A couple years ago Atiyah published a claimed proof that $S^6$ has no complex structure. I've heard murmurs and rumors that there are problems with the argument, but just a couple months ago he apparently published a follow-up which fleshes out the details. He writes:

In [1] I gave a proof of the long-standing conjecture that the 6-dimensional sphere has no complex structure. In this paper I will present the proof in a more transparent manner. I use the example of the 6-sphere to shed new light on many problems of physics. In the future I expect these ideas will provide a different perspective, with substantial benefits in all areas.

Unfortunately I can't find a version of the new paper in front of a pay wall, and even if I could I would wonder if he had addressed any of the possibly existent problems with the original. Does anyone have more information (or at least references)?

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    $\begingroup$ The update paper in question, "Understanding the 6-Dimensional Sphere", is published on pages 129–133 in the volume Foundations of Mathematics and Physics One Century After Hilbert: New Perspectives edited by Joseph Kouneiher. Sadly, copyright law does not permit me to give you a copy or even link to a site where it can be found, but this Wikipedia page can get you started if you wish to follow this path to the Dark Side. $\endgroup$ – Gro-Tsen Jul 1 '18 at 23:21
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    $\begingroup$ PS: this link works for the time being, but probably won't work for long since I believe Google books makes pages inaccessible as soon as too many people start reading it. $\endgroup$ – Gro-Tsen Jul 1 '18 at 23:24
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    $\begingroup$ Read literally, Atiyah's argument in this new paper would prove that there is no almost complex structure on the $6$-sphere, which, of course, is false. While I haven't gone through it in detail, there are several other unjustified claims that appear, on the face of them, to be false. For example, he supposedly constructs a nontrivial finite group $\Gamma$ that acts on the $6$-sphere and then claims that, because the group $\Gamma$ is 'intrinsic to the $6$-sphere', any (almost) complex structure on the $6$-sphere must be invariant under $\Gamma$. $\endgroup$ – Robert Bryant Jul 2 '18 at 0:16
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    $\begingroup$ @RobertBryant I was able to get my hands on a copy of the paper at last. It looks as though Atiyah claims that the index of the holomorphic Dirac operator on an almost complex manifold takes values in the representation ring of the quaternion group and that this group has a central $\mathbb{Z}/2$ factor which acts trivially if the almost complex structure is integrable. I don't understand this construction well enough to judge; do you claim that this doesn't actually distinguish between complex and almost complex structures? $\endgroup$ – Paul Siegel Jul 2 '18 at 11:08
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    $\begingroup$ @PaulSiegel: Yes, that is my claim. I can't see how integrability comes into it at all. Nothing that he says (or refers to in his reference [1]) actually uses integrability in any significant way. $\endgroup$ – Robert Bryant Jul 2 '18 at 11:28

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