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Heegaard splittings of graph manifolds. (English) Zbl 1415.57014

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 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.
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 \(M\).
Reviewer: Kun Du (Lanzhou)

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)

Citations:

Zbl 0168.44503
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References:

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