Topology of compact space forms from Platonic solids. II

Abstract

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 PSL 2 ( 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 constant curvature, 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, up to homeomorphism. Here we describe many topological properties of closed hyperbolic 3-manifolds arising from Platonic solids. As a consequence of our geometric and topological methods, we improve the distinction between the hyperbolic “Platonic” manifolds with the same homology, which up to this point was only known by computational means.