Comparison of approaches to characteristic classes of foliations

TL;DR

This paper compares different approaches to defining characteristic classes of foliations, showing how classes from Losik's framework relate to those in Čech-de Rham cohomology, with applications to codimension-one foliations.

Contribution

It establishes a mapping between Losik's characteristic classes and Čech-de Rham classes, elucidating their relationship and limitations within foliation theory.

Findings

The map from Losik's classes to Čech-de Rham classes is generally non-injective.

Similar relationships hold for exotic characteristic classes and diffeomorphism group actions.

Examples include detailed analysis of codimension-one foliations.

Abstract

It is shown that the characteristic classes of foliations that were defined by Losik and that take values in the de~Rham cohomology of the space of infinite order frames over the leaf space may be mapped to the characteristic classes with values in the \v{C}ech-de~Rham cohomology of the leaf space studied in details by Crainic and Moerdijk. This map is in general non-injective. All constructions are done using Losik's approach to Gelfand formal geometry. A similar result is obtained for the exotic characteristic classes as well as for the group actions of the diffeomorphisms. As illustrating examples, foliations of codimension one are discussed.