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Topology of boundary special generic maps into Euclidean spaces

Koki Iwakura

TL;DR

This work introduces boundary special generic maps as a natural generalization of special generic maps to manifolds with boundary, defining boundary definite fold points and analyzing the global topology via Reeb spaces. It proves a disk‑bundle decomposition of the source manifold along the Reeb space and provides precise classifications for when such maps exist into low‑dimensional targets ($\mathbb{R}$, $\mathbb{R}^2$, $\mathbb{R}^3$), revealing strong restrictions on the diffeomorphism types of the source. The paper then connects these boundary phenomena to non‑singular extension problems for maps of closed manifolds, deriving necessary conditions and constructing many new examples of special generic maps that do not arise as boundary special generic non‑extensions. Overall, the results illuminate how boundary singularities constrain global topology and offer new obstructions in the non‑singular extension program, with implications for classical objects such as lens spaces, Brieskorn spheres, and 2‑handlebodies.

Abstract

A boundary special generic map is a submersion from a compact, connected manifold with non-empty boundary into Euclidean space, whose restriction to the boundary has only boundary definite fold points as singular points. Such maps have been introduced by Shibata in the case of $3$-dimensional manifolds into the plane. In this paper, we generalize the definition and study its differential topological properties. As an application, we investigate the non-singular extension problem for maps of closed manifolds. In particular, by combining our results with known results on special generic maps, we obtain many new examples that do not admit a boundary special generic map as a non-singular extension.

Topology of boundary special generic maps into Euclidean spaces

TL;DR

This work introduces boundary special generic maps as a natural generalization of special generic maps to manifolds with boundary, defining boundary definite fold points and analyzing the global topology via Reeb spaces. It proves a disk‑bundle decomposition of the source manifold along the Reeb space and provides precise classifications for when such maps exist into low‑dimensional targets (, , ), revealing strong restrictions on the diffeomorphism types of the source. The paper then connects these boundary phenomena to non‑singular extension problems for maps of closed manifolds, deriving necessary conditions and constructing many new examples of special generic maps that do not arise as boundary special generic non‑extensions. Overall, the results illuminate how boundary singularities constrain global topology and offer new obstructions in the non‑singular extension program, with implications for classical objects such as lens spaces, Brieskorn spheres, and 2‑handlebodies.

Abstract

A boundary special generic map is a submersion from a compact, connected manifold with non-empty boundary into Euclidean space, whose restriction to the boundary has only boundary definite fold points as singular points. Such maps have been introduced by Shibata in the case of -dimensional manifolds into the plane. In this paper, we generalize the definition and study its differential topological properties. As an application, we investigate the non-singular extension problem for maps of closed manifolds. In particular, by combining our results with known results on special generic maps, we obtain many new examples that do not admit a boundary special generic map as a non-singular extension.

Paper Structure

This paper contains 16 sections, 23 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. On boundary special generic maps
  3. Basic terms in singularity theory
  4. Definition of boundary special generic maps
  5. Decomposition of the source manifolds
  6. Manifolds admiting boundary special generic maps into $\mathbb{R},\mathbb{R}^2$, and $\mathbb{R}^3$
  7. Case of functions
  8. Case of maps into $\mathbb{R}^2$
  9. Case of maps into $\mathbb{R}^3$
  10. Applications to non-singular extensions of maps
  11. Definition of non-singular extensions
  12. Definition of special generic maps
  13. Conditions for non-singular extensions
  14. Case for functions
  15. Case for maps into $\mathbb{R}^2$
  16. ...and 1 more sections

Key Result

Proposition 1

The Reeb space $W_F$ is a compact, connected, orientable $k$-dimensional manifold with boundary. Moreover, the Reeb map $\overline{F}\colon W_F\to\mathbb{R}^k$ is an immersion, and the quotient map $q_F\colon N\to W_F$ is a smooth map, in such a way that the restriction $q_F|_{S(F|_{\partial N})}\co

Figures (1)

  • Figure 1: A fold map $f\colon S^2\to\mathbb{R}^2$ that admits a non-singular extension but has no boundary special generic map as a non-singular extension.

Theorems & Definitions (43)

  • Definition 1: Singular point
  • Definition 2: Reeb space
  • Definition 3: Boundary definite fold point
  • Definition 4: Boundary special generic map
  • Remark 1
  • Proposition 1
  • Lemma 2.1
  • Proposition 2
  • Remark 2
  • Proposition 3
  • ...and 33 more