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Given a genus $g$ (where $g \geq 2$) closed surface $S$, and a pseudo-Anosov map $\varphi: S \to S$, one can construct the mapping torus, which turns out to be a hyperbolic $3$-manifold, which we'll call $M$. Inside $M$, one way to construct other closed embedded surfaces is via the following construction:

  • Take a (multi-)curve $\gamma$ in $S$ such that $\varphi(\gamma)$ is disjoint from $\gamma$.
  • (Post comment edit) The curves $\gamma$ and $\varphi(\gamma)$ bound a subsurface $S'$ with two boundary components, namely $\gamma$ and $\varphi(\gamma)$.
  • Take the union (in $M$) of $S'$ (where we think of $S'$ as the subsurface of the fiber over the point $0 \in S^1$) and $f_{[0,1]}(\gamma)$, where $f$ is the suspension flow of $\varphi$.

Questions:

  1. Is every embedded surface in $M$ homologous to a surface obtained via the above construction?
  2. If the answer to the above question is no, does the answer become yes if we are allowed to change the monodromy map $\varphi$ and the fiber $S$, while keeping the total space $M$ fixed?

P.S. I'm new to $3$-manifold topology, and thus am not sure if the construction I described has a well-known name already. If it does, someone could perhaps point that out in the comments, and I can edit this question accordingly.

EDIT: I realize that it might not be possible to realize surfaces of very high genus via this construction, since there might not be enough room on $S$, so maybe the question should be what are the surfaces that can be realized this way.

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  • $\begingroup$ That union, which I'll call $K$, is not a surface. Up to homeomorphism, $K$ is the quotient of the union of $S$ with $S^1 \times[0,1]$ where $S^1 \times 0$ is identified with $\gamma$ and $S^1 \times 1$ is identified with $\phi(\gamma)$. The topological space $K$ is not a 2-manifold near any point $x \in \gamma \cup \phi(\gamma)$, because the relative homology group $H_2(K,K-x;\mathbb Z)$ is isomorphic to $\mathbb Z^2$, not to $\mathbb Z$ as is required to be a 2-manifold. Intuitively, instead of just 2 surface sheets meeting near $x$, there are 3 of them. $\endgroup$
    – Lee Mosher
    Commented Jul 3, 2020 at 14:16
  • $\begingroup$ Yes, of course. I forgot to write down a step in the construction where I look at a subsurface $S'$ whose two boundary components are $\gamma$ and $\varphi(\gamma)$. That leads to the result space being a manifold with two boundary components, with a cylinder glued together to the two boundary components, which definitely is a surface $\endgroup$
    – sayantankhan
    Commented Jul 3, 2020 at 16:00
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    $\begingroup$ The surfaces you describe here are sometimes called "cut and cross-join surfaces". I believe they were first introduced in: Cooper, Long & Reid, Bundles and finite foliations. Invent. Math. 118 (1994), no. 2, 255–283, and further developed in Masters, Thick surfaces in hyperbolic 3-manifolds. Geom. Dedicata 119 (2006), 17–33. $\endgroup$
    – HJRW
    Commented Jul 5, 2020 at 16:05

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