{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:VRCNXOSC6OSICLL7JKM3O3K6WV","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":"6cdeda76712553f8e3da7284fd5e297311b4c3ae48bc32444ff01eb1695ed278","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-03-15T20:56:52Z","title_canon_sha256":"730cdf13afa9b89fbe3abcb994aae92ff7d1451830db18084243403b9920128f"},"schema_version":"1.0","source":{"id":"1003.3029","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.3029","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"arxiv_version","alias_value":"1003.3029v2","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.3029","created_at":"2026-05-18T01:13:05Z"},{"alias_kind":"pith_short_12","alias_value":"VRCNXOSC6OSI","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"VRCNXOSC6OSICLL7","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"VRCNXOSC","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:f9ac77ff8deffb8db712dfcbb174ba4d02580bc6cf4f43e9cc3b4cf3d30a69c5","target":"graph","created_at":"2026-05-18T01:13:05Z","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":"We prove that there is an algorithm which determines whether or not a given 2-polyhedron can be embedded into some integral homology 3-sphere.\n  This is a corollary of the following main result. Let $M$ be a compact connected orientable 3-manifold with boundary. Denote $G=\\Z$, $G=\\Z/p\\Z$ or $G=\\Q$. If $H_1(M;G)\\cong G^k$ and $\\bd M$ is a surface of genus $g$, then the minimal group $H_1(Q;G)$ for closed 3-manifolds $Q$ containing $M$ is isomorphic to $G^{k-g}$.\n  Another corollary is that for a graph $L$ the minimal number $\\rk H_1(Q;\\Z)$ for closed orientable 3-manifolds $Q$ containing $L\\tim","authors_text":"Dmitry Tonkonog","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-03-15T20:56:52Z","title":"Embedding 3-manifolds with boundary into closed 3-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3029","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:f3cfbc32be719469c519702720d6fe4a3f0236cefd86b5a9741124852ddfa790","target":"record","created_at":"2026-05-18T01:13:05Z","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":"6cdeda76712553f8e3da7284fd5e297311b4c3ae48bc32444ff01eb1695ed278","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-03-15T20:56:52Z","title_canon_sha256":"730cdf13afa9b89fbe3abcb994aae92ff7d1451830db18084243403b9920128f"},"schema_version":"1.0","source":{"id":"1003.3029","kind":"arxiv","version":2}},"canonical_sha256":"ac44dbba42f3a4812d7f4a99b76d5eb5528a161e20b38ac0a6c38eee756dc148","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac44dbba42f3a4812d7f4a99b76d5eb5528a161e20b38ac0a6c38eee756dc148","first_computed_at":"2026-05-18T01:13:05.334990Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:13:05.334990Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"78ZIbXHwhQVGX83ParJ/1YUhQl2l1Leq+vuQ0vt89c2aib4gZz2jmGbfv4x8hJdj4ppAllYrWv8xDhGm/NOOCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:13:05.335527Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.3029","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3cfbc32be719469c519702720d6fe4a3f0236cefd86b5a9741124852ddfa790","sha256:f9ac77ff8deffb8db712dfcbb174ba4d02580bc6cf4f43e9cc3b4cf3d30a69c5"],"state_sha256":"0a0f6983440e39e3073238532399a0a095cbc07b5facbdf4b9bf06467dff2100"}