References on almost complex structures on spheres - MathOverflow

References on almost complex structures on spheres

2010-05-22T06:24:34Z

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.

Answer by Spinorbundle for References on almost complex structures on spheres

2010-05-22T10:08:03Z

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)

Answer by Colin Tan for References on almost complex structures on spheres

2010-05-26T05:03:08Z

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".