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Jun 13, 2018 at 15:52 comment added Danu It appears that Atiyah has published another (attempt at a) proof in an article which is not on arXiv, but appears in the book "Foundations of Mathematics and Physics One Century after Hilbert", which was very recently released.
Dec 12, 2017 at 7:16 history edited Max Borovkov CC BY-SA 3.0
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Dec 11, 2017 at 11:55 comment added Neil Strickland (1) Part of Bott periodicity is that $\Omega^4(\mathbb{Z}\times BO)=\mathbb{Z}\times BSp$ and $\Omega^4(\mathbb{Z}\times BSp)=\mathbb{Z}\times BO$, so $KSp$ is the same as $KO$ up to a shift. (2) Atiyah is implicitly using $K$-theory with compact supports, for which $K(U)=\widetilde{K}(U\cup\{\infty\})$ almost by definition. And $S^6\simeq\mathbb{R}^6\cup\{\infty\}$.
Dec 11, 2017 at 0:16 comment added YangMills Honestly I cannot even see which part of Atiyah's argument would supposedly use that the almost complex structure is integrable.
Dec 10, 2017 at 20:56 history edited YCor
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Dec 10, 2017 at 20:27 comment added Max Borovkov @YCor, ok, I asked the last question only to have an updated information. However, first three questions might be still open.
Dec 10, 2017 at 20:15 comment added Denis Nardin I'm going out on a limb and say that the paper does not contain an actual proof. I don't know whether the author plans to publish a follow up with more details or not, but for now I still consider the problem open.
Dec 10, 2017 at 20:11 comment added YCor The last question "is this proof accepted as true in math society" makes a duplicate of mathoverflow.net/questions/263301. I'd suggest to stick to questions 1,2,3.
Dec 10, 2017 at 20:07 comment added David Roberts You should regard that paper as a sketch of a claimed proof that has not since appeared.
Dec 10, 2017 at 20:06 comment added YCor Links to past MO discussions: mathoverflow.net/questions/1973, mathoverflow.net/questions/253577, mathoverflow.net/questions/263301
Dec 10, 2017 at 20:00 history asked Max Borovkov CC BY-SA 3.0