Wrinkling of smooth mappings and its applications. I

Abstract

This paper, together with its sequel papers [EM1] and [EM2], contains a reexposition of the authors' theory of mappings with simple singularities (see [E1], [E2], [M1], [M2]). In addition, we give here new applications of the theory. The method of wrinkling of smooth mappings described in these papers provides an easier path to theorems from [E1] and [E2] about construction of mappings with prescribed singularities. It gives an alternative proof of Thurston's theorem about foliations of codimension >1 (see [Th] and [ME]). The method gives a new and, we think, simpler proof of K. Igusa's theorem on mappings without higher singularities (see [Ig]). Moreover, it allows us to remove all dimensional restrictions in this theorem. The idea of wrinkling ®rst appeared, in a somewhat implicit form, in S. Smale's work on regular homotopy (see [Sm]) and later was used by V. Poenaru (see [Po]). It was then transformed by M. Gromov (see [G1]) into a powerful general tool for solving partial di€erential relations. In the context of singularities the wrinkling was ®rst explicitly described by one of the authors (see [M1]). The three papers (the present one, [EM1] and [EM2]) have the following organization. We begin by introducing the notion of a wrinkled map. This is a map which can have fold and cusp singularities of very simple topology (see Fig. 1 and 1.2±1.3 below for the exact de®nition). Then we formulate the main technical result of the paper (see Theorem 1.5A below) and its globalized versions 1.6A and 1.6B which give an h-principle type description of Invent. math. 130, 345 ± 369 (1997)