{"paper":{"title":"Relative Schur multipliers and universal extensions of group homomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GR","authors_text":"Emmanuel D. Farjoun, Yoav Segev","submitted_at":"2014-10-22T16:04:21Z","abstract_excerpt":"In this note, starting with any group homomorphism $f\\colon\\Gamma\\to G$, which is surjective upon abelianization, we construct a universal central extension $u\\colon U\\twoheadrightarrow G,$ UNDER $\\Gamma$ with the same surjective property, such that for any central extension $m\\colon M\\twoheadrightarrow G,$ under $f,$ there is a unique homomorphism $U\\to M$ with the obvious commutation condition. The kernel of $u$ is the relative Schur multiplier group $H_2(G,\\Gamma;\\mathbb{Z})$ as defined in the paper. The case where $G$ is perfect corresponds to $\\Gamma=1$. This yields homological obstructio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.6090","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"}