Old and new results on density of stable mappings

TL;DR

This paper reviews the density of stable mappings, discussing Whitney's singularity theory, Thom-Mather stability, and recent advances, highlighting the conditions under which stable maps are dense and open questions about Lipschitz stability.

Contribution

It provides a comprehensive review of classical and recent results on the density of stable maps, including new insights into Thom-Mather maps and open problems on Lipschitz stability.

Findings

Density of proper stable maps in nice dimensions

Density of topologically stable maps in all dimensions

Open questions on Lipschitz stable map density

Abstract

Density of stable maps is the common thread of this paper. We review Whitney's contribution to singularities of differentiable mappings and Thom-Mather theories on $C^{\infty}$ and $C^{0}$-stability. Infinitesimal and algebraic methods are presented in order to prove Theorem A and Theorem B on density of proper stable and topologically stable mappings $f:N^{n}\to P^{p}.$ Theorem A states that the set of proper stable maps is dense in the set of all proper maps from $N$ to $P$, if and only if the pair $(n,p)$ is in \emph{nice dimensions,} while Theorem B shows that density of topologically stable maps holds for any pair $(n,p).$ A short review of results by du Plessis and Wall on the range in which proper smooth mappings are $C^{1}$- stable is given. A Thom-Mather map is a topologically stable map $f:N \to P$ whose associated $k$-jet map $j^{k}f:N \to P$ is transverse to the Thom-Mather stratification in $J^{k}(N,P).$ We give a detailed description of Thom-Mather maps for pairs $(n,p)$ in the boundary of the nice dimensions.The main open question on density of stable mappings is to determine the pairs $(n,p)$ for which Lipschitz stable mappings are dense. We discuss recent results by Nguyen, Ruas and Trivedi on this subject, formulating conjectures for the density of Lipschitz stable mappings in the boundary of the nice dimensions. At the final section, Damon's results relating $\mathcal{A}$-classification of map-germs and $\mathcal{K}_{V}$ classification of sections of the discriminant $V=\Delta(F)$ of a stable unfolding of $f$ are reviewed and open problems are discussed.