{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:Z3XGXP7KKKU2LJW6MUVHOWBMR3","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":"db53cb5b652628acdb386642995c8580ef2d19f9f12db0af6ec1a6f7dab5f4dd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2020-06-02T15:04:35Z","title_canon_sha256":"7912dfcc9cb745f99c265e56c8c6e3b1b3384ce377ada27944534a090983cebc"},"schema_version":"1.0","source":{"id":"2006.01689","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.01689","created_at":"2026-07-05T01:07:33Z"},{"alias_kind":"arxiv_version","alias_value":"2006.01689v1","created_at":"2026-07-05T01:07:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.01689","created_at":"2026-07-05T01:07:33Z"},{"alias_kind":"pith_short_12","alias_value":"Z3XGXP7KKKU2","created_at":"2026-07-05T01:07:33Z"},{"alias_kind":"pith_short_16","alias_value":"Z3XGXP7KKKU2LJW6","created_at":"2026-07-05T01:07:33Z"},{"alias_kind":"pith_short_8","alias_value":"Z3XGXP7K","created_at":"2026-07-05T01:07:33Z"}],"graph_snapshots":[{"event_id":"sha256:ad55b5cc723d9ee1018b16bbf1d55895b0c8ad0ee27ca939020df72b8ef96dc9","target":"graph","created_at":"2026-07-05T01:07:33Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2006.01689/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite ","authors_text":"Osamu Saeki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2020-06-02T15:04:35Z","title":"Reeb spaces of smooth functions on manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.01689","kind":"arxiv","version":1},"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:ff55e2ae8f4b46f41b9e651a1a86f8116c265761c8a00adb9c078ad9fb728a78","target":"record","created_at":"2026-07-05T01:07:33Z","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":"db53cb5b652628acdb386642995c8580ef2d19f9f12db0af6ec1a6f7dab5f4dd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2020-06-02T15:04:35Z","title_canon_sha256":"7912dfcc9cb745f99c265e56c8c6e3b1b3384ce377ada27944534a090983cebc"},"schema_version":"1.0","source":{"id":"2006.01689","kind":"arxiv","version":1}},"canonical_sha256":"ceee6bbfea52a9a5a6de652a77582c8ee897400694ebdd1dd0f37ca9c8dcec6b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ceee6bbfea52a9a5a6de652a77582c8ee897400694ebdd1dd0f37ca9c8dcec6b","first_computed_at":"2026-07-05T01:07:33.267349Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:07:33.267349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eVQd3xCS3apMduoOkrv0sbcDEz1Qmkzv23N6j2lnz/OXUfp8DgumnS5l3UDFLDM3tjTBcejMaE2B7planHZWDw==","signature_status":"signed_v1","signed_at":"2026-07-05T01:07:33.267722Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.01689","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ff55e2ae8f4b46f41b9e651a1a86f8116c265761c8a00adb9c078ad9fb728a78","sha256:ad55b5cc723d9ee1018b16bbf1d55895b0c8ad0ee27ca939020df72b8ef96dc9"],"state_sha256":"b4b678587fe8daca82e092a5976ed75d632d591b1be7f82f9aed04d3347c41a4"}