Isotopies vis-\`a-vis level-preserving embeddings

TL;DR

This paper presents a counterexample to the common claim in differential topology that the track of an isotopy of embeddings always results in a level-preserving embedding.

Contribution

It challenges a standard assertion by providing a simple counterexample, clarifying the limitations of level-preserving embeddings during isotopies.

Findings

Counterexample disproves the general claim

Level-preserving map need not be an embedding during isotopy

Clarifies misconceptions in differential topology literature

Abstract

Various standard texts on differential topology maintain that the level-preserving map defined by the track of an isotopy of embeddings is itself an embedding. This note describes a simple counterexample to this assertion.