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It is extensively used and cited the following statement due to Giroux:

Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and open book decompositions of $M$ up to positive stabilization.

Given such a contact structure, the existence of an open book supporting the contact structure is proven, for example, in Giroux's Géométrie de contact: de la dimension trois vers les dimensions supérieures.

I am asking for a complete proof of the uniqueness part of the result. At this point I would be surprised if somebody provided me with a link to a peer-reviewed paper containing a proof of the result (which is funny because this is widely acknowledged as a "theorem" inside and outside the field of contact geometry). Usual citations include the above paper (which does not contain a proof of the statement), some book that has been "in preparation" for years or even "transparencies from a seminar"!

So, links to detailed lecture notes or a proof itself will be appreciated. I know for example the existence of these Lectures on open book decompositions and contact structures by J. Etnyre, but they are somehow sketchy to my taste. I am not an expert in the field and I can't complete all the exercises left to the reader or fill in all the gaps in the "sketches of a proof".

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    $\begingroup$ According to John Etnyre, someone claimed that he has a written proof. But no-one saw it. $\endgroup$
    – Anubhav Mukherjee
    Commented Aug 23, 2018 at 22:05

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As far as I know, there is no publicly available written proof of uniqueness. Goodman's thesis pointed out by Chris proves neither uniqueness nor existence. What he did was to provide some of the first steps towards understanding the link between open books and tightness. Before that, he does sketch a proof of the open book theorem but, if I remember correctly, this sketch contains less information than what Giroux wrote in the ICM proceedings. In particular it entirely fails to cite Siebenmann's paper that Giroux cites twice in his uniqueness sketch and is the crucial starting point. This paper has been very hard to find for 30 years, but eventually got published as
Les bissections expliquent le théorème de Reidemeister-Singer: Un retour aux sources
Annales de la Faculté des sciences de Toulouse: Mathématiques, Série 6: Volume 24 (2015) no. 5

I'm almost certainly the mysterious person that Anubhav Mukherjee mentions in his comment, but writing a proof of this theorem is way beyond the scope of a mathoverflow answer, I'm sorry. I could probably answer more specific questions though.

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    $\begingroup$ Thank you for your answer. I have not yet look through Goodman’s thesis but, if what you say is true (which I would not be surprised if it were) my question is, why is it considered a Theorem? And who would get the credit after a complete written proof was carried out? $\endgroup$
    – Paul
    Commented Sep 1, 2018 at 19:32
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    $\begingroup$ It is considered a theorem because mathematical research is a complex sociological process involving large amounts of intuition and trust. Again this is a huge topic we cannot properly discuss here. And the publication status is not so relevant here. There are also published papers that are not trusted, although sometimes nobody has a specific error to point out. You can google "Coq mathcomp", "lean mathlib", "isabelle archive of formal proof" to get a sense of what is really verified in maths. But even there you'll need to convince yourself that definitions are correctly formalized. $\endgroup$
    – Patrick Massot
    Commented Sep 2, 2018 at 9:11
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    $\begingroup$ About gaps in this particular proof, it's hard to tell, because the available sketch is so sketchy that it can't really be wrong. I never managed to surprise Giroux when discussing subtle points of the proof with him. There are real mistakes in attempted detailed exposition that have been mentioned here, but it's a bit unfair to insist on this point. So we have direct evidence that it's not easy for "any expert" to work out the details. $\endgroup$
    – Patrick Massot
    Commented Sep 2, 2018 at 9:17
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    $\begingroup$ Here is an exercise illustrating the importance of carefully reading what Giroux wrote. On page 409 you read "On épaissit ensuite le 1-squelette L de ∆ en une surface compacte F (presque) tangente à ξ le long de L". (I just copied this sentence into DeepL translator, and the translation is perfect!). Try to find this "presque" in other accounts of the story. Then prove, using only the definition of a contact form, that it shouldn't have been removed. $\endgroup$
    – Patrick Massot
    Commented Sep 2, 2018 at 9:22
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    $\begingroup$ Thank you for your answers. I know that the vast majority of theorems haven’t been checked thoroughly down to the axioms. Rather the definition of theorem is “what the community agrees to be true” and the community tends to be tough. But still most theorems meet some standards (for example peer review process) which of course is not a total guarantee that they are true in a formal logic sense (recall Voevodsky famous non-theorem). But I agree that comments in MO are not the place for this discussion. $\endgroup$
    – Paul
    Commented Sep 2, 2018 at 16:56
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This might suffice for you, it is not published and only slightly longer than Etnyre's sketch, but without exercises. This has been shown in the PhD thesis of Noah Daniel Goodman (a student of Etnyre), specifically Theorem 3.4.4:

Contact Structures and Open Books

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    $\begingroup$ Thank you! Since my question is quite open and I did not know this source, I think it is fair that I up-vote you and check the answer (at least while I take the time to read the source). $\endgroup$
    – Paul
    Commented Aug 30, 2018 at 17:33
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I haven't read it carefully, but the new paper here by Licata-Vértesi appears to be (finally) a complete proof, albeit one that is different from the original.

EDIT: As the comments pointed out, the paper my original answer pointed to only proved it for tight contact structures. However, in a new paper here Licata-Vértesi prove the complete Giroux correspondence.

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    $\begingroup$ This is really cool. However, from the abstract (and the title), the proof is only for tight contact structures. $\endgroup$
    – Marco Golla
    Commented Sep 22, 2023 at 9:28
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    $\begingroup$ @MarcoGolla: Oh, that's a good point. I read it too quickly (and my mind passed over the tightness hypothesis due to the vague feeling that in most cases the tight case is the important one and the overtwisted case is easier, which I guess isn't true here). $\endgroup$
    – Andy Putman
    Commented Sep 22, 2023 at 17:07
  • $\begingroup$ In the abstract to their paper, Licata-Vértesi cite a paper by Breen-Honda-Huang that establishes the Giroux correspondence in all dimensions. $\endgroup$
    – Danny Ruberman
    Commented Aug 5 at 14:46

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