I will present the Turaev-Viro construction of an invariant of
3-manifolds. It is based
on a state sum associated to a triangulation of this manifold. This is
a sum over all
isomorphism class of simple object of a modular category. Then I will
say how one can modify this construction to fit with the context of
nilpotent representations of $U_q sl(2)$:
The problem is the existence of infinitly many non-isomorphic simple
representations.
The solution is to select some of these representations using a
cohomology class in
$H^1(M,C^*)$. Finally I will say how it leads to an invariant with
values in some
localization of the group ring $Z[H_1(M,Z)]$.