{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:N3ZKPDHYKHPXELRPMFR6Y7ZM73","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":"c2820cdd03625290db2489f94e4805d3b4cd13017f54162240a281c2a7aee0b7","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-02-12T17:31:30Z","title_canon_sha256":"1620ce0d35cfb1611f763e209e3f8cfc1f5ca153dce3441b25bde25c6711b81b"},"schema_version":"1.0","source":{"id":"1602.04131","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.04131","created_at":"2026-05-18T00:20:59Z"},{"alias_kind":"arxiv_version","alias_value":"1602.04131v1","created_at":"2026-05-18T00:20:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04131","created_at":"2026-05-18T00:20:59Z"},{"alias_kind":"pith_short_12","alias_value":"N3ZKPDHYKHPX","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"N3ZKPDHYKHPXELRP","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"N3ZKPDHY","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:11afb421ab09bebbf0ecce45d5b6d6ae5a8fd110e36b2e23623b5c5eca253d58","target":"graph","created_at":"2026-05-18T00:20:59Z","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":"The Rips complex at scale r of a set of points X in a metric space is the abstract simplicial complex whose faces are determined by finite subsets of X of diameter less than r. We prove that for X in the Euclidean 3-space R^3 the natural projection map from the Rips complex of X to its shadow in R^3 induces a surjection on fundamental groups. This partially answers a question of Chambers, de Silva, Erickson and Ghrist who studied this projection for subsets of R^2. We further show that Rips complexes of finite subsets of R^n are universal, in that they model all homotopy types of simplicial co","authors_text":"Adrien Vakili, Florian Frick, Michal Adamaszek","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-02-12T17:31:30Z","title":"On homotopy types of Euclidean Rips complexes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04131","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:bb441bde4c394ac50311ac8b342753da5b646323d1652ceeca7f2696a407b550","target":"record","created_at":"2026-05-18T00:20:59Z","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":"c2820cdd03625290db2489f94e4805d3b4cd13017f54162240a281c2a7aee0b7","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-02-12T17:31:30Z","title_canon_sha256":"1620ce0d35cfb1611f763e209e3f8cfc1f5ca153dce3441b25bde25c6711b81b"},"schema_version":"1.0","source":{"id":"1602.04131","kind":"arxiv","version":1}},"canonical_sha256":"6ef2a78cf851df722e2f6163ec7f2cfeff610ec282c45de5d6786e5e183773e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ef2a78cf851df722e2f6163ec7f2cfeff610ec282c45de5d6786e5e183773e5","first_computed_at":"2026-05-18T00:20:59.763623Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:59.763623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PNMoUmE/44IQAvlsNB57liGAd3izcM7FqVNq3BHII+rKABvXFMLCKFLjdOWZp4tio6bGnKRhszOY12TmCx3RDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:59.764048Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.04131","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb441bde4c394ac50311ac8b342753da5b646323d1652ceeca7f2696a407b550","sha256:11afb421ab09bebbf0ecce45d5b6d6ae5a8fd110e36b2e23623b5c5eca253d58"],"state_sha256":"a34656ca4f56d23fa709b4974447054198f92809df7b438e7baa3e140fb57d6e"}