Platonic polyhedra tune the 3-sphere: III. Harmonic analysis on octahedral spherical 3-manifolds

TL;DR

This paper develops harmonic analysis bases for three octahedral spherical 3-manifolds using group theory, enabling potential cosmic topology studies through CMB analysis.

Contribution

It constructs explicit harmonic analysis bases for octahedral spherical 3-manifolds using group representations, linking topology with harmonic analysis.

Findings

Constructed H-invariant polynomial bases for three manifolds

Provided tools for analyzing cosmic microwave background data

Linked topology of 3-manifolds with harmonic analysis methods

Abstract

From the homotopy groups of three distinct octahedral spherical 3-manifolds we construct the isomorphic groups H of deck transformations acting on the 3-sphere. The H-invariant polynomials on the 3-sphere constructed by representation theory span the bases for the harmonic analysis on three spherical manifolds. Analysis of the Cosmic Microwave Background in terms of these new bases can reveal a non-simple topology of the space part of space-time.