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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 $S^6$.

YCor
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David C
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    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
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    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
  • The forgetful map appears to be that you can treat a complex bundle as a real bundle, that gives your map from complex k-theory to this "KR" variant. Where is the sentence you are quoting? – Ryan Budney Oct 31 '16 at 20:45
  • This sentence is on p4 of Atiyah's preprint. – David C Oct 31 '16 at 21:02
  • The integers he's talking about is 6-dimensional complex k-theory of $S^6$, i.e. this group is infinite cyclic. – Ryan Budney Oct 31 '16 at 21:07
  • Yes of course, but do you know why it is trivial? – David C Oct 31 '16 at 21:08
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    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
  • @RyanBudney Can you explain a little bit more how to treat a complex bundle as a real bundle? You need to throw in a C_2-action and I really don't see how to do that (this is a bit embarassing because I should know KR very well). – Denis Nardin Nov 01 '16 at 00:20
  • @DenisNardin You go from $E$ to $E \oplus \overline{E}$; your involution is the obvious one that swaps factors. – mme Nov 01 '16 at 01:41
  • And the involution on the base space would be trivial. – David Roberts Nov 01 '16 at 01:42
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    I think basics of $KR$ theory is easy to learn from [1]. What is puzzling is the claim on p.5 of https://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 01 '16 at 02:56
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    Karoubi briefly treats KR-theory in "K-Theory: An Introduction" (page 177). The book is a good reference for K-Theory in general. – Tyrone Nov 01 '16 at 08:19
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    @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 01 '16 at 21:28
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    For your final question, the section titled "Ambient Space Construction" on this blog entry of mine is related. Basically it is a way to fix one conformal structure on $\mathbb{S}^6$. – Willie Wong Nov 03 '16 at 21:55
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    Regarding the map you wish to understand, I think that the KSp should be seen as coming from one specific KR. Btw https://web.archive.org/web/20230113145507/https://www.math.umd.edu/~jmr/NCGOA13.pdf should be rather useful, but I haven't been able to figure out the dimension correctly from there. Rosenberg uses a different grading to Atiyah. – David Roberts Nov 03 '16 at 22:38

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