{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:6VIZWUUZPWYH7WKY6YRYX5GLID","short_pith_number":"pith:6VIZWUUZ","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"},"canonical_sha256":"f5519b52997db07fd958f6238bf4cb40e823b7ccc743001ffe054ca6883c1806","source":{"kind":"arxiv","id":"1310.0105","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.0105","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"arxiv_version","alias_value":"1310.0105v1","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.0105","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"pith_short_12","alias_value":"6VIZWUUZPWYH","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6VIZWUUZPWYH7WKY","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6VIZWUUZ","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:6VIZWUUZPWYH7WKY6YRYX5GLID","target":"record","payload":{"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"},"canonical_sha256":"f5519b52997db07fd958f6238bf4cb40e823b7ccc743001ffe054ca6883c1806","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"},"source_kind":"arxiv","source_id":"1310.0105","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:11:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+kaHV9HMPhSkURepaxr/yTjyQXMJyvRt7HQ4WpJZbqUzwKJuJbskn1cR8Eh0gB/g54GzFG5fAA4o1t87NNUXBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:08:55.591690Z"},"content_sha256":"879879b96e2a053863d2273ddb62d143516db1907e075fdd1aaec7dcbf7aedfa","schema_version":"1.0","event_id":"sha256:879879b96e2a053863d2273ddb62d143516db1907e075fdd1aaec7dcbf7aedfa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:6VIZWUUZPWYH7WKY6YRYX5GLID","target":"graph","payload":{"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. The rational toral rank $r_0(X)$ of a space $X$ is the largest integer $r$ such that the torus $T^r$ can act continuously on a CW-complex in the rational homotopy type of $X$ with all its isotropy subgroups finite \\cite{H}. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0105","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:11:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tWr2a5F6FOJ08/AxESEO5xokLlz6JzyJXwGwqy4fEqpugc0bsj6fh8FNJt8Jym7EOOrKHB2K1uM7hYVXPcsaDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T21:08:55.592389Z"},"content_sha256":"f047317f67ed4d94eadfe93d6a1d1cf48aba7191b8474e067e3179cd5229385b","schema_version":"1.0","event_id":"sha256:f047317f67ed4d94eadfe93d6a1d1cf48aba7191b8474e067e3179cd5229385b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6VIZWUUZPWYH7WKY6YRYX5GLID/bundle.json","state_url":"https://pith.science/pith/6VIZWUUZPWYH7WKY6YRYX5GLID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6VIZWUUZPWYH7WKY6YRYX5GLID/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T21:08:55Z","links":{"resolver":"https://pith.science/pith/6VIZWUUZPWYH7WKY6YRYX5GLID","bundle":"https://pith.science/pith/6VIZWUUZPWYH7WKY6YRYX5GLID/bundle.json","state":"https://pith.science/pith/6VIZWUUZPWYH7WKY6YRYX5GLID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6VIZWUUZPWYH7WKY6YRYX5GLID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:6VIZWUUZPWYH7WKY6YRYX5GLID","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"13ac55753cf49a0ee44e91b467d4418d1eb11495c90962c9b7f4a18195bbaad6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-10-01T00:25:41Z","title_canon_sha256":"2b0d73232c7444ec82adfe5417e220c0fefb7416c1cf500be510bde0514a1f1d"},"schema_version":"1.0","source":{"id":"1310.0105","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.0105","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"arxiv_version","alias_value":"1310.0105v1","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.0105","created_at":"2026-05-18T03:11:45Z"},{"alias_kind":"pith_short_12","alias_value":"6VIZWUUZPWYH","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"6VIZWUUZPWYH7WKY","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"6VIZWUUZ","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:f047317f67ed4d94eadfe93d6a1d1cf48aba7191b8474e067e3179cd5229385b","target":"graph","created_at":"2026-05-18T03:11:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $X$ and $Y$ be simply connected CW complexes with finite rational cohomologies. The rational toral rank $r_0(X)$ of a space $X$ is the largest integer $r$ such that the torus $T^r$ can act continuously on a CW-complex in the rational homotopy type of $X$ with all its isotropy subgroups finite \\cite{H}. 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","authors_text":"Toshihiro Yamaguchi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-10-01T00:25:41Z","title":"Rational toral rank of a map"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.0105","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:879879b96e2a053863d2273ddb62d143516db1907e075fdd1aaec7dcbf7aedfa","target":"record","created_at":"2026-05-18T03:11:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"13ac55753cf49a0ee44e91b467d4418d1eb11495c90962c9b7f4a18195bbaad6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-10-01T00:25:41Z","title_canon_sha256":"2b0d73232c7444ec82adfe5417e220c0fefb7416c1cf500be510bde0514a1f1d"},"schema_version":"1.0","source":{"id":"1310.0105","kind":"arxiv","version":1}},"canonical_sha256":"f5519b52997db07fd958f6238bf4cb40e823b7ccc743001ffe054ca6883c1806","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f5519b52997db07fd958f6238bf4cb40e823b7ccc743001ffe054ca6883c1806","first_computed_at":"2026-05-18T03:11:45.606411Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:11:45.606411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cE7uTptW5OHZt6YQjsaujDhoJY73MDlX6iSOUwh1UoGTBcBImMVKNHpdQLt6EsrBNpcs7ox9N/PdWhAYl3ENAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:11:45.606889Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.0105","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:879879b96e2a053863d2273ddb62d143516db1907e075fdd1aaec7dcbf7aedfa","sha256:f047317f67ed4d94eadfe93d6a1d1cf48aba7191b8474e067e3179cd5229385b"],"state_sha256":"c0f1f7ffe98e1f8da32834c4f7e27f4dbf7ea83c053ff7ea42256f4bfc8ddfc5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kscTE6nlBh+2k2JjqYf+XFuqDCLPcD59/NEJ7fsZZCDIIA1WrIPKxk5r9FhVV/L/AmUiSY+ggJJZ2tzDI8PRDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T21:08:55.596311Z","bundle_sha256":"93bb1c9e7dccd8f74f03e8826e70a82ac528d5b6785547efbec839dd6398b163"}}