DISTRIBUTIONAL BETTI NUMBERS OF TRANSITIVE FOLIATIONS OF CODIMENSION ONE

  1. LÓPEZ, JESÚS A. ÁLVAREZ 2
  2. KORDYUKOV, YURI A. 1
  1. 1 Department of Mathematics, Ufa State Aviation Technical University, 12 K. Marx str., 450025 Ufa, Russia
  2. 2 Departamento de Xeometría e Topoloxía, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain
Actas:
Foliations: Geometry and Dynamics

ISBN: 978-981-4489-70-6

Ano de publicación: 2002

Tipo: Achega congreso

DOI: 10.1142/9789812778246_0007 GOOGLE SCHOLAR lock_openAcceso aberto editor

Resumo

Let be a transitive foliation of codimension one on a closed manifold M. This means that there is an infinitesimal transformation X of transverse to the leaves. The flow of X induces an ℝ-action on the reduced leafwise cohomology . By using leafwise Hodge theory, the trace of this action on each can be defined as a distribution on ℝ, which is called distributional Betti number because it is kind of a finite measure of the "size" of . So the corresponding distributional Euler characteristic, is a distribution on ℝ too. This is relevant because may be of infinite dimension, even when the leaves are dense, and its Euler characteristic makes no sense in general. The singularity at 0 of is expressed in terms of the Connes' Λ-Euler characteristic, where Λ is the holonomy invariant transverse measure of induced by the volume form dt on ℝ. Moreover the whole of is computed by showing a dynamical Lefschetz formula.