In proving that all closed, compact 3-manifolds $M$ admit Heegaard splittings one often takes a triangulation of $M$ and then takes as one side of the Heegaard splitting a regular neighborhood of the 1-skeleton. My question is: do all Heegaard splittings arise in this way?
Namely, given such a 3-manifold $M$ together with a Heegaard splitting $U \cup V = M$, does there exist some triangulation of $M$ such that $U$ is a regular neighborhood of the 1-skeleton of the triangulation?
Any triangulation of a 3-manifold contains at least one 3-simplex. As the 1-skeleton of 3-simplex is homotopy equivalent to a wedge of 4-circles, a regular neighborhood of the 1-skeleton of the whole 3-manifold is at least genus 4.
In general the genus is bounded by the number of 2-simplices as the face condition implies any two adjacent simplices have a 1-skeleton which is two one simplices glued along a single edge, and hence for $n$ 2-simplices the one-skeleton is homotopy equivalent to a wedge of $n$ circles and a regular neighborhood has genus at least $n$.