Suppose we have a 3 manifold $M$ with a codimension-1 foliation tangent to $\alpha=0$ for a 1-form $\alpha$. Then $\alpha \wedge d\alpha =0$, and $\exists$ a 1-form $\beta $ with $d\alpha =\alpha \wedge \beta$. The Godbillon-Vey invariant of the foliation is defined as the cohomology class of $\beta \wedge d\beta$. This class is independent of the choices made, at least up to sign.

The conjecture about this invariant is the following:

Conjecture: The Godbillon-Vey invariant is a topological invariant. That is, two topologically equivalent foliations have the same Godbillon-Vey class.

What are some updates on this conjecture?