{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2004:6PQEFWMWT5M65W3622HMZ7TOEG","short_pith_number":"pith:6PQEFWMW","canonical_record":{"source":{"id":"math/0406277","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2004-06-14T17:23:22Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"6c5d7adf24301b217d3ff1bdd53800207cb7bcb7ebe8d492b18ba70a0f5b29f2","abstract_canon_sha256":"bec57f32997c9386eb8986a3246d0a0200c742c05a793d48698578a4e87cd899"},"schema_version":"1.0"},"canonical_sha256":"f3e042d9969f59eedb7ed68eccfe6e21a137929ba5fa6d977a4063b24ef31351","source":{"kind":"arxiv","id":"math/0406277","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0406277","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0406277v2","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0406277","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"pith_short_12","alias_value":"6PQEFWMWT5M6","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"6PQEFWMWT5M65W36","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"6PQEFWMW","created_at":"2026-05-18T12:25:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2004:6PQEFWMWT5M65W3622HMZ7TOEG","target":"record","payload":{"canonical_record":{"source":{"id":"math/0406277","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2004-06-14T17:23:22Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"6c5d7adf24301b217d3ff1bdd53800207cb7bcb7ebe8d492b18ba70a0f5b29f2","abstract_canon_sha256":"bec57f32997c9386eb8986a3246d0a0200c742c05a793d48698578a4e87cd899"},"schema_version":"1.0"},"canonical_sha256":"f3e042d9969f59eedb7ed68eccfe6e21a137929ba5fa6d977a4063b24ef31351","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:00.061127Z","signature_b64":"/1/NC3zFxdayeD2s6C59XIj2S+2NQyPiKLUkq7kaULXS1IXXZC3tUjvG9s4srLLqZJEMuv88Lz7iFMnweRMYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3e042d9969f59eedb7ed68eccfe6e21a137929ba5fa6d977a4063b24ef31351","last_reissued_at":"2026-05-18T02:38:00.060654Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:00.060654Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0406277","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:38:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GyJAF+y/P1dRNNulT76t0aIK+2gMYV1QS2tcZbp039/SJEl9+IoiQGy4w+5b0oLOotm1HlPdh9+U1SGhGxStAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T10:11:18.156724Z"},"content_sha256":"e53dc838fdee17b87d867c0f9271ded2a9a64ea6a8ef4dc43a1f611ee711526e","schema_version":"1.0","event_id":"sha256:e53dc838fdee17b87d867c0f9271ded2a9a64ea6a8ef4dc43a1f611ee711526e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2004:6PQEFWMWT5M65W3622HMZ7TOEG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stabilization for the automorphisms of free groups with boundaries","license":"","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Allen Hatcher, Nathalie Wahl","submitted_at":"2004-06-14T17:23:22Z","abstract_excerpt":"The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms of homotopy equivalences of a graph fixing a subgraph. This is needed for the second author's recent work on the relationship between the infinite loop structures on the classifying spaces of mapping class groups of surfaces and automorphism groups of free groups, after stabilization and plus-construction. We show more generally that the homology groups of ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0406277","kind":"arxiv","version":2},"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":"2026-05-18T02:38:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"njG+5z67hSsA6HuwtSAPjkRCxDOXVuNDotZ7IBcI77cUuJSXEwd3cj9Oj/TdjjGYEFEruBwY410ncWq/Pi8KAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T10:11:18.157072Z"},"content_sha256":"bf24fccb5095bcd1791cee7b0e2f33eca6a21c5e2981035f8457c40080cf0948","schema_version":"1.0","event_id":"sha256:bf24fccb5095bcd1791cee7b0e2f33eca6a21c5e2981035f8457c40080cf0948"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6PQEFWMWT5M65W3622HMZ7TOEG/bundle.json","state_url":"https://pith.science/pith/6PQEFWMWT5M65W3622HMZ7TOEG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6PQEFWMWT5M65W3622HMZ7TOEG/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":"2026-06-12T10:11:18Z","links":{"resolver":"https://pith.science/pith/6PQEFWMWT5M65W3622HMZ7TOEG","bundle":"https://pith.science/pith/6PQEFWMWT5M65W3622HMZ7TOEG/bundle.json","state":"https://pith.science/pith/6PQEFWMWT5M65W3622HMZ7TOEG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6PQEFWMWT5M65W3622HMZ7TOEG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:6PQEFWMWT5M65W3622HMZ7TOEG","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":"bec57f32997c9386eb8986a3246d0a0200c742c05a793d48698578a4e87cd899","cross_cats_sorted":["math.AT"],"license":"","primary_cat":"math.GT","submitted_at":"2004-06-14T17:23:22Z","title_canon_sha256":"6c5d7adf24301b217d3ff1bdd53800207cb7bcb7ebe8d492b18ba70a0f5b29f2"},"schema_version":"1.0","source":{"id":"math/0406277","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0406277","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0406277v2","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0406277","created_at":"2026-05-18T02:38:00Z"},{"alias_kind":"pith_short_12","alias_value":"6PQEFWMWT5M6","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_16","alias_value":"6PQEFWMWT5M65W36","created_at":"2026-05-18T12:25:52Z"},{"alias_kind":"pith_short_8","alias_value":"6PQEFWMW","created_at":"2026-05-18T12:25:52Z"}],"graph_snapshots":[{"event_id":"sha256:bf24fccb5095bcd1791cee7b0e2f33eca6a21c5e2981035f8457c40080cf0948","target":"graph","created_at":"2026-05-18T02:38:00Z","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":"The homology groups of the automorphism group of a free group are known to stabilize as the number of generators of the free group goes to infinity, and this paper relativizes this result to a family of groups that can be defined in terms of homotopy equivalences of a graph fixing a subgraph. This is needed for the second author's recent work on the relationship between the infinite loop structures on the classifying spaces of mapping class groups of surfaces and automorphism groups of free groups, after stabilization and plus-construction. We show more generally that the homology groups of ma","authors_text":"Allen Hatcher, Nathalie Wahl","cross_cats":["math.AT"],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2004-06-14T17:23:22Z","title":"Stabilization for the automorphisms of free groups with boundaries"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0406277","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:e53dc838fdee17b87d867c0f9271ded2a9a64ea6a8ef4dc43a1f611ee711526e","target":"record","created_at":"2026-05-18T02:38:00Z","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":"bec57f32997c9386eb8986a3246d0a0200c742c05a793d48698578a4e87cd899","cross_cats_sorted":["math.AT"],"license":"","primary_cat":"math.GT","submitted_at":"2004-06-14T17:23:22Z","title_canon_sha256":"6c5d7adf24301b217d3ff1bdd53800207cb7bcb7ebe8d492b18ba70a0f5b29f2"},"schema_version":"1.0","source":{"id":"math/0406277","kind":"arxiv","version":2}},"canonical_sha256":"f3e042d9969f59eedb7ed68eccfe6e21a137929ba5fa6d977a4063b24ef31351","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f3e042d9969f59eedb7ed68eccfe6e21a137929ba5fa6d977a4063b24ef31351","first_computed_at":"2026-05-18T02:38:00.060654Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:00.060654Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/1/NC3zFxdayeD2s6C59XIj2S+2NQyPiKLUkq7kaULXS1IXXZC3tUjvG9s4srLLqZJEMuv88Lz7iFMnweRMYAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:00.061127Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0406277","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e53dc838fdee17b87d867c0f9271ded2a9a64ea6a8ef4dc43a1f611ee711526e","sha256:bf24fccb5095bcd1791cee7b0e2f33eca6a21c5e2981035f8457c40080cf0948"],"state_sha256":"f5a9bbc9883c1614fb453d84447aa0c896d68d18e3bc9e84b3fe9c45eaebe99a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/EtFd8InpxfH7shTnAPOZ3qoYikCK59hR4c0qKWBRYTLRDq6nlALJ9W8gYeJiI5RwAbV2f8Zo9eqxMehB8aDCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T10:11:18.159371Z","bundle_sha256":"b029f4d2c307083ff2070d00b8b663f8bcc73287c2a43d9f43e477318542dce6"}}