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**Heegaard splittings of \((\text{surface})\times I\) are standard.**
*(English)*
Zbl 0814.57010

C. 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
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\textit{M. Scharlemann} and \textit{A. Thompson}, Math. Ann. 295, No. 3, 549--564 (1993; Zbl 0814.57010)

### References:

[1] | [BO] Boileau, M., Otal, J.-P.: Sur les scindements de Heegaard du toreT 3. J. Diff. Geom.32, 209-233 (1990) · Zbl 0754.53012 |

[2] | [CG] Casson, A., Gordon, C. McA.: Reducing Heegaard splittings. Topology Appl.27, 275-283 (1990) · Zbl 0632.57010 |

[3] | [F] Frohman, C.: The topological uniqueness of triply periodic minimal surfaces inR 3. J. Differ. Geom.31, 277-283 (1990) · Zbl 0689.53002 |

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[5] | [FH] Frohman, C., Hass, J.: Unstable minimal surfaces and Heegaard splittings. Invent. Math.95, 529-540 (1989) · Zbl 0678.57009 |

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[8] | [H] Haken, W.: Some results on surfaces in 3-manifolds (Studies in Modern Topology, M. A. A., pp. 34-98) Englewood Cliffs: Prentice Hall 1968 |

[9] | [J] Jaco, W.: Adding a 2-handle to a 3-manifold: an application to propertyR. Proc. Am. Math. Soc.92, 288-292 (1984) · Zbl 0564.57009 |

[10] | [ST] Scharlemann, M., Thompson, A.: Detecting unknotted graphs in 3-space. J. Differ. Geom.34, 539-560 (1991) · Zbl 0751.05033 |

[11] | [W] Waldhausen, F.: Heegaard-Zerlegungen der 3-Sph?re. Topology7, 195-203 (1968) · Zbl 0157.54501 |

[12] | [Wu] Wu, Y.-Q.: A generalization of the handle addition theorem. Proc. Amer. Math. Soc.114, 237-242 (1992) · Zbl 0737.57007 |

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