Topology of Platonic Spherical Manifolds: From Homotopy to Harmonic Analysis

TL;DR

This paper performs harmonic analysis on four Platonic spherical 3-manifolds, linking their topological properties to spectral bases and exploring implications for cosmic microwave background studies.

Contribution

It develops a method to convert homotopies into deck operations, constructing specific harmonic bases for each manifold's topology, and applies these to cosmic topology.

Findings

Derived deck groups for each Platonic manifold

Constructed harmonic bases specific to each topology

Identified selection rules relevant for cosmic microwave background analysis

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.