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arXiv · 1504.01096

Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis

Peter Kramer
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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.

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Peter Kramer. 2015-04-05. Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis. https://doi.org/10.3390/sym7020305

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