Regularity at infinity of real mappings and a Morse-Sard theorem

TL;DR

This paper establishes a new Morse-Sard theorem for asymptotic critical values and a fibration theorem at infinity for semi-algebraic and $C^2$ mappings, linking various regularity conditions to control asymptotic behavior.

Contribution

It introduces a unified framework connecting different regularity conditions and proves new theorems on the asymptotic behavior of semi-algebraic and $C^2$ mappings.

Findings

Equivalence of three regularity conditions at infinity

New Morse-Sard theorem for asymptotic critical values

Fibration theorem at infinity for $C^2$ mappings

Abstract

We prove a new Morse-Sard type theorem for the asymptotic critical values of semi-algebraic mappings and a new fibration theorem at infinity for $C^2$ mappings. We show the equivalence of three different types of regularity conditions which have been used in the literature in order to control the asymptotic behaviour of mappings. The central role of our picture is played by the $t$-regularity and its bridge toward the $\rho$-regularity which implies topological triviality at infinity.