Topologically nontrivial counterexamples to Sard's theorem

Abstract

We prove the following dichotomy: if $n=2,3$ and $f\in C^1({\mathbb{S}}^{n+1},{\mathbb{S}}^n)$ is not homotopic to a constant map, then there is an open set $\Omega \subset {\mathbb{S}}^{n+1}$ such that $\mathrm{rank}df=n$ on $\Omega$ and $f(\Omega )$ is dense in ${\mathbb{S}}^n$, while for any $n\ge 4$, there is a map $f\in C^1({\mathbb{S}}^{n+1},{\mathbb{S}}^n)$ that is not homotopic to a constant map and such that $\mathrm{rank}df<n$ everywhere. The result in the case $n\ge 4$ answers a question of Larry Guth.