We introduce the (2+1)-spacetimes with compact space of genus g and with r
gravitating particles which arise by ``Minkowskian suspensions of flat or
hyperbolic cone surfaces'', by ``distinguished deformations'' of hyperbolic
suspensions and by ``patchworking'' of suspensions. Similarly to the
matter-free case, these spacetimes have nice properties with respect to the
canonical Cosmological Time Function. When the values of the masses are
sufficiently large and the cone points are suitably spaced, the distinguished
deformations of hyperbolic suspensions determine a relevant open subset of the
full parameter space; this open subset is homeomorphic to the product of an
Euclidean space of dimension 6g-6+2r with an open subset of the Teichm\"uller
Space of Riemann surfaces of genus g with r punctures. By patchworking of
suspensions one can produce examples of spacetimes which are not distinguished
deformations of any hyperbolic suspensions, although they have the same masses;
in fact, we will guess that they belong to different connected components of
the parameter space.Comment: 14 pages Late
