Non-diffeomorphic Reeb foliations and modified Godbillon-Vey class

The paper deals with a modified Godbillon-Vey class defined by Losik for codimension-one foliations. This characteristic class takes values in the cohomology of the second order frame bundle over the leaf space of the foliation. The definition of the Reeb foliation depends upon two real functions sa...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2022-02, Vol.300 (2), p.1335-1349
Main Authors: Bazaikin, Yaroslav V., Galaev, Anton S., Gumenyuk, Pavel
Format: Article
Language:English
Subjects:
Homology
Mathematics
Mathematics and Statistics
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Summary:The paper deals with a modified Godbillon-Vey class defined by Losik for codimension-one foliations. This characteristic class takes values in the cohomology of the second order frame bundle over the leaf space of the foliation. The definition of the Reeb foliation depends upon two real functions satisfying certain conditions. All these foliations are pairwise homeomorphic and have trivial Godbillon-Vey class. We show that the modified Godbillon-Vey is non-trivial for some Reeb foliations and it is trivial for some other Reeb foliations. In particular, the modified Godbillon-Vey class can distinguish non-diffeomorphic foliations and it provides more information than the classical Godbillon-Vey class. We also show that this class is non-trivial for some foliations on the two-dimensional surfaces.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-021-02828-1