2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221689In this paper we extend and Poincare dualize the concept of Euler structures, introduced by Turaev for manifolds with vanishing Euler-Poincare characteristic, to arbitrary manifolds. We use the Poincare dual concept, co-Euler structures, to remove all geometric ambiguities from the Ray-Singer torsion by providing a slightly modified object which is a topological invariant. We show that when the co-Euler structure is integral then the modified Ray-Singer torsion when regarded as a function on the variety of generically acyclic complex representations of the fundamental group of the manifold is the absolute value of a rational function which we call in this paper the Milnor-Turaev torsion.This is the version published by Geometry & Topology on 16 September 2006Differential GeometryGeometric Topology57R20, 58J52text