Advertisement

Lefschetz Theory on Manifolds with Singularities

  • Vladimir Nazaikinskii
  • Boris Sternin
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

The semiclassical method in Lefschetz theory is presented and applied to the computation of Lefschetz numbers of endomorphisms of elliptic complexes on manifolds with singularities. Two distinct cases are considered, one in which the endomorphism is geometric and the other in which the endomorphism is specified by Fourier integral operators associated with a canonical transformation. In the latter case, the problem includes a small parameter and the formulas are (semiclassically) asymptotic. In the first case, the parameter is introduced artificially and the semiclassical method gives exact answers. In both cases, the Lefschetz number is the sum of contributions of interior fixed points given (in the case of geometric endomorphisms) by standard formulas plus the contribution of fixed singular points. The latter is expressed as a sum of residues in the lower or upper half-plane of a meromorphic operator expression constructed from the conormal symbols of the operators involved in the problem.

Keywords

Lefschetz number singular manifold elliptic operator Fourier integral operator semiclassical method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M.F. Atiyah and R. Bott, A Lefschetz Fixed Point Formula for Elliptic Complexes I. Ann. of Math. 86 (1967), 374–407.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    S. Lefschetz, L’analysis situs et la géométrie algébrique. Gauthier-Villars, Paris, 1924.Google Scholar
  3. [3]
    V. Nazaikinskii, B.-W. Schulze, B. Sternin, and V. Shatalov, The Atiyah-Bott-Lefschetz Fixed Point Theorem for Manifolds with Conical Singularities. Ann. Global Anal. Geom. 17:5 (1999), 409–439.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    V. Nazaikinskii, B.-W. Schulze, and B. Sternin, A Semiclassical Quantization on Manifolds with Singularities and the Lefschetz Formula for Elliptic Operators. Preprint No. 98/19, Univ. Potsdam, Institut für Mathematik, Potsdam, 1998.Google Scholar
  5. [5]
    V. Nazaikinskii, Semiclassical Lefschetz Formulas on Smooth and Singular Manifolds. Russ. J. Math. Phys. 6:2 (1999), 202–213.zbMATHMathSciNetGoogle Scholar
  6. [6]
    B. Sternin and V. Shatalov, Atiyah-Bott-Lefschetz Fixed Point Theorem in Symplectic Geometry. Dokl. Akad. Nauk 348:2 (1996), 165–168.zbMATHMathSciNetGoogle Scholar
  7. [7]
    B. Sternin and V. Shatalov, The Fixed Point Lefschetz Theorem for Quantized Canonical Transformations. Funktsional. Anal. i Prilozhen. 32:4 (1998), 35–48.zbMATHMathSciNetGoogle Scholar
  8. [8]
    B. Sternin and V. Shatalov, Quantization of Symplectic Transformations and the Lefschetz Fixed Point Theorem. Preprint No. 92, Max-Planck-Institut für Mathematik, Bonn, 1994.Google Scholar
  9. [9]
    V.P. Maslov, Perturbation Theory and Asymptotic Methods. Moscow State University, Moscow, 1965.Google Scholar
  10. [10]
    V.P. Maslov, Operator Methods. Nauka, Moscow, 1973.zbMATHGoogle Scholar
  11. [11]
    A. Mishchenko, V. Shatalov, and B. Sternin, Lagrangian Manifolds and the Maslov Operator. Springer-Verlag, Berlin-Heidelberg, 1990.zbMATHGoogle Scholar
  12. [12]
    V. Nazaikinskii, B. Sternin, and V. Shatalov, Contact Geometry and Linear Differential Equations. Walter de Gruyter, Berlin-New York, 1992.Google Scholar
  13. [13]
    V. Nazaikinskii, B. Sternin, and V. Shatalov, Methods of Noncommutative Analysis. Theory and Applications. Walter de Gruyter, Berlin-New York, 1995.zbMATHGoogle Scholar
  14. [14]
    B.V. Fedosov, Trace Formula for Schrödinger Operator. Russ. J. Math. Phys. 1:4 (1993), 447–463.zbMATHMathSciNetGoogle Scholar
  15. [15]
    B. Sternin and V. Shatalov, Ten Lectures on Quantization. Preprint No. 22, Universit é de Nice-Sophia Antipolis, Nice, 1994.Google Scholar
  16. [16]
    B.-W. Schulze, Elliptic Complexes on Manifolds with Conical Singularities. In: Seminar Analysis of the Karl Weierstraß Institute 1986/87, number 106 in Teubner Texte zur Mathematik, Teubner, Leipzig, 1988. Pages 170–223.Google Scholar
  17. [17]
    B.-W. Schulze, Pseudodifferential Operators on Manifolds with Singularities. North-Holland, Amsterdam, 1991.Google Scholar
  18. [18]
    B.A. Plamenevskii, Algebras of Pseudodifferantial Operators. Kluwer, Dordrecht, 1989.Google Scholar
  19. [19]
    B.-W. Schulze, B. Sternin, and V. Shatalov, Differential Equations on Singular Manifolds. Semiclassical Theory and Operator Algebras. Wiley-VCH, Berlin-New York, 1998.zbMATHGoogle Scholar
  20. [20]
    M. Agranovich and M. Vishik, Elliptic Problems with Parameter and Parabolic Problems of General Type. Uspekhi Mat. Nauk 19:3 (1964), 53–161.zbMATHGoogle Scholar
  21. [21]
    M.A. Shubin, Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin-Heidelberg, 1985.zbMATHGoogle Scholar
  22. [22]
    M.S. Agranovich, B.Z. Katsenelenbaum, A.N. Sivov, and N.N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory. Wiley-VCH, Berlin, 1999.Google Scholar
  23. [23]
    A. Martinez, Microlocal Exponential Estimates and Applications to Tunneling. In: Microlocal Analysis and Spectral Theory, Kluwer, 1997. Pages 349–376.Google Scholar
  24. [24]
    L. Hörmander, The Spectral Function of an Elliptic Operator. Acta Math. 121 (1968), 193–218.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    R. Melrose, Transformation of Boundary Problems. Acta Math. 147 (1981), 149–236.zbMATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    V. Nazaikinskii, B.-W. Schulze, B. Sternin, and V. Shatalov, Quantization of Symplectic Transformations on Manifolds with Conical Singularities. Preprint No. 97/23, Univ. Potsdam, Institut für Mathematik, Potsdam, 1997.Google Scholar
  27. [27]
    V. Nazaikinskii, B.-W. Schulze, B. Sternin, and V. Shatalov, A Lefschetz Fixed Point Theorem on Manifolds with Conical Singularities. Preprint No. 97/20, Univ. Potsdam, Institut für Mathematik, Potsdam, 1997.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Vladimir Nazaikinskii
    • 1
  • Boris Sternin
    • 2
  1. 1.Institute for Problems in MechanicsRussian Academy of SciencesMoscowRussia
  2. 2.Independent University of MoscowMoscowRussia

Personalised recommendations