Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces and the Riemann zeta function

TL;DR

This paper derives explicit formulas for Laplacian determinants and Ray-Singer torsion on lens spaces, linking these to the Riemann zeta function and analyzing their growth with the fundamental group's size.

Contribution

It provides new explicit expressions for Laplacian determinants and torsion on lens spaces, and connects these to special values of the Riemann zeta function.

Findings

Determinants grow with the size of the fundamental group.

Explicit formulas for Ray-Singer torsion on lens spaces.

Identifies lens spaces with trivial torsion.

Abstract

We obtain explicit expressions for the determinants of the Laplacians on zero and one forms for an infinite class of three dimensional lens spaces $L(p,q)$. These expressions can be combined to obtain the Ray-Singer torsion of these lens spaces. As a consequence we obtain an infinite class of formulae for the Riemann zeta function $\zeta(3)$. The value of these determinants (and the torsion) grows as the size of the fundamental group of the lens space increases and this is also computed. The triviality of the torsion for just the three lens spaces $L(6,1)$, $L(10,3)$ and $L(12,5)$ is also noted. (postscript figures available as a compressed tar file)