Canonical Structure of Locally Homogeneous Systems on Compact Closed 3-Manifolds of Types E³, Nil and Sol

In this paper we investigate the canonical structure of diffeomorphism invariant phase spaces for spatially locally homogeneous spacetimes with 3-dimensional compact closed spaces. After giving a general algorithm to express the diffeomorphism-invariant phase space and the canonical structure of a locally homogeneous system in terms of those of a homogeneous system on a covering space and a moduli space, we completely determine the canonical structures and the Hamiltonians of locally homogeneous pure gravity systems on orientable compact closed 3-spaces of the Thurston-type $E^3$, $\Nil$ and $\Sol$ for all possible space topologies and invariance groups. We point out that in many cases the canonical structure becomes degenerate in the moduli sectors, which implies that the locally homogeneous systems are not canonically closed in general in the full diffeomorphism-invariant phase space of generic spacetimes with compact closed spaces.