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

Dan Burghelea and Stefan Haller

Geometry & Topology 10 (2006) 1185–1238

arXiv: math.DG/0310154


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.

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 2000
Primary: 57R20
Secondary: 58J52
Forward citations
Received: 16 December 2005
Revised: 27 March 2006
Accepted: 23 July 2006
Published: 16 September 2006
Proposed: Wolfgang Lück
Seconded: Bill Dwyer, Haynes Miller
Dan Burghelea
Dept. of Mathematics
The Ohio State University
231 West 18th Avenue
Columbus, OH 43210
Stefan Haller
Department of Mathematics
University of Vienna
Nordbergstrasse 15
A-1090, Vienna