  We generalize the concept of global moment maps to local moment maps, whose
different branches are labelled by the elements of the fundamental group of the
underlying symplectic manifold. These branches can be smoothly glued together
by employing fundamental-group-valued \u Cech cocycles on the phase space. In
the course of this work we prove a couple of theorems on the liftability of
group actions to symplectic covering spaces, and examine the possible
extensions of the original group by the fundamental group of the quotient phase
space. It it shown how the splitting of multiplets, this being a consequence of
the multiply-connectedness of the quotient phase space, can be described by
identification maps on a space of multiplets derived from a symplectic
universal covering manifold. The states that are identified in this process are
related by certain integrals over non-contractible loops in the quotient phase
space.
