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Lefschetz Distribution of Lie Foliations

  • Jesús A. Álvarez López
  • Yuri A. Kordyukov
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Let \( \mathcal{F} \) be a Lie foliation on a closed manifold M with structural Lie group G. Its transverse Lie structure can be considered as a transverse action Φ of G on (M,\( \mathcal{F} \)); i.e., an “action” which is defined up to leafwise homotopies. This Φ induces an action Φ* of G on the reduced leafwise cohomology \( \bar H\left( \mathcal{F} \right) \). By using leafwise Hodge theory, the supertrace of Φ* can be defined as a distribution L dis(\( \mathcal{F} \)) on G called the Lefschetz distribution of \( \mathcal{F} \). A distributional version of the Gauss-Bonett theorem is proved, which describes L dis(\( \mathcal{F} \) ) around the identity element. On any small enough open subset of G, L dis(\( \mathcal{F} \)) is described by a distributional version of the Lefschetz trace formula.

Keywords

Lie foliation Riemannian foliation leafwise reduced cohomology distributional trace Lefschetz distribution Λ-Euler characteristic Λ-Lefschetz number Lefschetz trace formula 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Jesús A. Álvarez López
    • 1
  • Yuri A. Kordyukov
    • 2
  1. 1.Departamento de Xeometría e Topoloxía Facultade de MatemáticasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Institute of MathematicsRussian Academy of SciencesUfaRussia

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