{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:4GS2ZVOM32PT5SGC7IHAYZKNFS","short_pith_number":"pith:4GS2ZVOM","canonical_record":{"source":{"id":"1212.2028","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-10T10:58:29Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"118773bc997fc984a51ace8b5082e5af4df73c1cf803c1b73272b6a2d9f747e6","abstract_canon_sha256":"1da67876f314ede251ab05d71fe4a9a14893c2d92f770724ffad4bc45877e534"},"schema_version":"1.0"},"canonical_sha256":"e1a5acd5ccde9f3ec8c2fa0e0c654d2c89b981ef9054427f952f0032cc0198b0","source":{"kind":"arxiv","id":"1212.2028","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2028","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2028v2","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2028","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"pith_short_12","alias_value":"4GS2ZVOM32PT","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4GS2ZVOM32PT5SGC","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4GS2ZVOM","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:4GS2ZVOM32PT5SGC7IHAYZKNFS","target":"record","payload":{"canonical_record":{"source":{"id":"1212.2028","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-10T10:58:29Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"118773bc997fc984a51ace8b5082e5af4df73c1cf803c1b73272b6a2d9f747e6","abstract_canon_sha256":"1da67876f314ede251ab05d71fe4a9a14893c2d92f770724ffad4bc45877e534"},"schema_version":"1.0"},"canonical_sha256":"e1a5acd5ccde9f3ec8c2fa0e0c654d2c89b981ef9054427f952f0032cc0198b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:04.665854Z","signature_b64":"ZB8bypSj0TYPpXO96QUX7dDNr4EcBPUw1LuNWmfLtm6aH5fGm3SnB2cvVShtB8wSU6ZFBH8LvqB+M0RDmX4BCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1a5acd5ccde9f3ec8c2fa0e0c654d2c89b981ef9054427f952f0032cc0198b0","last_reissued_at":"2026-05-18T03:38:04.665352Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:04.665352Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.2028","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:38:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V0D6QH+Bp2vIagZU2UOdEhO7wG45lPLwwtQWqMQSg3/EaCQ51GrAddYO8vqssS5s0yCJHnHe9Z5lWyDM24n5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T11:31:15.935698Z"},"content_sha256":"9a1b10156124ccaee83cd189a0926527fb6b546d1b707b0b3f97b4eceb73c9b0","schema_version":"1.0","event_id":"sha256:9a1b10156124ccaee83cd189a0926527fb6b546d1b707b0b3f97b4eceb73c9b0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:4GS2ZVOM32PT5SGC7IHAYZKNFS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Discrete Morse theory for moment-angle complexes of pairs (D^n,S^{n-1})","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Vladimir Grujic, Volkmar Welker","submitted_at":"2012-12-10T10:58:29Z","abstract_excerpt":"For a finite simplicial complex K and a CW-pair (X,A), there is an associated CW-complex Z_K(X,A), known as a polyhedral product. We apply discrete Morse theory to a particular CW-structure on the n-sphere moment-angle complexes Z_K(D^{n}, S^{n-1}). For the class of simplicial complexes with vertex-decomposable duals, we show that the associated n-sphere moment-angle complexes have the homotopy type of wedges of spheres. As a corollary we show that a sufficiently high suspension of any restriction of a simplicial complex with vertex-decomposable dual is homotopy equivalent to a wedge of sphere"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2028","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:38:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KPCgxlGObmPOsqba75QgoRjI8ZWxYUxjmHvuhttewC4UsST+sVlUhtFLFITqpvI0vFTMTdaeBSmJC8vkjaDJDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T11:31:15.936045Z"},"content_sha256":"5a79800d14cd2a2ffdda78ded5ced6d65754b4b3f0dab40e379fe6b0ab7c3dbc","schema_version":"1.0","event_id":"sha256:5a79800d14cd2a2ffdda78ded5ced6d65754b4b3f0dab40e379fe6b0ab7c3dbc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4GS2ZVOM32PT5SGC7IHAYZKNFS/bundle.json","state_url":"https://pith.science/pith/4GS2ZVOM32PT5SGC7IHAYZKNFS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4GS2ZVOM32PT5SGC7IHAYZKNFS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T11:31:15Z","links":{"resolver":"https://pith.science/pith/4GS2ZVOM32PT5SGC7IHAYZKNFS","bundle":"https://pith.science/pith/4GS2ZVOM32PT5SGC7IHAYZKNFS/bundle.json","state":"https://pith.science/pith/4GS2ZVOM32PT5SGC7IHAYZKNFS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4GS2ZVOM32PT5SGC7IHAYZKNFS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:4GS2ZVOM32PT5SGC7IHAYZKNFS","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":"1da67876f314ede251ab05d71fe4a9a14893c2d92f770724ffad4bc45877e534","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-10T10:58:29Z","title_canon_sha256":"118773bc997fc984a51ace8b5082e5af4df73c1cf803c1b73272b6a2d9f747e6"},"schema_version":"1.0","source":{"id":"1212.2028","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2028","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2028v2","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2028","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"pith_short_12","alias_value":"4GS2ZVOM32PT","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"4GS2ZVOM32PT5SGC","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"4GS2ZVOM","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:5a79800d14cd2a2ffdda78ded5ced6d65754b4b3f0dab40e379fe6b0ab7c3dbc","target":"graph","created_at":"2026-05-18T03:38:04Z","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":"For a finite simplicial complex K and a CW-pair (X,A), there is an associated CW-complex Z_K(X,A), known as a polyhedral product. We apply discrete Morse theory to a particular CW-structure on the n-sphere moment-angle complexes Z_K(D^{n}, S^{n-1}). For the class of simplicial complexes with vertex-decomposable duals, we show that the associated n-sphere moment-angle complexes have the homotopy type of wedges of spheres. As a corollary we show that a sufficiently high suspension of any restriction of a simplicial complex with vertex-decomposable dual is homotopy equivalent to a wedge of sphere","authors_text":"Vladimir Grujic, Volkmar Welker","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-10T10:58:29Z","title":"Discrete Morse theory for moment-angle complexes of pairs (D^n,S^{n-1})"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2028","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":"sha256:9a1b10156124ccaee83cd189a0926527fb6b546d1b707b0b3f97b4eceb73c9b0","target":"record","created_at":"2026-05-18T03:38:04Z","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":"1da67876f314ede251ab05d71fe4a9a14893c2d92f770724ffad4bc45877e534","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-10T10:58:29Z","title_canon_sha256":"118773bc997fc984a51ace8b5082e5af4df73c1cf803c1b73272b6a2d9f747e6"},"schema_version":"1.0","source":{"id":"1212.2028","kind":"arxiv","version":2}},"canonical_sha256":"e1a5acd5ccde9f3ec8c2fa0e0c654d2c89b981ef9054427f952f0032cc0198b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e1a5acd5ccde9f3ec8c2fa0e0c654d2c89b981ef9054427f952f0032cc0198b0","first_computed_at":"2026-05-18T03:38:04.665352Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:04.665352Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZB8bypSj0TYPpXO96QUX7dDNr4EcBPUw1LuNWmfLtm6aH5fGm3SnB2cvVShtB8wSU6ZFBH8LvqB+M0RDmX4BCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:04.665854Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.2028","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a1b10156124ccaee83cd189a0926527fb6b546d1b707b0b3f97b4eceb73c9b0","sha256:5a79800d14cd2a2ffdda78ded5ced6d65754b4b3f0dab40e379fe6b0ab7c3dbc"],"state_sha256":"e20b5f42445937864d56cdc39ca46c0590caa8d686f07e77a17244e56e7237b1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z9sHhVYMO6vSzIhBcQAzW5uutpO1fj/LVcfkPYT2B3yA3tH/1P9P/END5JxUhiKbyYnGjwkYDwgPkW/YvS3tBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T11:31:15.938305Z","bundle_sha256":"40e18b93c8eeb66622cc9d8812818d37e51f864c18d746863851e387ab6974fd"}}