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Under certain assumptions on $f$ we compute the group $\\pi_0\\mathcal{S}(f,\\partial B)$ of isotopy classes of such diffeomorphisms. In fact, those computations hold for functions $f:B\\to\\mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials $\\mathbb{R}^2\\to\\mathbb{R}$ without multiple factors.\n  Togethe"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.03528","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2019-01-11T09:46:18Z","cross_cats_sorted":[],"title_canon_sha256":"c0bd115551426e1607c016cd754526dfa3e4c758350d2d859d2aa3c0ef2c80c7","abstract_canon_sha256":"58bf4825151535e7015279dba4e27e69d0a7fae14ec0246937986320bce50313"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:31.001418Z","signature_b64":"IujYp8bKPAxYoxF2JHqlTyrlE4g/bFmBBQHbFSK61d5FzdvXqkfKfJlwjDJHv7h5yYurG3RxvZ5Os4MeeO2TBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2df026e08d6b66fbecd1e342960764c85a8f9c6cedef68e6a9d2a730057afe9d","last_reissued_at":"2026-05-17T23:56:31.001004Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:31.001004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homotopy properties of smooth functions on the M\\\"obius band","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Iryna Kuznietsova, Sergiy Maksymenko","submitted_at":"2019-01-11T09:46:18Z","abstract_excerpt":"Let $B$ be a M\\\"obius band and $f:B \\to \\mathbb{R}$ be a Morse map taking a constant value on $\\partial B$, and $\\mathcal{S}(f,\\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\\partial B$ and preserving $f$ in the sense that $f\\circ h = f$. 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In fact, those computations hold for functions $f:B\\to\\mathbb{R}$ whose germs at critical points are smoothly equivalent to homogeneous polynomials $\\mathbb{R}^2\\to\\mathbb{R}$ without multiple factors.\n  Togethe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.03528","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1901.03528","created_at":"2026-05-17T23:56:31.001063+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.03528v1","created_at":"2026-05-17T23:56:31.001063+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.03528","created_at":"2026-05-17T23:56:31.001063+00:00"},{"alias_kind":"pith_short_12","alias_value":"FXYCNYENNNTP","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_16","alias_value":"FXYCNYENNNTPX3GR","created_at":"2026-05-18T12:33:18.533446+00:00"},{"alias_kind":"pith_short_8","alias_value":"FXYCNYEN","created_at":"2026-05-18T12:33:18.533446+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB","json":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB.json","graph_json":"https://pith.science/api/pith-number/FXYCNYENNNTPX3GR4NBJMB3EZB/graph.json","events_json":"https://pith.science/api/pith-number/FXYCNYENNNTPX3GR4NBJMB3EZB/events.json","paper":"https://pith.science/paper/FXYCNYEN"},"agent_actions":{"view_html":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB","download_json":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB.json","view_paper":"https://pith.science/paper/FXYCNYEN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.03528&json=true","fetch_graph":"https://pith.science/api/pith-number/FXYCNYENNNTPX3GR4NBJMB3EZB/graph.json","fetch_events":"https://pith.science/api/pith-number/FXYCNYENNNTPX3GR4NBJMB3EZB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB/action/storage_attestation","attest_author":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB/action/author_attestation","sign_citation":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB/action/citation_signature","submit_replication":"https://pith.science/pith/FXYCNYENNNTPX3GR4NBJMB3EZB/action/replication_record"}},"created_at":"2026-05-17T23:56:31.001063+00:00","updated_at":"2026-05-17T23:56:31.001063+00:00"}