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 $S^6$.

suspectis going on. $\endgroup$ – Ryan Budney Oct 31 '16 at 21:22