Round fold maps of n--dimensional manifolds into ${\mathbb{R}}^{n-1}$

TL;DR

This paper classifies smooth n-dimensional closed manifolds that admit a special type of fold map into (n-1)-dimensional space, focusing on the structure of their critical value sets and equivalence classes.

Contribution

It characterizes and classifies n-dimensional closed manifolds admitting round fold maps into (n-1)-space, detailing the structure of critical value sets and equivalence classes.

Findings

Identifies manifolds admitting round fold maps into ${f R}^{n-1}$.

Classifies such maps up to $C^{inity}$ $ ext{A}$-equivalence.

Describes the structure of critical value sets as disjoint spheres.

Abstract

We determine those smooth $n$--dimensional closed manifolds with $n \geq 4$ which admit round fold maps into ${\mathbb{R}}^{n-1}$, i.e.\ fold maps whose critical value sets consist of disjoint spheres of dimension $n-2$ isotopic to concentric spheres. We also classify such round fold maps up to $C^{\infty}$ $\mathcal{A}$--equivalence.