Abstract
LetF be a flat vector bundle over a compact Riemannian manifoldM and letf :M → ℝ be a Morse function. Letg F be a smooth Euclidean metric onF, letg F t =e −2tf g F, and letρ RS (t) be the Ray-Singer analytic torsion ofF associated with the metricg F t . Assuming that ∇f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for logρ RS (t) fort→+∞ of the forma 0+a 1 t+blog(t/π)+o(1), where the coefficientb is a half-integer depending only on the Betti numbers ofF. In the case where all the critical values off are rational, we calculate the coefficientsa 0 anda 1 explicitly in terms of the spectral sequence of a filtration associated with the Morse function. These results are obtained as applications of a theorem by Bismut and Zhang.
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References
J.-M. Bismut, H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles, I, Comm. Math. Phys. 115 (1988), 49–78.
J.-M. Bismut, W. Zhang, An Extension of a Theorem by Cheeger and Müller, Astérisque 205, 1992.
J.-M. Bismut, W. Zhang, Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle, GAFA 4 (1994), 136–212.
R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, 1982.
M. Braverman, Witten deformation of Ray-Singer analytic torsion, Preprint.
D. Burghelea, L. Friedlander, T. Kappeler, Mayer-Vietoris type formula for determinants of elliptic differential operators, Journal of Funct. Anal. 107 (1992), 34–65.
D. Burghelea, L. Friedlander, T. Kappeler, Analytic torsion equals Reidemeister torsion, a new proof, Preprint (1992), Ohio State University.
D. Burghelea, L. Friedlander, T. Kappeler, Asymptotic expansion of the Witten deformation of the analytic torsion, preprint, 1994.
J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. 109 (1979), 259–300.
X. Dai, R.B. Melrose, Adiabatic limit of the analytic torsion, Preprint.
M. Farber, Singularities of analytic torsion, to appear in J. of Diff. Geom.
R. Forman, Hodge theory and spectral sequences, Topology 33 (1994), 591–611.
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966.
P. Kirk, E. Klassen, The spectral flow of the odd signature operators and higher Massey products, Preprint.
F.F. Knudsen, D. Mumford, The projectivity of the moduli spaces of stable curves, I: Preliminaries on “det” and “div”, Math. Scand. 39 (1976), 19–55.
V. Mathai, D. Quillen, Superconnections, Thom class, and equivariant differential forms, Topology 25 (1986), 85–110.
J. Milnor, Whitehead torsion, Bull. AMS 72 (1966), 358–426
J. Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. 76 (1962), 137–147.
W. Müller, Analytic torsion and R-torsion on Riemannian manifolds, Adv. in Math. 28 (1978), 233–305.
W. Müller, Analytic torsion and R-torsion for unimodular representations, Jour. of AMS 6 (1993), 721–753.
D. Ray, I. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145–210.
F. Rellich, Perturbation Theory of Eigenvalue Problems, Gordon and Breach, 1969.
R. Seeley, Complex powers of elliptic operators, Proc. Symp. Pure and Appl. Math. AMS 10 (1967), 288–307.
S. Smale, On gradient dynamical systems, Ann. of Math. 74 (1961), 199–206.
S. Smale, Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967), 747–817.
M. Shubin, Pseudodifferential Operators and Spectral Theory, Springer Verlag, Berlin, New York, 1980.
R. Thom, Sur une partition en cellules associée à une fonction sur une variété, C.R. Acad. Sc. Paris, Série A, 228 (1949), 661–692.
E. Witten, Supersymmetry and Morse Theory, J. of Diff. Geom. 17 (1982), 661–692.
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The research was supported by grant No. 449/94-1 from the Israel Academy of Sciences and Humanities.
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Braverman, M. Witten deformation of analytic torsion and the spectral sequence of a filtration. Geometric and Functional Analysis 6, 28–50 (1996). https://doi.org/10.1007/BF02246766
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DOI: https://doi.org/10.1007/BF02246766