The homotopy principle for maps with singularities of given $\mathscr{K}$-invariant class

Abstract

Let N and P be smooth manifolds of dimensions n and p respectively such that n≥p≥2 or n<p. Let $\mathscr{O}$(N,P) denote an open subspace of J^∞(N,P) which consists of all regular jets and singular jets of certain given $\mathscr{K}$-invariant class (including fold jets if n≥p). An $\mathscr{O}$-regular map f:N→P refers to a smooth map such that j^∞f(N)⊂\mathscr{O}(N,P). We will prove that a continuous section s of $\mathscr{O}$(N,P) over N has an $\mathscr{O}$-regular map f such that s and j^∞f are homotopic as sections. As an application we will prove this homotopy principle for maps with $\mathscr{K}$-simple singularities of given class.