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In this paper we extend and Poincare dualize the concept of Euler structures,introduced by Turaev for manifolds with vanishing Euler-Poincarecharacteristic, to arbitrary manifolds. We use the Poincare dual concept,co-Euler structures, to remove all geometric ambiguities from the Ray-Singertorsion by providing a slightly modified object which is a topologicalinvariant. We show that when the co-Euler structure is integral then themodified Ray-Singer torsion when regarded as a function on the variety ofgenerically acyclic complex representations of the fundamental group of themanifold is the absolute value of a rational function which we call in thispaper the Milnor-Turaev torsion.Comment: This is the version published by Geometry & Topology on 16 September  200

Euler structures, the variety of representations and the Milnor–Turaev torsion