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It is known that $S^6$ admits an almost complex structure which is not integrable, recently Michael Atiyah posted a proof of non-existent of a complex structure on this sphere.

I am curious now about the other spheres i.e what about $S^{4}$ and $S^{2n}$ for $n\geq4$?

Is there any good introduction reference where one at the undergraduate level can read about that?

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    Related: [here](http://mathoverflow.net/questions/11664/complex-structure-on-sn), [here](http://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere) and [here](http://mathoverflow.net/questions/253577/atiyahs-paper-on-complex-structures-on-s6) – 2017-02-08
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    what you will need: some knowledge of characteristic classes and k theory. atiyah's proof is most likely flawed, by the way. – 2017-02-08

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