{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:GQG3GWPZK6J56YESDWZ3XI7UF4","short_pith_number":"pith:GQG3GWPZ","canonical_record":{"source":{"id":"2405.04846","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2024-05-08T06:49:04Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"e54980ef58f0ae23dca90b8e2cfc5a7c2349cdd8dfb0b1b78c84af902aa27075","abstract_canon_sha256":"c3372f9dd819591a52c53152285212083bb67b85dec9fed1822539da39f301a7"},"schema_version":"1.0"},"canonical_sha256":"340db359f95793df60921db3bba3f42f23ec8eaad3f8808b8a0279a1ca0cc7ff","source":{"kind":"arxiv","id":"2405.04846","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.04846","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"arxiv_version","alias_value":"2405.04846v1","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.04846","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"pith_short_12","alias_value":"GQG3GWPZK6J5","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"pith_short_16","alias_value":"GQG3GWPZK6J56YES","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"pith_short_8","alias_value":"GQG3GWPZ","created_at":"2026-07-05T08:16:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:GQG3GWPZK6J56YESDWZ3XI7UF4","target":"record","payload":{"canonical_record":{"source":{"id":"2405.04846","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2024-05-08T06:49:04Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"e54980ef58f0ae23dca90b8e2cfc5a7c2349cdd8dfb0b1b78c84af902aa27075","abstract_canon_sha256":"c3372f9dd819591a52c53152285212083bb67b85dec9fed1822539da39f301a7"},"schema_version":"1.0"},"canonical_sha256":"340db359f95793df60921db3bba3f42f23ec8eaad3f8808b8a0279a1ca0cc7ff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:16:55.018497Z","signature_b64":"dkYsORN/uAGhNYy6EQRHzTwj0XsxtW5lewEjjYumUQeCp0Dg5niicBdoINm8pi5X92vagwDHTl9IQGtPKEgHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"340db359f95793df60921db3bba3f42f23ec8eaad3f8808b8a0279a1ca0cc7ff","last_reissued_at":"2026-07-05T08:16:55.018043Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:16:55.018043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2405.04846","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":"2026-07-05T08:16:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6f6j/OS6Oq8KCaxyDCCDajAXo0GLsh0hI4QeLTNF/KahLe2CDaD0ODyb71RMlQxyLdkEn6INkFdDqxQq7r33Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T22:38:46.241565Z"},"content_sha256":"4f74a0320ca482b3a93f312d95092ec9b9b73241a56b19eec69ceccd945c95d9","schema_version":"1.0","event_id":"sha256:4f74a0320ca482b3a93f312d95092ec9b9b73241a56b19eec69ceccd945c95d9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:GQG3GWPZK6J56YESDWZ3XI7UF4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Expansion and torsion homology of 3-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.GT","authors_text":"Jonathan Zung","submitted_at":"2024-05-08T06:49:04Z","abstract_excerpt":"A Riemannian manifold is a called a good rational expander in dimension $i$ if every $i$-cycle bounds a rational $i+1$-chain of comparatively small volume. We construct 3-manifolds which are good expanders in all dimensions. On the other hand, we show that expanders must be topologically complicated: they must have lots of torsion homology. We also give some applications to topological overlap problems, constructing examples of 3-manifolds with large width over $\\mathbb R^2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.04846","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.04846/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:16:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WXkTWIvC+oYBgDyVm6T8+nE965KXhjTtyJJJQuC8vQBYN+Gf3hhJ+EJRkLiawMioJ20L9frluGCIa6NQzCX+Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T22:38:46.242061Z"},"content_sha256":"3e1410ccfa03d4d73ff21f89f8ebfd5521a5b120d6f954d6531b5005bc958465","schema_version":"1.0","event_id":"sha256:3e1410ccfa03d4d73ff21f89f8ebfd5521a5b120d6f954d6531b5005bc958465"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GQG3GWPZK6J56YESDWZ3XI7UF4/bundle.json","state_url":"https://pith.science/pith/GQG3GWPZK6J56YESDWZ3XI7UF4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GQG3GWPZK6J56YESDWZ3XI7UF4/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":"2026-08-20T22:38:46Z","links":{"resolver":"https://pith.science/pith/GQG3GWPZK6J56YESDWZ3XI7UF4","bundle":"https://pith.science/pith/GQG3GWPZK6J56YESDWZ3XI7UF4/bundle.json","state":"https://pith.science/pith/GQG3GWPZK6J56YESDWZ3XI7UF4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GQG3GWPZK6J56YESDWZ3XI7UF4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:GQG3GWPZK6J56YESDWZ3XI7UF4","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":"c3372f9dd819591a52c53152285212083bb67b85dec9fed1822539da39f301a7","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2024-05-08T06:49:04Z","title_canon_sha256":"e54980ef58f0ae23dca90b8e2cfc5a7c2349cdd8dfb0b1b78c84af902aa27075"},"schema_version":"1.0","source":{"id":"2405.04846","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2405.04846","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"arxiv_version","alias_value":"2405.04846v1","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2405.04846","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"pith_short_12","alias_value":"GQG3GWPZK6J5","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"pith_short_16","alias_value":"GQG3GWPZK6J56YES","created_at":"2026-07-05T08:16:55Z"},{"alias_kind":"pith_short_8","alias_value":"GQG3GWPZ","created_at":"2026-07-05T08:16:55Z"}],"graph_snapshots":[{"event_id":"sha256:3e1410ccfa03d4d73ff21f89f8ebfd5521a5b120d6f954d6531b5005bc958465","target":"graph","created_at":"2026-07-05T08:16:55Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2405.04846/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A Riemannian manifold is a called a good rational expander in dimension $i$ if every $i$-cycle bounds a rational $i+1$-chain of comparatively small volume. We construct 3-manifolds which are good expanders in all dimensions. On the other hand, we show that expanders must be topologically complicated: they must have lots of torsion homology. We also give some applications to topological overlap problems, constructing examples of 3-manifolds with large width over $\\mathbb R^2$.","authors_text":"Jonathan Zung","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2024-05-08T06:49:04Z","title":"Expansion and torsion homology of 3-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.04846","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":"sha256:4f74a0320ca482b3a93f312d95092ec9b9b73241a56b19eec69ceccd945c95d9","target":"record","created_at":"2026-07-05T08:16:55Z","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":"c3372f9dd819591a52c53152285212083bb67b85dec9fed1822539da39f301a7","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2024-05-08T06:49:04Z","title_canon_sha256":"e54980ef58f0ae23dca90b8e2cfc5a7c2349cdd8dfb0b1b78c84af902aa27075"},"schema_version":"1.0","source":{"id":"2405.04846","kind":"arxiv","version":1}},"canonical_sha256":"340db359f95793df60921db3bba3f42f23ec8eaad3f8808b8a0279a1ca0cc7ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"340db359f95793df60921db3bba3f42f23ec8eaad3f8808b8a0279a1ca0cc7ff","first_computed_at":"2026-07-05T08:16:55.018043Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:16:55.018043Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dkYsORN/uAGhNYy6EQRHzTwj0XsxtW5lewEjjYumUQeCp0Dg5niicBdoINm8pi5X92vagwDHTl9IQGtPKEgHAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:16:55.018497Z","signed_message":"canonical_sha256_bytes"},"source_id":"2405.04846","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f74a0320ca482b3a93f312d95092ec9b9b73241a56b19eec69ceccd945c95d9","sha256:3e1410ccfa03d4d73ff21f89f8ebfd5521a5b120d6f954d6531b5005bc958465"],"state_sha256":"b3d5f6efb413864d7f51a7b4f3e6b197f6a0a5a75671b1ec891025940f98105f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TabsnbppgTqS2ma6XBZw2CfX/c/5lHbUcf8mMWrE2vG9iyOE5X54LjgZMJrkumNVuJPDzKKZHVeCS2r3ZRG4Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T22:38:46.248728Z","bundle_sha256":"ad8b4da6693c6dff5451e7576b74de76f1baa2672da43e1ffca06232aff90dfe"}}