Heegaard splittings of graph manifolds (Q5920208)

A closed $3$-manifold $M$ is said to be a graph manifold if its Jaco-Shalen-Johansson decomposition admits only Seifert-fibered pieces. These manifolds were classified by \textit{F. Waldhausen} [Invent. Math. 3, 308--333 (1967; Zbl 0168.44503)] and are completely determined by a normalized weighted graph. In the paper under review, the authors provide an explicit method to construct a Heegaard diagram of any graph manifold by using a plumbing graph. Also, the authors give some examples. Although the method does not provide the minimal Heegaard splitting of a graph manifold, the method can be used to decrease the Heegaard genus of a graph manifold. \par Theorem 2.1. Let $M$ be a manifold associated to a plumbing graph $(\Gamma, g, e, o)$. Fix a cocycle representing $o$ (described as a choice of a sign for each edge) and a spanning tree $T$ of $\Gamma$. Then, the method described in $(G1)-(G7)$ provides an explicit Heegaard diagram of $M$.