{"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"}