Relative homotopy groups and Serre fibrations for polynomial maps

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.