Relative homotopy groups and Serre fibrations for polynomial maps

Abstract

Let 𝑓 be a polynomial map from ℝ^𝑚 to ℝ^𝑛 with 𝑚 > 𝑛 > 0 and 𝑡₀ be a regular value of 𝑓. For a small open ball 𝐷_{𝑡₀} centered at 𝑡₀, we show that the map 𝑓 : 𝑓⁻¹(𝐷_{𝑡₀}) → 𝐷_{𝑡₀} is a Serre fibration if and only if 𝑓 is a Serre fibration over a finite number of certain simple arcs starting at 𝑡₀. We characterize the fibration 𝑓 : 𝑓⁻¹(𝐷_{𝑡₀}) → 𝐷_{𝑡₀} by relative homotopy groups defined for these arcs and use it to prove the assertion.