{"paper":{"title":"Branched covering representation of non-orientable $4$-manifolds","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Daniele Zuddas, Riccardo Piergallini, Valentina Bais","submitted_at":"2025-09-11T10:08:16Z","abstract_excerpt":"We show that every closed connected non-orientable PL $4$-manifold $X$ is a simple branched covering of $\\RP^4$. We also show that $X$ is a simple branched covering of the twisted $S^3$-bundle $S^1 \\simtimes S^3$ if and only if the first Stiefel--Whitney class $w_1(X)$ admits an integral lift. In both cases, the degree of the covering can be any number $d \\geq 4$, provided that $d$ has the same parity of the Stiefel--Whitney number $w_1^4[X]$ in the case of $\\RP^4$. Moreover, the branch set can be assumed to be non-singular if $d \\geq 5$ and to have just nodal singularities if $d=4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.09319","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.09319/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}