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Is it true that when the first fundamental group of a topological space $X$ is isomorphic to $\mathbb{Z}$ then $X$ is homeomorphic to $S^1 \times Y$ where the first fundamental group of $Y$ is trivial?

With a discussion with my of friends, the above question turned into (!) finding a topological space $X$ s.t. there is no quotient space obtained from $X$ being homeomorphic to $S^1.$

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  • $\begingroup$ A little modification! $\endgroup$
    – Ehsan M. Kermani
    Commented Mar 16, 2011 at 15:04
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    $\begingroup$ I suggest you ask this question on math.stackexchange.com as the counterexample is well-known to students of topology. $\endgroup$
    – S. Carnahan ♦
    Commented Mar 16, 2011 at 15:17

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The M"obius band is a counterexample (to the original and current versions of the stated question).

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  • $\begingroup$ Yes, it's been edited sir. I'm new to this field. Can you elaborate it more, please? $\endgroup$
    – Ehsan M. Kermani
    Commented Mar 16, 2011 at 15:11

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