Geometry & Topology 10 (2006) 1185–1238

DOI: 10.2140/gt.2006.10.1185

arXiv: math.DG/0310154

Abstract

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 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.

Keywords

Euler structure, co-Euler structure, combinatorial torsion, analytic torsion, theorem of Bismut–Zhang, Chern–Simons theory, geometric regularization, mapping torus, rational function

Mathematical Subject Classification

Primary: 57R20

Secondary: 58J52

References

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Publication

Received: 16 December 2005
Revised: 27 March 2006
Accepted: 23 July 2006
Published: 16 September 2006
Proposed: Wolfgang Lück
Seconded: Bill Dwyer, Haynes Miller

Authors