{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:6SOK5JR6T3CSWCIQ4DFMFU3A47","short_pith_number":"pith:6SOK5JR6","canonical_record":{"source":{"id":"1807.02613","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-07-07T05:09:56Z","cross_cats_sorted":[],"title_canon_sha256":"a167fcf5c4e8c48cacadc2e1eb4fbd07437913bd9887322e5512fd5b6587b05a","abstract_canon_sha256":"a06ce918bee10614cf68f4aeefd57e3802aca506834d87daa08abfc4cc520ff6"},"schema_version":"1.0"},"canonical_sha256":"f49caea63e9ec52b0910e0cac2d360e7cbb5882ed330739d0ff2f5791299fb1b","source":{"kind":"arxiv","id":"1807.02613","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.02613","created_at":"2026-05-17T23:57:15Z"},{"alias_kind":"arxiv_version","alias_value":"1807.02613v2","created_at":"2026-05-17T23:57:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.02613","created_at":"2026-05-17T23:57:15Z"},{"alias_kind":"pith_short_12","alias_value":"6SOK5JR6T3CS","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"6SOK5JR6T3CSWCIQ","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"6SOK5JR6","created_at":"2026-05-18T12:32:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:6SOK5JR6T3CSWCIQ4DFMFU3A47","target":"record","payload":{"canonical_record":{"source":{"id":"1807.02613","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-07-07T05:09:56Z","cross_cats_sorted":[],"title_canon_sha256":"a167fcf5c4e8c48cacadc2e1eb4fbd07437913bd9887322e5512fd5b6587b05a","abstract_canon_sha256":"a06ce918bee10614cf68f4aeefd57e3802aca506834d87daa08abfc4cc520ff6"},"schema_version":"1.0"},"canonical_sha256":"f49caea63e9ec52b0910e0cac2d360e7cbb5882ed330739d0ff2f5791299fb1b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:15.069699Z","signature_b64":"8wOStHokcrAFcVcMxMhnuhKVkN1Pg1voPyJOu/iOq9rUVSILYULfMiBNbWaN3UVsbPvezZb1SnVQzRhycmRzBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f49caea63e9ec52b0910e0cac2d360e7cbb5882ed330739d0ff2f5791299fb1b","last_reissued_at":"2026-05-17T23:57:15.069208Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:15.069208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.02613","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":"2026-05-17T23:57:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u1uOvhbfYj0R9tQW0JoeGML1pPuIcohR00hk23TB1R4Ubvd5/7ZVRkkWs4hyt7dRid1Tl8F3hdHmH1K6SsrkAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:40:29.896869Z"},"content_sha256":"73533cf63a7a3fa85636054caf04662a750412934c2bbbc287f9f386f2f6d848","schema_version":"1.0","event_id":"sha256:73533cf63a7a3fa85636054caf04662a750412934c2bbbc287f9f386f2f6d848"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:6SOK5JR6T3CSWCIQ4DFMFU3A47","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The homotopy groups of a homotopy group completion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Daniel A. Ramras","submitted_at":"2018-07-07T05:09:56Z","abstract_excerpt":"Let $M$ be a topological monoid with homotopy group completion $\\Omega BM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $\\pi_k (\\Omega BM)$ is the quotient of the monoid of free homotopy classes $[S^k, M]$ by its submonoid of nullhomotopic maps.\n  We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of point-wise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of disc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02613","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":"2026-05-17T23:57:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dJUqhH8TTEPDEwdDfGAFJ9pDZILTLa5Ke5S74Txc72KuLIWNyewZ/Znj9XWuyMt/nTyJGQs372/Y9YBLIdLJDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T22:40:29.897214Z"},"content_sha256":"b2bb66d4566d047fab4f88508331c08c132978a1e02654d2b4e3df2ebc0c9d9e","schema_version":"1.0","event_id":"sha256:b2bb66d4566d047fab4f88508331c08c132978a1e02654d2b4e3df2ebc0c9d9e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6SOK5JR6T3CSWCIQ4DFMFU3A47/bundle.json","state_url":"https://pith.science/pith/6SOK5JR6T3CSWCIQ4DFMFU3A47/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6SOK5JR6T3CSWCIQ4DFMFU3A47/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":"2026-06-24T22:40:29Z","links":{"resolver":"https://pith.science/pith/6SOK5JR6T3CSWCIQ4DFMFU3A47","bundle":"https://pith.science/pith/6SOK5JR6T3CSWCIQ4DFMFU3A47/bundle.json","state":"https://pith.science/pith/6SOK5JR6T3CSWCIQ4DFMFU3A47/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6SOK5JR6T3CSWCIQ4DFMFU3A47/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6SOK5JR6T3CSWCIQ4DFMFU3A47","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":"a06ce918bee10614cf68f4aeefd57e3802aca506834d87daa08abfc4cc520ff6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-07-07T05:09:56Z","title_canon_sha256":"a167fcf5c4e8c48cacadc2e1eb4fbd07437913bd9887322e5512fd5b6587b05a"},"schema_version":"1.0","source":{"id":"1807.02613","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.02613","created_at":"2026-05-17T23:57:15Z"},{"alias_kind":"arxiv_version","alias_value":"1807.02613v2","created_at":"2026-05-17T23:57:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.02613","created_at":"2026-05-17T23:57:15Z"},{"alias_kind":"pith_short_12","alias_value":"6SOK5JR6T3CS","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"6SOK5JR6T3CSWCIQ","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"6SOK5JR6","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:b2bb66d4566d047fab4f88508331c08c132978a1e02654d2b4e3df2ebc0c9d9e","target":"graph","created_at":"2026-05-17T23:57:15Z","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 $M$ be a topological monoid with homotopy group completion $\\Omega BM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $\\pi_k (\\Omega BM)$ is the quotient of the monoid of free homotopy classes $[S^k, M]$ by its submonoid of nullhomotopic maps.\n  We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of point-wise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of disc","authors_text":"Daniel A. Ramras","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-07-07T05:09:56Z","title":"The homotopy groups of a homotopy group completion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02613","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":"sha256:73533cf63a7a3fa85636054caf04662a750412934c2bbbc287f9f386f2f6d848","target":"record","created_at":"2026-05-17T23:57:15Z","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":"a06ce918bee10614cf68f4aeefd57e3802aca506834d87daa08abfc4cc520ff6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2018-07-07T05:09:56Z","title_canon_sha256":"a167fcf5c4e8c48cacadc2e1eb4fbd07437913bd9887322e5512fd5b6587b05a"},"schema_version":"1.0","source":{"id":"1807.02613","kind":"arxiv","version":2}},"canonical_sha256":"f49caea63e9ec52b0910e0cac2d360e7cbb5882ed330739d0ff2f5791299fb1b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f49caea63e9ec52b0910e0cac2d360e7cbb5882ed330739d0ff2f5791299fb1b","first_computed_at":"2026-05-17T23:57:15.069208Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:57:15.069208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8wOStHokcrAFcVcMxMhnuhKVkN1Pg1voPyJOu/iOq9rUVSILYULfMiBNbWaN3UVsbPvezZb1SnVQzRhycmRzBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:57:15.069699Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.02613","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73533cf63a7a3fa85636054caf04662a750412934c2bbbc287f9f386f2f6d848","sha256:b2bb66d4566d047fab4f88508331c08c132978a1e02654d2b4e3df2ebc0c9d9e"],"state_sha256":"9a8f1f1d58256f508c987056fbcc798f69ea100dfc9f60105e170f1a3c9ef989"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M8o/3rebBbXn4dORjYFzdZdHiSM7GrgbMUPZGVY7V7dHUcIpaM0WySPUhQRscLKhAwV5nKPHxIPIbISvSqSvBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T22:40:29.899190Z","bundle_sha256":"61daadd3db10a9041bb23b76abb6dba6ebb43eefb55cd4d49157bf9917d330de"}}