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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.

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.

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 $\gamma$.
• Take the union (in $M$) of $S \setminus \{\gamma, \varphi(\gamma)\}$ (where we think of $S$ as 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.

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.

Given a genus $g$ (where $g \geq 2$) closed surface, and a pseudo-Anosov map $\varphi$, 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 $\gamma$.
• Take the union (in $M$) of $S \setminus \{\gamma, \varphi(\gamma)\}$ (where we think of $S$ as 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.

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 $\gamma$.
• Take the union (in $M$) of $S \setminus \{\gamma, \varphi(\gamma)\}$ (where we think of $S$ as 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.