and some graphical data for the resulting determinants and torsion. The torsion studied in this paper has its origins in the 1930’s, cf. Franz \[1\], where it was combinatorially defined and used to distinguish various lens spaces from one another. Given a manifold $M$ and a representation of its fundamental group $\pi_1(M)$ in a flat bundle $E$, this Reidemeister-Franz torsion is a real number which is defined as a particular product of ratio’s of volume elements $V^i$ constructed from the cohomology groups $H^i(M;E)$.

Since volume elements are essentially determinants then, for any alternative definition of a determinant, an alternative definition of the torsion can be given. Now if one uses de Rham cohomology to compute $H^i(M;E)$ then these determinants become determinants of Laplacians $\Delta^E_p$ on $p$-forms with coefficients in $E$. But zeta functions for elliptic operators can be used to give finite values to such infinite dimensional determinants and so an analytic definition of the torsion results and this is the analytic torsion of Ray and Singer \[2,3,4\] given in the 1970’s; furthermore this torsion was proved by them to be independent of the Riemannian metric used to define the Laplacian’s $\Delta^E_p$.

This analytic torsion coincided, for the case of 