Constructing near-embeddings of codimension one manifolds with countable dense singular sets

TL;DR

This paper constructs simple examples of continuous maps from closed (n-1)-manifolds to closed n-manifolds with countable dense singular sets that can still be approximated by embeddings, expanding understanding of near-embeddings.

Contribution

It provides explicit examples of near-embeddings with dense singular sets in higher dimensions, demonstrating their approximation properties.

Findings

Existence of near-embeddings with dense singular sets in all dimensions n ≥ 3

Such maps can be approximated arbitrarily closely by embeddings

Examples are simple and explicit, broadening the class of known near-embeddings.

Abstract

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an embedding, i.e. $f$ is a {\sl near-embedding}.