Maps on 3-manifolds given by surgery (Q442039)

Consider a smooth 3-manifold \(M\). Any continuous map from \(M\) to the plane can be deformed to obtain a smooth, stable map. Hence, theoretically, there is an abundance of stable maps from \(M\) to the plane. Describing one of them is still challenging, and this is the goal of the present paper.NEWLINENEWLINELet \(M\) be given by integral surgery along a link \(L\) in the 3-sphere. Starting with this description the authors construct a sequence of stable maps \(f_1,\ldots,f_6\) with more and more desirable properties. The steps of deformations to obtain one from the preceding one are illustrated by several clear pictures. Along the way the authors keep track of the number and complexity of the singularities and the fibers. One of their main technical tools is the Stein factorization of a stable map.NEWLINENEWLINEAn immediate consequence of such a construction is a collection of upper bounds on the minimal number of certain singularities of the map, as well as on the minimal number of connected components of the fibers. In particular they show that certain locally stable singularities can always be eliminated. As an additional result, the construction of the paper leads to upper bounds on a version of the Thurston-Bennequin number of negative Legendrian knots.NEWLINENEWLINEThe authors compare their upper bounds with known lower bounds on the topological complexity of the singularities of the map.