{"paper":{"title":"Non-isotopic surfaces in $T^4\\#(S^2\\times S^2)$: an example","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"Infinitely many embedded tori in T^4 # (S^2 × S^2) with a common geometric dual are homotopic, diffeomorphic but not isotopic even after stabilizations.","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jianfeng Lin, Yue Wu","submitted_at":"2026-04-07T12:42:01Z","abstract_excerpt":"We prove that there exist infinitely many embedded tori with a common geometric dual in $T^4\\#(S^2\\times S^2)$ that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations. These surfaces are obtained by applying the Norman trick to a fixed immersed surface, using non-homotopic tubing arcs. The isotopy classes of these surfaces are distinguished by homotopy classes of the 2-handles (relative to the boundary) in the complement of the image of the $0$- and $1$-handles."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that there exist infinitely many embedded tori with a common geometric dual in T^4#(S^2×S^2) that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The isotopy classes of these surfaces are distinguished by homotopy classes of the 2-handles (relative to the boundary) in the complement of the image of the 0- and 1-handles, assuming this homotopy invariant is preserved under isotopy and that the Norman trick with non-homotopic arcs produces the claimed embedded surfaces.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Infinitely many non-isotopic embedded tori with a common geometric dual exist in T^4#(S^2×S^2), built via the Norman trick on a fixed immersed surface using non-homotopic tubing arcs and distinguished by homotopy classes of 2-handles in the complement.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Infinitely many embedded tori in T^4 # (S^2 × S^2) with a common geometric dual are homotopic, diffeomorphic but not isotopic even after stabilizations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a0103964d7f93c15e3bc27ef557bb64421fb26795d962795418bfba1a4692d38"},"source":{"id":"2604.05805","kind":"arxiv","version":2},"verdict":{"id":"10af683d-5cc6-47b4-a842-b88b513412a0","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T18:37:23.786348Z","strongest_claim":"We prove that there exist infinitely many embedded tori with a common geometric dual in T^4#(S^2×S^2) that are homotopic, diffeomorphic, but not isotopic to each other, even after arbitrary many external stabilizations.","one_line_summary":"Infinitely many non-isotopic embedded tori with a common geometric dual exist in T^4#(S^2×S^2), built via the Norman trick on a fixed immersed surface using non-homotopic tubing arcs and distinguished by homotopy classes of 2-handles in the complement.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The isotopy classes of these surfaces are distinguished by homotopy classes of the 2-handles (relative to the boundary) in the complement of the image of the 0- and 1-handles, assuming this homotopy invariant is preserved under isotopy and that the Norman trick with non-homotopic arcs produces the claimed embedded surfaces.","pith_extraction_headline":"Infinitely many embedded tori in T^4 # (S^2 × S^2) with a common geometric dual are homotopic, diffeomorphic but not isotopic even after stabilizations."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.05805/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"}