  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.
