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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 
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© 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

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