  A thorough classification of the topologies of compact homogeneous universes
is given except for the hyperbolic spaces, and their global degrees of freedom
are completely worked out. To obtain compact universes, spatial points are
identified by discrete subgroups of the isometry group of the generalized
Thurston geometries, which are related to the Bianchi and the
Kantowski-Sachs-Nariai universes. Corresponding to this procedure their total
degrees of freedom are shown to be categorised into those of the universal
covering space and the Teichm\"uller parameters. The former are given by
constructing homogeneous metrics on simply connected manifold. The
Teichm\"uller spaces are also given by explicitly constructing expressions for
the discrete subgroups of the isometry group.
