{"paper":{"title":"A product involving the $\\beta$-family in stable homotopy theory","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Wending Li, Xiugui Liu","submitted_at":"2009-11-05T18:21:38Z","abstract_excerpt":"In the stable homotopy groups\n  $\\pi_{q(p^n+p^m+1)-3}(S)$ of the sphere spectrum $S$ localized at the prime $p$ greater than three, J. Lin constructed an essential family $\\xi_{m,n}$ for $n \\geq m + 2 >5$. In this paper, the authors show that the composite $\\xi_{m,n}\\beta_{s}\\in \\pi_{q(p^n+p^m+sp+s)-5}(S)$ for $2 \\leq s < p$ is non-trivial, where $q=2(p-1)$ and $\\beta_s \\in \\pi_{q(sp+s-1)-2}(S)$ is the known $\\beta$-family. We show our result by explicit combinatorial analysis of the (modified) May spectral sequence."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.1093","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":""},"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"}