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Denote by $S(f,r)$, $(0\\leq r \\leq \\infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $S(f,\\infty) = \\cdots = S(f,1)$ for all such $f$ and that $S(f,1) \\not= S(f,0)$ if and only if $f$ is a product of at least two distinct irreducible over $\\mathbb{R}$ quadratic forms."},"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":"0806.0159","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2008-06-01T17:16:37Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"776721156a343bdde996b022f94dda03fb188fc2dfb446da7e57fc2c1b4834db","abstract_canon_sha256":"09cf0c41e6715b2d4908ab26d3e508ef71fc1d6e3a26b9b522d5d41db3b7c946"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:48.588333Z","signature_b64":"Aqg/emejB8bYR6E1fEWZ8aqqGpy2OR8KSak2UyuDbLDyZY90kOIKxXQDHd8kMYS47mAnOzcOv4yE2q5trOCiDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"441cc6741c000357876b75cac5a69833d65d96271cbf950a587c093ba5d1dfc2","last_reissued_at":"2026-05-18T01:23:48.587687Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:48.587687Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Connected components of partition preserving diffeomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.DS","authors_text":"Sergiy Maksymenko","submitted_at":"2008-06-01T17:16:37Z","abstract_excerpt":"Let $f:\\mathbb{R}^2 \\to \\mathbb{R}$ be a real homogeneous polynomial and $S(f)$ be the group of diffeomorphisms $h:\\mathbb{R}^2 \\to \\mathbb{R}^2$ preserving $f$, i.e. $f \\circ h = f$. Denote by $S(f,r)$, $(0\\leq r \\leq \\infty)$, the identity path component of $S(f)$ with respect to the weak Whitney $C^{r}_{W}$-topology. We prove that $S(f,\\infty) = \\cdots = S(f,1)$ for all such $f$ and that $S(f,1) \\not= S(f,0)$ if and only if $f$ is a product of at least two distinct irreducible over $\\mathbb{R}$ quadratic forms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.0159","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0806.0159","created_at":"2026-05-18T01:23:48.587775+00:00"},{"alias_kind":"arxiv_version","alias_value":"0806.0159v3","created_at":"2026-05-18T01:23:48.587775+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.0159","created_at":"2026-05-18T01:23:48.587775+00:00"},{"alias_kind":"pith_short_12","alias_value":"IQOMM5A4AABV","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"IQOMM5A4AABVPB3L","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"IQOMM5A4","created_at":"2026-05-18T12:25:57.157939+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/IQOMM5A4AABVPB3LOXFMLJUYGP","json":"https://pith.science/pith/IQOMM5A4AABVPB3LOXFMLJUYGP.json","graph_json":"https://pith.science/api/pith-number/IQOMM5A4AABVPB3LOXFMLJUYGP/graph.json","events_json":"https://pith.science/api/pith-number/IQOMM5A4AABVPB3LOXFMLJUYGP/events.json","paper":"https://pith.science/paper/IQOMM5A4"},"agent_actions":{"view_html":"https://pith.science/pith/IQOMM5A4AABVPB3LOXFMLJUYGP","download_json":"https://pith.science/pith/IQOMM5A4AABVPB3LOXFMLJUYGP.json","view_paper":"https://pith.science/paper/IQOMM5A4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0806.0159&json=true","fetch_graph":"https://pith.science/api/pith-number/IQOMM5A4AABVPB3LOXFMLJUYGP/graph.json","fetch_events":"https://pith.science/api/pith-number/IQOMM5A4AABVPB3LOXFMLJUYGP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IQOMM5A4AABVPB3LOXFMLJUYGP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IQOMM5A4AABVPB3LOXFMLJUYGP/action/storage_attestation","attest_author":"https://pith.science/pith/IQOMM5A4AABVPB3LOXFMLJUYGP/action/author_attestation","sign_citation":"https://pith.science/pith/IQOMM5A4AABVPB3LOXFMLJUYGP/action/citation_signature","submit_replication":"https://pith.science/pith/IQOMM5A4AABVPB3LOXFMLJUYGP/action/replication_record"}},"created_at":"2026-05-18T01:23:48.587775+00:00","updated_at":"2026-05-18T01:23:48.587775+00:00"}