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An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$
It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
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