Lefschetz distribution of Lie foliations

2007/03/26 by Jesus A. Alvarez Lopez, Lopez, Jesus A. Alvarez, Yuri A. Kordyukov +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.math/0703753

40 pages, LaTeX 2e

arxiv created 2007/03/26 · arxiv updated 2009/12/01

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 H(\mathcal F). 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.

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