TY - JOUR

T1 - Stable maps and branched shadows of 3-manifolds

AU - Ishikawa, Masaharu

AU - Koda, Yuya

N1 - Funding Information:
M. Ishikawa is supported by the Grant-in-Aid for Scientific Research (C), JSPS KAKENHI Grant Number 25400078. Y. Koda is supported by the Grant-in-Aid for Young Scientists (B), JSPS KAKENHI Grant Number 26800028.
Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - 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.

AB - 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.

KW - 57M27

KW - 57N70

KW - 57R45

KW - 58K15

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U2 - 10.1007/s00208-016-1403-4

DO - 10.1007/s00208-016-1403-4

M3 - Article

AN - SCOPUS:84962149011

SN - 0025-5831

VL - 367

SP - 1819

EP - 1863

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -