Relative homotopy groups and Serre fibrations for polynomial maps

TL;DR

This paper characterizes when polynomial maps from real space to real space are Serre fibrations over small neighborhoods of regular values, using relative homotopy groups and simple arc structures.

Contribution

It provides a new criterion for Serre fibrations of polynomial maps based on relative homotopy groups and simple arc analysis.

Findings

Serre fibration condition characterized by simple arcs

Use of relative homotopy groups for polynomial maps

Fibration over small neighborhoods determined by arc structure

Abstract

Let $f$ be a polynomial map from $\mathbb R^m$ to $\mathbb R^n$ with $m>n>0$ and $t_0$ be a regular value of $f$. For a small open ball $D_{t_0}$ centered at $t_0$, we show that the map $f:f^{-1}(D_{t_0})\to D_{t_0}$ is a Serre fibration if and only if $f$ is a Serre fibration over a finite number of certain simple arcs starting at $t_0$. We characterize the fibration $f:f^{-1}(D_{t_0})\to D_{t_0}$ by relative homotopy groups defined for these arcs and use it to prove the assertion.