{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:WDZ3F5UMDMMFFK7H6HF7HMPZGD","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":"2903dc601a66e4e6c58021770b7c729525869af9ec867ee7d627a84d9b6a353d","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-08-08T16:27:35Z","title_canon_sha256":"457ba99cf01fa688ee0de6678e3c5b3bb6475f1d8c198922f1c87cec3f38169e"},"schema_version":"1.0","source":{"id":"1408.1903","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.1903","created_at":"2026-05-18T02:41:52Z"},{"alias_kind":"arxiv_version","alias_value":"1408.1903v4","created_at":"2026-05-18T02:41:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.1903","created_at":"2026-05-18T02:41:52Z"},{"alias_kind":"pith_short_12","alias_value":"WDZ3F5UMDMMF","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"WDZ3F5UMDMMFFK7H","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"WDZ3F5UM","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:b7ad0aa62023a21e4ead31b0ed9ffa552a8d48ecc6509bfb1b2090fc1a9c86f2","target":"graph","created_at":"2026-05-18T02:41:52Z","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 a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homological stability for the moduli spaces of manifolds of dimension $2n > 4$, with respect to forming connected sums with $S^{n}\\times S^{n}$.","authors_text":"Nathan Perlmutter","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-08-08T16:27:35Z","title":"Homological Stability For The Moduli Spaces of Products of Spheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1903","kind":"arxiv","version":4},"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:a1050446cac6fb5264ab802e3041c857cd031f1eb1c62dcbd3605df5259e1916","target":"record","created_at":"2026-05-18T02:41:52Z","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":"2903dc601a66e4e6c58021770b7c729525869af9ec867ee7d627a84d9b6a353d","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-08-08T16:27:35Z","title_canon_sha256":"457ba99cf01fa688ee0de6678e3c5b3bb6475f1d8c198922f1c87cec3f38169e"},"schema_version":"1.0","source":{"id":"1408.1903","kind":"arxiv","version":4}},"canonical_sha256":"b0f3b2f68c1b1852abe7f1cbf3b1f930ced21d5597ba56210a7161d89e1c7f4d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0f3b2f68c1b1852abe7f1cbf3b1f930ced21d5597ba56210a7161d89e1c7f4d","first_computed_at":"2026-05-18T02:41:52.099083Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:52.099083Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tssBQ0NUnDUtiX9LQz8P6B3+wcSffp2mBlDDZx91lDWJF8aPnFP2usWu3hUM6kw++Xwvw5BDzy8Kzz+Y5RSaDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:52.099525Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.1903","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a1050446cac6fb5264ab802e3041c857cd031f1eb1c62dcbd3605df5259e1916","sha256:b7ad0aa62023a21e4ead31b0ed9ffa552a8d48ecc6509bfb1b2090fc1a9c86f2"],"state_sha256":"61d29aa1f5dbd9675e75310f4e89617f03008d635087699aa7c0a40eccec9903"}