Stable maps and branched shadows of 3-manifolds

TL;DR

This paper introduces the stable map complexity for 3-manifolds, linking it to branched shadows, and characterizes hyperbolic links with minimal stable map complexity, connecting it to shadow complexity and hyperbolic volume estimates.

Contribution

It defines stable map complexity for 3-manifolds and proves its equivalence to branched shadow vertex count, providing new characterizations of hyperbolic links and insights into shadow and volume complexities.

Findings

Stable map complexity equals minimal branched shadow vertices.

Characterization of hyperbolic links with stable map complexity 1.

Observation of the relation between stable map complexity, shadow complexity, and hyperbolic volume.

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.