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I'm looking for a proof of that the only spheres with almost complex structure are $S^2$ and $S^6$. I've googled "almost complex structure sphere", but all I get is comments saying that "this fact is well-known".

Are there good write-ups on this topic? Thanks in advance.

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    $\begingroup$ References are given at the beginning of Chapter VI of the book "Almost complex and complex structures" by C.C. Hsiung. It uses characteristic classes and cohomology operations to get obstructions. $\endgroup$
    – BCnrd
    Commented May 22, 2010 at 6:42
  • $\begingroup$ One must be careful with the last chapters of the book BCnrd mentioned, I think Dr. Bryant has said on this site that there is an error in them $\endgroup$
    – user74900
    Commented Apr 29, 2018 at 7:21

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I think this "well known fact" was proved first by Borel and Serre,

Borel, A., Serre, J. P.: Groupes de Lie et puissances réduites de Steenrod. Amer. J. Math.75, 409–448 (1953)

For a more detailed timeline, see Differential Geometry: Geometry in mathematical physics and related topics by Greene and Yau (p.100).

Or as BCnrd suggested Almost complex and complex structures by C. C. Hsiung (Chapter VI)

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Peter May's A Concise Introduction to Algebraic Topology has a proof sketch on pages 207-209. It is in subsection 4 called "The Chern character; almost complex structures on spheres" in chapter 24 "An Introduction to K-theory".

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Konstantis & Parton - Almost complex structures on spheres is a self-contained paper which explains the proof very clearly.

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