Is $S^2$ homeomorphic to $X \times X$?

Question

It can be shown that $S^1$ is not homeomorphic to the direct product of two spaces $X \times X$ by considering fundamental groups and then deriving a contradiction.

I was wondering whether this was true for general $n$? I.e, is $S^n \not\cong X \times X$ for some topological space $X$?

Thank you.

Answer 1

You can apply Künneth formula instead : if $S^n \cong X \times X$, then $1+t^n = p(t)^2$ for some polynomial $p(t) \in \Bbb Z[t]$ which is easily seen to be impossible.