Haken proved that an incompressible surface in a triangulated irreducible 3-manifold is isotopic to a surface which is normal with respect to the triangulation (Theorie der Normalflächen. Acta Math. 105 1961 245–375).
While normal surfaces are tremendously useful, I want my surfaces to be isotopic into the two-skeleton and I am unsure when I can conclude that they indeed are isotopic into the two-skeleton. I suspect that there are triangulated 3-manifolds out there containing incompressible surfaces which are not isotopic into the two-skeleton, but I am hoping that if the 3-manifold has a metric and the tetrahedra are "small enough" compared to the injectivity radius of the 3-manifold, then incompressible surfaces can be isotoped into the two-skeleton.
Can someone point me to a reference or explain whether or not there are reasonable properties of a triangulation which imply that incompressible surfaces are isotopic into the two-skeleton?