  For the abelian Chern-Simons field theory, we consider the quantum functional
integration over the Deligne-Beilinson cohomology classes and we derive the
main properties of the observables in a generic closed orientable 3-manifold.
We present an explicit path-integral non-perturbative computation of the
Chern-Simons links invariants in the case of the torsion-free 3-manifolds
$S^3$, $S^1 \times S^2$ and $S^1 \times \Sigma_g$.
