Round fold maps on $3$--manifolds

TL;DR

This paper characterizes which closed orientable 3-manifolds admit special types of fold maps into the plane, linking the existence of these maps to the manifold's structure as a graph manifold.

Contribution

It provides a complete characterization of closed orientable 3-manifolds that admit round fold maps and directed round fold maps into the plane, extending previous results.

Findings

A closed orientable 3-manifold admits a round fold map into the plane iff it is a graph manifold.

Characterization of graph manifolds admitting directed round fold maps into the plane.

Extension of the characterization for simple stable maps to more general fold maps.

Abstract

We show that a closed orientable 3--dimensional manifold admits a round fold map into the plane, i.e. a fold map whose critical value set consists of disjoint simple closed curves isotopic to concentric circles, if and only if it is a graph manifold, generalizing the characterization for simple stable maps into the plane. Furthermore, we also give a characterization of closed orientable graph manifolds that admit directed round fold maps into the plane, i.e.\ round fold maps such that the number of regular fiber components of a regular value increases toward the central region in the plane.