Losik classes for codimension-one foliations

TL;DR

This paper explores characteristic classes for codimension-one foliations derived from Gelfand-Fuchs cohomology, revealing conditions for their non-triviality and their relation to dynamical properties and geometric structures.

Contribution

It introduces new characteristic classes for foliations based on Losik's approach, showing their non-triviality in cases where classical classes like Godbillon-Vey are trivial, and links their properties to dynamical systems.

Findings

Non-triviality of classes for Reeb foliations.

Classes are trivial for large classes of holonomy-free foliations.

Classes serve as obstructions to certain geometric structures.

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 non-triviality in terms of dynamical properties of generators of the holonomy groups are found. The non-triviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class, e.g, 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.