{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:L72U2RYQ67JAGNLFRAHKXKLVRV","short_pith_number":"pith:L72U2RYQ","schema_version":"1.0","canonical_sha256":"5ff54d4710f7d2033565880eaba9758d45301e4c77c0f6e419f3cc192c164ce8","source":{"kind":"arxiv","id":"1007.3175","version":2},"attestation_state":"computed","paper":{"title":"Discrete Morse Theory for Manifolds with Boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CO"],"primary_cat":"math.AT","authors_text":"Bruno Benedetti","submitted_at":"2010-07-19T15:11:46Z","abstract_excerpt":"We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain \"Relative Morse Inequalities\" relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman's Sphere Theorem. The main corollaries of our work are:\n  -- For each d \\ge 3 and for each k \\ge 0, there is a PL d-sphere on which any discrete Morse function has more than k critical (d-1)-cells. (This solves a problem by Chari.)\n  -- For fixed d and k, the"},"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":"1007.3175","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-07-19T15:11:46Z","cross_cats_sorted":["math.AC","math.CO"],"title_canon_sha256":"1a8151c579154fed14ecfcd22fd3b64ddfa258856e049b07ce1fc1a7abc78205","abstract_canon_sha256":"0f5a3166e7a7a7e14ba64bc0f0f91d90c1f66817ab7cc51c23cdec6d3ab1b0e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:59.081429Z","signature_b64":"j0mc0uHHex0FN5sAthQqWlA55dWdKai1Wru0Rs7wZgPmRolQmgtJ+mvfRmVVXIyiIEOonBrmP3kZmPusIEtvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ff54d4710f7d2033565880eaba9758d45301e4c77c0f6e419f3cc192c164ce8","last_reissued_at":"2026-05-18T04:39:59.080872Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:59.080872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Discrete Morse Theory for Manifolds with Boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CO"],"primary_cat":"math.AT","authors_text":"Bruno Benedetti","submitted_at":"2010-07-19T15:11:46Z","abstract_excerpt":"We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain \"Relative Morse Inequalities\" relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman's Sphere Theorem. The main corollaries of our work are:\n  -- For each d \\ge 3 and for each k \\ge 0, there is a PL d-sphere on which any discrete Morse function has more than k critical (d-1)-cells. (This solves a problem by Chari.)\n  -- For fixed d and k, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.3175","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1007.3175","created_at":"2026-05-18T04:39:59.080979+00:00"},{"alias_kind":"arxiv_version","alias_value":"1007.3175v2","created_at":"2026-05-18T04:39:59.080979+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.3175","created_at":"2026-05-18T04:39:59.080979+00:00"},{"alias_kind":"pith_short_12","alias_value":"L72U2RYQ67JA","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"L72U2RYQ67JAGNLF","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"L72U2RYQ","created_at":"2026-05-18T12:26:10.704358+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/L72U2RYQ67JAGNLFRAHKXKLVRV","json":"https://pith.science/pith/L72U2RYQ67JAGNLFRAHKXKLVRV.json","graph_json":"https://pith.science/api/pith-number/L72U2RYQ67JAGNLFRAHKXKLVRV/graph.json","events_json":"https://pith.science/api/pith-number/L72U2RYQ67JAGNLFRAHKXKLVRV/events.json","paper":"https://pith.science/paper/L72U2RYQ"},"agent_actions":{"view_html":"https://pith.science/pith/L72U2RYQ67JAGNLFRAHKXKLVRV","download_json":"https://pith.science/pith/L72U2RYQ67JAGNLFRAHKXKLVRV.json","view_paper":"https://pith.science/paper/L72U2RYQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1007.3175&json=true","fetch_graph":"https://pith.science/api/pith-number/L72U2RYQ67JAGNLFRAHKXKLVRV/graph.json","fetch_events":"https://pith.science/api/pith-number/L72U2RYQ67JAGNLFRAHKXKLVRV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L72U2RYQ67JAGNLFRAHKXKLVRV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L72U2RYQ67JAGNLFRAHKXKLVRV/action/storage_attestation","attest_author":"https://pith.science/pith/L72U2RYQ67JAGNLFRAHKXKLVRV/action/author_attestation","sign_citation":"https://pith.science/pith/L72U2RYQ67JAGNLFRAHKXKLVRV/action/citation_signature","submit_replication":"https://pith.science/pith/L72U2RYQ67JAGNLFRAHKXKLVRV/action/replication_record"}},"created_at":"2026-05-18T04:39:59.080979+00:00","updated_at":"2026-05-18T04:39:59.080979+00:00"}