{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:FL5DWRSZWG63KMS67VLFMNLPZW","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":"b2012b09584bd3700f471eb85abdf808e97840c56d01a70e29a8df4c95c15da7","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-10-27T22:30:24Z","title_canon_sha256":"f94b6057ff9b7e36512046370715ec0296700b16656dbc64464ff8449a9e3f0f"},"schema_version":"1.0","source":{"id":"0910.5262","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.5262","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"arxiv_version","alias_value":"0910.5262v2","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.5262","created_at":"2026-05-18T03:59:14Z"},{"alias_kind":"pith_short_12","alias_value":"FL5DWRSZWG63","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_16","alias_value":"FL5DWRSZWG63KMS6","created_at":"2026-05-18T12:25:59Z"},{"alias_kind":"pith_short_8","alias_value":"FL5DWRSZ","created_at":"2026-05-18T12:25:59Z"}],"graph_snapshots":[{"event_id":"sha256:cc00fd3f4c288f24430af01f12d9e87a9a51068100d76eed8da34c81e588dd2d","target":"graph","created_at":"2026-05-18T03:59:14Z","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":"We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product ","authors_text":"Takuya Sakasai","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-10-27T22:30:24Z","title":"Lagrangian mapping class groups from a group homological point of view"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.5262","kind":"arxiv","version":2},"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":"sha256:41612d02f15ab0d3b74cae7d7cbfa0056a91d41ce2efd8669dedea735243a5de","target":"record","created_at":"2026-05-18T03:59:14Z","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":"b2012b09584bd3700f471eb85abdf808e97840c56d01a70e29a8df4c95c15da7","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2009-10-27T22:30:24Z","title_canon_sha256":"f94b6057ff9b7e36512046370715ec0296700b16656dbc64464ff8449a9e3f0f"},"schema_version":"1.0","source":{"id":"0910.5262","kind":"arxiv","version":2}},"canonical_sha256":"2afa3b4659b1bdb5325efd5656356fcda4d3b4cfb9b4667bd95f2925214d8645","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2afa3b4659b1bdb5325efd5656356fcda4d3b4cfb9b4667bd95f2925214d8645","first_computed_at":"2026-05-18T03:59:14.147197Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:14.147197Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mpdvvQNLa2Dz3dlG/ckGNoyyGEDH5aItfTbOWVAk2qHyUnfjgwWjblieDolU6u17f/1Zfsm2z05RslvfHoagCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:14.147914Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.5262","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41612d02f15ab0d3b74cae7d7cbfa0056a91d41ce2efd8669dedea735243a5de","sha256:cc00fd3f4c288f24430af01f12d9e87a9a51068100d76eed8da34c81e588dd2d"],"state_sha256":"aef000517955a0116236821dbc2bf3cf70bfd41c5924b77a63211c3e882bca7d"}