Topology of compact space forms from Platonic solids. I.
type: Item
instance of: scholarly article
title: Topology of compact space forms from Platonic solids. I.
zbMATH Open document ID: 1165.57014
author: Alberto Cavicchioli
author: Fulvia Spaggiari
author: A. I. Telloni
published in: Topology and its Applications
publication date: 2009-03-06T00:00:00Z
review text: 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.
reviewed by: Ivan Izmestiev
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 5526279
zbMATH Keywords: 3-manifolds
zbMATH Keywords: Platonic solids
zbMATH Keywords: spines
zbMATH Keywords: branched coverings
zbMATH Keywords: knots
MaRDI profile type: Publication
full work available at URL: j.topol.2008.10.015
OpenAlex ID: W4212818395
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DOI: 10.1016/J.TOPOL.2008.10.015
description: scientific article; zbMATH DE number 5526279