{"paper":{"title":"$\\mathbb{K}$-framings and $\\mathbb{K}$-quadratic forms on surfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit.","cross_cats":["math.AT"],"primary_cat":"math.GT","authors_text":"Nariya Kawazumi","submitted_at":"2026-04-30T07:34:00Z","abstract_excerpt":"We introduce the notions of $\\mathbb{K}$-framings, based $\\mathbb{K}$-framings and relative $\\mathbb{K}$-framings of a compact connected oriented surface $\\Sigma$ for any commutative ring $\\mathbb{K}$ with unit, and a map which maps a based loop on $\\Sigma$ to a homology class of its unit tangent bundle $U\\Sigma$, which recovers Johnson's lifting in the case $\\mathbb{K} = \\mathbb{Z}/2$. This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring $\\mathbb{K}$ with unit. If the genus of $\\Sigma$ is positive, we have a bijection"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring K with unit. If the genus of Σ is positive, we have a bijection between the set of K-framings and the set of some twisted cocycles of the mapping class group of the surface Σ.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The constructions of K-framings and the stated bijection with twisted cocycles of the mapping class group are assumed to hold for every commutative ring K with unit on any compact connected oriented surface of positive genus (and the relation to the extended first Johnson homomorphism when the boundary is non-empty and connected).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"K-framings generalize Johnson's quadratic form-spin structure correspondence to any commutative ring K, yielding bijections with twisted cocycles of the mapping class group for positive-genus surfaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a38e03d42dd77dc213c39393106b5056363ebde921ac22da28938a5d6d91072e"},"source":{"id":"2604.27531","kind":"arxiv","version":2},"verdict":{"id":"f7b4ea68-3718-407e-ae7f-8b72ffd8ec3d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T07:48:28.808013Z","strongest_claim":"This generalizes the correspondence between a quadratic form and a spin structure established by Johnson to any commutative ring K with unit. If the genus of Σ is positive, we have a bijection between the set of K-framings and the set of some twisted cocycles of the mapping class group of the surface Σ.","one_line_summary":"K-framings generalize Johnson's quadratic form-spin structure correspondence to any commutative ring K, yielding bijections with twisted cocycles of the mapping class group for positive-genus surfaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The constructions of K-framings and the stated bijection with twisted cocycles of the mapping class group are assumed to hold for every commutative ring K with unit on any compact connected oriented surface of positive genus (and the relation to the extended first Johnson homomorphism when the boundary is non-empty and connected).","pith_extraction_headline":"K-framings on oriented surfaces generalize the quadratic form-spin structure correspondence to any commutative ring K with unit."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.27531/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-20T22:34:05.035571Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:12:50.537610Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"6104c32cd57796fde7f31ee990d0ebb753b861b9fd9b331e55dc59514737ffd7"},"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"}