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Let's consider $S^1$-bundle $E$ over a 2-manifold $M$. How many isotopy classes of embeddings of the torus $\mathbb{T}^2$ in $E$?

For each free homotopy classes $\gamma$ of mappings of the circle into $M$ with "trivial monodromy" I can construct the embedding of torus $\gamma \times S^1$ into E. So, I think that is all possible isotopic classes. Is it true? It would be great if you will show me explicit isotopy.

More general question: What about calculation of isotopy classes of embeddings of the torus $\mathbb{T}^2$ in any 3-manifold?

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    $\begingroup$ I do not think your intuition can be right when $M=\mathbb{T}^2$. $\endgroup$
    – Benoît Kloeckner
    Commented Sep 14, 2013 at 19:29
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    $\begingroup$ To have a reasonable answer you have to assume that the tori are incompressible. Then your guess is right if the base is hyperbolic and wrong if it is torus. $\endgroup$
    – Misha
    Commented Sep 14, 2013 at 19:34

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As stated, your question is rather hopeless. Consider the 3-sphere, which also happens to admit the Hopf fibration over the 2-sphere. Then classifying 2-tori in the 3-sphere is essentially equivalent to classifying all knots. Clearly, the vertical tori as in your questions are insufficient.

However, if you assume that your tori are incompressible, then in general compact 3-manifold, a classification is given by the JSJ theory. In particular, for Seifert manifolds with hyperbolic base, all incompressible tori are vertical.

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  • $\begingroup$ Yea, for trivial bundle $E = D^2 \times S^1$ we have very many classes... $\endgroup$
    – Gleb
    Commented Sep 15, 2013 at 6:22

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