Is there a counterexample to the following assertion?:


Let $p\_1:E\_1\to B\_1$ and $p\_2:E\_2\to B\_2$ be fibrations with the same fiber $\mathbb S ^1$ such that $E\_1$ and $E\_2$ are homeomorphic (and both connected). Then $B\_1$ and $B\_2$ are homotopic. 



A homeomorphism between total spaces with same fiber and base spaces not homotopic
$S^1\times S^3$ fibers both over $S^3$ (obvious) and $S^1\times S^2$ (identity cross Hopf). BTW, you don't say "homotopic" about spaces.


