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I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions.

Consider the isomorphism on the 4th page: "$KSp(\mathbb{R}^6) \rightarrow KR^{(7,1)}(pt) = \mathbb{Z}_2$"

My questions are as follows:

1) Atiyah says that the $KSp(\mathbb{R}^6)$ is "the stable homotopy group determined by Bott". But as far as I understand $KSp(\mathbb{R}^6)$ is the K-theory of vector bundles over base space $B$, s.t. the isomorphisms from definition of vector bundle $\pi^{-1}(b) \mapsto \{b\} \times \mathbb{R}^n$ lie in sympletic group $Sp(n)$. How is it connected with each other? Or how should I think about $KSp(\mathbb{R}^6)$?

2)Moreover, after that Atiyah claims that $KSp(\mathbb{R}^6)$ is "just the reduced $KSp$ of $\mathbb{S}^6$". How could one see it?

3)Next, Atiyah says that "There are natural forgetful maps from complex K-theory to KR theory and in dimension 6 the integers go to 0, so a hypothetical complex structure on $\mathbb{S}^6$" would give an even element. But we know one almost complex structure J(0) which is odd..." Here I can't understand the meaning of "even and odd". In what terms should I talk about it?

I will be grateful if someone makes these things more clear to me. Anyway, is this proof accepted as true in math society?

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    $\begingroup$ Links to past MO discussions: mathoverflow.net/questions/1973, mathoverflow.net/questions/253577, mathoverflow.net/questions/263301 $\endgroup$
    – YCor
    Dec 10 '17 at 20:06
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    $\begingroup$ You should regard that paper as a sketch of a claimed proof that has not since appeared. $\endgroup$ Dec 10 '17 at 20:07
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    $\begingroup$ 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. $\endgroup$ Dec 10 '17 at 20:15
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    $\begingroup$ (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 $S^6\simeq\mathbb{R}^6\cup\{\infty\}$. $\endgroup$ Dec 11 '17 at 11:55
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    $\begingroup$ 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. $\endgroup$
    – Danu
    Jun 13 '18 at 15:52

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