{"paper":{"title":"Brieskorn spheres and rational homology ball symplectic fillings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Brieskorn spheres admit no rational homology ball symplectic fillings for their contact structures except in listed cases.","cross_cats":["math.SG"],"primary_cat":"math.GT","authors_text":"Alberto Cavallo, Antonio Alfieri, Irena Matkovi\\v{c}","submitted_at":"2026-05-13T17:39:50Z","abstract_excerpt":"Given a canonically oriented Brieskorn sphere $Y=\\Sigma(a_1,...,a_n)$, we confirm some statements conjectured by Gompf. More specifically, we obstruct the existence of rational homology ball symplectic fillings for any contact structure on $-Y$ if $n=3$, and when there is no half convex Giroux torsion for $n>3$. Furthermore, we show that the same result holds for the Milnor fillable structure on $Y$ with the possible exception of $\\Sigma(3,4,5),$ $\\Sigma(2,5,7)$ and $\\Sigma(2,3,6k+1)$ for $k\\geq1$. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correctio"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we obstruct the existence of rational homology ball symplectic fillings for any contact structure on −Y if n=3, and when there is no half convex Giroux torsion for n>3. Furthermore, we show that the same result holds for the Milnor fillable structure on Y with the possible exception of Σ(3,4,5), Σ(2,5,7) and Σ(2,3,6k+1) for k≥1. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The canonical orientation on the Brieskorn sphere Y and the precise definition of half-convex Giroux torsion are assumed to hold in the stated cases; the obstruction relies on the correction term vanishing or the absence of torsion without additional verification for all contact structures.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Brieskorn spheres Σ(a1,...,an) obstruct rational homology ball symplectic fillings for any contact structure on -Y when n=3 or without half convex Giroux torsion for n>3, with limited exceptions for Milnor fillable cases, and those with vanishing correction terms have at most two fillable structures","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Brieskorn spheres admit no rational homology ball symplectic fillings for their contact structures except in listed cases.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"eb404feacd88815448f9a708562a7dba3552563ac605bf7d8448602010adb709"},"source":{"id":"2605.13812","kind":"arxiv","version":1},"verdict":{"id":"4d0e692f-d4dc-4e1f-870e-20e2d340aef7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:31:58.970647Z","strongest_claim":"we obstruct the existence of rational homology ball symplectic fillings for any contact structure on −Y if n=3, and when there is no half convex Giroux torsion for n>3. Furthermore, we show that the same result holds for the Milnor fillable structure on Y with the possible exception of Σ(3,4,5), Σ(2,5,7) and Σ(2,3,6k+1) for k≥1. Along the way, we determine every canonically oriented Brieskorn sphere with vanishing correction term carrying at most two fillable structures, up to isotopy.","one_line_summary":"Brieskorn spheres Σ(a1,...,an) obstruct rational homology ball symplectic fillings for any contact structure on -Y when n=3 or without half convex Giroux torsion for n>3, with limited exceptions for Milnor fillable cases, and those with vanishing correction terms have at most two fillable structures","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The canonical orientation on the Brieskorn sphere Y and the precise definition of half-convex Giroux torsion are assumed to hold in the stated cases; the obstruction relies on the correction term vanishing or the absence of torsion without additional verification for all contact structures.","pith_extraction_headline":"Brieskorn spheres admit no rational homology ball symplectic fillings for their contact structures except in listed cases."},"references":{"count":28,"sample":[{"doi":"","year":null,"title":"P. Aceto, D. McCoy and J. H. Park,A survey on embeddings of3-manifolds in definite4-manifolds, arXiv:2407.03692","work_id":"f9fe6a17-943a-4a03-98d4-3b032b79ab93","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"S. Akbulut and K. Larson,Brieskorn spheres bounding rational balls, Proc. Am. Math. Soc.,146(2018), no. 4, pp. 1817–1824","work_id":"c1b24e3c-320c-4edb-933b-9ac8f68884e8","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"A. Berman and R. Plemmons,Nonnegative matrices in the mathematical sciences, Volume 9 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994","work_id":"061db4ef-e2c5-4467-bb3b-b848afebaae8","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"M. Bhupal and A. Stipsicz,Weighted homogeneous singularities and rational homology disk smoothings, Am. J. Math.,133(2011), no. 5, pp. 1259–1297","work_id":"f5d789f0-a684-4779-93ac-9db488133d98","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Fillable structures on negative-definite Seifert fibred spaces","work_id":"23bc730a-2bd8-4c76-936b-2172efd3700b","ref_index":5,"cited_arxiv_id":"2604.28174","is_internal_anchor":true}],"resolved_work":28,"snapshot_sha256":"7189781bde4e86c6c449e235dd9898a5614f034c989c48190440e9a6b60fd35f","internal_anchors":2},"formal_canon":{"evidence_count":2,"snapshot_sha256":"22a4d83cb6b8acb4b60194e2d65437fdd444f46d1f24d46ae0e9b64eb1a3358d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}