I am looking for the simplest explicit example of a $3$-dimensional manifold $M$ which is a surface fiber bundle over the circle $S^1$ in two different ways -- with fibers $F_1$ and $F_2$ respectively. Can I take $M=F_1 \times S^1$, where $F_1$ is a punctured torus? (Alternatively $F_1$ be closed of genus $2$.) What would be $F_2$ and the explicit monodromy on $F_2$? (If I understand it correctly, $F_2$ can be a double cover of $F_1$.)
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@IvanIzmestiev: Can it be done for punctured torus? I know that the monodromy will be a periodic homeomorphism of F_2. Which one? – Adam May 19 '17 at 13:42
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Yes, it can be done for the punctured torus by the construction given in the answer to that question. The monodromy is the covering monodromy of $F_2 \to F_1$. – Ivan Izmestiev May 19 '17 at 13:54
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Yet another answers to the same question: https://mathoverflow.net/questions/241822/homeomorphic-but-non-conjugate-mapping-tori/241832#241832 – ThiKu May 20 '17 at 10:21