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Simplifying generic smooth maps to the 2-sphere and to the plane

Osamu Saeki

TL;DR

The paper develops a Cerf-theoretic framework to deform generic smooth maps $f: M^n \to S^2$ (and to $\mathbb{R}^2$) so that the resulting $C^\infty$ stable maps have highly controlled singularities. It proves that for $n$ odd, all folds can be restricted to absolute index $\frac{n-1}{2}$ with no cusps, and for $n$ even, folds have index $\frac{n-2}{2}$ with at most one cusp, while preserving an embedding on the singular-set image in the even case; it also yields a map to $\mathbb{R}^2$ with at most one cusp and an embedding on the singular set. The results yield constructive open-book extensions in odd dimensions and, for $n \ge 7$ odd and suitable $k$, a fold-free map into $\mathbb{R}^2$ avoiding folds of low absolute indices for $k$-connected $n$-manifolds. Collectively, these contributions provide new, explicit tools for simplifying and controlling singularities of low-dimensional target maps and connect to open-book structures in topology.

Abstract

We study how to construct explicit deformations of generic smooth maps from closed $n$--dimensional manifolds $M$ with $n \geq 2$ to the $2$--sphere $S^2$ and show that every smooth map $M \to S^2$ is homotopic to a $C^\infty$ stable map with at most one cusp point and with only folds of the middle absolute index. Furthermore, if $n$ is even, such a $C^\infty$ stable map can be so constructed that the restriction to the singular point set is a topological embedding. As a corollary, we show that for $n \geq 2$ even, there always exists a $C^\infty$ stable map $M \to \mathbf{R}^2$ with at most one cusp point such that the restriction to the singular point set is a topological embedding. As another corollary, we give a new proof to the existence of an open book structure on odd dimensional manifolds which extends a given one on the boundary, originally due to Quinn. Finally, using the open book structure thus constructed, we show that $k$--connected $n$--dimensional manifolds always admit a fold map into $\mathbf{R}^2$ without folds of absolute indices $i$ with $1 \leq i \leq k$, for $n \geq 7$ odd and $1 \leq k \leq (n-5)/2$.

Simplifying generic smooth maps to the 2-sphere and to the plane

TL;DR

The paper develops a Cerf-theoretic framework to deform generic smooth maps (and to ) so that the resulting stable maps have highly controlled singularities. It proves that for odd, all folds can be restricted to absolute index with no cusps, and for even, folds have index with at most one cusp, while preserving an embedding on the singular-set image in the even case; it also yields a map to with at most one cusp and an embedding on the singular set. The results yield constructive open-book extensions in odd dimensions and, for odd and suitable , a fold-free map into avoiding folds of low absolute indices for -connected -manifolds. Collectively, these contributions provide new, explicit tools for simplifying and controlling singularities of low-dimensional target maps and connect to open-book structures in topology.

Abstract

We study how to construct explicit deformations of generic smooth maps from closed --dimensional manifolds with to the --sphere and show that every smooth map is homotopic to a stable map with at most one cusp point and with only folds of the middle absolute index. Furthermore, if is even, such a stable map can be so constructed that the restriction to the singular point set is a topological embedding. As a corollary, we show that for even, there always exists a stable map with at most one cusp point such that the restriction to the singular point set is a topological embedding. As another corollary, we give a new proof to the existence of an open book structure on odd dimensional manifolds which extends a given one on the boundary, originally due to Quinn. Finally, using the open book structure thus constructed, we show that --connected --dimensional manifolds always admit a fold map into without folds of absolute indices with , for odd and .
Paper Structure (8 sections, 12 theorems, 12 equations, 20 figures)

This paper contains 8 sections, 12 theorems, 12 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Generic maps into surfaces
  3. Diagrams and their moves
  4. Stable maps into $S^2$
  5. Maps into $\mathbb{R}^2$
  6. Relation to open book structures
  7. Elimination of indefinite folds of certain indices
  8. Singular point set

Key Result

Theorem 4.3

Let $M$ be a closed connected $n$--dimensional manifold with $n \geq 3$. Then, every map $M \to S^2$ is homotopic to a stable map $f : M \to S^2$ with the following properties.

Figures (20)

  • Figure 1: Normal orientation for $f|_{S(f)}$, depicted as thick downward arrow
  • Figure 2: Indices and normal orientations around the image of a cusp
  • Figure 3: Absolute indices and normal orientations around cusp images
  • Figure 4: Moves of type I
  • Figure 5: Moves of type II
  • ...and 15 more figures

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • ...and 39 more