Stable maps and branched shadows of 3-manifolds
Stable maps and branched shadows of 3-manifolds
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DOI：
10.1007/s00208-016-1403-4
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发表时间：
2014-03
影响因子：
1.4
通讯作者：
M. Ishikawa;Yuya Koda
中科院分区：
文献类型：
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作者：
M. Ishikawa;Yuya Koda
在20世纪90年代初，Turaev引入了阴影的概念，作为4和3-流形的组合表示。后来，Costantino-Thurston揭示了三维流形到真实的平面的稳定映射的Stein分解与流形的阴影之间的强关系。事实上，阴影可以局部地看作是稳定映射的Stein分解。本文定义了紧致可定向3-流形的稳定映射复杂性的概念，并在一定的权下，计算了稳定映射到真实的平面上的余维为2的奇异纤维的最小数目，证明了这个数目等于它的分支阴影的最小顶点数.在此基础上，我们给出了三维球面中外部具有稳定映射复杂度1的双曲链环的Dehn手术的完整刻画，并利用双曲体积的估计给出了关于稳定映射复杂度和阴影复杂度的一致性的观察.
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.