  We consider the $U(1)$ Chern-Simons gauge theory defined in a general closed
oriented 3-manifold $M$; the functional integration is used to compute the
normalized partition function and the expectation values of the link
holonomies. The nonperturbative path-integral is defined in the space of the
gauge orbits of the connections which belong to the various inequivalent $U(1)$
principal bundles over $M$; the different sectors of the configuration space
are labelled by the elements of the first homology group of $M$ and are
characterized by appropriate background connections. The gauge orbits of flat
connections, whose classification is also based on the homology group, control
the extent of the nonperturbative contributions to the mean values. The
functional integration is achieved in any 3-manifold $M$, and the corresponding
path-integral invariants turn out to be strictly related with the abelian
Reshetikhin-Turaev surgery invariants.
