I have some questions about the $n$-sphere:
I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$?
I have the same question for the $n$-spheres admitting an almost complex structures $(n=2,6)$. Is there a general reason to conclude it doesn't have one for any other $n$?
And finally, (this is a homework question):
Can $S^4$ have a Lorentz metric?