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