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

Koki Iwakura

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.

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

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 -dimensional manifolds that serve as the source manifolds of non-singular extensions.

Paper Structure

This paper contains 5 sections, 6 theorems, 25 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Examples
  4. Main theorem and its proof
  5. Properties of the source manifolds of extensions

Key Result

Proposition 2.5

Let $N$ be a compact $3$-dimensional manifold with non-empty boundary, and $F\colon N\to\mathbb{R}^{2}$ be a submersion, which is a stable fold map on $\partial N$. Let $p\in\partial N$. Then, there exist local coordinates $(t, x_{1},x_{2})$ of $N$ around $p$ and $(y_{1},y_{2})$ of $\mathbb{R}^{2}$ where $t=0$ and $t>0$ correspond to $\partial N$ and ${\rm{Int}} N$, respectively. $\blacktrianglel

Figures (12)

  • Figure 1: The figure on the left- (resp. right-) hand side represents $F$ around $p\in S_{{\rm{I}}}(F)$ (resp. $p\in S_{{\rm{II}}}(F)$).
  • Figure 3: The submersion $g_1$ and the signed graph $G_{f_1}$.
  • Figure 4: The pairing map $\delta_1$ from $\mathcal{R}_{f_1}$ to $\mathcal{M}_{f_1}$.
  • Figure 5: The submersion $g_2$ and the signed graph $G_{f_2}$.
  • Figure 6: The pairing map $\delta_2$ from $\mathcal{R}_{f_2}$ to $\mathcal{M}_{f_2}$.
  • ...and 7 more figures

Theorems & Definitions (24)

  • Remark 2.1
  • Definition 2.2: Horizontal
  • Definition 2.3: Non-singular extension
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8: Signed graph
  • Remark 2.9
  • Remark 2.10
  • ...and 14 more