Topologically nontrivial counterexamples to Sard's theorem, II

My talk is based on my recent joint work with Pawel Goldstein and Pekka Pankka. We prove the following dichotomy:

if $n=2,3$ and $f\in C^1(S^{n+1},S^n)$ is not homotopic to a constant map,

then there is an open set $\Omega\subset S^{n+1}$ such that ${\rm rank}\, Df=n$ on $\Omega$ and $f(\Omega)$ is dense in $S^n$, while

for any $n\geq 4$, there is a map $f\in C^1(S^{n+1},S^n)$ that is not homotopic to a constant map and such that

${\rm rank} Df<n$ everywhere. The proofs are based on a mixture of methods: generalized Hopf invariant, Hodge decomposition, and 

the Freudenthal suspension theorem. In the first talk I will discuss the case $n=2,3$ and in the second one the case $n\geq 4$.