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Precomposing the circle bundle immersions with their universal covering maps, we get for n>0 immersions g_n of the 3-sphere into 4-space. In this note, we compute the Smale invariants of g_n. The computation is carried out by (partially) resolving the singularities of the natural singular map of the punctured complex projective plane which extends g_n.\n  As an application, we determine "},"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":"0903.0238","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-03-02T09:04:22Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"56922954c98c1c975221eaa7e352204366796ba4143dcceee6dab42b7883119c","abstract_canon_sha256":"aad6595ffa34443f00413ffa5e5c4df9614a85286b6b3d66c60b6ae8281ce62d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:48.336176Z","signature_b64":"T7Qj3JT+AsLV2cfMLQNxdz4JQlWHCgVtWE0aXXTAPXYEW+wHinj9W9wACauxkM6ZPa6dtVOsoKXzVZ87hjuODA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"581bc902895a73fdaf0c15077516497347def4cea62c374d00997ea454ffe696","last_reissued_at":"2026-05-18T02:16:48.335444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:48.335444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Singular Seifert surfaces and Smale invariants for a family of 3-sphere immersions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Masamichi Takase, Tobias Ekholm","submitted_at":"2009-03-02T09:04:22Z","abstract_excerpt":"A self-transverse immersion of the 2-sphere into 4-space with algebraic number of self intersection points equal to -n induces an immersion of the circle bundle over the 2-sphere of Euler class 2n into 4-space. 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