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?

noton arXiv, but appears in the book "Foundations of Mathematics and Physics One Century after Hilbert", which was very recently released. $\endgroup$3more comments