A Thom Isotopy Theorem for nonproper semialgebraic maps

TL;DR

This paper extends the Thom Isotopy Theorem to nonproper semialgebraic maps, providing a new understanding of their topological behavior in the context of semialgebraic geometry.

Contribution

It establishes a version of the Thom Isotopy Theorem applicable to nonproper semialgebraic maps, broadening the scope of topological invariance results.

Findings

Proves a Thom Isotopy Theorem for nonproper semialgebraic maps

Demonstrates topological invariance under certain conditions for these maps

Extends classical results to a broader class of semialgebraic maps

Abstract

We prove a version of the Thom Isotopy Theorem for nonproper semialgebraic maps $f\colon X\rightarrow \mathbb{R}^m$, where $X \subset\mathbb{R}^n$ is a semialgebraic set and $f$ is the restriction to $X$ of a smooth semialgebraic map $F:\mathbb{R}^n\to \mathbb{R}^m$.