Euler structures, the variety of representationsand the Milnor--Turaev torsion

In this paper we extend, and Poincaré dualize, the concept of Euler structures, introduced by Turaev for manifolds with vanishing Euler– Poincaré characteristic, to arbitrary manifolds. We use the Poincaré 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 the modified Ray–Singer torsion when regarded as a function on the variety of complex representations of the fundamental group of the manifold is actually the absolute value of a rational function which we call in this paper the Milnor–Turaev torsion. As a consequence one obtains for a closed manifold M and an acyclic unimodular representation of its fundamental group ρ, a computable numerical invariant S(M,ρ) with value in R/πZ cf. section 4.4. Some examples and applications are also discussed, see sections 4.3, 4.5, and 4.6.