{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2008:MZCFKV7H63PMLZVA5PJXGH4GGC","short_pith_number":"pith:MZCFKV7H","schema_version":"1.0","canonical_sha256":"66445557e7f6dec5e6a0ebd3731f863082914870ad1852209776df23b7fd6197","source":{"kind":"arxiv","id":"0810.4531","version":5},"attestation_state":"computed","paper":{"title":"The loop cohomology of a space with the polynomial cohomology algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Samson Saneblidze","submitted_at":"2008-10-24T19:22:49Z","abstract_excerpt":"Given a simply connected space $X$ with the cohomology $H^*(X;{\\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(\\Omega X;{\\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\\mathbb Z}_2).$ As a consequence we obtain that $H^*(\\Omega X;{\\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\\ast}(X;{\\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel."},"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":"0810.4531","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2008-10-24T19:22:49Z","cross_cats_sorted":[],"title_canon_sha256":"05ca36049ca2f3a0c2cbe957b556cfb4c5955a9693d874e43082e2e2e4e2f02c","abstract_canon_sha256":"85dc66302f47e7eefaa3e0acb95905849d9df78aa6b4ff122fc02f915fa90cc5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:45.987371Z","signature_b64":"8CLXUmhl/OKz8+z0Y0MFdpM7rbZ86aiMHAEBZjytl5cyKBZodT9NnPSm9D0cB7jciJ3PFnlLNa9FApnlPlI+BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66445557e7f6dec5e6a0ebd3731f863082914870ad1852209776df23b7fd6197","last_reissued_at":"2026-05-18T04:09:45.986976Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:45.986976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The loop cohomology of a space with the polynomial cohomology algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Samson Saneblidze","submitted_at":"2008-10-24T19:22:49Z","abstract_excerpt":"Given a simply connected space $X$ with the cohomology $H^*(X;{\\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(\\Omega X;{\\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\\mathbb Z}_2).$ As a consequence we obtain that $H^*(\\Omega X;{\\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\\ast}(X;{\\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0810.4531","kind":"arxiv","version":5},"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":"0810.4531","created_at":"2026-05-18T04:09:45.987032+00:00"},{"alias_kind":"arxiv_version","alias_value":"0810.4531v5","created_at":"2026-05-18T04:09:45.987032+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0810.4531","created_at":"2026-05-18T04:09:45.987032+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZCFKV7H63PM","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZCFKV7H63PMLZVA","created_at":"2026-05-18T12:25:57.157939+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZCFKV7H","created_at":"2026-05-18T12:25:57.157939+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/MZCFKV7H63PMLZVA5PJXGH4GGC","json":"https://pith.science/pith/MZCFKV7H63PMLZVA5PJXGH4GGC.json","graph_json":"https://pith.science/api/pith-number/MZCFKV7H63PMLZVA5PJXGH4GGC/graph.json","events_json":"https://pith.science/api/pith-number/MZCFKV7H63PMLZVA5PJXGH4GGC/events.json","paper":"https://pith.science/paper/MZCFKV7H"},"agent_actions":{"view_html":"https://pith.science/pith/MZCFKV7H63PMLZVA5PJXGH4GGC","download_json":"https://pith.science/pith/MZCFKV7H63PMLZVA5PJXGH4GGC.json","view_paper":"https://pith.science/paper/MZCFKV7H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0810.4531&json=true","fetch_graph":"https://pith.science/api/pith-number/MZCFKV7H63PMLZVA5PJXGH4GGC/graph.json","fetch_events":"https://pith.science/api/pith-number/MZCFKV7H63PMLZVA5PJXGH4GGC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZCFKV7H63PMLZVA5PJXGH4GGC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZCFKV7H63PMLZVA5PJXGH4GGC/action/storage_attestation","attest_author":"https://pith.science/pith/MZCFKV7H63PMLZVA5PJXGH4GGC/action/author_attestation","sign_citation":"https://pith.science/pith/MZCFKV7H63PMLZVA5PJXGH4GGC/action/citation_signature","submit_replication":"https://pith.science/pith/MZCFKV7H63PMLZVA5PJXGH4GGC/action/replication_record"}},"created_at":"2026-05-18T04:09:45.987032+00:00","updated_at":"2026-05-18T04:09:45.987032+00:00"}