{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:S4FAKUITYMJEFW7MTZIUFSC2VI","short_pith_number":"pith:S4FAKUIT","canonical_record":{"source":{"id":"math/0701127","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2007-01-04T08:15:48Z","cross_cats_sorted":["math.AT","math.GN"],"title_canon_sha256":"9c1c6ea9723d78db03c751ad568169edf3d668302b6bd842e0a5f55322285c99","abstract_canon_sha256":"1d57b0e75531e36a8c454aea22e3b69343ec237fe62a4c8fb5b771c73da78a15"},"schema_version":"1.0"},"canonical_sha256":"970a055113c31242dbec9e5142c85aaa2a0858c33541981003409c75cadf57c1","source":{"kind":"arxiv","id":"math/0701127","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0701127","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"arxiv_version","alias_value":"math/0701127v1","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0701127","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"pith_short_12","alias_value":"S4FAKUITYMJE","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"S4FAKUITYMJEFW7M","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"S4FAKUIT","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:S4FAKUITYMJEFW7MTZIUFSC2VI","target":"record","payload":{"canonical_record":{"source":{"id":"math/0701127","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2007-01-04T08:15:48Z","cross_cats_sorted":["math.AT","math.GN"],"title_canon_sha256":"9c1c6ea9723d78db03c751ad568169edf3d668302b6bd842e0a5f55322285c99","abstract_canon_sha256":"1d57b0e75531e36a8c454aea22e3b69343ec237fe62a4c8fb5b771c73da78a15"},"schema_version":"1.0"},"canonical_sha256":"970a055113c31242dbec9e5142c85aaa2a0858c33541981003409c75cadf57c1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:26.639383Z","signature_b64":"cf8SAGFmMWFUpNtV+wWFw+M4lcLJfQD9ggT2XAmYP1DMTRRYAfFAT2DQVYA92p1yuAqWWl6TvRu9FvNT6g/1BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"970a055113c31242dbec9e5142c85aaa2a0858c33541981003409c75cadf57c1","last_reissued_at":"2026-05-18T02:41:26.638832Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:26.638832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0701127","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":"2026-05-18T02:41:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CRGkmpqWU4Ki1Sp0u3ijwoWZ+R8oN2+lSz8qRlNQhNetUAvRSzXK32HaPVw6bMSbUXD2H0PAQ1tW9RMQ28QXBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:08:26.246339Z"},"content_sha256":"d2cd9771478bfe839175a92861b5bf5d3bdbcd839b3f81d7197b29694914a853","schema_version":"1.0","event_id":"sha256:d2cd9771478bfe839175a92861b5bf5d3bdbcd839b3f81d7197b29694914a853"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:S4FAKUITYMJEFW7MTZIUFSC2VI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maps to the projective plane","license":"","headline":"","cross_cats":["math.AT","math.GN"],"primary_cat":"math.GT","authors_text":"Jerzy Dydak, Michael Levin","submitted_at":"2007-01-04T08:15:48Z","abstract_excerpt":"We prove the projective plane $\\rp^2$ is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by A.N.Dranishnikov) in extension theory. One of the main tools is the computation of the fundamental group of the function space $\\Map(\\rp^n,\\rp^{n+1})$ (based at inclusion) as being isomorphic to either $\\Z_4$ or $\\Z_2\\oplus\\Z_2$ for $n\\ge 1$. Double surgery and the above fact yield the proof."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701127","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":"2026-05-18T02:41:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vcCZWYSynH3U6RwlaoEgpOLHQy7Z1QOawGiEz/YlXri6PIoyY6TQApqaDL6137fLBY/54HcbdvEmBo2wOzCkBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T23:08:26.246684Z"},"content_sha256":"6492cca32b61dc30cfc88e84b9bd521defbbe389159a07650259ea9fea2d707f","schema_version":"1.0","event_id":"sha256:6492cca32b61dc30cfc88e84b9bd521defbbe389159a07650259ea9fea2d707f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S4FAKUITYMJEFW7MTZIUFSC2VI/bundle.json","state_url":"https://pith.science/pith/S4FAKUITYMJEFW7MTZIUFSC2VI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S4FAKUITYMJEFW7MTZIUFSC2VI/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":"2026-06-02T23:08:26Z","links":{"resolver":"https://pith.science/pith/S4FAKUITYMJEFW7MTZIUFSC2VI","bundle":"https://pith.science/pith/S4FAKUITYMJEFW7MTZIUFSC2VI/bundle.json","state":"https://pith.science/pith/S4FAKUITYMJEFW7MTZIUFSC2VI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S4FAKUITYMJEFW7MTZIUFSC2VI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:S4FAKUITYMJEFW7MTZIUFSC2VI","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":"1d57b0e75531e36a8c454aea22e3b69343ec237fe62a4c8fb5b771c73da78a15","cross_cats_sorted":["math.AT","math.GN"],"license":"","primary_cat":"math.GT","submitted_at":"2007-01-04T08:15:48Z","title_canon_sha256":"9c1c6ea9723d78db03c751ad568169edf3d668302b6bd842e0a5f55322285c99"},"schema_version":"1.0","source":{"id":"math/0701127","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0701127","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"arxiv_version","alias_value":"math/0701127v1","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0701127","created_at":"2026-05-18T02:41:26Z"},{"alias_kind":"pith_short_12","alias_value":"S4FAKUITYMJE","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"S4FAKUITYMJEFW7M","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"S4FAKUIT","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:6492cca32b61dc30cfc88e84b9bd521defbbe389159a07650259ea9fea2d707f","target":"graph","created_at":"2026-05-18T02:41:26Z","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 the projective plane $\\rp^2$ is an absolute extensor of a finite-dimensional metric space $X$ if and only if the cohomological dimension mod 2 of $X$ does not exceed 1. This solves one of the remaining difficult problems (posed by A.N.Dranishnikov) in extension theory. One of the main tools is the computation of the fundamental group of the function space $\\Map(\\rp^n,\\rp^{n+1})$ (based at inclusion) as being isomorphic to either $\\Z_4$ or $\\Z_2\\oplus\\Z_2$ for $n\\ge 1$. Double surgery and the above fact yield the proof.","authors_text":"Jerzy Dydak, Michael Levin","cross_cats":["math.AT","math.GN"],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2007-01-04T08:15:48Z","title":"Maps to the projective plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701127","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:d2cd9771478bfe839175a92861b5bf5d3bdbcd839b3f81d7197b29694914a853","target":"record","created_at":"2026-05-18T02:41:26Z","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":"1d57b0e75531e36a8c454aea22e3b69343ec237fe62a4c8fb5b771c73da78a15","cross_cats_sorted":["math.AT","math.GN"],"license":"","primary_cat":"math.GT","submitted_at":"2007-01-04T08:15:48Z","title_canon_sha256":"9c1c6ea9723d78db03c751ad568169edf3d668302b6bd842e0a5f55322285c99"},"schema_version":"1.0","source":{"id":"math/0701127","kind":"arxiv","version":1}},"canonical_sha256":"970a055113c31242dbec9e5142c85aaa2a0858c33541981003409c75cadf57c1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"970a055113c31242dbec9e5142c85aaa2a0858c33541981003409c75cadf57c1","first_computed_at":"2026-05-18T02:41:26.638832Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:26.638832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cf8SAGFmMWFUpNtV+wWFw+M4lcLJfQD9ggT2XAmYP1DMTRRYAfFAT2DQVYA92p1yuAqWWl6TvRu9FvNT6g/1BA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:26.639383Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0701127","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d2cd9771478bfe839175a92861b5bf5d3bdbcd839b3f81d7197b29694914a853","sha256:6492cca32b61dc30cfc88e84b9bd521defbbe389159a07650259ea9fea2d707f"],"state_sha256":"686602c71c49990d7634aac91b85bceea460e610e69ac596888d8699bae6a23c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I6qFtw3BpXpQmgMYK+51JUZ17hO/bzW0z/WKF1K3u71A5A6nOx3eupfOv3OXGTd8VVuEL6l36R4Jmxo2cddwDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T23:08:26.248591Z","bundle_sha256":"6cc2de4dfbc3da1d3934cc87846aa156fc5d905dbbe078e4e7544d46d0b7b486"}}