{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:QFAQS7LWSSCGDIYDVW2VLOWS7E","short_pith_number":"pith:QFAQS7LW","canonical_record":{"source":{"id":"math/0503085","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2005-03-05T05:22:33Z","cross_cats_sorted":[],"title_canon_sha256":"cef154a0e309a627e25e6decdce4b74aa2d4f93f1ab5de3946a51b33f9f1561a","abstract_canon_sha256":"7b40ea14e6b18c8d5a4222c16a1b70ea58bc0379732adb34ff8271e62bba1775"},"schema_version":"1.0"},"canonical_sha256":"8141097d76948461a303adb555bad2f90da44749e61479ccab62beb0cf503ed4","source":{"kind":"arxiv","id":"math/0503085","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0503085","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"arxiv_version","alias_value":"math/0503085v1","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0503085","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"pith_short_12","alias_value":"QFAQS7LWSSCG","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"pith_short_16","alias_value":"QFAQS7LWSSCGDIYD","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"pith_short_8","alias_value":"QFAQS7LW","created_at":"2026-07-04T14:39:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:QFAQS7LWSSCGDIYDVW2VLOWS7E","target":"record","payload":{"canonical_record":{"source":{"id":"math/0503085","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AT","submitted_at":"2005-03-05T05:22:33Z","cross_cats_sorted":[],"title_canon_sha256":"cef154a0e309a627e25e6decdce4b74aa2d4f93f1ab5de3946a51b33f9f1561a","abstract_canon_sha256":"7b40ea14e6b18c8d5a4222c16a1b70ea58bc0379732adb34ff8271e62bba1775"},"schema_version":"1.0"},"canonical_sha256":"8141097d76948461a303adb555bad2f90da44749e61479ccab62beb0cf503ed4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:39:45.977661Z","signature_b64":"m1WXiQtVS/fI7GeJ/f3+NDWDGNFIsDvWe3UrX2W+o8epaUanO4S3xllfpqs+iWDHdnTaHZCV+iqBf4KbHO6iBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8141097d76948461a303adb555bad2f90da44749e61479ccab62beb0cf503ed4","last_reissued_at":"2026-07-04T14:39:45.977329Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:39:45.977329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0503085","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":"2026-07-04T14:39:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S6K4h9+VLJPOqKtTO6Ca0ZPX8funeuw+gsuhx4gj3KQOcSmTIarQFTKR14DEtvOo3LlN6B7a8SZ7WHR2GnyLCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T14:14:53.038030Z"},"content_sha256":"8af819585a3e2e03067144268fc5b051a7df13c0362a30dd19d8a05a8f4839d2","schema_version":"1.0","event_id":"sha256:8af819585a3e2e03067144268fc5b051a7df13c0362a30dd19d8a05a8f4839d2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:QFAQS7LWSSCGDIYDVW2VLOWS7E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$({\\Bbb Z}_2)^k$-actions with $w(F)=1$","license":"","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Zhi L\\\"u","submitted_at":"2005-03-05T05:22:33Z","abstract_excerpt":"Suppose that $(\\Phi, M^n)$ is a smooth $({\\Bbb Z}_2)^k$-action on a closed smooth $n$-dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each connected component of the fixed point set $F$ vanish in positive dimension. This paper shows that if $\\dim M^n>2^k\\dim F$ and each $p$-dimensional part $F^p$ possesses the linear independence property, then $(\\Phi, M^n)$ bounds equivariantly, and in particular, $2^k\\dim F$ is the best possible upper bound of $\\dim M^n$ if $(\\Phi, M^n)$ is nonbounding."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0503085","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0503085/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":"2026-07-04T14:39:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ss8CeQTgQQFVyk6NE8l17l1ZvbZnJoje9TYymXTId23f54lwQ62UJuSly7RbmcmDyqJ3y5lKW06k9sPkxCmTCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-04T14:14:53.039112Z"},"content_sha256":"dabc203c213c50773dfa9b9803bce95b41212736c1885c98273fbc138825a377","schema_version":"1.0","event_id":"sha256:dabc203c213c50773dfa9b9803bce95b41212736c1885c98273fbc138825a377"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QFAQS7LWSSCGDIYDVW2VLOWS7E/bundle.json","state_url":"https://pith.science/pith/QFAQS7LWSSCGDIYDVW2VLOWS7E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QFAQS7LWSSCGDIYDVW2VLOWS7E/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":"2026-08-04T14:14:53Z","links":{"resolver":"https://pith.science/pith/QFAQS7LWSSCGDIYDVW2VLOWS7E","bundle":"https://pith.science/pith/QFAQS7LWSSCGDIYDVW2VLOWS7E/bundle.json","state":"https://pith.science/pith/QFAQS7LWSSCGDIYDVW2VLOWS7E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QFAQS7LWSSCGDIYDVW2VLOWS7E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:QFAQS7LWSSCGDIYDVW2VLOWS7E","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":"7b40ea14e6b18c8d5a4222c16a1b70ea58bc0379732adb34ff8271e62bba1775","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2005-03-05T05:22:33Z","title_canon_sha256":"cef154a0e309a627e25e6decdce4b74aa2d4f93f1ab5de3946a51b33f9f1561a"},"schema_version":"1.0","source":{"id":"math/0503085","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0503085","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"arxiv_version","alias_value":"math/0503085v1","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0503085","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"pith_short_12","alias_value":"QFAQS7LWSSCG","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"pith_short_16","alias_value":"QFAQS7LWSSCGDIYD","created_at":"2026-07-04T14:39:45Z"},{"alias_kind":"pith_short_8","alias_value":"QFAQS7LW","created_at":"2026-07-04T14:39:45Z"}],"graph_snapshots":[{"event_id":"sha256:dabc203c213c50773dfa9b9803bce95b41212736c1885c98273fbc138825a377","target":"graph","created_at":"2026-07-04T14:39:45Z","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/math/0503085/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Suppose that $(\\Phi, M^n)$ is a smooth $({\\Bbb Z}_2)^k$-action on a closed smooth $n$-dimensional manifold such that all Stiefel-Whitney classes of the tangent bundle to each connected component of the fixed point set $F$ vanish in positive dimension. This paper shows that if $\\dim M^n>2^k\\dim F$ and each $p$-dimensional part $F^p$ possesses the linear independence property, then $(\\Phi, M^n)$ bounds equivariantly, and in particular, $2^k\\dim F$ is the best possible upper bound of $\\dim M^n$ if $(\\Phi, M^n)$ is nonbounding.","authors_text":"Zhi L\\\"u","cross_cats":[],"headline":"","license":"","primary_cat":"math.AT","submitted_at":"2005-03-05T05:22:33Z","title":"$({\\Bbb Z}_2)^k$-actions with $w(F)=1$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0503085","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":"sha256:8af819585a3e2e03067144268fc5b051a7df13c0362a30dd19d8a05a8f4839d2","target":"record","created_at":"2026-07-04T14:39:45Z","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":"7b40ea14e6b18c8d5a4222c16a1b70ea58bc0379732adb34ff8271e62bba1775","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2005-03-05T05:22:33Z","title_canon_sha256":"cef154a0e309a627e25e6decdce4b74aa2d4f93f1ab5de3946a51b33f9f1561a"},"schema_version":"1.0","source":{"id":"math/0503085","kind":"arxiv","version":1}},"canonical_sha256":"8141097d76948461a303adb555bad2f90da44749e61479ccab62beb0cf503ed4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8141097d76948461a303adb555bad2f90da44749e61479ccab62beb0cf503ed4","first_computed_at":"2026-07-04T14:39:45.977329Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:39:45.977329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m1WXiQtVS/fI7GeJ/f3+NDWDGNFIsDvWe3UrX2W+o8epaUanO4S3xllfpqs+iWDHdnTaHZCV+iqBf4KbHO6iBg==","signature_status":"signed_v1","signed_at":"2026-07-04T14:39:45.977661Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0503085","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8af819585a3e2e03067144268fc5b051a7df13c0362a30dd19d8a05a8f4839d2","sha256:dabc203c213c50773dfa9b9803bce95b41212736c1885c98273fbc138825a377"],"state_sha256":"81e40e59e4fff70f0f8fd31f88fb0502bbd52c2a096094554e46493d2e0d8a7d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fVfR/hblLsIAVkE/nu4vyNsNaktX3QD7Yicx8HhSOyqUSc5ErYuiXV0lv5DSw+eazMXVm8/TCWJWgCyucu4FBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-04T14:14:53.048751Z","bundle_sha256":"2f45ba198f8b94f7282d3eadc96ecf1ca8a1f51734ee63489ebb08d44c9c8d9a"}}