{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:XBQC5X6MXGGG2E4CXHP5OQ3ISJ","short_pith_number":"pith:XBQC5X6M","canonical_record":{"source":{"id":"1003.5595","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-03-29T16:22:40Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"c0b8d851a1ceb0e2da79f5afd73041f26e0ec7fdc21a09a8ab87cbe667f15552","abstract_canon_sha256":"4ce003feb3f7bb94a1a7dabfdcdfad0692e3d0c3b92d8c99b56d116333578087"},"schema_version":"1.0"},"canonical_sha256":"b8602edfccb98c6d1382b9dfd7436892621c8fcb9171648d23936c47e00eef66","source":{"kind":"arxiv","id":"1003.5595","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.5595","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"arxiv_version","alias_value":"1003.5595v1","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.5595","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"pith_short_12","alias_value":"XBQC5X6MXGGG","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XBQC5X6MXGGG2E4C","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XBQC5X6M","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:XBQC5X6MXGGG2E4CXHP5OQ3ISJ","target":"record","payload":{"canonical_record":{"source":{"id":"1003.5595","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-03-29T16:22:40Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"c0b8d851a1ceb0e2da79f5afd73041f26e0ec7fdc21a09a8ab87cbe667f15552","abstract_canon_sha256":"4ce003feb3f7bb94a1a7dabfdcdfad0692e3d0c3b92d8c99b56d116333578087"},"schema_version":"1.0"},"canonical_sha256":"b8602edfccb98c6d1382b9dfd7436892621c8fcb9171648d23936c47e00eef66","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:50.908232Z","signature_b64":"I5ZvR6CxDnf2Jv5tYXpQAjkA+zg5QSm23d7tpGREdH5c65rwIsyIdUxEWSPSfuQIaTABO4SjK6dDz44J9p/3Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8602edfccb98c6d1382b9dfd7436892621c8fcb9171648d23936c47e00eef66","last_reissued_at":"2026-05-18T02:41:50.907634Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:50.907634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.5595","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":"2026-05-18T02:41:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tKB/WLNJoS6ESyUAaR7pQbWkADeVgufOLVsoAL3Mo1H/ByGd47wys4jZHnDrvY/MOXeGKRt4xkYaMM1gwp4+CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T18:15:36.080979Z"},"content_sha256":"b0d6caa3cbdb8df28817c5957518d72c689bd8ad573d5bcdf1e0dbddeb6ca74d","schema_version":"1.0","event_id":"sha256:b0d6caa3cbdb8df28817c5957518d72c689bd8ad573d5bcdf1e0dbddeb6ca74d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:XBQC5X6MXGGG2E4CXHP5OQ3ISJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dyer-Lashof operations on Tate cohomology of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AT","authors_text":"Martin Langer","submitted_at":"2010-03-29T16:22:40Z","abstract_excerpt":"Let k be the field with p>0 elements, and let G be a finite group. By exhibiting an E-infinity-operad action on Hom(P,k) for a complete projective resolution P of the trivial kG-module k, we obtain power operations of Dyer-Lashof type on Tate cohomology H*(G; k). Our operations agree with the usual Steenrod operations on ordinary cohomology. We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups G. We also show that the operations in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.5595","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":"2026-05-18T02:41:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tJ7JDZ5ZWfhhrI5ogN48Jjsxw5GH3+3S0/XS+XrrDrbGxC/gSsp2if59ZdMA9PAnoJ5fDKMC25K+naI7dyR6Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T18:15:36.081345Z"},"content_sha256":"c0a59f76100778ce03fed8ab9bf6f96f01b73cae86aeccbb1f756a17e5c4a746","schema_version":"1.0","event_id":"sha256:c0a59f76100778ce03fed8ab9bf6f96f01b73cae86aeccbb1f756a17e5c4a746"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XBQC5X6MXGGG2E4CXHP5OQ3ISJ/bundle.json","state_url":"https://pith.science/pith/XBQC5X6MXGGG2E4CXHP5OQ3ISJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XBQC5X6MXGGG2E4CXHP5OQ3ISJ/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":"2026-06-01T18:15:36Z","links":{"resolver":"https://pith.science/pith/XBQC5X6MXGGG2E4CXHP5OQ3ISJ","bundle":"https://pith.science/pith/XBQC5X6MXGGG2E4CXHP5OQ3ISJ/bundle.json","state":"https://pith.science/pith/XBQC5X6MXGGG2E4CXHP5OQ3ISJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XBQC5X6MXGGG2E4CXHP5OQ3ISJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XBQC5X6MXGGG2E4CXHP5OQ3ISJ","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":"4ce003feb3f7bb94a1a7dabfdcdfad0692e3d0c3b92d8c99b56d116333578087","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-03-29T16:22:40Z","title_canon_sha256":"c0b8d851a1ceb0e2da79f5afd73041f26e0ec7fdc21a09a8ab87cbe667f15552"},"schema_version":"1.0","source":{"id":"1003.5595","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.5595","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"arxiv_version","alias_value":"1003.5595v1","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.5595","created_at":"2026-05-18T02:41:50Z"},{"alias_kind":"pith_short_12","alias_value":"XBQC5X6MXGGG","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XBQC5X6MXGGG2E4C","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XBQC5X6M","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:c0a59f76100778ce03fed8ab9bf6f96f01b73cae86aeccbb1f756a17e5c4a746","target":"graph","created_at":"2026-05-18T02:41:50Z","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":"Let k be the field with p>0 elements, and let G be a finite group. By exhibiting an E-infinity-operad action on Hom(P,k) for a complete projective resolution P of the trivial kG-module k, we obtain power operations of Dyer-Lashof type on Tate cohomology H*(G; k). Our operations agree with the usual Steenrod operations on ordinary cohomology. We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups G. We also show that the operations in","authors_text":"Martin Langer","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-03-29T16:22:40Z","title":"Dyer-Lashof operations on Tate cohomology of finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.5595","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:b0d6caa3cbdb8df28817c5957518d72c689bd8ad573d5bcdf1e0dbddeb6ca74d","target":"record","created_at":"2026-05-18T02:41:50Z","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":"4ce003feb3f7bb94a1a7dabfdcdfad0692e3d0c3b92d8c99b56d116333578087","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-03-29T16:22:40Z","title_canon_sha256":"c0b8d851a1ceb0e2da79f5afd73041f26e0ec7fdc21a09a8ab87cbe667f15552"},"schema_version":"1.0","source":{"id":"1003.5595","kind":"arxiv","version":1}},"canonical_sha256":"b8602edfccb98c6d1382b9dfd7436892621c8fcb9171648d23936c47e00eef66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b8602edfccb98c6d1382b9dfd7436892621c8fcb9171648d23936c47e00eef66","first_computed_at":"2026-05-18T02:41:50.907634Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:50.907634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I5ZvR6CxDnf2Jv5tYXpQAjkA+zg5QSm23d7tpGREdH5c65rwIsyIdUxEWSPSfuQIaTABO4SjK6dDz44J9p/3Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:50.908232Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.5595","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0d6caa3cbdb8df28817c5957518d72c689bd8ad573d5bcdf1e0dbddeb6ca74d","sha256:c0a59f76100778ce03fed8ab9bf6f96f01b73cae86aeccbb1f756a17e5c4a746"],"state_sha256":"ef99b92947b6c83a812c6518309bb6b28dee06b1daeb693018d0d8e981f46c37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"olVgeYZ6NGE9hH5LbawfJprFKAB3471QoQV/ZsAj1SKLoCjatUecDMeHVbNpM8qMiPLpwqHSaHVl4G/udHU0Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T18:15:36.083368Z","bundle_sha256":"fe31a31ba0d9f1830710ff11effaa7c5b1371e49613665b5729f2157ac56e2d7"}}