Approach to low-dimensional topology from singularity theory: Evaluation of hyperbolic volume by stable mapping

This is an introductory article on studies of the evaluation of the Gromov norm of a 3-dimensional manifold, especially from a point of view of singularity theory. We will explain that the Gromov norm can be evaluated by the number of certain singular fibers of stable maps from the 3-dimensional manifold to the real plane under a certain condition. Polyhedrons obtained as the Stein factorizations of stable maps and Turaev’s shadows play important roles. The condition to have the evaluation is given by the gleams of shadows. The article starts from a basic observation of the same problem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional manifolds.