Linking between singular locus and regular fibers

TL;DR

This paper investigates the relationship between the singular locus and regular fibers in generic maps from 3-manifolds to the plane, focusing on links that can serve as singular point sets with prescribed regular fibers.

Contribution

It characterizes which links in the complement of a given null-cobordant framed link can be realized as singular sets of generic maps with specified regular fibers.

Findings

Identifies links in the complement of L that can be singular sets.

Analyzes the linking behavior between singular points and regular fibers.

Provides conditions for realizing certain links as singular sets.

Abstract

Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an oriented framed regular fiber. Then, we study the linking behavior between the singular point set and regular fibers for generic maps of $M$ into $\mathbb{R}^2$.