TOPOLOGY OF COMPACT SPACE FORMS FROM PLATONIC SOLIDS. II

The problem of classifying, up to isometry, the orientable 3-manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004) 253–263]. His work completes the classification begun by Best [L.A. Best, On torsion-free discrete subgroups of PSL2(C) with compact orbit space, Canad. J. Math. 23 (1971) 451–460], Lorimer [P.J. Lorimer, Four dodecahedral spaces, Pacific J. Math. 156 (2) (1992) 329–335], Prok [I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (2) (1998) 497–515; I. Prok, Fundamental tilings with marked cubes in spaces of constantcurvature, Acta Math. Hungar. 71 (1–2) (1996) 1–14], and Richardson and Rubinstein [J. Richardson, J.H. Rubinstein, Hyperbolic manifolds from a regular polyhedron, preprint].In a previous paper we investigated the topology of closed orientable 3-manifolds from Platonic solids in the spherical and Euclidean cases, and completely classified them, upto homeomorphism. Here we describe many topological properties of closed hyperbolic 3-manifolds arising from Platonic solids. As a consequence of our geometric and topologicalmethods, we improve the distinction between the hyperbolic “Platonic” manifolds with the same homology, which up to this point was only known by computational means.