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Is there a space $X$ such that $S^1$ is homeomorphic to $X \times X$?

If there is it would need to have fundamental group $\mathbb{Z}$.

$\pi_1(X \times X)=\pi_1(X) \times \pi_1(X)$

What can I do now?

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  • $\begingroup$ Observe that $\mathbb{Z}$ cannot be $G \times G$ for any $G$. This is quite easy. $\endgroup$ Apr 2 '17 at 22:27
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Is there a group $ G $ such that $ \mathbf Z \cong G \times G $?

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  • $\begingroup$ Don't think so but how can I prove this formally? $\endgroup$
    – Walter
    Apr 2 '17 at 22:28
  • $\begingroup$ @Walter When is $ G \times G $ cyclic? $\endgroup$
    – Ege Erdil
    Apr 2 '17 at 22:29
  • $\begingroup$ Not sure, maybe gcd is 1? $\endgroup$
    – Walter
    Apr 2 '17 at 22:34
  • $\begingroup$ The gcd of what and what? $\endgroup$
    – Ege Erdil
    Apr 2 '17 at 22:35
  • $\begingroup$ Could you finish your answer please. i don't really understand the answer on the linked question $\endgroup$
    – Walter
    Apr 19 '17 at 21:28

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