Platonic polyhedra tune the 3-sphere: Harmonic analysis on simplices

TL;DR

This paper develops harmonic analysis methods on spherical manifolds derived from Platonic polyhedra, enabling the study of cosmic topology and the effects of multiply connected spaces on cosmic microwave background data.

Contribution

It generalizes harmonic analysis on simplicial spherical manifolds of dimensions 1 to 3, extending previous work on the Poincare dodecahedral space.

Findings

Harmonic analysis on simplicial spherical manifolds constructed from Platonic polyhedra.

Reduction of group representations reveals selection rules for harmonic functions.

Application to cosmic microwave background data for detecting cosmic topology.

Abstract

A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of reduction of group representations. A basis for the harmonic analysis on the (n-1)-sphere is given by the spherical harmonics which transform according to irreducible representations of the orthogonal group. The deck transformations form a subgroup, and so the representations of the orthogonal group can be reduced to those of this subgroup. Upon reducing to the identity representation of the subgroup, the reduced subset of spherical harmonics becomes periodic on the tiling and tunes the harmonic analysis on the (n-1)-sphere to the manifold. A particular class of spherical 3-manifolds arises from the Platonic polyhedra. The harmonic analysis on the Poincare dodecahedral 3-manifold was analyzed along these lines. For comparison we construct here the harmonic analysis on simplicial spherical manifolds of dimension n=1,2,3. Harmonic analysis applied to the cosmic microwave background by selection rules can provide evidence for multiply connected cosmic topologies.