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](http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=79845&vfpref=html&r=29&mx-pid=141106)). 


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?



When are incompressible surfaces isotopic into a two-skeleton?
Unfortunately, you can't always expect every incompressible surface to be isotoped into the $2$-skeleton. There are only finitely many subsets of the $2$-skeleton, but there can be infinitely many nonparallel incompressible surfaces. This is simplest in the $3$-torus, but there are many other examples. Jaco's stair construction gives incompressible surfaces of arbitrarily high genus in a surface (of positive genus) times a circle. 


