Suppose we have a compact three-manifold $M$, a codimension-one foliation $F$ of $M$, and a one-form on $\alpha$, chosen so that $F$ is tangent to $\alpha = 0$. We deduce that $\alpha \wedge d\alpha = 0$. Also, there is some one-form $\beta$ so that $d\alpha = \alpha \wedge \beta$.


The *Godbillon-Vey invariant* of $F$ is defined as the cohomology class of $\beta \wedge d\beta$. This class is independent of the choices made (at least up to sign).


Then we have the following question which is raised by Etienne Ghys:


**Conjecture:** The Godbillon-Vey invariant is a topological invariant.


See the paper É. Ghys, L’invariant de Godbillon–Vey, Astérisque (1989) 177–178, Exp. No. 706, Séminaire
Bourbaki, Vol. 1988/89, 155–181


That is, two topologically equivalent foliations have the same Godbillon-Vey class.



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The current situation of the Godbillon-Vey invariant conjecture
Check out:


*Hilsum, Michel*, [**Functions with bounded variation and the class of Godbillon-Vey**](https://doi.org/10.1093/qmath/hav013), Q. J. Math. 66, No. 2, 547-562 (2015). [ZBL1401.57040](https://zbmath.org/?q=an:1401.57040).


