The extra-nice dimensions

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