{"paper":{"title":"Rigidity of self-maps of $V_{n,2}$ and manifolds tangentially homotopy equivalent to $V_{n,2} \\times S^k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For most n, self-maps of the Stiefel manifold V_{n,2} homotopic to almost diffeomorphisms are determined, and manifolds tangentially homotopy equivalent to V_{n,2} × S^k are classified up to almost diffeomorphism for k=3,5 or 7 to n-3 (k ≠ ","cross_cats":["math.GT"],"primary_cat":"math.AT","authors_text":"Sagnik Biswas","submitted_at":"2026-04-17T12:00:46Z","abstract_excerpt":"We study two problems concerning the Stiefel manifolds $V_{n,2}$ and their products with spheres. First, we address a rigidity problem: we determine, for most values of~$n$, all self-maps of $V_{n,2}$ that are homotopic to an almost diffeomorphism. Second, we classify smooth closed manifolds tangentially homotopy equivalent to $V_{n,2} \\times S^k$ up to almost diffeomorphism, for $k = 3, 5$ or $7 \\leq k, k \\neq 2^i - 2 \\ \\text{and} \\ Dim(V_{n,2} \\times S^k) \\neq 2^i - 2$. Our method is to find explicit inverses in the structure set via normal invariants of specific tangential homotopy equivale"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We determine, for most values of n, all self-maps of V_{n,2} that are homotopic to an almost diffeomorphism. We classify smooth closed manifolds tangentially homotopy equivalent to V_{n,2} × S^k up to almost diffeomorphism, for k = 3, 5 or 7 ≤ k ≤ n-3, k ≠ 2^i - 2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That explicit inverses in the structure set can be found via normal invariants of specific tangential homotopy equivalences under the stated conditions on k and n, without additional obstructions arising in the general case.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Rigidity results for self-maps of V_{n,2} and classification of tangentially homotopy equivalent manifolds to V_{n,2} x S^k up to almost diffeomorphism for certain k.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For most n, self-maps of the Stiefel manifold V_{n,2} homotopic to almost diffeomorphisms are determined, and manifolds tangentially homotopy equivalent to V_{n,2} × S^k are classified up to almost diffeomorphism for k=3,5 or 7 to n-3 (k ≠ ","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d08712810506dd7cf9cb031adbfcf5c4be5450381d0b9a1ceb0cc9865b4d53a4"},"source":{"id":"2604.15984","kind":"arxiv","version":2},"verdict":{"id":"fcb92136-1fae-4a34-8173-6fc9d48d6054","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T07:27:30.677325Z","strongest_claim":"We determine, for most values of n, all self-maps of V_{n,2} that are homotopic to an almost diffeomorphism. We classify smooth closed manifolds tangentially homotopy equivalent to V_{n,2} × S^k up to almost diffeomorphism, for k = 3, 5 or 7 ≤ k ≤ n-3, k ≠ 2^i - 2.","one_line_summary":"Rigidity results for self-maps of V_{n,2} and classification of tangentially homotopy equivalent manifolds to V_{n,2} x S^k up to almost diffeomorphism for certain k.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That explicit inverses in the structure set can be found via normal invariants of specific tangential homotopy equivalences under the stated conditions on k and n, without additional obstructions arising in the general case.","pith_extraction_headline":"For most n, self-maps of the Stiefel manifold V_{n,2} homotopic to almost diffeomorphisms are determined, and manifolds tangentially homotopy equivalent to V_{n,2} × S^k are classified up to almost diffeomorphism for k=3,5 or 7 to n-3 (k ≠ "},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.15984/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"}