Generic Projections in the Semi-nice Dimensions

Abstract

In his celebrated paper ‘Generic Projections’, Mather obtained the key result that a generic projection to an affine subspace of a smooth submanifold in Euclidean space is jet-transverse to any ‘modular’ submanifold of (multi-) jet space. He also gave an explicit stratification by modular submanifolds, and used it to conclude that the projection, if in the nice dimensions, is generically (C^∞-) stable. In this article, we extend the result to the semi-nice dimensions (where only C⁰-stability is obtained), using the stratification given in our book. We first recall the definitions of the nice and semi-nice dimensions, review the main known results which involve them, and proceed to the statement of our main results. Next we discuss the condition of modularity, and present a number of methods for establishing modularity of particular strata. Finally, we show that all the strata needed for the main result are covered by these methods.