{"paper":{"title":"Topologically nontrivial counterexamples to Sard's theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CA","authors_text":"Pawe{\\l} Goldstein, Pekka Pankka, Piotr Haj{\\l}asz","submitted_at":"2018-04-20T15:03:03Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07658","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}