{"paper":{"title":"Smooth functions that split a Klein bottle into two M\\\"obius bands","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT","math.DG"],"primary_cat":"math.GT","authors_text":"Bohdan Mazhar, Sergiy Maksymenko","submitted_at":"2025-08-27T07:25:12Z","abstract_excerpt":"Given a compact surface $M$, consider the right action $\\mathcal{C}^{\\infty}(M)\\times\\mathcal{D}(M)\\to\\mathcal{C}^{\\infty}(M)$, $(f, h) \\mapsto f\\circ h$, of the group $\\mathcal{D}(M)$ of $\\mathcal{C}^{\\infty}$ diffeomorphisms of $M$ on the space $\\mathcal{C}^{\\infty}(M)$ of $\\mathcal{C}^{\\infty}$ functions on $M$. For $f\\in\\mathcal{C}^{\\infty}(M)$ denote by $\\mathcal{O}(f)$ its orbit, and by $\\mathcal{O}_f(f)$ the path component of $\\mathcal{O}(f)$ containing $f$.\n  The paper continues a series of computations by many authors of homotopy types of orbits $\\mathcal{O}_f(f)$ of smooth functions "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.19636","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.19636/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}