{"paper":{"title":"On the distance between homotopy classes in $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Itai Shafrir","submitted_at":"2017-07-18T17:47:38Z","abstract_excerpt":"For every $p\\in(1,\\infty)$ there is a natural notion of topological degree for maps in $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)$ which allows us to write that space as a disjoint union of classes, $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)=\\bigcup_{d\\in{\\mathbb Z}}\\mathcal{E}_d$. For every pair $d_1,d_2\\in {\\mathbb Z}$, we show that the distance $\\text{Dist}_{W^{1/p,p}}({\\mathcal\n  E}_{d_1}, {\\mathcal E}_{d_2}):=\\sup_{f\\in{\\mathcal E}_{d_1}}\\ \\inf_{g\\in{\\mathcal E}_{d_2}}\\ d_{W^{1/p,p}}(f, g)$ equals the minimal $W^{1/p,p}$-energy in $\\mathcal{E}_{d_1-d_2}$. In the special case $p=2$ we deduce fro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.05764","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"}