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As a rational homotopical condition to be a toral map preserving almost free toral actions for a map $f:X\\to Y$, we define the rational toral rank $r_0(f)$ of $f$, which is a natural invariant with $r_0(id_X)=r_0(X)$ for the identity map $id_X$ of $X$. We will see some properties of it by Su"},"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":"1310.0105","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-10-01T00:25:41Z","cross_cats_sorted":[],"title_canon_sha256":"2b0d73232c7444ec82adfe5417e220c0fefb7416c1cf500be510bde0514a1f1d","abstract_canon_sha256":"13ac55753cf49a0ee44e91b467d4418d1eb11495c90962c9b7f4a18195bbaad6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:11:45.606889Z","signature_b64":"cE7uTptW5OHZt6YQjsaujDhoJY73MDlX6iSOUwh1UoGTBcBImMVKNHpdQLt6EsrBNpcs7ox9N/PdWhAYl3ENAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f5519b52997db07fd958f6238bf4cb40e823b7ccc743001ffe054ca6883c1806","last_reissued_at":"2026-05-18T03:11:45.606411Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:11:45.606411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational toral rank of a map","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Toshihiro Yamaguchi","submitted_at":"2013-10-01T00:25:41Z","abstract_excerpt":"Let $X$ and $Y$ be simply connected CW complexes with finite rational cohomologies. 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