{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:KHR6OFD7J7HZXTUNTL2VTBHPD5","short_pith_number":"pith:KHR6OFD7","schema_version":"1.0","canonical_sha256":"51e3e7147f4fcf9bce8d9af55984ef1f6ec41ba52842652229f65d099f784430","source":{"kind":"arxiv","id":"1207.3066","version":4},"attestation_state":"computed","paper":{"title":"Morse theory for manifolds with boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Andr\\'as N\\'emethi, Andrew Ranicki, Maciej Borodzik","submitted_at":"2012-07-12T19:09:43Z","abstract_excerpt":"We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of manifolds with boundary splits as a union of left product cobordisms and right product cobordisms."},"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":"1207.3066","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-07-12T19:09:43Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"95823b76788258f2adba36037de52e91b865bd63ae2f71ed1c15a7342aa07763","abstract_canon_sha256":"a37894823a65512b3a03b4b7ce7377e11f8be637d28cda32de6ccfc8d7c2fa3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:47.357799Z","signature_b64":"a5PFMNjHTVqHms71TumgR19dbyymGyr0wDKe9jrKgLEpUdH8o0ybRUF2kybcW/5n/OB38w/U2ijM9//gv4rFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51e3e7147f4fcf9bce8d9af55984ef1f6ec41ba52842652229f65d099f784430","last_reissued_at":"2026-05-18T01:15:47.357254Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:47.357254Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Morse theory for manifolds with boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Andr\\'as N\\'emethi, Andrew Ranicki, Maciej Borodzik","submitted_at":"2012-07-12T19:09:43Z","abstract_excerpt":"We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of manifolds with boundary splits as a union of left product cobordisms and right product cobordisms."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.3066","kind":"arxiv","version":4},"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":"1207.3066","created_at":"2026-05-18T01:15:47.357358+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.3066v4","created_at":"2026-05-18T01:15:47.357358+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.3066","created_at":"2026-05-18T01:15:47.357358+00:00"},{"alias_kind":"pith_short_12","alias_value":"KHR6OFD7J7HZ","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"KHR6OFD7J7HZXTUN","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"KHR6OFD7","created_at":"2026-05-18T12:27:11.947152+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/KHR6OFD7J7HZXTUNTL2VTBHPD5","json":"https://pith.science/pith/KHR6OFD7J7HZXTUNTL2VTBHPD5.json","graph_json":"https://pith.science/api/pith-number/KHR6OFD7J7HZXTUNTL2VTBHPD5/graph.json","events_json":"https://pith.science/api/pith-number/KHR6OFD7J7HZXTUNTL2VTBHPD5/events.json","paper":"https://pith.science/paper/KHR6OFD7"},"agent_actions":{"view_html":"https://pith.science/pith/KHR6OFD7J7HZXTUNTL2VTBHPD5","download_json":"https://pith.science/pith/KHR6OFD7J7HZXTUNTL2VTBHPD5.json","view_paper":"https://pith.science/paper/KHR6OFD7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.3066&json=true","fetch_graph":"https://pith.science/api/pith-number/KHR6OFD7J7HZXTUNTL2VTBHPD5/graph.json","fetch_events":"https://pith.science/api/pith-number/KHR6OFD7J7HZXTUNTL2VTBHPD5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KHR6OFD7J7HZXTUNTL2VTBHPD5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KHR6OFD7J7HZXTUNTL2VTBHPD5/action/storage_attestation","attest_author":"https://pith.science/pith/KHR6OFD7J7HZXTUNTL2VTBHPD5/action/author_attestation","sign_citation":"https://pith.science/pith/KHR6OFD7J7HZXTUNTL2VTBHPD5/action/citation_signature","submit_replication":"https://pith.science/pith/KHR6OFD7J7HZXTUNTL2VTBHPD5/action/replication_record"}},"created_at":"2026-05-18T01:15:47.357358+00:00","updated_at":"2026-05-18T01:15:47.357358+00:00"}