{"paper":{"title":"Homotopy groups of spheres and Lipschitz homotopy groups of Heisenberg groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.GT","authors_text":"Armin Schikorra, Jeremy T. Tyson, Piotr Hajlasz","submitted_at":"2013-01-21T20:45:37Z","abstract_excerpt":"We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, $\\pi_m^{Lip}(H_n)$, in terms of properties of the classical homotopy group of the sphere, $\\pi_m(S^n)$. As an application we provide a new simplified proof of the fact that $\\pi_n^{Lip}(H_n)\\neq 0$, $n=1,2,...$, and we prove a new result that $\\pi_{4n-1}^{Lip}(H_{2n})\\neq 0$ for $n=1,2,...$ The last result is based on a new generalization of the Hopf invariant. We also prove that Lipschitz mappings are not dense in the Sobolev space $W^{1,p}(M,H_{2n})$ when $dim M\\geq 4n$ and $4n-1\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.4978","kind":"arxiv","version":3},"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"}