Topologically nontrivial counterexamples to Sard's theorem

TL;DR

This paper constructs counterexamples to Sard's theorem on spheres, showing that for certain dimensions, nontrivial maps can have dense images with full rank on open sets or be rank-deficient everywhere, challenging classical assumptions.

Contribution

It establishes a dimension-dependent dichotomy for smooth maps between spheres, providing the first known counterexamples in higher dimensions and answering a question posed by Larry Guth.

Findings

For n=2,3, nontrivial maps have dense images with full rank on open sets.

For n≥4, there exist nontrivial maps with rank less than n everywhere.

The results demonstrate topologically nontrivial maps can violate classical Sard's theorem expectations.

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 $n\geq 4$, 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.