### 抜粋

In the early 1990s, Turaev introduced the notion of shadows as a combinatorial presentation of both 4 and 3-manifolds. Later, Costantino–Thurston revealed a strong relation between the Stein factorizations of stable maps of 3-manifolds into the real plane and the shadows of the manifolds. In fact, a shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the real plane, and prove that this number equals the minimal number of vertices of its branched shadows. In consequence, we give a complete characterization of hyperbolic links in the 3-sphere whose exteriors have stable map complexity 1 in terms of Dehn surgeries, and also give an observation concerning the coincidence of the stable map complexity and shadow complexity using estimations of hyperbolic volumes.

元の言語 | English |
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ページ（範囲） | 1819-1863 |

ページ数 | 45 |

ジャーナル | Mathematische Annalen |

巻 | 367 |

発行部数 | 3-4 |

DOI | |

出版物ステータス | Published - 2017 4 1 |

外部発表 | Yes |

### ASJC Scopus subject areas

- Mathematics(all)

## フィンガープリント Stable maps and branched shadows of 3-manifolds' の研究トピックを掘り下げます。これらはともに一意のフィンガープリントを構成します。

## これを引用

*Mathematische Annalen*,

*367*(3-4), 1819-1863. https://doi.org/10.1007/s00208-016-1403-4