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We introduce the (2 + 1)-spacetimes with compact space of genus g
greater than or equal to 0 and r gravitating particles which arise by
three kinds of construction called: (a) the Minkowskian suspension of
flat or hyperbolic cone surfaces; (b) the distinguished deformation of
hyperbolic suspensions; (c) the 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 UxR(6g-6+2r), where U is a non empty open set
of the Teichmuller space T-g(r). By patchworking of suspensions one can
produce examples of spacetimes which are not distinguished deformations
of any hyperbolic suspensions, although they have the same topology and
same masses; in fact, we will guess that they belong to different
connected components of the parameter space
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