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

Details Euler structures, the variety of representations and the Milnor-Turaev torsion Euler structures, the variety of representations and the Milnor-Turaev torsion

2003-10-10

arxiv.org/abs/math/0310154v7

Geom. Topol. 10 (2006) 1185-1238

Mathematics

Differential Geometry

This is the version published by Geometry & Topology on 16 September 2006

Scientific paper

10.2140/gt.2006.10.1185

In 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.

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