# On a construction of stable maps from 3-manifolds into surfaces

## TL;DR

This paper presents a visual method to construct stable maps from 3-manifolds, specifically from the 3-sphere and other closed orientable 3-manifolds, into surfaces, with controlled singularities and fiber types.

## Contribution

It introduces a new visual construction technique for stable maps from 3-manifolds into surfaces, linking links in the 3-sphere to specific stable map properties.

## Key findings

- Constructed stable maps with no cusp points
- Set of definite fold points coincides with the given link
- Obtained stable maps into the 2-sphere for all closed orientable 3-manifolds

## Abstract

For any link in the $3$-sphere, we give a visual construction of a stable map $f$ from the $3$-sphere into the real plane enjoying the following properties; $f$ has no cusp point, the set of definite fold points of $f$ coincides with the given link and $f$ only has certain type of fibers containing two indefinite fold points. As a corollary, we obtain a similar stable map from every closed orientable $3$-manifold into the $2$-sphere.