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Our main result is that if $M$ is a closed, orientable 3-manifold, $n\\ge 2$, and $G\\cong \\pi_1(M)/\\pi_1(M)^{(n)}$ is finite, then the cup product pairing $H^2(G)\\otimes H^2(G)\\to H^4(G)$ has cyclic image $C$, and the pairing $H^2(G)\\otimes H^2(G)\\stackrel{\\smile}{\\longrightarrow} C$ is isomorphic to the linking pairing $H_1(M)_{\\textrm{Tors}}\\otimes H_1(M)_{\\textrm{Tors}}\\to \\mathbb{Q}/"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1405.4351","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-05-17T04:45:14Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"d986f66cdd338f5b84ba5cd313893d348f479e0304435874d5e6c9a8ef27aa03","abstract_canon_sha256":"1344988f044589c850a036025869aba9eb0caaa6d24980d3df1a973fb506b740"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:56.026020Z","signature_b64":"ggGT1zTK/3vlzF6zwtObL2wMMLjQBkEFykJQZuEq5pxH+WUxvby0IVuxPnQgFsR13WIA+sc8HXzhxcB6PjKpBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9046a50047c0e576f29024cce90227e7065bfc02b50cc244872b33938b5a419c","last_reissued_at":"2026-05-18T01:21:56.025277Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:56.025277Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On finite derived quotients of 3-manifold groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Will Cavendish","submitted_at":"2014-05-17T04:45:14Z","abstract_excerpt":"This paper studies the set of finite groups appearing as $\\pi_1(M)/\\pi_1(M)^{(n)}$, where $M$ is a closed, orientable 3-manifold and $\\pi_1(M)^{(n)}$ denotes the $n$-th term of the derived series of $\\pi_1(M)$. Our main result is that if $M$ is a closed, orientable 3-manifold, $n\\ge 2$, and $G\\cong \\pi_1(M)/\\pi_1(M)^{(n)}$ is finite, then the cup product pairing $H^2(G)\\otimes H^2(G)\\to H^4(G)$ has cyclic image $C$, and the pairing $H^2(G)\\otimes H^2(G)\\stackrel{\\smile}{\\longrightarrow} C$ is isomorphic to the linking pairing $H_1(M)_{\\textrm{Tors}}\\otimes H_1(M)_{\\textrm{Tors}}\\to \\mathbb{Q}/"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4351","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1405.4351","created_at":"2026-05-18T01:21:56.025406+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.4351v1","created_at":"2026-05-18T01:21:56.025406+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.4351","created_at":"2026-05-18T01:21:56.025406+00:00"},{"alias_kind":"pith_short_12","alias_value":"SBDKKACHYDSX","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"SBDKKACHYDSXN4UQ","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"SBDKKACH","created_at":"2026-05-18T12:28:49.207871+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44","json":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44.json","graph_json":"https://pith.science/api/pith-number/SBDKKACHYDSXN4UQETGOSARH44/graph.json","events_json":"https://pith.science/api/pith-number/SBDKKACHYDSXN4UQETGOSARH44/events.json","paper":"https://pith.science/paper/SBDKKACH"},"agent_actions":{"view_html":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44","download_json":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44.json","view_paper":"https://pith.science/paper/SBDKKACH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.4351&json=true","fetch_graph":"https://pith.science/api/pith-number/SBDKKACHYDSXN4UQETGOSARH44/graph.json","fetch_events":"https://pith.science/api/pith-number/SBDKKACHYDSXN4UQETGOSARH44/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44/action/storage_attestation","attest_author":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44/action/author_attestation","sign_citation":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44/action/citation_signature","submit_replication":"https://pith.science/pith/SBDKKACHYDSXN4UQETGOSARH44/action/replication_record"}},"created_at":"2026-05-18T01:21:56.025406+00:00","updated_at":"2026-05-18T01:21:56.025406+00:00"}