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. 

>What are some updates on this conjecture?