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Open AccessArticle

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
Symmetry 2015, 7(2), 305-326; https://doi.org/10.3390/sym7020305
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)
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
MDPI and ACS Style

Kramer, P. Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis. Symmetry 2015, 7, 305-326. https://doi.org/10.3390/sym7020305

AMA Style

Kramer P. Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis. Symmetry. 2015; 7(2):305-326. https://doi.org/10.3390/sym7020305

Chicago/Turabian Style

Kramer, Peter. 2015. "Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis" Symmetry 7, no. 2: 305-326. https://doi.org/10.3390/sym7020305

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