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

TL;DR

This paper demonstrates how to explicitly deform smooth maps from closed manifolds to the 2-sphere or plane into stable maps with minimal singularities, providing new insights into their structure and applications.

Contribution

It constructs explicit homotopies to stable maps with controlled singularities and introduces new proofs for open book decompositions and fold maps on manifolds.

Findings

Every smooth map to S^2 is homotopic to a stable map with at most one cusp.

For even n, such maps can restrict to topological embeddings on singular sets.

Existence of fold maps into R^2 with controlled singularities on certain manifolds.

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 $1 \leq k \leq (n-5)/2$.