Stable maps and branched shadows of 3-manifolds

2014/03/03 by Ishikawa, Masaharu, Koda, Yuya · 2 citations
#57M27 #57N70 #57R45 #58K15 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1403.0596

Abstract

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.

Cited by

• On a construction of stable maps from 3-manifolds into surfaces
• Analysis of Hopf solitons as generalized fold maps

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• Stable maps and hyperbolic links
• Shadow-complexity and trisection genus
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• Four-manifolds with shadow-complexity one
• On a construction of stable maps from 3-manifolds into surfaces