Stable maps and branched shadows of 3-manifolds

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5 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1819-1863
Number of pages45
JournalMathematische Annalen
Issue number3-4
Publication statusPublished - 2017 Apr 1
Externally publishedYes


  • 57M27
  • 57N70
  • 57R45
  • 58K15

ASJC Scopus subject areas

  • General Mathematics


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