Timeline for The current situation of the Godbillon-Vey invariant conjecture
Current License: CC BY-SA 4.0
33 events
| when toggle format | what | by | license | comment | |
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| Aug 9, 2023 at 20:37 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| S Aug 9, 2023 at 6:38 | history | bounty ended | Ali Taghavi | ||
| S Aug 9, 2023 at 6:38 | history | notice removed | Ali Taghavi | ||
| Aug 9, 2023 at 6:38 | vote | accept | Ali Taghavi | ||
| Aug 8, 2023 at 21:11 | answer | added | Ian Agol | timeline score: 4 | |
| S Aug 6, 2023 at 6:29 | history | bounty started | Ali Taghavi | ||
| S Aug 6, 2023 at 6:29 | history | notice added | Ali Taghavi | Draw attention | |
| Aug 5, 2023 at 10:47 | comment | added | Ali Taghavi | @SamNead thank you for the reference by Ghys. I think Daniel Asimov meant two smooth foliation which are not smooth equivalent but are topological equivalent | |
| Aug 4, 2023 at 6:12 | comment | added | Ali Taghavi | @DanielAsimov BTW i think that there are some works on Godbilon Vey invariant of Lie Groups by Robert Roussarie | |
| Aug 1, 2023 at 22:07 | comment | added | Ali Taghavi | @DanielAsimov May be there is a counterexample in the Novikov book or Toender book? Thank you for your chalenging question. (Existence of two foliation topological equivalent but not smooth equivalent) | |
| Aug 1, 2023 at 21:48 | comment | added | Ali Taghavi | @DanielAsimov I think I was mistaken, Sorry. Because an smooth equivalent need not preserve the derivative of Poincare return map. My mistake initiated from the following confusion: Two smooth equivalent singularity have conjugate(similar) linear part. So do you agree my argument and hence your argumenyt do not work? | |
| Aug 1, 2023 at 19:50 | comment | added | Ali Taghavi | @DanielAsimov yes this gives topological equivalent non smooth equivalent foliation. Thank you. | |
| Aug 1, 2023 at 18:51 | comment | added | Daniel Asimov | 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. | |
| Aug 1, 2023 at 18:29 | comment | added | Daniel Asimov | Never mind, I see what you mean by a 1-dimensional transversal. | |
| Aug 1, 2023 at 18:19 | comment | added | Daniel Asimov | I also do not understand "Holonomy map defined on a 1 dimensional transversal". Don't you need a closed curve along a leaf to get a holonomy map? | |
| Aug 1, 2023 at 17:24 | comment | added | Ali Taghavi | @DanielAsimov Sorry for my mistake | |
| Aug 1, 2023 at 17:15 | comment | added | Ali Taghavi | @DanielAsimov yes Thank you I was mistaken. It is inded $PSL(2,\mathbb{Z})$ | |
| Aug 1, 2023 at 15:44 | comment | added | Daniel Asimov | Ali Taghavi: Re the comment mentioning the action of SL(2,ℝ): That should be SL(2,ℤ) instead. | |
| Aug 1, 2023 at 9:45 | comment | added | Sam Nead | @Ali Taghavi - regarding your requested counterexample (immediately above): do you require both foliations be smooth? There are "easy" examples where one foliation is smooth and the other is not. See Figure 2 and Section 3 of Osculating curves: around the Tait-Kneser Theorem by Ghys, Tabachnikov, and Timorin. | |
| Aug 1, 2023 at 8:45 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Aug 1, 2023 at 8:45 | comment | added | Ali Taghavi | @SamNead Thank you for your edit | |
| Aug 1, 2023 at 8:44 | comment | added | Ali Taghavi | I would appreciate if you let me know a precise counter example of two topological equivalent but non smooth equivalent foliations if there exist any. | |
| Aug 1, 2023 at 8:42 | comment | added | Ali Taghavi | Two Kronecker foliations of tori with slops $\alpha, \beta$ are not topological equiivalent if $\alpha , \beta$ do not lie on the same orbits of the action of $SL(2,\mathbb{R}$ via action $\pmatrix{a&b\\c&d}.z=\frac{az+b}{cz+d}$. Now Consider the product foliations by $F_1\times S^1$ and $F_2\times S^1$. In this way, can we obtain two topological equivalent foliation of $\mathbb{T}^3$ which are not smoothly equivalent? | |
| Aug 1, 2023 at 8:37 | comment | added | Ali Taghavi | As anotnter her possible counter example I would like to pose the following question: | |
| Aug 1, 2023 at 8:36 | comment | added | Ali Taghavi | @DanielAsimov On the other hand I am thinking to some other possible counter example: First We know that there are examples which show that Hartman Grobman is valid topologicaly but not in C^1 class(Around singularity). So I wonder can we introduce a vector field around singularity at $0\in \mathbb{R}^4$ which is topologically linearaizable but not smoothly. Then after a Blow up of singularity can we obtain two 0n1 dimensional foliation of $S^3$ which are topological equivalent but not smooth equivalent? | |
| Aug 1, 2023 at 8:32 | comment | added | Ali Taghavi | @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. | |
| Aug 1, 2023 at 7:09 | history | edited | Sam Nead | CC BY-SA 4.0 |
typesetting, formatting
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| Aug 1, 2023 at 2:24 | comment | added | Daniel Asimov | I was thinking of M compact, and I assume this question is, too. | |
| Aug 1, 2023 at 2:05 | comment | added | Daniel Asimov | Are there examples of smooth foliations on a 3-manifold M that are topologically equivalent but not diffeomorphic? | |
| Aug 1, 2023 at 2:02 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Added "at least up to sign"
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| Aug 1, 2023 at 1:52 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Fixed multiple infelicities in the language of this question.
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| Jul 31, 2023 at 13:28 | history | edited | Ali Taghavi | CC BY-SA 4.0 |
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| Jul 31, 2023 at 9:26 | history | asked | Ali Taghavi | CC BY-SA 4.0 |