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.
Part of this work was done while both authors enjoyed the hospitality of the Max Planck Institute for Mathematics in Bonn. A previous version was written while the second author enjoyed the hospitality of the Ohio State University. The second author was partially supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austrian Science Fund), project number P17108-N04.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry. II, Math. Proc. Cambridge Philos. Soc. 78(1975), 405–432.
J.M. Bismut and W. Zhang, An extension of a theorem by Cheeger and Müller, Astérisque 205, Société Mathématique de France, 1992.
R. Bott and L.W. Tu, Differential forms in algebraic topology. Graduate texts in Mathematics 82. Springer Verlag, New York-Berlin, 1982.
M. Braverman and T. Kappeler, Refined analytic torsion, preprint math.DG/0505537.
D. Burghelea, Removing Metric Anomalies from Ray-Singer Torsion, Lett. Math. Phys. 47(1999), 149–158.
D. Burghelea, L. Friedlander and T. Kappeler, Relative Torsion, Commun. Contemp. Math. 3(2001), 15–85.
D. Burghelea and S. Haller, Euler structures, the variety of representations and the Milnor-Turaev torsion, Geom. and Topol. 10(2006) 1185–1238.
D. Burghelea and S. Haller, Laplace transform, dynamics and spectral geometry (version 1), to appear in Journal of Topology, Vol 1, 2008 preprint math.DG/0405037.
D. Burghelea and S. Haller, Dynamics, Laplace transform, and spectral geometry (version 2), preprint math.DG/0508216.
D. Burghelea and S. Haller, Complex valued Ray-Singer torsion, J. Funct. Anal. Vol 248 (2007) 27–78. preprint math.DG/0604484.
D. Burghelea and S. Haller, Complex valued Ray-Singer torsion II, preprint math.DG/0604484.
D. Burghelea and S. Haller, Dynamical torsion, a non-commutative version of a theorem of Hutchings-Lee and Pajitnov, in preparation.
S. Dineen, Complex Analysis in Locally Convex Spaces. Mathematics Studies 57, North-Holland, 1981.
M. Farber, Singularities of the analytic torsion, J. Differential Geom. 41(1995), 528–572.
M. Farber and V. Turaev, Poincaré-Reidemeister metric, Euler structures and torsion, J. Reine Angew. Math. 520(2000), 195–225.
R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics 52. Springer Verlag, New York-Heidelberg, 1977.
M. Hutchings, Reidemeister torsion in generalized Morse Theory, Forum Math. 14(2002), 209–244.
B. Jiang, Estimation of the number of periodic orbits, Pac. J. Math. 172(1996), 151–185.
A. Kriegl and P.W. Michor, The convenient setting of global analysis. Mathematical Surveys and Monographs 53, American Mathematical Society, Providence, RI, 1997.
J. Marcsik, Analytic torsion and closed one forms. Thesis, Ohio State University, 1995.
V. Mathai and D. Quillen, Superconnections, Thom Classes, and Equivariant differential forms, Topology 25(1986), 85–110.
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72(1966), 358–426.
W. Müller, private communication.
L. Nicolaescu, The Reidemeister Torsion of 3-manifolds. De Gruyter Studies in Mathematics 30, Walter de Gruyter & Co., Berlin, 2003.
D. Quillen, Determinants of Cauchy-Riemann operators on Riemann surfaces, Functional Anal. Appl. 19(1985), 31–34. Translation of Funktsional. Anal. i Prilozhen. 19(1985), 37–41.
D.B. Ray and I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7(1971), 145–210.
N. Steenrod, The topology of fiber bundles. Reprint of the 1957 edition. Princeton University Press, Princeton, NJ, 1999.
G. Su and W. Zhang, A Cheeger-Müller theorem for symmetric bilinear torsion, preprint math.DG/0610577.
V. Turaev, Reidemeister torsion in Knot theory, Uspekhi Mat. Nauk 41(1986), 119–182.
V. Turaev, Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Math. USSR-Izv. 34(1990), 627–662.
V. Turaev, Introduction to combinatorial torsion. Notes taken by Felix Schlenk. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2001.
V. Turaev, Torsion of 3-dimensional manifolds. Progress in Mathematics 208. Birkhäuser Verlag, Basel-Boston-Berlin, 2002.
G.W. Whitehead, Elements of homotopy theory. Graduate Texts in Mathematics 61. Springer Verlag, New York-Berlin, 1978.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Burghelea, D., Haller, S. (2008). Torsion, as a Function on the Space of Representations. In: Burghelea, D., Melrose, R., Mishchenko, A.S., Troitsky, E.V. (eds) C*-algebras and Elliptic Theory II. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8604-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-7643-8604-7_2
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-8603-0
Online ISBN: 978-3-7643-8604-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)