Bound for the genus of the incompressible surfaces that can be embedded in a closed compact orientable irreducible 3-manifold

Question

Let $M$ be a closed compact orientable irreducible 3-manifold. I would like to know if there exists a bound $n\in \mathbb{N}$ such that every incompressible surface embedded in $M$ has genus lesser than $n$.

I think this bound exists for Seifert manifolds, since the incompressible surfaces are isotopic to the base space or to a torus. However, I don't know if it is true in general.

Thanks in advance.

Answer 1

This is not true, no such bound exists in general.

For example, if $M$ is a closed 3-manifold that fibers over the circle with fiber a closed, oriented surface $S$ of genus $\ge 2$, and if the rank of $H_2(M;\mathbb R)$ is $\ge 2$, then according to the theory of the Thurston norm on $H_2(M;\mathbb R)$ there exist other fibrations of $M$ over the circle whose fibers have arbitrarily large genus.