Heegaard splittings of \((\text{surface})\times I\) are standard.

*(English)*Zbl 0814.57010C. Frohman and J. Hass have shown [Invent. Math. 95, No. 3, 529-540 (1989; Zbl 0678.57009)] that genus three Heegaard splittings of the 3-torus are standard. M. Boileau and J.-P. Otal [J. Differ. Geom. 32, No. 1, 209-233 (1990; Zbl 0754.53012)] generalize this result to show that all Heegaard splittings of the 3-torus are standard. A crucial part of Boileau-Otal’s argument is to show that all Heegaard splittings of a torus crossed with an interval are standard. We generalize this part of their paper to prove that all Heegaard splittings of a closed orientable genus \(g\) surface crossed with an interval are standard. Many of our arguments are based on theirs; we differ substantially in Sect. 4, which allows us to obtain the more general result.

The paper is organized as follows: Section 1 begins with a discussion of compression bodies and their spines. In Sect. 2 we discuss Heegaard splittings and state the main theorem (2.11). The proof of the main theorem begins in Sect. 3, where we prove a lemma which splits the remainder of the proof into two cases. These cases are considered in Sect. 4 and Sect. 5. In Sect. 6 we exploit the main theorem to generalize a theorem of Frohman.

Pitts and Rubinstein have announced related results, using different methods.

The paper is organized as follows: Section 1 begins with a discussion of compression bodies and their spines. In Sect. 2 we discuss Heegaard splittings and state the main theorem (2.11). The proof of the main theorem begins in Sect. 3, where we prove a lemma which splits the remainder of the proof into two cases. These cases are considered in Sect. 4 and Sect. 5. In Sect. 6 we exploit the main theorem to generalize a theorem of Frohman.

Pitts and Rubinstein have announced related results, using different methods.

##### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

57Q37 | Isotopy in PL-topology |

57Q35 | Embeddings and immersions in PL-topology |

57M15 | Relations of low-dimensional topology with graph theory |

##### Keywords:

Heegaard splittings of a closed orientable genus \(g\) surface crossed with an interval; compression bodies; spines##### References:

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