Maps on 3-manifolds given by surgery

TL;DR

This paper constructs stable maps from 3-manifolds obtained by surgery to the plane, providing bounds on singularities and fiber components based on the link properties.

Contribution

It introduces a method to produce stable maps for 3-manifolds from surgery diagrams and establishes bounds related to the link structure.

Findings

Upper bounds for crossings and singularities

Bounds on connected components of fibers

Construction of stable maps from surgery presentations

Abstract

Suppose that the 3-manifold M is given by integral surgery along a link L in S^3. In the following we construct a stable map from M to the plane, whose singular set is canonically oriented. We obtain upper bounds for the minimal numbers of crossings and non-simple singularities and of connected components of fibers of stable maps from M to the plane in terms of properties of L.