{"paper":{"title":"The Rosenberg $ \\mathbb{S}^{1} $-Stability Conjecture for $ \\chi(X) = 0 $","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jie Xu","submitted_at":"2026-07-09T15:54:14Z","abstract_excerpt":"Let $ X $ be a closed, oriented manifold with $ \\dim X \\geqslant 5 $. In this article, we show that 2006 Rosenberg's $ \\mathbb{S}^{1} $-stability holds when $ X $ has zero Euler characteristic. The 2006 Rosenberg-Stolz Conjecture for $ X \\times \\mathbb{R} $ also follows under the same assumption, provided that the Riemannian metric $ g $ on $ X \\times \\mathbb{R} $ is complete, is of bounded curvature, and whose smallest eigenvalue is uniformly bounded below by some positive constant. We then show a $ \\mathbb{T}^{n} $-stability theorem with the same hypothesis of $ X $."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.08621","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.08621/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"}