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**Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds.**
*(English)*
Zbl 1071.57015

R. H. Fox showed that any compact connected \(3\)-dimensional submanifold \(M\) of the \(3\)-sphere \(S^3\) is homeomorphic to the complement of a disjoint union of handlebodies in \(S^3\) [Ann. Math. (2) 49, 462–470 (1948; Zbl 0032.12502)]. This is generalized in the paper under review.

Let \(N\) be a closed orientable irreducible non-Haken \(3\)-manifold, and \(M\) be a connected compact \(3\)-submanifold of \(N\) with non-empty boundary. Then \(M\) is homeomorphic to a submanifold of \(N\) whose complement is homeomorphic to a disjoint union of handlebodies or to \(N \sharp\) (disjoint union of handlebodies), where \(\sharp\) denotes connected sum. Moreover, when the boundary \(\partial M\) is connected, keeping the above conditions, \(M\) can be reimbedded in \(N\) so that there is a complete collection of reducing spheres \({\mathbb S}\) for \(\partial M\) with \(M-{\mathbb S}\) being a union of solid tori and \(\partial\)-irreducible manifolds. A sphere in \(N\) is called a reducing sphere for \(\partial M\) if it intersects \(\partial M\) in an essential circle. A collection of reducing spheres \({\mathbb S}\) is called complete if the number of non-planar components of \(\partial M - {\mathbb S}\) is no less than that for any other collection containing \({\mathbb S}\).

In the course of the proof, the theorem below is shown. Let \(H\) be a handlebody of genus \(g\) imbedded in a closed orientable irreducible non-Haken \(3\)-manifold \(N\). If \(\partial H\) is compressible in the complement of \(H\), then either (1) the Heegaard genus of \(N\) is less than or equal to \(g\), (2) \(\partial H\) has a reducing sphere or (3) \(H\) has an unknotted core. The last condition (3) means that there are a proper disk \(D\) in \(H\) and a circle \(c\) in \(\partial H\) such that \(c\) intersects \(\partial D\) in a single point and \(c\) bounds a disk in \(N\) transverse to \(\partial H\).

Let \(N\) be a closed orientable irreducible non-Haken \(3\)-manifold, and \(M\) be a connected compact \(3\)-submanifold of \(N\) with non-empty boundary. Then \(M\) is homeomorphic to a submanifold of \(N\) whose complement is homeomorphic to a disjoint union of handlebodies or to \(N \sharp\) (disjoint union of handlebodies), where \(\sharp\) denotes connected sum. Moreover, when the boundary \(\partial M\) is connected, keeping the above conditions, \(M\) can be reimbedded in \(N\) so that there is a complete collection of reducing spheres \({\mathbb S}\) for \(\partial M\) with \(M-{\mathbb S}\) being a union of solid tori and \(\partial\)-irreducible manifolds. A sphere in \(N\) is called a reducing sphere for \(\partial M\) if it intersects \(\partial M\) in an essential circle. A collection of reducing spheres \({\mathbb S}\) is called complete if the number of non-planar components of \(\partial M - {\mathbb S}\) is no less than that for any other collection containing \({\mathbb S}\).

In the course of the proof, the theorem below is shown. Let \(H\) be a handlebody of genus \(g\) imbedded in a closed orientable irreducible non-Haken \(3\)-manifold \(N\). If \(\partial H\) is compressible in the complement of \(H\), then either (1) the Heegaard genus of \(N\) is less than or equal to \(g\), (2) \(\partial H\) has a reducing sphere or (3) \(H\) has an unknotted core. The last condition (3) means that there are a proper disk \(D\) in \(H\) and a circle \(c\) in \(\partial H\) such that \(c\) intersects \(\partial D\) in a single point and \(c\) bounds a disk in \(N\) transverse to \(\partial H\).

Reviewer: Chuichiro Hayashi (Tokyo)

### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

### Citations:

Zbl 0032.12502
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\textit{M. Scharlemann} and \textit{A. Thompson}, Proc. Am. Math. Soc. 133, No. 6, 1573--1580 (2005; Zbl 1071.57015)

### References:

[1] | A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275 – 283. · Zbl 0632.57010 |

[2] | Ralph H. Fox, On the imbedding of polyhedra in 3-space, Ann. of Math. (2) 49 (1948), 462 – 470. · Zbl 0032.12502 |

[3] | John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. · Zbl 0345.57001 |

[4] | W. Menasco and A. Thompson, Compressing handlebodies with holes, Topology 28 (1989), no. 4, 485 – 494. · Zbl 0683.57004 |

[5] | Martin Scharlemann and Abigail Thompson, Thin position for 3-manifolds, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 231 – 238. · Zbl 0818.57013 |

[6] | Friedhelm Waldhausen, Heegaard-Zerlegungen der 3-Sphäre, Topology 7 (1968), 195 – 203 (German). · Zbl 0157.54501 |

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