Bifurcation values of $C^{\infty} functions

TL;DR

This paper explores methods for trivializing smooth functions using trivializations and extends these techniques to functions on hypersurfaces satisfying certain regularity conditions.

Contribution

It introduces a way to construct trivializations of smooth functions via trivializations of their domain and extends these methods to functions on hypersurfaces with $ ho_0$-regularity.

Findings

Method for trivializing functions via domain trivialization

Extension of trivialization techniques to hypersurfaces

Applicable to functions satisfying $ ho_0$-regularity condition

Abstract

We show how one can use a trivialization of a function $f : \mathbb{R}^n\to\mathbb{R} $ to construct a trivialization of $f$ in $\mathbb{R}^n$. Additionally we adopt a method for trivialising functions which satisfy the $\rho_0$-regularity condition to the case of functions defined on hypersurfaces of the form $M=g^{-1}(0)$.