Abstract
For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of π1 (M) on a finite dimensional vector space to a representation on aA-Hilbert moduleW of finite type whereA is a finite von Neumann algebra. If (M,W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, theL 2-analytic andL 2-Reidemeister torsions are equal.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Astérisque 32–33 (1976), 43–72.
J.P. Bismut, W. Zhang, An extension of a theorem by Cheeger and Müller, Astérisque 205 (1992), 1–223.
J.P. Bismut, W. Zhang, Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, GAFA 4 (1994), 136–212.
L. Boutet de Monve, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11–51.
D. Burghelea, L. Friedlander, T. Kappeler, Asymptotic expansion of the Witten deformation of the analytic torsion, J. Funct. Anal., to appear.
D. Burghelea, L. Friedlander, T. Kappeler, Mayer-Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107 (1992), 34–66.
D. Burghelea, L. Friedlander, T. Kappeler, Torsions for manifolds with boundary and glueing formulas, preprint.
A.L. Carey, V. Mathai,L 2-torsion invariants, J. Funct. Anal. 110 (1992), 377–409.
J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. 109 (1979), 259–300.
J. Cohen, Von Neumann dimension and the homology of covering spaces, Quart. J. of Math. Oxford 30 (1970), 133–142.
H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators, Text and Monographs in Physics, Springer Verlag, 1987.
J. Dixmier, Von Neumann Algebras, North-Holland, Amsterdam, 1981.
J. Dodziuk, De Rham-Hodge theory forL 2 cohomology of infinite coverings, Topology 16 (1977), 157–165.
A.V. Efremov, Combinatorial and analytic Novikov-Shubin invariants, preprint.
A.V. Efremov, Cellular decomposition and Novikov-Shubin invariants, Russ. Math. Surveys 46 (1991), 219–222.
A.T. Fomenko, A.S. Miscenko, The index of elliptic operators overC*-algebras, Math. USSR Izvestija 15 (1980), 87–112.
L. Friedlander, The asymptotic of the determinant function for a class of operators, Proc. AMS 107 (1989), 169–178.
B. Fuglede, R.V. Kadison, Determinant theory in finite factors, Ann. of Math. 55 (1952), 520–530.
P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish, Wilmington, 1984.
D. Gong,L 2-analytic Torsions, Equivariant Cyclic Cohomology and the Novikov Conjecture, Ph.D. Thesis S.U.N.Y (Stony Brook), 1992.
M. Gromov, M.A. Shubin, Von Neumann spectra near zero, GAFA 1 (1991), 375–404.
B. Helffer, J. Sjöstrand, Puits multiples en mécanique semi-classique, IV Etude du complexe de Witten, Comm. in PDE 10 (1985), 245–340.
B. Helffer, J. Sjöstrand, Multiple wells in the semi-classical limit I, Comm. in PDE 9 (1984), 337–408.
L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer Verlag, New York, 1985.
Y. Lee, Mayer-Vietoris type formula for determinants, preprint.
S. Levendorskii, Degenerate Elliptic Equations, Kluwer Academic Publishers, Dordrecht, 1993.
J. Lott, Heat kernels on covering spaces and topological invariants, J. of Diff. Geo. 35 (1992), 471–510.
J. Lott, W. Lück,L 2-topological invariants of 3-manifolds, preprint.
W. Lück, Analytic and algebraic torsion for manifolds with boundary and symmetries, J. of Diff. Geo. 37 (1993), 263–322.
W. Lück,L 2-torsion and 3-manifolds, preprint.
W. Lück, ApproximatingL 2-invariants by their finite dimensional analogues, GAFA 4 (1994), 455–481.
W. Lück, M. Rothenberg, Reidemeister torsion and the K-theory of von Neumann algebras, K-theory 5 (1991), 213–264.
G. Luke, Pseudodifferential operators on Hilbert bundles, J. of Diff. Equ. 12 (1972), 566–589.
V. Mathai,L 2 analytic torsion, J. of Funct. Anal. 107 (1992), 369–386.
J. Milnor, Whitehead torsion, Bull. AMS 72 (1966), 358–426.
J. Milnor, Lectures onh-cobordism Theorem, Princeton University Press, 1965.
N. Mohamma, Algebra of pseudodifferentialC*-operators, Rendiconti Inst. Mat. Trieste 20 (1988), 203–214.
W. Müller, Analytic torsion andR-torsion on Riemannian manifolds, Adv. in Math. 28 (1978), 233–305.
S.P. Novikov, M.A. Shubin, Morse theory and von NeumannII 1-factors, Dokl. Akad. Nauk SSSR 289 (1986), 289–292.
S.P. Novikov, M.A. Shubin, Morse theory and von Neumann invariants on non-simply connected manifold, Uspekhi Nauk 41 (1986), 222–223.
M. Poźniak, Triangulation of smooth compact manifolds and Morse theory, Warwick preprint 11 (1990).
D.B. Ray, I. Singer,R-Torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145–210.
R. Seeley, Complex powers of elliptic operators, Proc. Symp. Pure and Appl. Math. AMS 10 (1967), 288–307.
R. Seeley, Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math. 91 (1969), 963–983.
M. Shubin, Pseudodifferential Operators and Spectral Theory, Springer Verlag, New York, 1980.
M. Shubin, Pseudodifferential almost-periodic operators and von Neumann algebras, Trans. Moscow Math. Soc. 35 (1976), 103–166.
M. Shubin, Semiclassical asymptotics on covering manifolds and Morse inequalities, GAFA 6 (1996), 370–409.
I.M. Singer, Some remarks on operator theory and index theory, in “K-Theory and Operator Algebras”, Proceedings 1975, Springer Lecture Notes in Mathematics 575 (1977), 128–138.
A. Voros, Spectral function, special functions and Selberg zeta function, Comm. Math. Phys. 110 (1987), 439–465.
E. Witten, Supersymmetry and Morse theory, J. of Diff. Geom. 17 (1982), 661–692.
Author information
Authors and Affiliations
Additional information
The first three authors were supported by NSF. The first two authors wish to thank the Erwin-Schrödinger-Institute, Vienna, for hospitality and support during the summer of 1993 when part of this work was done.
Rights and permissions
About this article
Cite this article
Burghelea, D., Kappeler, T., McDonald, P. et al. Analytic and Reidemeister torsion for representations in finite type Hilbert modules. Geometric and Functional Analysis 6, 751–859 (1996). https://doi.org/10.1007/BF02246786
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02246786