{"paper":{"title":"Intersection of subgroups in free groups and homotopy groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GR","authors_text":"Hans-Joachim Baues, Roman Mikhailov","submitted_at":"2008-04-12T07:36:35Z","abstract_excerpt":"We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group $\\pi_3$. This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the computation of $\\pi_2$. Let $K$ be a two-dimensional CW-complex with subcomplexes $K_1,K_2,K_3$ such that $K=K_1\\cup K_2\\cup K_3$ and $K_1\\cap K_2\\cap K_3$ is the 1-skeleton $K^1$ of $K$. We construct a natural homomorphism of $\\pi_1(K)$-modules $$ \\pi_3(K)\\to \\frac{R_1\\cap R_2\\cap R_3}{[R_1,R_2\\cap R_3][R_2,R_3\\cap R_1][R_3,R_1\\cap R_2]}, $$ where $R_i=ker\\{\\pi_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0804.1999","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"}