Laplacian

The Hodge Laplacian pairs a coboundary with its adjoint. On a cochain complex with whose spaces carry inner products, the coboundary has an adjoint , and

It is self-adjoint and positive semidefinite. Hodge decomposition splits each space orthogonally as with the harmonic part , so the kernel of the Laplacian is cohomology. Two data make it well-defined. A complex gives , and an inner product gives the adjoint ; without both there is no .

Example. De Rham

The prototype is the de Rham complex of differential forms on a closed Riemannian manifold. The metric and the Hodge star supply , so is the Hodge–de Rham Laplacian, and on functions () it is . Harmonic forms represent de Rham cohomology, the analytic content of Hodge theory.

Example. Discrete

On a graph the map is the signed incidence, a discrete gradient, and is the graph Laplacian. On a cellular-sheaf with Hilbertian stalks the restriction maps assemble a coboundary , and is the sheaf Laplacian, well-defined precisely when the stalks carry inner products and . A connection twists the same construction by parallel transport.