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**Local detection of strongly irreducible Heegaard splittings.**
*(English)*
Zbl 0926.57018

Let \(M\) be a compact \(3\)-manifold containing a Heegaard surface \(S\) that splits \(M\) into two handlebodies \(H_1\) and \(H_2\) (or, if \(M\) has nonempty boundary, into two compression bodies). A. J. Casson and C. McA. Gordon [ibid. 27, No. 3, 275-283 (1987; Zbl 0632.57010)] introduced the notion of a strongly irreducible Heegaard splitting. This means that any pair of essential discs, one in \(H_1\) and one in \(H_2\), must have intersecting boundaries. Such splittings are prevalent. For example, in a non-Haken \(3\)-manifold, any splitting either is strongly irreducible or is reducible, and consequently any minimal genus splitting is strongly irreducible.

In this paper, the author shows that for a strongly irreducible Heegaard splitting, the possible intersections of \(S\) with a \(3\)-ball or a solid torus are significantly constrained. If \(B\) is a \(3\)-ball in \(M\) (transverse to \(S\)) for which the planar surfaces \(\partial B\cap H_i\) are incompressible, then \(S\cap B\) is connected, planar, and unknotted in \(B\) (the latter means that it is parallel to a submanifold of \(\partial B\)). If \(V\) is a solid torus such that \(\partial V\) intersects \(S\) in parallel essential curves, not bounding discs in \(V\), then \(S\) intersects \(V\) in a collection of boundary-parallel annuli and possibly one other component, obtained from one or two annuli by attaching a tube along an arc parallel to a subarc of \(\partial V\). If the latter sort of component is in \(V\), then \(S-V\) is incompressible in \(M-V\).

In this paper, the author shows that for a strongly irreducible Heegaard splitting, the possible intersections of \(S\) with a \(3\)-ball or a solid torus are significantly constrained. If \(B\) is a \(3\)-ball in \(M\) (transverse to \(S\)) for which the planar surfaces \(\partial B\cap H_i\) are incompressible, then \(S\cap B\) is connected, planar, and unknotted in \(B\) (the latter means that it is parallel to a submanifold of \(\partial B\)). If \(V\) is a solid torus such that \(\partial V\) intersects \(S\) in parallel essential curves, not bounding discs in \(V\), then \(S\) intersects \(V\) in a collection of boundary-parallel annuli and possibly one other component, obtained from one or two annuli by attaching a tube along an arc parallel to a subarc of \(\partial V\). If the latter sort of component is in \(V\), then \(S-V\) is incompressible in \(M-V\).

Reviewer: D.McCullough (Norman)

### MSC:

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

57M99 | General low-dimensional topology |

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\textit{M. Scharlemann}, Topology Appl. 90, No. 1--3, 135--147 (1998; Zbl 0926.57018)

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### References:

[1] | Bonahon, F.; Otal, J. P., Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup., 16, 451-466 (1983) · Zbl 0545.57002 |

[2] | Casson, A.; Gordon, C. McA., Reducing Heegaard splittings, Topology Appl., 27, 275-283 (1987) · Zbl 0632.57010 |

[3] | Frohman, C., The topological uniqueness of triply periodic minimal surfaces in \(R^3\), J. Differential Geom., 31, 277-283 (1990) · Zbl 0689.53002 |

[4] | Haken, W., Some results on surfaces in 3-manifolds, (Studies in Modern Topology (1968), Prentice-Hall), 34-98 · Zbl 0194.24902 |

[6] | Scharlemann, M.; Thompson, A., Thin position and Heegaard splittings of the 3-sphere, J. Differential Geom., 39, 343-357 (1994) · Zbl 0820.57005 |

[7] | Scharlemann, M.; Thompson, A., Thin position for 3-manifolds, (Contemp. Math., 164 (1994), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 231-238 · Zbl 0818.57013 |

[8] | Waldhausen, Heegaard-Zerlegungen der 3-Sphäre, Topology, 7, 195-203 (1968) · Zbl 0157.54501 |

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