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G&T Monographs |
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Euler structures, the variety of representations and the
Milnor–Turaev torsion
Dan Burghelea and Stefan Haller
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Geometry & Topology 10 (2006)
1185–1238
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Abstract
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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.
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Keywords
Euler structure, co-Euler structure,
combinatorial torsion, analytic torsion, theorem of
Bismut–Zhang, Chern–Simons theory, geometric
regularization, mapping torus, rational function
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Mathematical Subject Classification
Primary: 57R20
Secondary: 58J52
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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
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