Given a 3-manifold $M$ with a triangulation $T$, will every essential surface in $M$ be a fundamental one? If not, then what are the conditions on $T$ so that these essential surfaces become fundamental?
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3$\begingroup$ I would explain the meaning of "essential", since it depends on the context. In general, there are finitely many fundamental surfaces, while essential surfaces are often infinite in number, for most notions of "essential" and for many manifolds M. $\endgroup$– Bruno MartelliCommented Sep 12, 2017 at 19:15
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Since you start by mentioning a triangulation, I assume that by "fundamental surface" you mean "fundamental normal surface". As Bruno says, in any triangulation of any compact three-manifold there are only finitely many fundamental normal surfaces.
So consider the three-torus $\mathbb{T}^3$. This is obtained by taking the usual three-space $\mathbb{R}^3$ and quotienting out by the integer lattice $\mathbb{Z}^3$. Any rational plane through the origin descends to give an essential two-torus inside of $\mathbb{T}^3$. Thus the answer to your first question is "no".
This also shows that there is no possible condition on the triangulation $T$ that would satisfy your second question.
However, you may be interested in instead placing conditions on the three-manifold $M$. One idea here would be to somehow ensure that there are only finitely many essential surfaces. This could be done by assuming that $M$ is non-Haken.
A different approach might be to change your focus from essential surfaces to an-annular surfaces.
