# Atiyah's paper on complex structures on $S^6$

M. Atiyah has posted a preprint on arXiv on the non-existence of complex structure on the sphere $S^6$.

https://arxiv.org/abs/1610.09366

It relies on the topological $K$-theory $KR$ and in particular on the forgetful map from topological complex $K$-theory to $KR$.

Question: what are the nice references to learn about $KR$?

Edit : Thank you very much for the comments and suggestions, M. Atiyah's paper "K-theory and reality." Quart. J. Math., Oxford (2), 17 (1966),367-86 is a fantastic paper. However I have more questions.

Question 1: how to build the morphism $$KSp(\mathbb{R}^6)\rightarrow K^{7,1}(pt)?$$

Question 2: why do we use $\mathbb{R}^{7,1}$?

In fact I do not understand the sentence "The 6-sphere then appears naturally as the base of the light-cone". And why it suffices to look at this particular model of $S^6$.

• It looks like you implicitly do not accept Atiyah's reference [1] in his paper. What isn't "nice" about that? – Ryan Budney Oct 31 '16 at 20:05
• I am reading [1], it is beautifully written. However as I am a complete amateur, I would like to understand the relationships between topological complex K-theory and KR-theory. In particular this sentence: "There are natural forgetful maps from complex K-theory to KR-theory and in dimension 6 the integers go to 0". – David C Oct 31 '16 at 20:22
• I think it's just a straight-forward computation, isn't it? I've only scanned the paper and this is my first time thinking about $KR$, but I suspect what's going on is that the generator for complex k-theory is the bundle with Euler class $1$. In $KR$ theory this is twice the generator, because the generator is one where the fixed-point set of the involution is a Moebius bundle. I haven't thought about this in detail but that's what I suspect is going on. – Ryan Budney Oct 31 '16 at 21:22
• I think basics of $KR$ theory is easy to learn from [1]. What is puzzling is the claim on p.5 of arxiv.org/abs/1610.09366 that "Because of the Atiyah-Singer theory, the linear algebra acquires a topological meaning, and that is embodied in $KR$ theory." – Igor Belegradek Nov 1 '16 at 2:56
• @RyanBudney I suspect that the sentence "integers goes to 0" means integrable ACS goes to 0 in KR, while non-integrable ones goes to 1. It seems that he was using this map to separate integrable and non-integrable ACSs, it cannot be done if it is trivial. I feel that this might be the most crucial step, but do not have a clear idea what's going on... – Mingcong Zeng Nov 1 '16 at 21:28