at.algebraic topology - complex structure on S^n

It is known that $S^4$ doesn't even have an almost complex structure, and the case for $S^6$ 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 $dim_{mathbb{Q}}H^2(X,mathbb{Q})=0$ has $chi(X)=0$, but the four sphere doesn't. (It follows from the index theorem, here's a quick reference, first result.)