{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JVIOPDOBYEEVVXMRYLJ2J3F6SR","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":"02bb9ed20ca075d8f2caf79b0c9b9e1e2278db1359def425e368eaf3e72d006b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-05T18:31:11Z","title_canon_sha256":"6c4c156d16b7ddf0223490d76b5f0e76ae7a3df1705a158dc5ed3459de0de711"},"schema_version":"1.0","source":{"id":"1506.02006","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.02006","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"arxiv_version","alias_value":"1506.02006v3","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.02006","created_at":"2026-05-18T00:50:31Z"},{"alias_kind":"pith_short_12","alias_value":"JVIOPDOBYEEV","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JVIOPDOBYEEVVXMR","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JVIOPDOB","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:7ae82b9e557dc153b209c7e2842ae1252337121ff254c3e55bf890e22156dcca","target":"graph","created_at":"2026-05-18T00:50:31Z","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":"Following an earlier similar conjecture of Kellendonk and Putnam, Giordano, Putnam and Skau conjectured that all minimal, free ${\\mathbb Z}^d$ actions on Cantor sets admit \"small cocycles.\" These represent classes in $H^1$ that are mapped to small vectors in ${\\mathbb R}^d$ by the Ruelle-Sullivan (RS) map. We show that there exist ${\\mathbb Z}^d$ actions where no such small cocycles exist, and where the image of $H^1$ under RS is ${\\mathbb Z}^d$. Our methods involve tiling spaces and shape deformations, and along the way we prove a relation between the image of RS and the set of \"virtual eigen","authors_text":"Alex Clark, Lorenzo Sadun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-05T18:31:11Z","title":"Small cocycles, fine torus fibrations, and a ${\\mathbb Z}^2$ subshift with neither"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02006","kind":"arxiv","version":3},"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":"sha256:2d6bd00d20b5dae8e5ea20faa62dea722437840ef0f08fb4708e628ddd4b8abc","target":"record","created_at":"2026-05-18T00:50:31Z","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":"02bb9ed20ca075d8f2caf79b0c9b9e1e2278db1359def425e368eaf3e72d006b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-06-05T18:31:11Z","title_canon_sha256":"6c4c156d16b7ddf0223490d76b5f0e76ae7a3df1705a158dc5ed3459de0de711"},"schema_version":"1.0","source":{"id":"1506.02006","kind":"arxiv","version":3}},"canonical_sha256":"4d50e78dc1c1095add91c2d3a4ecbe9470700007ac310710a2cb492cf96fb334","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4d50e78dc1c1095add91c2d3a4ecbe9470700007ac310710a2cb492cf96fb334","first_computed_at":"2026-05-18T00:50:31.287447Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:50:31.287447Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z557QZlKRlMs8a55ZumhdYoYkJdZQ+8niYiluZ0y4kEoy7fnd+zd4VyVVuhI1mc6G36MbA3Sr8ioRZ4WbZknCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:50:31.287998Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.02006","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d6bd00d20b5dae8e5ea20faa62dea722437840ef0f08fb4708e628ddd4b8abc","sha256:7ae82b9e557dc153b209c7e2842ae1252337121ff254c3e55bf890e22156dcca"],"state_sha256":"7dc17c30f0fb269c89ab90ebb76590842f54b1d8047da09b7942e4bf6b677942"}