{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:444FKXCMM2MADT2Z46P3RRR62J","short_pith_number":"pith:444FKXCM","canonical_record":{"source":{"id":"1312.7450","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-12-28T16:51:56Z","cross_cats_sorted":[],"title_canon_sha256":"c738b37ec54df420dfa06dbdc70f49aa8c2be7f60fecb7fcf83f61b1edc02817","abstract_canon_sha256":"369501e771196884bebd286ee5a8d4f3297f3a72116b483af7331d314805d913"},"schema_version":"1.0"},"canonical_sha256":"e738555c4c669801cf59e79fb8c63ed25339085cec2f145df3cd92906083a4f8","source":{"kind":"arxiv","id":"1312.7450","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.7450","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"arxiv_version","alias_value":"1312.7450v2","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.7450","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"pith_short_12","alias_value":"444FKXCMM2MA","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"444FKXCMM2MADT2Z","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"444FKXCM","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:444FKXCMM2MADT2Z46P3RRR62J","target":"record","payload":{"canonical_record":{"source":{"id":"1312.7450","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-12-28T16:51:56Z","cross_cats_sorted":[],"title_canon_sha256":"c738b37ec54df420dfa06dbdc70f49aa8c2be7f60fecb7fcf83f61b1edc02817","abstract_canon_sha256":"369501e771196884bebd286ee5a8d4f3297f3a72116b483af7331d314805d913"},"schema_version":"1.0"},"canonical_sha256":"e738555c4c669801cf59e79fb8c63ed25339085cec2f145df3cd92906083a4f8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:31.003383Z","signature_b64":"HwSV666K/xTKZqbZbh4WXmYlnDkygoFtnP+JJ1vlt14SCoynkWRCNj3PbHi1lIpUusCA8nrDaPwGS9g1hSD7BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e738555c4c669801cf59e79fb8c63ed25339085cec2f145df3cd92906083a4f8","last_reissued_at":"2026-05-18T01:19:31.002726Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:31.002726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.7450","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":"2026-05-18T01:19:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"546jSXw65CWzhJpl4gqzWzSsJD+AQpf0Yi28me5rxNbArx8ynlSDGxe7dzEG6BiS3WQEXyCLtukr3XN+0m3iAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T09:48:29.726502Z"},"content_sha256":"a217d7601eeedb670e682d2c585d78be2d6525cefc52bf9a02e55448d5111d97","schema_version":"1.0","event_id":"sha256:a217d7601eeedb670e682d2c585d78be2d6525cefc52bf9a02e55448d5111d97"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:444FKXCMM2MADT2Z46P3RRR62J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Classifying spaces of twisted loop groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Thomas Baird","submitted_at":"2013-12-28T16:51:56Z","abstract_excerpt":"We study the classifying space of a twisted loop group $L_{\\sigma}G$ where $G$ is a compact Lie group and $\\sigma$ is an automorphism of $G$ of finite order modulo inner automorphisms. Equivalently, we study the $\\sigma$-twisted adjoint action of $G$ on itself. We derive a formula for the cohomology ring $H^*(BL_{\\sigma}G)$ and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.7450","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":"2026-05-18T01:19:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WtYmlQnHTvImZNHgrN2nfXA+CElUBk/95L1P0Y6Zwi6gQbvqAdeT/nXlcauy8wQZhGTOng8Man6V27dU78GJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T09:48:29.726878Z"},"content_sha256":"2cd31aa66f73ff0ad105a6da0b6e0c9dfe6cb37df6c695b601986687f5a54e90","schema_version":"1.0","event_id":"sha256:2cd31aa66f73ff0ad105a6da0b6e0c9dfe6cb37df6c695b601986687f5a54e90"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/444FKXCMM2MADT2Z46P3RRR62J/bundle.json","state_url":"https://pith.science/pith/444FKXCMM2MADT2Z46P3RRR62J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/444FKXCMM2MADT2Z46P3RRR62J/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":"2026-06-07T09:48:29Z","links":{"resolver":"https://pith.science/pith/444FKXCMM2MADT2Z46P3RRR62J","bundle":"https://pith.science/pith/444FKXCMM2MADT2Z46P3RRR62J/bundle.json","state":"https://pith.science/pith/444FKXCMM2MADT2Z46P3RRR62J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/444FKXCMM2MADT2Z46P3RRR62J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:444FKXCMM2MADT2Z46P3RRR62J","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":"369501e771196884bebd286ee5a8d4f3297f3a72116b483af7331d314805d913","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-12-28T16:51:56Z","title_canon_sha256":"c738b37ec54df420dfa06dbdc70f49aa8c2be7f60fecb7fcf83f61b1edc02817"},"schema_version":"1.0","source":{"id":"1312.7450","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.7450","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"arxiv_version","alias_value":"1312.7450v2","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.7450","created_at":"2026-05-18T01:19:31Z"},{"alias_kind":"pith_short_12","alias_value":"444FKXCMM2MA","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"444FKXCMM2MADT2Z","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"444FKXCM","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:2cd31aa66f73ff0ad105a6da0b6e0c9dfe6cb37df6c695b601986687f5a54e90","target":"graph","created_at":"2026-05-18T01:19:31Z","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 study the classifying space of a twisted loop group $L_{\\sigma}G$ where $G$ is a compact Lie group and $\\sigma$ is an automorphism of $G$ of finite order modulo inner automorphisms. Equivalently, we study the $\\sigma$-twisted adjoint action of $G$ on itself. We derive a formula for the cohomology ring $H^*(BL_{\\sigma}G)$ and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.","authors_text":"Thomas Baird","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-12-28T16:51:56Z","title":"Classifying spaces of twisted loop groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.7450","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:a217d7601eeedb670e682d2c585d78be2d6525cefc52bf9a02e55448d5111d97","target":"record","created_at":"2026-05-18T01:19:31Z","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":"369501e771196884bebd286ee5a8d4f3297f3a72116b483af7331d314805d913","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2013-12-28T16:51:56Z","title_canon_sha256":"c738b37ec54df420dfa06dbdc70f49aa8c2be7f60fecb7fcf83f61b1edc02817"},"schema_version":"1.0","source":{"id":"1312.7450","kind":"arxiv","version":2}},"canonical_sha256":"e738555c4c669801cf59e79fb8c63ed25339085cec2f145df3cd92906083a4f8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e738555c4c669801cf59e79fb8c63ed25339085cec2f145df3cd92906083a4f8","first_computed_at":"2026-05-18T01:19:31.002726Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:31.002726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HwSV666K/xTKZqbZbh4WXmYlnDkygoFtnP+JJ1vlt14SCoynkWRCNj3PbHi1lIpUusCA8nrDaPwGS9g1hSD7BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:31.003383Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.7450","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a217d7601eeedb670e682d2c585d78be2d6525cefc52bf9a02e55448d5111d97","sha256:2cd31aa66f73ff0ad105a6da0b6e0c9dfe6cb37df6c695b601986687f5a54e90"],"state_sha256":"7402e9aaea52adfdfeeed10abe484f47a87e5e8dc434bc65efa4a70ff66d46fa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7ME4plxiWg1tI6c/RIeh/ixOl08Y5BzaBF6vEStHlBVIvAS/uTWB8c6lP8DnYY8B/1io5YkxSBqoUttz9k1UCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T09:48:29.729220Z","bundle_sha256":"2468f58bd4b76e22f105503b7919c43c03a8809f083d4237a53963142312be40"}}