Stratified {{mathcal {C}}}^p-semialgebraic triviality

Anna Valette

doi:10.1007/s00229-020-01227-w

Abstract

We show that a stratified submersion between stratified semialgebraic sets is locally {{mathcal {C}}}^p-semialgebraically trivial outside the set of stratified generalized critical values.

Highlights

One of the main issues in singularity theory is the study of stability of functions and mappings
Recall that a local semialgebraic trivialization of the map f : X → Y at a point y ∈ Y is a semialgebraic homeomorphism h : f −1(Uy) → T × Uy, where Uy is a neighbourhood of y in Y and T is a semialgebraic set, such that π ◦ h = f, π : T × Uy → Uy coincides with the canonical projection onto Uy
We say that the map f : X → Y is locally C p-semialgebraically stratified trivial if it is locally trivial and if there exists a stratification of X such that each local trivialization h = (h y, f ) : f −1(Uy) → f −1(y) × Y induces on every stratum SjaCp semialgebraic diffeomorphism h|Sj : S j → (S j ∩ f −1(y)) × Y

Summary

Introduction

One of the main issues in singularity theory is the study of stability of functions and mappings. Simon [10] that f admits local C p trivializations (not necessarily semialgebraic) on the preimage of the complement of the set K0( f ) ∪ K∞( f ), where K0( f ) is the set of critical values and K∞( f ) is the set of asymptotic critical values This is a special case of Rabier’s fibration theorem [11]. In an interesting paper [7], introduced the so called t-regularity condition and compared it with other regularity conditions In particular they proved that in the case of C1 semialgebraic mappings f : Rn → Rk the notions of asymptotic critical point and t-regular point.

Basic definitions

Stratified generalized critical values

First Isotopy Lemma