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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 Aug 23 '18 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 Sep 1 '18 at 19:32
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    $\begingroup$ Is there a chance that, for example, there is a gap that nobody has dealt with before? $\endgroup$ – Paul Sep 1 '18 at 19:48
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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 Sep 2 '18 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 Sep 2 '18 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 Sep 2 '18 at 9:22
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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 Aug 30 '18 at 17:33

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