On deformations of foliations and characteristic classes

TL;DR

This paper explores characteristic classes for foliation deformations, introduces a differential graded algebra framework, and reveals new classes beyond classical invariants, with implications for understanding foliation invariants.

Contribution

It introduces a new DGA framework for foliation characteristic classes, extending classical classes and analyzing their deformation properties.

Findings

Some classes cannot be described by classical invariants.

Non-trivial deformations produce new characteristic classes.

The cohomology of the DGA reveals novel structural properties.

Abstract

We study characteristic classes for deformations of foliations. Those classes include known classes such as the Godbillon--Vey class and the Fuks--Lodder--Kotschick class. We introduce a certain differential graded algebra (DGA for short) which recovers the Bott vanishing and some formulae by Heitsch. Some basic properties and structures of the cohomology of those DGA's are discussed. In particular, it is shown that at the level of the cohomology of DGA, there are some classes which cannot be described by the Godbillon--Vey class and the Fuks--Lodder--Kotschick class. It is also shown that if a certain type of characteristic classes admit non-trivial deformations in examples, then they yield another kind of classes which admit also non-trivial deformations.