Ray-Singer torsion of compact 3-manifolds with finite abelian fundamental group

Question

Is the Ray-Singer analytic torsion for an arbitrary compact 3-manifold with finite Abelian fundamental group equivalent to the Ray-Singer analytic torsion of S^3 mod some direct product of Z_N's? It seems like Thurston's elliptization conjecture implies the answer is yes, but my understanding in this area is very limited. I am a physicist so please forgive me if I am being imprecise with terminology. References would also be very helpful.

Answer 1

A good reference for elliptic 3-manifolds is Thurston's book, "Three Dimensional Geometry and Topology, Vol 1." Theorem 4.4.14 classifies the possible fundamental groups of elliptic 3-manifolds. In particular, if the group is Abelian then the manifold is a lens space. To be explicit, this means the manifold is homeomorphic to the quotient $\{(w,z)| w,z \in \mathbb{C},|w|^2+|z|^2=1\}/ (w,z) \tilde{} (\zeta_p w, \zeta_p ^q z)$ where $\zeta_p$ is a primitive pth root of unity and $p,q$ are relatively prime. Also, in this case, the lens space denoted by $L(p,q)$.

It seems that Charles Nash and D. J. O’ Connor's paper, Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces and the Riemann zeta function computes the Ray-Singer analytic torsion for lens spaces of the form $L(p,1)$.