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Symmetry 2015, 7(2), 305-326; doi:10.3390/sym7020305

Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis

Institut für Theoretische Physik der Universität Tübingen, Tübingen 72076, Germany
Academic Editor: Louis H. Kauffman
Received: 3 January 2015 / Revised: 14 March 2015 / Accepted: 19 March 2015 / Published: 31 March 2015
(This article belongs to the Special Issue Diagrams, Topology, Categories and Logic)
View Full-Text   |   Download PDF [315 KB, uploaded 31 March 2015]   |  

Abstract

We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology, the three-sphere is tiled into copies of a fundamental domain under the corresponding deck group. We employ the point symmetry of each Platonic manifold to construct its fundamental domain as a spherical orbifold. While the three-sphere supports an orthonormal complete basis for harmonic analysis formed by Wigner polynomials, a given spherical orbifold leads to a selection of a specific subbasis. The resulting selection rules find applications in cosmic topology, probed by the cosmic microwave background. View Full-Text
Keywords: topology of Platonic three-manifolds; spherical orbifolds; harmonic analysis on orbifolds topology of Platonic three-manifolds; spherical orbifolds; harmonic analysis on orbifolds
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Kramer, P. Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis. Symmetry 2015, 7, 305-326.

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