{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:IONI4KHD7Q47WO5O7J54QCMYLB","short_pith_number":"pith:IONI4KHD","canonical_record":{"source":{"id":"0706.0790","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2007-06-06T09:25:12Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"77bf3ce93e7e3eb0703a9838c94bbad42ed0f61b935f6161233f47afd94d0d60","abstract_canon_sha256":"39c88c4b30bbe6edaf6b3c6ce6f0a8fb332389e2e2cba33f19cf021dca815749"},"schema_version":"1.0"},"canonical_sha256":"439a8e28e3fc39fb3baefa7bc80998584a6ccc8a27f6174242d8dabfc62aef3d","source":{"kind":"arxiv","id":"0706.0790","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0706.0790","created_at":"2026-05-18T03:34:12Z"},{"alias_kind":"arxiv_version","alias_value":"0706.0790v2","created_at":"2026-05-18T03:34:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0706.0790","created_at":"2026-05-18T03:34:12Z"},{"alias_kind":"pith_short_12","alias_value":"IONI4KHD7Q47","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"IONI4KHD7Q47WO5O","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"IONI4KHD","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:IONI4KHD7Q47WO5O7J54QCMYLB","target":"record","payload":{"canonical_record":{"source":{"id":"0706.0790","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2007-06-06T09:25:12Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"77bf3ce93e7e3eb0703a9838c94bbad42ed0f61b935f6161233f47afd94d0d60","abstract_canon_sha256":"39c88c4b30bbe6edaf6b3c6ce6f0a8fb332389e2e2cba33f19cf021dca815749"},"schema_version":"1.0"},"canonical_sha256":"439a8e28e3fc39fb3baefa7bc80998584a6ccc8a27f6174242d8dabfc62aef3d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:34:12.640652Z","signature_b64":"meFdZP/NWs+AZSCSR+ezR7WIUH4CGh+M2pZVH8a56Q0g3cX9ApDcibvOT/yvWgG7d/S7mpLJ9N5wqULWq8mWBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"439a8e28e3fc39fb3baefa7bc80998584a6ccc8a27f6174242d8dabfc62aef3d","last_reissued_at":"2026-05-18T03:34:12.640049Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:34:12.640049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0706.0790","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":"2026-05-18T03:34:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S5peuZS7uxI5OdMB4ErRfUOb+RtsblxcAxjs/rBZO2r03/Fm34vWE5Jfio7dYPhSGBi4rxh2/5ZkS6FGqfSCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:57:35.601272Z"},"content_sha256":"97c0b0c921887e8b0f948cbbedbdb0bfa6719bc9001584d877a453dee31c9c9e","schema_version":"1.0","event_id":"sha256:97c0b0c921887e8b0f948cbbedbdb0bfa6719bc9001584d877a453dee31c9c9e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:IONI4KHD7Q47WO5O7J54QCMYLB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Free Actions of Finite Groups on $S^n \\times S^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Ian Hambleton, Ozgun Unlu","submitted_at":"2007-06-06T09:25:12Z","abstract_excerpt":"Let $p$ be an odd prime. We construct a non-abelian extension $\\Gamma$ of $S^1$ by $Z/p \\times Z/p$, and prove that any finite subgroup of $\\Gamma$ acts freely and smoothly on $S^{2p-1} \\times S^{2p-1}$. In particular, for each odd prime $p$ we obtain free smooth actions of infinitely many non-metacyclic rank two $p$-groups on $S^{2p-1} \\times S^{2p-1}$. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0706.0790","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":"2026-05-18T03:34:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p/+Utn3R/Q7ukCnY/1nNu/VPj86KXkYop9pFLOmAPY8Ln8xEBWpC4fkl12RHNnH/z8/iyoGm0mxWXyTO1ftxAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T00:57:35.601641Z"},"content_sha256":"cc1bcda759ed4aaf90c8f97854a60dd5c2376cee24200b6f0a65f02b0853c0d6","schema_version":"1.0","event_id":"sha256:cc1bcda759ed4aaf90c8f97854a60dd5c2376cee24200b6f0a65f02b0853c0d6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IONI4KHD7Q47WO5O7J54QCMYLB/bundle.json","state_url":"https://pith.science/pith/IONI4KHD7Q47WO5O7J54QCMYLB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IONI4KHD7Q47WO5O7J54QCMYLB/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":"2026-05-31T00:57:35Z","links":{"resolver":"https://pith.science/pith/IONI4KHD7Q47WO5O7J54QCMYLB","bundle":"https://pith.science/pith/IONI4KHD7Q47WO5O7J54QCMYLB/bundle.json","state":"https://pith.science/pith/IONI4KHD7Q47WO5O7J54QCMYLB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IONI4KHD7Q47WO5O7J54QCMYLB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:IONI4KHD7Q47WO5O7J54QCMYLB","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":"39c88c4b30bbe6edaf6b3c6ce6f0a8fb332389e2e2cba33f19cf021dca815749","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2007-06-06T09:25:12Z","title_canon_sha256":"77bf3ce93e7e3eb0703a9838c94bbad42ed0f61b935f6161233f47afd94d0d60"},"schema_version":"1.0","source":{"id":"0706.0790","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0706.0790","created_at":"2026-05-18T03:34:12Z"},{"alias_kind":"arxiv_version","alias_value":"0706.0790v2","created_at":"2026-05-18T03:34:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0706.0790","created_at":"2026-05-18T03:34:12Z"},{"alias_kind":"pith_short_12","alias_value":"IONI4KHD7Q47","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"IONI4KHD7Q47WO5O","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"IONI4KHD","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:cc1bcda759ed4aaf90c8f97854a60dd5c2376cee24200b6f0a65f02b0853c0d6","target":"graph","created_at":"2026-05-18T03:34:12Z","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 $p$ be an odd prime. We construct a non-abelian extension $\\Gamma$ of $S^1$ by $Z/p \\times Z/p$, and prove that any finite subgroup of $\\Gamma$ acts freely and smoothly on $S^{2p-1} \\times S^{2p-1}$. In particular, for each odd prime $p$ we obtain free smooth actions of infinitely many non-metacyclic rank two $p$-groups on $S^{2p-1} \\times S^{2p-1}$. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres.","authors_text":"Ian Hambleton, Ozgun Unlu","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2007-06-06T09:25:12Z","title":"Free Actions of Finite Groups on $S^n \\times S^n$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0706.0790","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":"sha256:97c0b0c921887e8b0f948cbbedbdb0bfa6719bc9001584d877a453dee31c9c9e","target":"record","created_at":"2026-05-18T03:34:12Z","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":"39c88c4b30bbe6edaf6b3c6ce6f0a8fb332389e2e2cba33f19cf021dca815749","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2007-06-06T09:25:12Z","title_canon_sha256":"77bf3ce93e7e3eb0703a9838c94bbad42ed0f61b935f6161233f47afd94d0d60"},"schema_version":"1.0","source":{"id":"0706.0790","kind":"arxiv","version":2}},"canonical_sha256":"439a8e28e3fc39fb3baefa7bc80998584a6ccc8a27f6174242d8dabfc62aef3d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"439a8e28e3fc39fb3baefa7bc80998584a6ccc8a27f6174242d8dabfc62aef3d","first_computed_at":"2026-05-18T03:34:12.640049Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:12.640049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"meFdZP/NWs+AZSCSR+ezR7WIUH4CGh+M2pZVH8a56Q0g3cX9ApDcibvOT/yvWgG7d/S7mpLJ9N5wqULWq8mWBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:12.640652Z","signed_message":"canonical_sha256_bytes"},"source_id":"0706.0790","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97c0b0c921887e8b0f948cbbedbdb0bfa6719bc9001584d877a453dee31c9c9e","sha256:cc1bcda759ed4aaf90c8f97854a60dd5c2376cee24200b6f0a65f02b0853c0d6"],"state_sha256":"9e9a1c53a4d6bc69304482b7c2ad1db4982bbc75eb394d0369a630387a0ac617"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xyr0ylU8oKkDqXGChZ5xC1Rub1scINTBJtXJfYWHE6Cs6/ddgyBncWq9iAqXNow+1fuGgQtMiHHYXwIdvpoeDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T00:57:35.603745Z","bundle_sha256":"8ca67114980e6323ab70176fab16ace2be0f7d874ef4f080048df1debae71dd2"}}