Topologically Nontrivial Counterexamples to Sard’s Theorem

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 $\operatorname{rank} df=n$ on $\Omega $ and $f(\Omega )$ is dense in $\mathbb{S}^n$, while for any $n\geq 4$, 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 $\operatorname{rank} df<n$ everywhere. The result in the case $n\geq 4$ answers a question of Larry Guth.