Let M be a closed manifold, ρ a representation of π 1 (M ) on an A-Hilbert model W of finite type (A a finite von Neumann algebra) and µ a Hermitian structure on the flat bundle E → M associated to ρ. The relative torsion, first introduced by Carey, Mathai and Mishchenko, [CMM], associates to any pair (g, τ ), consisting of a Riemannian metric g on M and a generalized triangulation τ = (h, g ′ ), a numerical invariant, R(M, ρ, µ, g, τ ).Unlike the analytic torsion T an , associated to (M, ρ, µ, g), or the Reidemeister torsion T Reid , associated to F = (M, ρ, µ, g, τ ), which are defined only when the pair (M, ρ) is of determinant class, R is always defined and when (M, ρ) is of determinant class is equal to the quotient T an /T Reid . The purpose of this paper is to prove the following Theorem:Theorem (i) There exists a density α F on M \ Cr(h), which is a local quantity so that if µ is parallel in a smooth neighborhood of Cr(h) then α F vanishes on the neighborhood of the critical points and log R = M\Cr(h) α F . (ii) If µ is parallel then R = 1.An exact formula for R is also provided. This theorem can be viewed as an extension of our result [BFKM] which says that in the case where (M, ρ) is of determinant class and µ is parallel then R = 1.
