{"paper":{"title":"Quadratic Functions of Cocycles and Pin Structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Greg Brumfiel, John Morgan","submitted_at":"2018-08-30T19:00:26Z","abstract_excerpt":"We construct a natural bijective correspondence between equivalence classes of Pin$^-$ structures on a compact simplicial $n$-manifold $M^n$, possibly with boundary, and $\\mathbb{Z}/4$-valued 'quadratic functions' $Q$ defined on degree $n-1$ relative $\\mathbb{Z}/2$ cocycles, $Q \\colon Z^{n-1}(M^n, \\partial M^n ; \\mathbb{Z} /2) \\to \\mathbb{Z}/4$. The 'quadratic' property of $Q(p+q)$ and the values $Q(dc)$ on coboundaries are expressed in terms of higher $\\cup_i$ products of Steenrod. For $n = 2$ the results extend old results relating Pin$^-$ structures on closed surfaces to quadratic refinemen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.10484","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"}