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Mathematische Zeitschrift

, Volume 273, Issue 3–4, pp 1085–1117 | Cite as

On the gluing formula for the analytic torsion

Article

Abstract

In this paper, we derive the Cheeger–Müller/Bismut–Zhang theorem for manifolds with boundary and the gluing formula for the analytic torsion of flat vector bundles in full generality, i.e., we do not assume that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary.

Keywords

Vector Bundle Product Structure Canonical Isomorphism Morse Function Euler Class 
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References

  1. 1.
    Banyaga, A., Hurtubise, D.: Lectures on Morse homology. In: Kluwer Texts in the Mathematical Sciences, 29. Kluwer Academic Publishers Group, Dordrecht (2004)Google Scholar
  2. 2.
    Bismut J.-M., Gillet H., Soulé C.: Analytic torsion and holomorphic determinant bundles.I. Comm. Math. Phys. 115, 49–78 (1988)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bismut J.-M., Goette S.: Families torsion and Morse functions. Astérisque 275, 293 (2001)MathSciNetGoogle Scholar
  4. 4.
    Bismut J.-M., Zhang W.: An extension of a theorem by Cheeger and Müller. Astérisque 205, 236 (1992)Google Scholar
  5. 5.
    Bismut J.-M., Zhang W.: Milnor and Ray–Singer metrics on the equivariant determinant of a flat vector bundle. Geom. Funct. Anal. 4, 136–212 (1994)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Brüning, J., Ma, X.: An anomaly formula for Ray–Singer metrics on manifolds with boundary. Geom. Funct. Anal. 16, 767–837 (2006); announced in C.R. Math. Acad. Sci. Paris, 335, 603–608 (2002)Google Scholar
  7. 7.
    Cheeger J.: Analytic torsion and the heat equation. Ann. of Math. 109, 259–322 (1979)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Goette S.: Torsion invariants for families. Astérisque 328, 161–206 (2009)MathSciNetGoogle Scholar
  9. 9.
    Hassell A.: Analytic surgery and analytic torsion. Comm. Anal. Geom. 6, 255–289 (1998)MathSciNetMATHGoogle Scholar
  10. 10.
    Hartmann L., Spreafico M.: The analytic torsion of a cone over a sphere. J. Math. Pures Appl. (9) 93(4), 408–435 (2010)MathSciNetMATHGoogle Scholar
  11. 11.
    Lott J., Rothenberg M.: Analytic torsion for group actions. J. Diff. Geom. 34, 431–481 (1991)MathSciNetMATHGoogle Scholar
  12. 12.
    Lück W.: Analytic and topological torsion for manifolds with boundary and symmetry. J. Diff. Geom. 37, 263–322 (1993)MATHGoogle Scholar
  13. 13.
    Ma, X., Zhang, W.: An anomaly formula for L 2-analytic torsions on manifolds with boundary. In: Analysis, Geometry and Topology of Elliptic Operators. pp. 235–262. World Sci. Publ., Hackensack (2006)Google Scholar
  14. 14.
    Magnus, W., Oberhettinger, F., Soni, R.: Formulas and theorems for the special functions of mathematical physics. Third enlarged edition. (Die Grundlehren der mathematischen Wissenschaften, Band 52) Springer-Verlag New York, Inc., New York (1966)Google Scholar
  15. 15.
    Milnor J.: Whitehead torsion. Bull. Am. Math. Soc. 72, 358–426 (1966)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Müller W.: Analytic torsion and R-torsion of Riemannian manifolds. Adv. Math. 28, 233–305 (1978)MATHCrossRefGoogle Scholar
  17. 17.
    Müller W.: Analytic torsion and R-torsion for unimodular representations. JAMS 6, 721–753 (1993)MATHGoogle Scholar
  18. 18.
    Ray D.B., Singer I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. in Math. 7, 145–210 (1971)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Seeley R.T.: Analytic extension of the trace associated with elliptic boundary problems. Amer. J. Math. 91, 963–983 (1969)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Smale S.: Differentiable dynamical systems. Bull. Amer. Math. Soc. 73, 747–817 (1967)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Vertman B.: Refined analytic torsion on manifolds with boundary. Geom. Topol. 13(4), 1989–2027 (2009)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Vishik S.M.: Generalized Ray–Singer conjecture I: a manifold with a smooth boundary. Comm. Math. Phys. 167, 1–102 (1995)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.UFR de MathématiquesUniversité Paris Diderot-Paris 7Paris Cedex 13France

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