The extra-nice dimensions

TL;DR

This paper introduces the concept of extra-nice dimensions, characterizes them through properties of stable germs and discriminants, and explores their relation to the simplicity of certain map germs, extending Mather's work on nice dimensions.

Contribution

It defines and characterizes the extra-nice dimensions, linking them to the density of pseudo-isotopies and properties of stable germs, and addresses the simplicity of $ ext{A}_e$-codimension 2 germs.

Findings

Extra-nice dimensions are characterized by hyperplane sections of stable germs having $ ext{A}_e$-codimension 1.

A sufficient condition for $ ext{A}_e$-codimension 2 germs to be simple is provided.

The boundary of the extra-nice dimensions is established, and a question by Wall on non-simple map codimension is answered.

Abstract

We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in $C^{\infty}(N\times[0,1],P)$, also known as pseudo-isotopies, is dense if and only if the pair of dimensions $(\dim N, \dim P)$ is in the extra-nice dimensions. This result is parallel to Mather's characterization of the nice dimensions as the pairs $(n,p)$ for which stable maps are dense. The extra-nice dimensions are characterized by the property that discriminants of stable germs in one dimension higher have $\mathcal A_e$-codimension 1 hyperplane sections. They are also related to the simplicity of $\mathcal A_e$-codimension 2 germs. We give a sufficient condition for any $\mathscr A_e$-codimension 2 germ to be simple and give an example of a corank 2 codimension 2 germ in the nice dimensions which is not simple. Then we establish the boundary of the extra-nice dimensions. Finally we answer a question posed by Wall about the codimension of non-simple maps.