Topology of compact space forms from Platonic solids. I.
This is the first in a series of two papers where the authors determine the topological types of all constant curvature 3-manifolds that can be glued from a Platonic solid. Here a Platonic solid in the spherical or hyperbolic 3-space is a convex polyhedron with the combinatorics and the symmetry group of a Platonic solid in \({\mathbb R}^3\). A complete list of geometric manifolds from Platonic solids was obtained in the works of \textit{B. Everitt} [Topology Appl. 138, No. 1--3, 253--263 (2004; Zbl 1048.57008)], \textit{L. A. Best} [Can. J. Math. 23, 451--460 (1971; Zbl 0214.50304)], \textit{P. J. Lorimer} [Pac. J. Math. 156, No.~2, 329--335 (1992; Zbl 0770.57011)], \textit{I. Prok} [Beitr. Algebra Geom. 39, No.~2, 497--515 (1998; Zbl 0926.52021)], and \textit{J. Richardson} and \textit{J. H. Rubinstein} [Hyperbolic manifolds from a regular polyhedron, preprint]. In the present paper, all spherical and Euclidean manifolds under question are identified as lens spaces or Seifert manifolds. A theorem describing topological types of the hyperbolic manifolds is stated; its proof is postponed to a forthcoming paper.
scientific article; zbMATH DE number 5526279
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