{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:B7CQVA2ADGXRVGHIOHK6UIXKIQ","short_pith_number":"pith:B7CQVA2A","canonical_record":{"source":{"id":"1006.3269","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-06-16T16:46:30Z","cross_cats_sorted":[],"title_canon_sha256":"2e395b545de9bcd7ff3a3677f27ecaadf85d11ab91f00d91fa8d1a0ccc2c039d","abstract_canon_sha256":"9b1095002addf1ced8a6458e41d535ff9e85ae970cdb6c6f32a95b7b70e8afe0"},"schema_version":"1.0"},"canonical_sha256":"0fc50a834019af1a98e871d5ea22ea441094a29af564b73428a04e2629631fdb","source":{"kind":"arxiv","id":"1006.3269","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.3269","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"arxiv_version","alias_value":"1006.3269v1","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.3269","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"pith_short_12","alias_value":"B7CQVA2ADGXR","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"B7CQVA2ADGXRVGHI","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"B7CQVA2A","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:B7CQVA2ADGXRVGHIOHK6UIXKIQ","target":"record","payload":{"canonical_record":{"source":{"id":"1006.3269","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-06-16T16:46:30Z","cross_cats_sorted":[],"title_canon_sha256":"2e395b545de9bcd7ff3a3677f27ecaadf85d11ab91f00d91fa8d1a0ccc2c039d","abstract_canon_sha256":"9b1095002addf1ced8a6458e41d535ff9e85ae970cdb6c6f32a95b7b70e8afe0"},"schema_version":"1.0"},"canonical_sha256":"0fc50a834019af1a98e871d5ea22ea441094a29af564b73428a04e2629631fdb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:38.678445Z","signature_b64":"Un0iMuhwbWG6I4nyxKZCTG/tV3siGjkhtqwIB1jymrxdR1xKObrj4T9ku0FYoMVSpT4ETWdqEAipe9fqUhoFCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fc50a834019af1a98e871d5ea22ea441094a29af564b73428a04e2629631fdb","last_reissued_at":"2026-05-18T04:20:38.677686Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:38.677686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1006.3269","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":"2026-05-18T04:20:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kNP96QyNTVn8c/xx9WnggsLgp9Za99Q8qOJxkUUzYKAKOcrzvEXgG2sLy/n7QQKpwB5hQGOnwswO+mpYVT/jAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T01:00:41.302386Z"},"content_sha256":"ec81c671158aec4f646fcebbe8d4113381c6536f2266db45b9f1321bb7faeda7","schema_version":"1.0","event_id":"sha256:ec81c671158aec4f646fcebbe8d4113381c6536f2266db45b9f1321bb7faeda7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:B7CQVA2ADGXRVGHIOHK6UIXKIQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the structure of the commutator subgroup of certain homeomorphism groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ilona Michalik, Tomasz Rybicki","submitted_at":"2010-06-16T16:46:30Z","abstract_excerpt":"An important theorem of Ling states that if $G$ is any factorizable non-fixing group of homeomorphisms of a paracompact space then its commutator subgroup $[G,G]$ is perfect. This paper is devoted to further studies on the algebraic structure (e.g. uniform perfectness, uniform simplicity) of $[G,G]$ and $[\\tilde G,\\tilde G]$, where $\\tilde G$ is the universal covering group of $G$. In particular, we prove that if $G$ is bounded factorizable non-fixing group of homeomorphisms then $[G,G]$ is uniformly perfect (Corollary 3.4). The case of open manifolds is also investigated. Examples of homeomor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.3269","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":"2026-05-18T04:20:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kvRUFx+sjNdoFuy6nqzsJWC+WTYZ+iW2snpOU2m0g26j4EDvgawbbsFjvqblN5WZU7+ewNDuCMc0buXKPR9aAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T01:00:41.302729Z"},"content_sha256":"97debe895ad105db7a221f4f4b0c5e3d1c08ac13c18a49327a00f9d86c576c4e","schema_version":"1.0","event_id":"sha256:97debe895ad105db7a221f4f4b0c5e3d1c08ac13c18a49327a00f9d86c576c4e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B7CQVA2ADGXRVGHIOHK6UIXKIQ/bundle.json","state_url":"https://pith.science/pith/B7CQVA2ADGXRVGHIOHK6UIXKIQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B7CQVA2ADGXRVGHIOHK6UIXKIQ/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":"2026-06-20T01:00:41Z","links":{"resolver":"https://pith.science/pith/B7CQVA2ADGXRVGHIOHK6UIXKIQ","bundle":"https://pith.science/pith/B7CQVA2ADGXRVGHIOHK6UIXKIQ/bundle.json","state":"https://pith.science/pith/B7CQVA2ADGXRVGHIOHK6UIXKIQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B7CQVA2ADGXRVGHIOHK6UIXKIQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:B7CQVA2ADGXRVGHIOHK6UIXKIQ","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":"9b1095002addf1ced8a6458e41d535ff9e85ae970cdb6c6f32a95b7b70e8afe0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-06-16T16:46:30Z","title_canon_sha256":"2e395b545de9bcd7ff3a3677f27ecaadf85d11ab91f00d91fa8d1a0ccc2c039d"},"schema_version":"1.0","source":{"id":"1006.3269","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.3269","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"arxiv_version","alias_value":"1006.3269v1","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.3269","created_at":"2026-05-18T04:20:38Z"},{"alias_kind":"pith_short_12","alias_value":"B7CQVA2ADGXR","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"B7CQVA2ADGXRVGHI","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"B7CQVA2A","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:97debe895ad105db7a221f4f4b0c5e3d1c08ac13c18a49327a00f9d86c576c4e","target":"graph","created_at":"2026-05-18T04:20:38Z","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":"An important theorem of Ling states that if $G$ is any factorizable non-fixing group of homeomorphisms of a paracompact space then its commutator subgroup $[G,G]$ is perfect. This paper is devoted to further studies on the algebraic structure (e.g. uniform perfectness, uniform simplicity) of $[G,G]$ and $[\\tilde G,\\tilde G]$, where $\\tilde G$ is the universal covering group of $G$. In particular, we prove that if $G$ is bounded factorizable non-fixing group of homeomorphisms then $[G,G]$ is uniformly perfect (Corollary 3.4). The case of open manifolds is also investigated. Examples of homeomor","authors_text":"Ilona Michalik, Tomasz Rybicki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-06-16T16:46:30Z","title":"On the structure of the commutator subgroup of certain homeomorphism groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.3269","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:ec81c671158aec4f646fcebbe8d4113381c6536f2266db45b9f1321bb7faeda7","target":"record","created_at":"2026-05-18T04:20:38Z","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":"9b1095002addf1ced8a6458e41d535ff9e85ae970cdb6c6f32a95b7b70e8afe0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2010-06-16T16:46:30Z","title_canon_sha256":"2e395b545de9bcd7ff3a3677f27ecaadf85d11ab91f00d91fa8d1a0ccc2c039d"},"schema_version":"1.0","source":{"id":"1006.3269","kind":"arxiv","version":1}},"canonical_sha256":"0fc50a834019af1a98e871d5ea22ea441094a29af564b73428a04e2629631fdb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fc50a834019af1a98e871d5ea22ea441094a29af564b73428a04e2629631fdb","first_computed_at":"2026-05-18T04:20:38.677686Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:20:38.677686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Un0iMuhwbWG6I4nyxKZCTG/tV3siGjkhtqwIB1jymrxdR1xKObrj4T9ku0FYoMVSpT4ETWdqEAipe9fqUhoFCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:20:38.678445Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.3269","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec81c671158aec4f646fcebbe8d4113381c6536f2266db45b9f1321bb7faeda7","sha256:97debe895ad105db7a221f4f4b0c5e3d1c08ac13c18a49327a00f9d86c576c4e"],"state_sha256":"3aa58799f489a4abcd94d047d19fcb88243df416c9a3b5f03ab563794dcec55d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0yYNa6EvjNJv/17uDdAC5+5F7NH73J2CnIpyyU9qG1mnOZuwns9mNnUo3kWnaRWN9PFpSpEPRAtsAKjnzemPBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T01:00:41.304585Z","bundle_sha256":"effc93f1519560a04ebace081d5f7c192cd14564055082fc156c6426c9392274"}}