{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7IOO5ZKGGQLPJQFZHAUBDR6VID","short_pith_number":"pith:7IOO5ZKG","canonical_record":{"source":{"id":"1105.5948","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-05-30T11:47:44Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"f2b78213046adef8420ccc3595720966c97d355bcf1e6368692f7ef8fa721a91","abstract_canon_sha256":"91aad40e7e4e319291611036d95da08140d88ce5fd6af6b3acb61dfa7404fddf"},"schema_version":"1.0"},"canonical_sha256":"fa1ceee5463416f4c0b9382811c7d540ed0476dce5829bfed3b82d4126c45cd3","source":{"kind":"arxiv","id":"1105.5948","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.5948","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"arxiv_version","alias_value":"1105.5948v1","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.5948","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"pith_short_12","alias_value":"7IOO5ZKGGQLP","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7IOO5ZKGGQLPJQFZ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7IOO5ZKG","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7IOO5ZKGGQLPJQFZHAUBDR6VID","target":"record","payload":{"canonical_record":{"source":{"id":"1105.5948","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-05-30T11:47:44Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"f2b78213046adef8420ccc3595720966c97d355bcf1e6368692f7ef8fa721a91","abstract_canon_sha256":"91aad40e7e4e319291611036d95da08140d88ce5fd6af6b3acb61dfa7404fddf"},"schema_version":"1.0"},"canonical_sha256":"fa1ceee5463416f4c0b9382811c7d540ed0476dce5829bfed3b82d4126c45cd3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:28:06.527620Z","signature_b64":"pIM2Kb8XTV1jOZ07EJQEh2KZYEpxhjxozw/9XRvqhZc7P5OsCknv5azCgpSmfjU7inNzjee5O2urRSAT4ZQiAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa1ceee5463416f4c0b9382811c7d540ed0476dce5829bfed3b82d4126c45cd3","last_reissued_at":"2026-05-18T03:28:06.526891Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:28:06.526891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1105.5948","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":"2026-05-18T03:28:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AJ2lJWD6Px+CwpkbzhZvEvj0t6Uy/RhAFx0D0b/T0MMOEVUhGFI+UuqnkOWRVqEe7AIvXJsb/Iyy+hXPpSkYBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T05:59:32.929761Z"},"content_sha256":"e72b0af390dc1c871d8eb60a58a42484bbd34acb0e78d2cfb0168634539a9f3a","schema_version":"1.0","event_id":"sha256:e72b0af390dc1c871d8eb60a58a42484bbd34acb0e78d2cfb0168634539a9f3a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7IOO5ZKGGQLPJQFZHAUBDR6VID","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cohomology of measurable laminations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AT","authors_text":"Carlos Meni\\~no Cot\\'on","submitted_at":"2011-05-30T11:47:44Z","abstract_excerpt":"A new notion of cohomology is introduced for MT-spaces, which are measurable and topological spaces whose measurable structure may not agree with the Borel $\\sigma$-algebra of their topology. The main examples of MTspaces are measurable foliations. This is a singular version of the measurable simplicial cohomology defined by Heitsch and Lazarov for foliations and extended by Bermudez for MT-spaces. Basic topics of algebraic topology are adapted, and applications to the theory of foliations are given. Moreover we introduce a new notion of singular L2-cohomology for MT-spaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.5948","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":"2026-05-18T03:28:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s4btGYja/qAsR1QTHyvNiQnz7EV53kKi4AlVot8+sMdhzANoIuFx/lEc7k9STAbLdu09dlsDVag80vKLlYc/CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T05:59:32.930095Z"},"content_sha256":"acce63505e16862bd82f00c552a5cd11714f0dc529c3c6487d7877c3ddf58fa0","schema_version":"1.0","event_id":"sha256:acce63505e16862bd82f00c552a5cd11714f0dc529c3c6487d7877c3ddf58fa0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7IOO5ZKGGQLPJQFZHAUBDR6VID/bundle.json","state_url":"https://pith.science/pith/7IOO5ZKGGQLPJQFZHAUBDR6VID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7IOO5ZKGGQLPJQFZHAUBDR6VID/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":"2026-06-27T05:59:32Z","links":{"resolver":"https://pith.science/pith/7IOO5ZKGGQLPJQFZHAUBDR6VID","bundle":"https://pith.science/pith/7IOO5ZKGGQLPJQFZHAUBDR6VID/bundle.json","state":"https://pith.science/pith/7IOO5ZKGGQLPJQFZHAUBDR6VID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7IOO5ZKGGQLPJQFZHAUBDR6VID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7IOO5ZKGGQLPJQFZHAUBDR6VID","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":"91aad40e7e4e319291611036d95da08140d88ce5fd6af6b3acb61dfa7404fddf","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-05-30T11:47:44Z","title_canon_sha256":"f2b78213046adef8420ccc3595720966c97d355bcf1e6368692f7ef8fa721a91"},"schema_version":"1.0","source":{"id":"1105.5948","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.5948","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"arxiv_version","alias_value":"1105.5948v1","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.5948","created_at":"2026-05-18T03:28:06Z"},{"alias_kind":"pith_short_12","alias_value":"7IOO5ZKGGQLP","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7IOO5ZKGGQLPJQFZ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7IOO5ZKG","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:acce63505e16862bd82f00c552a5cd11714f0dc529c3c6487d7877c3ddf58fa0","target":"graph","created_at":"2026-05-18T03:28:06Z","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":"A new notion of cohomology is introduced for MT-spaces, which are measurable and topological spaces whose measurable structure may not agree with the Borel $\\sigma$-algebra of their topology. The main examples of MTspaces are measurable foliations. This is a singular version of the measurable simplicial cohomology defined by Heitsch and Lazarov for foliations and extended by Bermudez for MT-spaces. Basic topics of algebraic topology are adapted, and applications to the theory of foliations are given. Moreover we introduce a new notion of singular L2-cohomology for MT-spaces.","authors_text":"Carlos Meni\\~no Cot\\'on","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-05-30T11:47:44Z","title":"Cohomology of measurable laminations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.5948","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:e72b0af390dc1c871d8eb60a58a42484bbd34acb0e78d2cfb0168634539a9f3a","target":"record","created_at":"2026-05-18T03:28:06Z","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":"91aad40e7e4e319291611036d95da08140d88ce5fd6af6b3acb61dfa7404fddf","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2011-05-30T11:47:44Z","title_canon_sha256":"f2b78213046adef8420ccc3595720966c97d355bcf1e6368692f7ef8fa721a91"},"schema_version":"1.0","source":{"id":"1105.5948","kind":"arxiv","version":1}},"canonical_sha256":"fa1ceee5463416f4c0b9382811c7d540ed0476dce5829bfed3b82d4126c45cd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa1ceee5463416f4c0b9382811c7d540ed0476dce5829bfed3b82d4126c45cd3","first_computed_at":"2026-05-18T03:28:06.526891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:28:06.526891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pIM2Kb8XTV1jOZ07EJQEh2KZYEpxhjxozw/9XRvqhZc7P5OsCknv5azCgpSmfjU7inNzjee5O2urRSAT4ZQiAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:28:06.527620Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.5948","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e72b0af390dc1c871d8eb60a58a42484bbd34acb0e78d2cfb0168634539a9f3a","sha256:acce63505e16862bd82f00c552a5cd11714f0dc529c3c6487d7877c3ddf58fa0"],"state_sha256":"427010b9864b4c66d531991a8c09bea1bcb59dcbe0b0891c6872c873cb27696b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2pgiEd/O5VDveeXecg7DmqYtpRrbVuMhJr+DQ9zawHUfkHCT+fJ92woPqpmWhBOviS8LJqI7zd1zYxGiYYgvCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T05:59:32.931985Z","bundle_sha256":"d2b4eead04fa8cc68941907e93f35bae2b21dcf1cc6ad76e3352efee21b55896"}}