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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?

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    $\begingroup$ Are there examples of smooth foliations on a 3-manifold M that are topologically equivalent but not diffeomorphic? $\endgroup$
    – Daniel Asimov
    Commented Aug 1, 2023 at 2:05
  • $\begingroup$ I was thinking of M compact, and I assume this question is, too. $\endgroup$
    – Daniel Asimov
    Commented Aug 1, 2023 at 2:24
  • $\begingroup$ @DanielAsimov Dear Prof. Asimov, Thank you for your attention to my question and your edit. I think a smooth invariant of a Reeb foliation of $S^3$ is the derivative of Holonomy map defined on a 1 dimensional transversal. So If we introduce two different Reeb foliations $F_1, F_2 $ with different $p_1'(0), p_2'(0)$, the derivative of Poincare or Holonomy map so they are not smooth equivalent. $\endgroup$
    – Ali Taghavi
    Commented Aug 1, 2023 at 8:32
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    $\begingroup$ Ali Taghavi: Re the comment mentioning the action of SL(2,ℝ): That should be SL(2,ℤ) instead. $\endgroup$
    – Daniel Asimov
    Commented Aug 1, 2023 at 15:44
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    $\begingroup$ I now see that it is easy to find two topologically equivalent, oriented smooth foliations on the 2-torus that are not smoothly equivalent: Omitting details: Consider two (1,0) unit vector fields V_1, V_2 — each with two closed orbits — with V_1, V_2 having holonomy maps with distinct derivatives. Then multiplying the torus and each leaf by S^1 gives such examples of smooth codimension-1 foliations on the 3-torus. $\endgroup$
    – Daniel Asimov
    Commented Aug 1, 2023 at 18:51

1 Answer 1

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Check out:

Hilsum, Michel, Functions with bounded variation and the class of Godbillon-Vey, Q. J. Math. 66, No. 2, 547-562 (2015). ZBL1401.57040.

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    $\begingroup$ Thank you very much for your very helpful answer and very intetesting paper a partial solutions to a problem raised by E. Ghys $\endgroup$
    – Ali Taghavi
    Commented Aug 9, 2023 at 6:38
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    $\begingroup$ @AliTaghavi maybe you could update your question with a reference to Ghys’ paper where the question was posed $\endgroup$
    – Ian Agol
    Commented Aug 9, 2023 at 9:36
  • $\begingroup$ yes that is a good idea. I will update it ASAP $\endgroup$
    – Ali Taghavi
    Commented Aug 9, 2023 at 11:56
  • $\begingroup$ I did updated the question. Thank you for your suggestion $\endgroup$
    – Ali Taghavi
    Commented Aug 9, 2023 at 20:38

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