Topology of Platonic spherical manifolds: from homotopy to harmonic analysis (Q2406199)

Summary: We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies [\textit{B. Everitt}, Topology Appl. 138, No. 1--3, 253--263 (2004; Zbl 1048.57008)], 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.