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 $n\geq 4$, 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 $\mathrm{ rank}\, df<n$ everywhere. The result in the case $n\geq 4$ answers a question of Larry Guth.
Categories: math.CA math.AT