  Turaev's 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.
