{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:BERNG6ZK47AP7IZ43BXCOAWUFI","short_pith_number":"pith:BERNG6ZK","schema_version":"1.0","canonical_sha256":"0922d37b2ae7c0ffa33cd86e2702d42a168e264394469c882802e49def1b82cf","source":{"kind":"arxiv","id":"1011.2549","version":3},"attestation_state":"computed","paper":{"title":"Hopf algebras and homotopy invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Jelena Grbic, Victor Buchstaber","submitted_at":"2010-11-11T03:21:21Z","abstract_excerpt":"In this paper we explore new relations between Algebraic Topology and the theory of Hopf Algebras. For an arbitrary topological space $X$, the loop space homology $H_*(\\Omega\\Sigma X; \\coefZ)$ is a Hopf algebra. We introduce a new homotopy invariant of a topological space $X$ taking for its value the isomorphism class (over the integers) of the Hopf algebra $H_*(\\Omega\\Sigma X; \\coefZ)$. This invariant is trivial if and only if the Hopf algebra $H_*(\\Omega\\Sigma X; \\coefZ)$ is isomorphic to a Lie-Hopf algebra, that is, to a primitively generated Hopf algebra. We show that for a given $X$ these"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1011.2549","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2010-11-11T03:21:21Z","cross_cats_sorted":[],"title_canon_sha256":"29b107b21b9648bf00f8ddf5bddad5bd160db148fe9c247eb54d4c58798141bc","abstract_canon_sha256":"6db3b7e8fb43bf5ddd1ea733b95100bc0f861a12cae9d400371aa86c4089064c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:14.623205Z","signature_b64":"JexLR3gVU90DmxNFxgBQ99DQCgWXqFhRR7yeG9Hl2XFKsfeq5QQrdsspZfZHwo1yDxT7n70mgayG2aEfxx1yDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0922d37b2ae7c0ffa33cd86e2702d42a168e264394469c882802e49def1b82cf","last_reissued_at":"2026-05-18T03:40:14.622607Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:14.622607Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hopf algebras and homotopy invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Jelena Grbic, Victor Buchstaber","submitted_at":"2010-11-11T03:21:21Z","abstract_excerpt":"In this paper we explore new relations between Algebraic Topology and the theory of Hopf Algebras. For an arbitrary topological space $X$, the loop space homology $H_*(\\Omega\\Sigma X; \\coefZ)$ is a Hopf algebra. We introduce a new homotopy invariant of a topological space $X$ taking for its value the isomorphism class (over the integers) of the Hopf algebra $H_*(\\Omega\\Sigma X; \\coefZ)$. This invariant is trivial if and only if the Hopf algebra $H_*(\\Omega\\Sigma X; \\coefZ)$ is isomorphic to a Lie-Hopf algebra, that is, to a primitively generated Hopf algebra. We show that for a given $X$ these"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.2549","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1011.2549","created_at":"2026-05-18T03:40:14.622718+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.2549v3","created_at":"2026-05-18T03:40:14.622718+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.2549","created_at":"2026-05-18T03:40:14.622718+00:00"},{"alias_kind":"pith_short_12","alias_value":"BERNG6ZK47AP","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"BERNG6ZK47AP7IZ4","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"BERNG6ZK","created_at":"2026-05-18T12:26:05.355336+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI","json":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI.json","graph_json":"https://pith.science/api/pith-number/BERNG6ZK47AP7IZ43BXCOAWUFI/graph.json","events_json":"https://pith.science/api/pith-number/BERNG6ZK47AP7IZ43BXCOAWUFI/events.json","paper":"https://pith.science/paper/BERNG6ZK"},"agent_actions":{"view_html":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI","download_json":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI.json","view_paper":"https://pith.science/paper/BERNG6ZK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.2549&json=true","fetch_graph":"https://pith.science/api/pith-number/BERNG6ZK47AP7IZ43BXCOAWUFI/graph.json","fetch_events":"https://pith.science/api/pith-number/BERNG6ZK47AP7IZ43BXCOAWUFI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI/action/storage_attestation","attest_author":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI/action/author_attestation","sign_citation":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI/action/citation_signature","submit_replication":"https://pith.science/pith/BERNG6ZK47AP7IZ43BXCOAWUFI/action/replication_record"}},"created_at":"2026-05-18T03:40:14.622718+00:00","updated_at":"2026-05-18T03:40:14.622718+00:00"}