge decomposition
/ HOJ /
Take any differential form on a compact manifold. The Hodge decomposition says you can always split it cleanly into three orthogonal pieces: an 'exact' part (the derivative of something), a 'coexact' part (the co-derivative of something), and a 'harmonic' part that is the topologically essential remainder. It is the form-valued analogue of writing a vector field as a gradient plus a curl plus a harmonic remainder — the Helmholtz decomposition that physicists know — and it is what makes harmonic forms into canonical representatives of cohomology.
Precisely, on a compact oriented Riemannian manifold the space of smooth k-forms decomposes as an L^2-orthogonal direct sum Omega^k = (image of d) + (image of d^*) + (harmonic k-forms), that is Omega^k = d Omega^{k-1} (+) d^* Omega^{k+1} (+) H^k, where H^k = kernel of the Hodge Laplacian. The three summands are mutually orthogonal in the L^2 inner product. The payoff is the Hodge theorem: a closed form (d omega = 0) is the sum of an exact form and a unique harmonic form, so the map sending a harmonic form to its cohomology class is an isomorphism H^k (harmonic) ~= H^k_dR(M). Concretely, to find the harmonic representative of a de Rham class you take any closed form in it and subtract off the exact part d(something) chosen to make the result coclosed — the leftover is the unique norm-minimizing, metric-dependent harmonic representative. This is how a topological invariant (a Betti number) gets computed by solving an elliptic equation.
Hodge decomposition is one of the great unifications in geometry: it makes de Rham cohomology concrete, it is the foundation of Hodge theory on Kahler manifolds (where it refines further by (p,q)-type), and it powers the Bochner technique and index theory. The honest caveats are real. The decomposition needs compactness — on noncompact or incomplete manifolds the three-way splitting can fail or require L^2 and boundary conditions, and there are several inequivalent 'L^2 Hodge theories' in that setting. The harmonic representative depends on the chosen metric (the cohomology class does not), so 'the' harmonic form is metric-relative. And the cohomology it computes is real (or complex) de Rham cohomology: Hodge theory is blind to the torsion that integral cohomology sees, so it never recovers the full integral cohomology of a space.
On a compact surface a 1-form omega splits as df + d^*(g dA) + h, where df is the gradient part, the middle is the 'rotational' part, and h is harmonic; the dimension of the harmonic piece is 2g (twice the genus), recovering the first Betti number purely from the surface's shape.
On a genus-g surface the harmonic 1-forms form a 2g-dimensional space, equal to b_1.
Hodge theory computes only real/complex cohomology; it is blind to torsion, so a claim like 'Hodge gives the full cohomology of the space' is wrong whenever integral torsion is present — de Rham and singular agree only over a field.