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Stable maps and branched shadows of 3-manifolds

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Abstract

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.

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Acknowledgments

The authors wish to express their gratitude to Francesco Costantino, Bruno Martelli, and Osamu Saeki for very helpful suggestions and comments. This work was carried out while Y. Koda was visiting Università di Pisa as a JSPS Postdoctoral Fellow for Research Abroad. He is grateful to the university and its staff for the warm hospitality. Finally, the authors would like to thank the anonymous referee for his or her valuable comments and suggestions which helped them improve the exposition.

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Masaharu Ishikawa

  2. Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan

    Yuya Koda

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  1. Masaharu Ishikawa
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  2. Yuya Koda
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Correspondence to Yuya Koda.

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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.

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Ishikawa, M., Koda, Y. Stable maps and branched shadows of 3-manifolds. Math. Ann. 367, 1819–1863 (2017). https://doi.org/10.1007/s00208-016-1403-4

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