Non-diffeomorphic Reeb foliations and modified Godbillon-Vey class

TL;DR

This paper explores a modified Godbillon-Vey class for codimension-one foliations, demonstrating its ability to distinguish non-diffeomorphic Reeb foliations and providing new insights beyond the classical class.

Contribution

It introduces a modified Godbillon-Vey class that can detect differences between Reeb foliations not distinguished by the classical class.

Findings

Modified Godbillon-Vey class is non-trivial for some Reeb foliations.

The class can distinguish non-diffeomorphic foliations.

It is non-trivial for certain foliations on 2D surfaces.

Abstract

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.