On deformations of foliations and characteristic classes

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.