{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:6BQHHGDLSMHJOGEJQ5Z3YY3SGR","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7e02518b416e1f1e028054c0bc71300174a89ecec6387145649a243bd6aade88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-09-11T13:09:51Z","title_canon_sha256":"4275a9c09a78e8023518132c15f084adcedae10fa7e482101a75e82e070fc9f8"},"schema_version":"1.0","source":{"id":"1509.03490","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.03490","created_at":"2026-05-18T01:26:17Z"},{"alias_kind":"arxiv_version","alias_value":"1509.03490v2","created_at":"2026-05-18T01:26:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.03490","created_at":"2026-05-18T01:26:17Z"},{"alias_kind":"pith_short_12","alias_value":"6BQHHGDLSMHJ","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"6BQHHGDLSMHJOGEJ","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"6BQHHGDL","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:a039f06b8daa6bf66f3e200178569ed96ab662b532748bde9f651ee8a40228ff","target":"graph","created_at":"2026-05-18T01:26:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Given a Morse function f on a closed manifold M with distinct critical values, and given a field F, there is a canonical complex, called the Morse-Barannikov complex, which is equivalent to any Morse complex associated with f and whose form is simple. In particular, the homology of M with coefficients in F is immediately readable on this complex. The bifurcation theory of this complex in a generic one-parameter family of functions will be investigated. Applications to the boundary manifolds will be given.","authors_text":"Francois Laudenbach (LMJL)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-09-11T13:09:51Z","title":"On an article by S. A. Barannikov"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03490","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4a701408f1d15fa4063cdfb2e9cc12fd8416b4a2d6816dc333754589b90cd266","target":"record","created_at":"2026-05-18T01:26:17Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7e02518b416e1f1e028054c0bc71300174a89ecec6387145649a243bd6aade88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-09-11T13:09:51Z","title_canon_sha256":"4275a9c09a78e8023518132c15f084adcedae10fa7e482101a75e82e070fc9f8"},"schema_version":"1.0","source":{"id":"1509.03490","kind":"arxiv","version":2}},"canonical_sha256":"f06073986b930e9718898773bc6372344244d34da1086ad700a9433167cc9b20","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f06073986b930e9718898773bc6372344244d34da1086ad700a9433167cc9b20","first_computed_at":"2026-05-18T01:26:17.827295Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:26:17.827295Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5sl4MIHuJxvlmAP7tBLtwX+01Z3E1HbYkaq0MYkPz6aCZwnJPs5N1xQ+qeNDTzMCxIk1S88NeLMzOrnXh983Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T01:26:17.827911Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.03490","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a701408f1d15fa4063cdfb2e9cc12fd8416b4a2d6816dc333754589b90cd266","sha256:a039f06b8daa6bf66f3e200178569ed96ab662b532748bde9f651ee8a40228ff"],"state_sha256":"9a3f2927d3c183442f108fd8d2415ea79b325872892efdaaec228697eadeb192"}