{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:PJRSP7XOGMINA4ELFVKI2P7ARC","short_pith_number":"pith:PJRSP7XO","canonical_record":{"source":{"id":"1908.05543","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-08-15T13:52:39Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"1eab77aa6c5f298b0822d4a55007d525049803ec3f6fbd7e0fdcb8afa1fbc2ef","abstract_canon_sha256":"caf74065b168fd3f7062aca99c08f326bf575eac8d07499f5e9f5da6a3354e53"},"schema_version":"1.0"},"canonical_sha256":"7a6327feee3310d0708b2d548d3fe0889327e99a95b0a81873cc22a6c8c642d0","source":{"kind":"arxiv","id":"1908.05543","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05543","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05543v3","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05543","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"pith_short_12","alias_value":"PJRSP7XOGMIN","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"pith_short_16","alias_value":"PJRSP7XOGMINA4EL","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"pith_short_8","alias_value":"PJRSP7XO","created_at":"2026-07-05T00:57:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:PJRSP7XOGMINA4ELFVKI2P7ARC","target":"record","payload":{"canonical_record":{"source":{"id":"1908.05543","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-08-15T13:52:39Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"1eab77aa6c5f298b0822d4a55007d525049803ec3f6fbd7e0fdcb8afa1fbc2ef","abstract_canon_sha256":"caf74065b168fd3f7062aca99c08f326bf575eac8d07499f5e9f5da6a3354e53"},"schema_version":"1.0"},"canonical_sha256":"7a6327feee3310d0708b2d548d3fe0889327e99a95b0a81873cc22a6c8c642d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:57:33.735276Z","signature_b64":"6RAa3ysOQjgkscAelpDsKC9DRXtUu36dkQFP/SprY0St74pPXhRPYbuvs7TZXmfOnUuhP0EnOLYvj2aR5aE1DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a6327feee3310d0708b2d548d3fe0889327e99a95b0a81873cc22a6c8c642d0","last_reissued_at":"2026-07-05T00:57:33.734071Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:57:33.734071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.05543","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:57:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RrvlN1S72r8JdP3jkqL+RXJxjPvTLEWBokUUv1IoYvSVaMXc0UkW7a0BFlBLBfrlK/R9/hbx4GmbIAv4AtgcDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:40:55.639328Z"},"content_sha256":"9f162e6735d59b32310a4f38e2e2c4743a07c23ecc98606a7102b6ca45020426","schema_version":"1.0","event_id":"sha256:9f162e6735d59b32310a4f38e2e2c4743a07c23ecc98606a7102b6ca45020426"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:PJRSP7XOGMINA4ELFVKI2P7ARC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Brown's Criterion and classifying spaces for families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GR","authors_text":"Eduardo Mart\\'inez-Pedroza, Luis Jorge S\\'anchez Salda\\~na","submitted_at":"2019-08-15T13:52:39Z","abstract_excerpt":"Let $G$ be a group and $\\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\\mathcal{F}$, and for all $H\\in \\mathcal{F}$ the fixed point subspace $X^H$ is contractible. The group $G$ is of type $\\mathcal{F}\\text{-}\\mathrm{F}_{n}$ if it admits a model for $E_\\mathcal{F} G$ with $n$-skeleton with compact orbit space. The main result of the article provides is a characterization of $\\mathcal{F}\\text{-}\\mathrm{F}_{n}$ analogue to Brown's criterion for $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05543","kind":"arxiv","version":3},"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/1908.05543/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":"2026-07-05T00:57:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gad6svHmHnU20priz19kdcl20k4K5BPNdqItI5gVmElaCq5mgaow7iEOXaP+caz1Swdbg61KX/1JRgwgdkBCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T05:40:55.640537Z"},"content_sha256":"f93fa5e3ad9037457a7ce0faed0174e6de40106eb3ec93f9ececd34351606682","schema_version":"1.0","event_id":"sha256:f93fa5e3ad9037457a7ce0faed0174e6de40106eb3ec93f9ececd34351606682"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PJRSP7XOGMINA4ELFVKI2P7ARC/bundle.json","state_url":"https://pith.science/pith/PJRSP7XOGMINA4ELFVKI2P7ARC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PJRSP7XOGMINA4ELFVKI2P7ARC/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":"2026-08-16T05:40:55Z","links":{"resolver":"https://pith.science/pith/PJRSP7XOGMINA4ELFVKI2P7ARC","bundle":"https://pith.science/pith/PJRSP7XOGMINA4ELFVKI2P7ARC/bundle.json","state":"https://pith.science/pith/PJRSP7XOGMINA4ELFVKI2P7ARC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PJRSP7XOGMINA4ELFVKI2P7ARC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:PJRSP7XOGMINA4ELFVKI2P7ARC","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":"caf74065b168fd3f7062aca99c08f326bf575eac8d07499f5e9f5da6a3354e53","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-08-15T13:52:39Z","title_canon_sha256":"1eab77aa6c5f298b0822d4a55007d525049803ec3f6fbd7e0fdcb8afa1fbc2ef"},"schema_version":"1.0","source":{"id":"1908.05543","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05543","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05543v3","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05543","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"pith_short_12","alias_value":"PJRSP7XOGMIN","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"pith_short_16","alias_value":"PJRSP7XOGMINA4EL","created_at":"2026-07-05T00:57:33Z"},{"alias_kind":"pith_short_8","alias_value":"PJRSP7XO","created_at":"2026-07-05T00:57:33Z"}],"graph_snapshots":[{"event_id":"sha256:f93fa5e3ad9037457a7ce0faed0174e6de40106eb3ec93f9ececd34351606682","target":"graph","created_at":"2026-07-05T00:57:33Z","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/1908.05543/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $G$ be a group and $\\mathcal{F}$ be a family of subgroups closed under conjugation and subgroups. A model for the classifying space $E_{\\mathcal{F}} G$ is a $G$-CW-complex $X$ such that every isotropy group belongs to $\\mathcal{F}$, and for all $H\\in \\mathcal{F}$ the fixed point subspace $X^H$ is contractible. The group $G$ is of type $\\mathcal{F}\\text{-}\\mathrm{F}_{n}$ if it admits a model for $E_\\mathcal{F} G$ with $n$-skeleton with compact orbit space. The main result of the article provides is a characterization of $\\mathcal{F}\\text{-}\\mathrm{F}_{n}$ analogue to Brown's criterion for $","authors_text":"Eduardo Mart\\'inez-Pedroza, Luis Jorge S\\'anchez Salda\\~na","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-08-15T13:52:39Z","title":"Brown's Criterion and classifying spaces for families"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05543","kind":"arxiv","version":3},"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":"sha256:9f162e6735d59b32310a4f38e2e2c4743a07c23ecc98606a7102b6ca45020426","target":"record","created_at":"2026-07-05T00:57:33Z","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":"caf74065b168fd3f7062aca99c08f326bf575eac8d07499f5e9f5da6a3354e53","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-08-15T13:52:39Z","title_canon_sha256":"1eab77aa6c5f298b0822d4a55007d525049803ec3f6fbd7e0fdcb8afa1fbc2ef"},"schema_version":"1.0","source":{"id":"1908.05543","kind":"arxiv","version":3}},"canonical_sha256":"7a6327feee3310d0708b2d548d3fe0889327e99a95b0a81873cc22a6c8c642d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7a6327feee3310d0708b2d548d3fe0889327e99a95b0a81873cc22a6c8c642d0","first_computed_at":"2026-07-05T00:57:33.734071Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:57:33.734071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6RAa3ysOQjgkscAelpDsKC9DRXtUu36dkQFP/SprY0St74pPXhRPYbuvs7TZXmfOnUuhP0EnOLYvj2aR5aE1DA==","signature_status":"signed_v1","signed_at":"2026-07-05T00:57:33.735276Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.05543","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9f162e6735d59b32310a4f38e2e2c4743a07c23ecc98606a7102b6ca45020426","sha256:f93fa5e3ad9037457a7ce0faed0174e6de40106eb3ec93f9ececd34351606682"],"state_sha256":"addc04f05543a4f279a74f82deee23042493c91d55956b95179ff3bea37ea569"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"09vrtNQeI+8Z2YTh6TyuYIlnYf8zcv/+5MA8FGn1g0bD78vCjZTfI+9LBcpzIDvR/3FuJrwfQkhVFcDLkJNUBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T05:40:55.706185Z","bundle_sha256":"b8102c5a77a34c94b91dadf49dc7e2d99529a05ac2e3d3a214789d5370b8698e"}}