{"paper":{"title":"A Lower Bound on the Self-intersections of Fold Singularities","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A sharp lower bound exists on the number of self-intersections of fold singularities for maps from oriented surfaces to the plane.","cross_cats":[],"primary_cat":"math.GT","authors_text":"Joshua Drouin, Liam Kahmeyer","submitted_at":"2026-05-13T04:37:54Z","abstract_excerpt":"For an oriented surface $S$, the singular set of a fold map $f:S\\rightarrow \\mathbb{R}^2$ is a collection of smooth curves, also known as fold singularities. We construct a sharp lower bound on the number of self-intersections of such fold singularities. This is done by first establishing a sharp lower bound on the number of self-intersections of the boundary of a surface immersed in $\\mathbb{R}^2$. We then construct a sharp lower bound for the number of self-intersections of the singular set of a simple stable fold map of a surface to $\\mathbb{R}^2$ by viewing the connected components of the "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct a sharp lower bound on the number of self-intersections of such fold singularities.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The map is a simple stable fold map, and the connected components of the singular set can be viewed as boundary components of smaller surface components to which the immersed-boundary bound applies directly.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A sharp lower bound is established on the self-intersections of fold singularities for stable maps from oriented surfaces to R^2 by reducing the problem to bounds on immersed surface boundaries.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A sharp lower bound exists on the number of self-intersections of fold singularities for maps from oriented surfaces to the plane.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a10b7f76de4b1233ca50e3f3515e07e42c35057f7bdbc277e2420e18f7c3cbda"},"source":{"id":"2605.12989","kind":"arxiv","version":1},"verdict":{"id":"376b8fbd-8347-4af5-ac4c-25827ae271b5","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T02:18:24.550501Z","strongest_claim":"We construct a sharp lower bound on the number of self-intersections of such fold singularities.","one_line_summary":"A sharp lower bound is established on the self-intersections of fold singularities for stable maps from oriented surfaces to R^2 by reducing the problem to bounds on immersed surface boundaries.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The map is a simple stable fold map, and the connected components of the singular set can be viewed as boundary components of smaller surface components to which the immersed-boundary bound applies directly.","pith_extraction_headline":"A sharp lower bound exists on the number of self-intersections of fold singularities for maps from oriented surfaces to the plane."},"references":{"count":17,"sample":[{"doi":"","year":1970,"title":"On singularities of folding type.Mathematics of the USSR-Izvestiya, 4(5):1119–1134, Oct 1970","work_id":"5a424931-afb7-409f-a278-7e99b634c686","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"Singularities, expanders and topology of maps","work_id":"52788031-9c96-4f0d-9984-1525001c52f6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"Minimal number of self-intersections of the boundary of an immersed surface in the plane","work_id":"1747f4cd-4d03-40bc-91d3-ec9f8eb50128","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1960,"title":"Quelques remarques sur les applications diff´ erentiables d’une surface dans le plan.Annales de l’institut Fourier, 10:47–60","work_id":"46162ec6-4be9-4c93-997d-a3ba93938497","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"Stable maps of surfaces into the plane.Topology and its Applications, 107(3):307–316, Nov 2000","work_id":"2edb3875-4480-4043-976e-5fbc1096375f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"0853107d8768c4477d7c5934c2781660b0f0fdd4fbec1b604556ec24ac742908","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"}