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 $S^1.$



Is $X$ homeomorphic to $S^1 \times Y$?
The M"obius band is a counterexample (to the original and current versions of the stated question).


