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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_"},"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":"0804.1999","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2008-04-12T07:36:35Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"c0c5e412bfb506a52c4887f92558a591c892f5bec6b1d49c3073dea3aa8f895c","abstract_canon_sha256":"d5df0cd725358ba977f4b0a33e6fa9e73c3cfc47dd34f84f23fc7be5c49bedc5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:36.901058Z","signature_b64":"CrXS7iLo6SCcv2zd4brUXFTqOCtEkL/wONNuWIV6Juf1GxT/G83k/QElEoJgMG7hLYOfiaq3MghxHc7GmNjvCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccdc9592b7441c049b2810190ab1e309e1422aa07922a626289d748044dbf6fd","last_reissued_at":"2026-05-18T04:41:36.900641Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:36.900641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0804.1999","created_at":"2026-05-18T04:41:36.900702+00:00"},{"alias_kind":"arxiv_version","alias_value":"0804.1999v1","created_at":"2026-05-18T04:41:36.900702+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0804.1999","created_at":"2026-05-18T04:41:36.900702+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZTOJLEVXIQOA","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZTOJLEVXIQOAJGZI","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZTOJLEVX","created_at":"2026-05-18T12:25:58.018023+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/ZTOJLEVXIQOAJGZICAMQVMPDBH","json":"https://pith.science/pith/ZTOJLEVXIQOAJGZICAMQVMPDBH.json","graph_json":"https://pith.science/api/pith-number/ZTOJLEVXIQOAJGZICAMQVMPDBH/graph.json","events_json":"https://pith.science/api/pith-number/ZTOJLEVXIQOAJGZICAMQVMPDBH/events.json","paper":"https://pith.science/paper/ZTOJLEVX"},"agent_actions":{"view_html":"https://pith.science/pith/ZTOJLEVXIQOAJGZICAMQVMPDBH","download_json":"https://pith.science/pith/ZTOJLEVXIQOAJGZICAMQVMPDBH.json","view_paper":"https://pith.science/paper/ZTOJLEVX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0804.1999&json=true","fetch_graph":"https://pith.science/api/pith-number/ZTOJLEVXIQOAJGZICAMQVMPDBH/graph.json","fetch_events":"https://pith.science/api/pith-number/ZTOJLEVXIQOAJGZICAMQVMPDBH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZTOJLEVXIQOAJGZICAMQVMPDBH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZTOJLEVXIQOAJGZICAMQVMPDBH/action/storage_attestation","attest_author":"https://pith.science/pith/ZTOJLEVXIQOAJGZICAMQVMPDBH/action/author_attestation","sign_citation":"https://pith.science/pith/ZTOJLEVXIQOAJGZICAMQVMPDBH/action/citation_signature","submit_replication":"https://pith.science/pith/ZTOJLEVXIQOAJGZICAMQVMPDBH/action/replication_record"}},"created_at":"2026-05-18T04:41:36.900702+00:00","updated_at":"2026-05-18T04:41:36.900702+00:00"}