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In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and realized I lacked the background to understand. The preprint inspired interest in KR-theory and a few postings on MathOverflow (Atiyah's paper on complex structures on $S^6$; https://mathoverflow.net/a/253490/5792), but a Google search does not seem to turn up any exposition or commentary later than November. A paper so brief on such an important topic has, however, no doubt been thoroughly read and reread by competent experts since that time.

  1. Has an exposition with more detail been published anywhere?
  2. Is the argument of the paper currently thorougly understood? If not,
  3. What parts are currently understood?
  4. What remains to be more closely explained?
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    $\begingroup$ The consensus among the experts I have spoken to is that the proof is incorrect. For instance, it does not use the fact that the almost complex structure is integrable in a serious way. $\endgroup$
    – Andy Putman
    Commented Feb 28, 2017 at 3:43
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    $\begingroup$ @GregFriedman: It is well-known that the 6-sphere admits an almost complex structure, but the known ones are not integrable, so the problem is to determine whether there is an integrable one. Thus, it would be essential to use the integrability hypothesis to address the problem. $\endgroup$
    – Robert Bryant
    Commented Feb 28, 2017 at 9:35
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    $\begingroup$ @SamHopkins: Unfortunately, recent claims of existence of integrable complex structures on the 6-sphere have known serious problems. One such claim (actually published) does not provide convincing justification that the claimed construction works (or even enough detail to actually specify the 'construction' completely), another makes claims that are demonstrably false. $\endgroup$
    – Robert Bryant
    Commented Feb 28, 2017 at 9:42
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    $\begingroup$ @SebastianGoette: In the most recent version that I have seen, the attempts to define even and odd were unconvincing and seemed to be mixing two different concepts. In any case, the assertion that an integrable almost complex structure would necessarily be even is made without any supporting evidence that I can detect. $\endgroup$
    – Robert Bryant
    Commented Feb 28, 2017 at 20:38
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    $\begingroup$ Here is Atiyah’s argument: there exists a differential operator on a complex sphere. If it is integrable, the operator is even. But the symbol of the operator depends only on the almost complex structure, and can be computed to be odd. The problem with the argument is the first step: he does not say what the differential operator is. I would file this under "not an argument." $\endgroup$
    – Ben Wieland
    Commented Mar 1, 2017 at 20:16

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