Stable maps and branched shadows of 3-manifolds

As used in a paper of Costantino and D. Thurston, Turaev's shadow can be regarded locally as the Stein factorization of a stable map. In [1], we introduced 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 proved that this number equals its branched shadow complexity. In consequence, we see that the hyperbolic volume is bounded from above and below by the stable map complexity, which is a direct corollary of an observation of Costantino and Thurston and an inequality obtained by Futer, Kalfagianni and Purcell.
This is a joint work with Yuya Koda in Hiroshima University. Partially supported by the Grant-in-Aid for Scientific Research (C), JSPS KAKENHI Grant Number 16K05140.
References:

• M. Ishikawa, Y. Koda, Stable maps and branched shadows of 3-manifolds. arXiv:math/1403.0596.