{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:TUAX7ULSLHAJC4OUT6WNA4YEID","short_pith_number":"pith:TUAX7ULS","canonical_record":{"source":{"id":"1203.5882","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-03-27T07:15:19Z","cross_cats_sorted":[],"title_canon_sha256":"f83f95d2c0debce4d8394823a160f99d603686dfb485616ea4c29fc4a4bfb2cd","abstract_canon_sha256":"6a12e1a5d24934bc8a8229a11c95b2e1ac034c220e8a47ef6f4aaec71bf14919"},"schema_version":"1.0"},"canonical_sha256":"9d017fd17259c09171d49facd0730440d1700990b1341088a6ce5b71353e1e8a","source":{"kind":"arxiv","id":"1203.5882","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.5882","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"arxiv_version","alias_value":"1203.5882v1","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5882","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"pith_short_12","alias_value":"TUAX7ULSLHAJ","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TUAX7ULSLHAJC4OU","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TUAX7ULS","created_at":"2026-05-18T12:27:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:TUAX7ULSLHAJC4OUT6WNA4YEID","target":"record","payload":{"canonical_record":{"source":{"id":"1203.5882","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-03-27T07:15:19Z","cross_cats_sorted":[],"title_canon_sha256":"f83f95d2c0debce4d8394823a160f99d603686dfb485616ea4c29fc4a4bfb2cd","abstract_canon_sha256":"6a12e1a5d24934bc8a8229a11c95b2e1ac034c220e8a47ef6f4aaec71bf14919"},"schema_version":"1.0"},"canonical_sha256":"9d017fd17259c09171d49facd0730440d1700990b1341088a6ce5b71353e1e8a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:13.168744Z","signature_b64":"/8on+2ggo6LfXezWJCIGkQGreS8h/ijJa2lHOw+Dj55+khvm1E/SufLzu4ShgQ2UYIN1rVjDJGoQSlhSmZ9zAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d017fd17259c09171d49facd0730440d1700990b1341088a6ce5b71353e1e8a","last_reissued_at":"2026-05-18T03:59:13.168248Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:13.168248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1203.5882","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":"2026-05-18T03:59:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QdmAKVaMUkDqn2lrYWnzvu6NhF7uyPV3PZpmcAGoeys5yCWM6m++PNaJ6SJuTMDSOADP4IO3UTgK1qeoGtPKDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:17:44.614048Z"},"content_sha256":"b27b7ab390728fdd711cb1960f5c60ff1a03e2dcf08d0b4fc2e2a9ad84f411ea","schema_version":"1.0","event_id":"sha256:b27b7ab390728fdd711cb1960f5c60ff1a03e2dcf08d0b4fc2e2a9ad84f411ea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:TUAX7ULSLHAJC4OUT6WNA4YEID","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Constructing homologically trivial actions on products of spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Ergun Yalcin, Ozgun Unlu","submitted_at":"2012-03-27T07:15:19Z","abstract_excerpt":"We prove that if a finite group $G$ has a representation with fixity $f$, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of $f+1$ spheres. This shows, in particular, that every finite group acts freely and homologically trivially on some finite CW-complex homotopy equivalent to a product of spheres."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5882","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":"2026-05-18T03:59:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6ixFMmWKd5sKi9C7rrVF3vxEPAc8w2UNF/LT3e+dQzY08TjY0ZFl211V5hofDr1rrN1b4Dp9ikCQCau9wAsnCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T00:17:44.614466Z"},"content_sha256":"d279e7eaec6f78805f060eb5265daa42118a090b841afc34bea63d51e7d43f93","schema_version":"1.0","event_id":"sha256:d279e7eaec6f78805f060eb5265daa42118a090b841afc34bea63d51e7d43f93"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TUAX7ULSLHAJC4OUT6WNA4YEID/bundle.json","state_url":"https://pith.science/pith/TUAX7ULSLHAJC4OUT6WNA4YEID/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TUAX7ULSLHAJC4OUT6WNA4YEID/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":"2026-05-26T00:17:44Z","links":{"resolver":"https://pith.science/pith/TUAX7ULSLHAJC4OUT6WNA4YEID","bundle":"https://pith.science/pith/TUAX7ULSLHAJC4OUT6WNA4YEID/bundle.json","state":"https://pith.science/pith/TUAX7ULSLHAJC4OUT6WNA4YEID/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TUAX7ULSLHAJC4OUT6WNA4YEID/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:TUAX7ULSLHAJC4OUT6WNA4YEID","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":"6a12e1a5d24934bc8a8229a11c95b2e1ac034c220e8a47ef6f4aaec71bf14919","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-03-27T07:15:19Z","title_canon_sha256":"f83f95d2c0debce4d8394823a160f99d603686dfb485616ea4c29fc4a4bfb2cd"},"schema_version":"1.0","source":{"id":"1203.5882","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1203.5882","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"arxiv_version","alias_value":"1203.5882v1","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5882","created_at":"2026-05-18T03:59:13Z"},{"alias_kind":"pith_short_12","alias_value":"TUAX7ULSLHAJ","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TUAX7ULSLHAJC4OU","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TUAX7ULS","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:d279e7eaec6f78805f060eb5265daa42118a090b841afc34bea63d51e7d43f93","target":"graph","created_at":"2026-05-18T03:59:13Z","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 prove that if a finite group $G$ has a representation with fixity $f$, then it acts freely and homologically trivially on a finite CW-complex homotopy equivalent to a product of $f+1$ spheres. This shows, in particular, that every finite group acts freely and homologically trivially on some finite CW-complex homotopy equivalent to a product of spheres.","authors_text":"Ergun Yalcin, Ozgun Unlu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-03-27T07:15:19Z","title":"Constructing homologically trivial actions on products of spheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5882","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:b27b7ab390728fdd711cb1960f5c60ff1a03e2dcf08d0b4fc2e2a9ad84f411ea","target":"record","created_at":"2026-05-18T03:59:13Z","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":"6a12e1a5d24934bc8a8229a11c95b2e1ac034c220e8a47ef6f4aaec71bf14919","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2012-03-27T07:15:19Z","title_canon_sha256":"f83f95d2c0debce4d8394823a160f99d603686dfb485616ea4c29fc4a4bfb2cd"},"schema_version":"1.0","source":{"id":"1203.5882","kind":"arxiv","version":1}},"canonical_sha256":"9d017fd17259c09171d49facd0730440d1700990b1341088a6ce5b71353e1e8a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d017fd17259c09171d49facd0730440d1700990b1341088a6ce5b71353e1e8a","first_computed_at":"2026-05-18T03:59:13.168248Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:13.168248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/8on+2ggo6LfXezWJCIGkQGreS8h/ijJa2lHOw+Dj55+khvm1E/SufLzu4ShgQ2UYIN1rVjDJGoQSlhSmZ9zAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:13.168744Z","signed_message":"canonical_sha256_bytes"},"source_id":"1203.5882","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b27b7ab390728fdd711cb1960f5c60ff1a03e2dcf08d0b4fc2e2a9ad84f411ea","sha256:d279e7eaec6f78805f060eb5265daa42118a090b841afc34bea63d51e7d43f93"],"state_sha256":"168a8d4900d5d740c83fa0812548519524984e74ee33c5e5e558ac254b825560"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RkqQFGdPv+c3SEVFL8ZYWYUUdtFi8QJmfJw8thtvNQWvNGl1KpwldqNHi6GfUAM8lObfKewqVZ4G6bqZHzDHBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T00:17:44.617542Z","bundle_sha256":"2f983edd428fbfc5409ac603caea818ada600d4e2d01d0d60b0f4e0f900156a3"}}