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LOSIK CLASSES FOR CODIMENSION-ONE FOLIATIONS

Part of: Global differential geometry Differential topology

Published online by Cambridge University Press:  08 January 2021

Yaroslav V. Bazaikin  and
Anton S. Galaev Open the ORCID record for Anton S. Galaev [Opens in a new window]
Yaroslav V. Bazaikin
Affiliation:
Sobolev Institute of Mathematics Novosibirsk, Russia and University of Hradec Králové, Faculty of Science, Rokitanského 62, 500 03 Hradec Králové, Czech Republic (bazaikin@math.nsc.ru)
Anton S. Galaev
Affiliation:
University of Hradec Králové, Faculty of Science, Rokitanského 62, 500 03 Hradec Králové, Czech Republic (anton.galaev@uhk.cz)
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Abstract

Following Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

Keywords

foliationleaf space of foliationcharacteristic classes of foliationGelfand-Fuchs cohomologyGodbillon-Vey-Losik classChern-Losik classDuminy-Losik classtransverse connectionfoliation almost without holonomydynamical systemsergodic theoryconjugacy of diffeomorphisms

MSC classification

Primary: 53C12: Foliations (differential geometric aspects) 57R32: Classifying spaces for foliations; Gelfand-Fuks cohomology
Secondary: 57R30: Foliations; geometric theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 4 , July 2022 , pp. 1391 - 1419
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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