Non-singular extensions of horizontal stable fold maps from surfaces to the plane

TL;DR

This paper investigates conditions for extending horizontal stable fold maps from surfaces to the plane without singularities, using combinatorial pairing maps and topological invariants.

Contribution

It introduces a pairing map criterion for non-singular extensions and computes topological invariants of source manifolds.

Findings

Existence of a non-singular extension is equivalent to the existence of a pairing map.

Computed Euler characteristics of source manifolds.

Determined fundamental groups of source manifolds.

Abstract

In this paper, we study the non-singular extension problem of horizontal stable fold maps. This problem asks what conditions ensure the existence of a submersion whose restriction to the boundary coincides with a given map, called a non-singular extension. By defining a combinatorial object called a pairing map, we prove that the existence of a non-singular extension is equivalent to the existence of a pairing map. Furthermore, to facilitate the application of the main theorem, we compute the Euler characteristics and the fundamental groups of compact $3$-dimensional manifolds that serve as the source manifolds of non-singular extensions.