References on almost complex structures on spheres - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T18:36:32Z http://mathoverflow.net/feeds/question/25557 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25557/references-on-almost-complex-structures-on-spheres References on almost complex structures on spheres Colin Tan 2010-05-22T06:24:34Z 2010-05-26T05:03:08Z <p>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".</p> <p>Are there good write-ups on this topic? Thanks in advance.</p> http://mathoverflow.net/questions/25557/references-on-almost-complex-structures-on-spheres/25566#25566 Answer by Spinorbundle for References on almost complex structures on spheres Spinorbundle 2010-05-22T10:08:03Z 2010-05-22T10:08:03Z <p>I think this "well known fact" was proved first by Borel and Serre,</p> <p>Borel, A., Serre, J. P.: Groupes de Lie et puissances réduites de Steenrod. Amer. J. Math.75, 409–448 (1953)</p> <p>For a more detailed timeline, see <a href="http://books.google.de/books?id=1s-UePH5KfAC&amp;pg=PA100&amp;lpg=PA100&amp;dq=almost+complex+structure+on+spheres+S%5E2+S%5E6&amp;source=bl&amp;ots=HZDRuFRPB6&amp;sig=T_jsFRQlEIhPPBTvfGHOZp5426I&amp;hl=de&amp;ei=9qn3S9ShO6KemwPUgenHAg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=6&amp;ved=0CC8Q6AEwBTgU#v=onepage&amp;q&amp;f=false" rel="nofollow">Differential Geometry: Geometry in mathematical physics and related topics</a> by Greene and Yau (p.100).</p> <p>Or as BCnrd suggested <a href="http://books.google.de/books?id=-yz03cskUOgC&amp;pg=PA233&amp;lpg=PA233&amp;dq=almost+complex+structure+on+spheres+S%5E2+S%5E6&amp;source=bl&amp;ots=KxGtjBRHCP&amp;sig=AANscUt-KM5KILqIaPMJ1zRId94&amp;hl=de&amp;ei=9qn3S9ShO6KemwPUgenHAg&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=3&amp;ved=0CCEQ6AEwAjgU#v=onepage&amp;q&amp;f=false" rel="nofollow">Almost complex and complex structures</a> by C. C. Hsiung (Chapter VI)</p> http://mathoverflow.net/questions/25557/references-on-almost-complex-structures-on-spheres/25970#25970 Answer by Colin Tan for References on almost complex structures on spheres Colin Tan 2010-05-26T05:03:08Z 2010-05-26T05:03:08Z <p>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".</p>