Tuesday, 27 March 2012

at.algebraic topology - complex structure on S^n

It is known that S4 doesn't even have an almost complex structure, and the case for S6 is open. The lack of almost complex structure can be proved a number of ways, one way is by showing that an almost complex, compact, four manifold with dimmathbbQH2(X,mathbbQ)=0 has chi(X)=0, but the four sphere doesn't. (It follows from the index theorem, here's a quick reference, first result.)

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