Heegaard splittings of graph manifolds

Abstract

Let $M$ be a totally orientable graph manifold with characteristic submanifold $\mathcal{T}$ and let $M=V{\cup }_{S}W$ be a Heegaard splitting. We prove that $S$ is standard. In particular, $S$ is the amalgamation of strongly irreducible Heegaard splittings. The splitting surfaces $S_i$ of these strongly irreducible Heegaard splittings have the property that for each vertex manifold $N$ of $M$, $S_i\cap N$ is either horizontal, pseudohorizontal, vertical or pseudovertical.