The extra-nice dimensions

Abstract

We define the extra-nice dimensions and prove that the subset of locally stable 1-parameter families in C^{infty }(Ntimes [0,1],P) 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 {mathscr {A}}_e-codimension 1 hyperplane sections. They are also related to the simplicity of {mathscr {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.