MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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. – BCnrd May 22 '10 at 6:42
up vote 6 down vote accepted

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)

share|cite|improve this answer
Thank you Spinorbundle. – user2529 May 22 '10 at 12:25

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".

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.