  A quantum deformation of 3-dimensional lattice gauge theory is defined by
applying the Reshetikhin-Turaev functor to a Heegaard diagram associated to a
given cell complex. In the root-of-unity case, the construction is carried out
with a modular Hopf algebra. In the topological (weak-coupling) limit, the
gauge theory partition function gives a 3-fold invariant, coinciding in the
simplicial case with the Turaev-Viro one. We discuss bounded manifolds as well
as links in manifolds. By a dimensional reduction, we obtain a q-deformed gauge
theory on Riemann surfaces and find a connection with the algebraic
Alekseev-Grosse-Schomerus approach.
