Analytic torsion on spherical factors and tessellations

TL;DR

This paper computes the analytic torsion on spherical space forms and tessellations, introducing an improved spectral technique that simplifies calculations and reveals algebraic number results, including for icosahedral space with quaternion twisting.

Contribution

It presents an enhanced method for computing analytic torsion on spherical factors, extending Ray's technique and applying it to complex tessellations with singularities.

Findings

Analytic torsion values are algebraic numbers.

Quaternion twisting in icosahedral space relates torsion to fundamental units.

Torsions on tessellations include contributions from edge conical singularities.

Abstract

The analytic torsion is computed on fixed-point free and non fixed-point free factors (tessellations) of the three--sphere. We repeat the standard computation on spherical space forms (Clifford-Klein spaces) by an improved technique. The transformation to a simpler form of the spectral expression of the torsion on spherical factors effected by Ray is shown to be more general than his derivation implies. It effectively allows the eigenvalues to be considered as squares of integers, and applies also to trivial twistings. The analytic torsions compute to algebraic numbers, as expected. In the case of icosahedral space, the quaternion twisting gives a torsion proportional to the fundamental unit of Q(5^(1/2)). As well as a direct calculation, the torsions are obtained from the lens space values by a character inducing procedure.On tessellations, terms occur due to edge conical singularities.