  The notion of $q$-deformed lattice gauge theory is introduced. If the
deformation parameter is a root of unity, the weak coupling limit of a 3-$d$
partition function gives a topological invariant for a corresponding
3-manifold. It enables us to define the generalized Turaev-Viro invariant for
cell complexes. It is shown that this invariant is determined by an action of a
fundamental group on a universal covering of a complex. A connection with
invariants of framed links in a manifold is also explored. A model giving a
generating function of all simplicial complexes weighted with the invariant is
investigated.
