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We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem:\n  Theorem. Let M be a connected orientable closed smooth (n+1)-manifold, n>2. Then the map deg:\\pi^n(M)\\to H_1(M;Z) is\n  1-to-1 (i.e., bijective), if the product w_2(M) x r_2 H_2(M;Z) is nonzero, where r_2 is the mod2 reducti"},"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":"0808.1209","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2008-08-08T13:21:12Z","cross_cats_sorted":[],"title_canon_sha256":"65d71dfc13b9d755fe3e4999b587dba8979c8aebc0e13920aa9c399beb907a07","abstract_canon_sha256":"954125c68a977fa5b92dcbcab4e3e69433cfeca50c1d1689ae7245d5667c62de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:06.380111Z","signature_b64":"7MJB8NrMaSPcRLWXMJ+pSfYyd29Y2ZAOt7sVrFdGFSHh4sHgR6Olvd68u/kSKA8Z413z5lCY7crtGblF9GDPBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"958e397ba14eb01078570b20664eba332447839d76b26de8752ea954605dbdda","last_reissued_at":"2026-05-18T03:25:06.379427Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:06.379427Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Pontryagin-Steenrod-Wu theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Du\\v{s}an Repov\\v{s}, Fulvia Spaggiari, Mikhail Skopenkov","submitted_at":"2008-08-08T13:21:12Z","abstract_excerpt":"This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] is the fundamental class. We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem:\n  Theorem. Let M be a connected orientable closed smooth (n+1)-manifold, n>2. Then the map deg:\\pi^n(M)\\to H_1(M;Z) is\n  1-to-1 (i.e., bijective), if the product w_2(M) x r_2 H_2(M;Z) is nonzero, where r_2 is the mod2 reducti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0808.1209","kind":"arxiv","version":2},"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":"0808.1209","created_at":"2026-05-18T03:25:06.379521+00:00"},{"alias_kind":"arxiv_version","alias_value":"0808.1209v2","created_at":"2026-05-18T03:25:06.379521+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0808.1209","created_at":"2026-05-18T03:25:06.379521+00:00"},{"alias_kind":"pith_short_12","alias_value":"SWHDS65BJ2YB","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_16","alias_value":"SWHDS65BJ2YBA6CX","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_8","alias_value":"SWHDS65B","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/SWHDS65BJ2YBA6CXBMQGMTV2GM","json":"https://pith.science/pith/SWHDS65BJ2YBA6CXBMQGMTV2GM.json","graph_json":"https://pith.science/api/pith-number/SWHDS65BJ2YBA6CXBMQGMTV2GM/graph.json","events_json":"https://pith.science/api/pith-number/SWHDS65BJ2YBA6CXBMQGMTV2GM/events.json","paper":"https://pith.science/paper/SWHDS65B"},"agent_actions":{"view_html":"https://pith.science/pith/SWHDS65BJ2YBA6CXBMQGMTV2GM","download_json":"https://pith.science/pith/SWHDS65BJ2YBA6CXBMQGMTV2GM.json","view_paper":"https://pith.science/paper/SWHDS65B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0808.1209&json=true","fetch_graph":"https://pith.science/api/pith-number/SWHDS65BJ2YBA6CXBMQGMTV2GM/graph.json","fetch_events":"https://pith.science/api/pith-number/SWHDS65BJ2YBA6CXBMQGMTV2GM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SWHDS65BJ2YBA6CXBMQGMTV2GM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SWHDS65BJ2YBA6CXBMQGMTV2GM/action/storage_attestation","attest_author":"https://pith.science/pith/SWHDS65BJ2YBA6CXBMQGMTV2GM/action/author_attestation","sign_citation":"https://pith.science/pith/SWHDS65BJ2YBA6CXBMQGMTV2GM/action/citation_signature","submit_replication":"https://pith.science/pith/SWHDS65BJ2YBA6CXBMQGMTV2GM/action/replication_record"}},"created_at":"2026-05-18T03:25:06.379521+00:00","updated_at":"2026-05-18T03:25:06.379521+00:00"}