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Torsion, as a Function on the Space of Representations

  • Dan Burghelea
  • Stefan Haller
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex-valued Ray-Singer torsion, the Milnor-Turaev torsion, and the dynamical torsion. They are associated essentially to a closed smooth manifold equipped with a (co)Euler structure and a Riemannian metric in the first case, a smooth triangulation in the second case, and a smooth flow of type described in Section 2 in the third case. In this paper we define these functions, describe some of their properties and calculate them in some case. We conjecture that they are essentially equal and have analytic continuation to rational functions on the variety of representations. We discuss what we know to be true. As particular cases of our torsion, we recognize familiar rational functions in topology such as the Lefschetz zeta function of a diffeomorphism, the dynamical zeta function of closed trajectories, and the Alexander polynomial of a knot. A numerical invariant derived from Ray-Singer torsion and associated to two homotopic acyclic representations is discussed in the last section.

Keywords

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

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Dan Burghelea
    • 1
  • Stefan Haller
    • 2
  1. 1.Dept. of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of MathematicsUniversity of ViennaViennaAustria

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