{"paper":{"title":"Revisiting the $\\beta_1$-action on the $3$-primary stable homotopy groups of spheres","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.AT","authors_text":"Jack Morgan Davies","submitted_at":"2025-11-04T07:21:10Z","abstract_excerpt":"Let $\\beta_1$ be the first $3$-torsion class in the stable homotopy groups of spheres in even degree. Toda showed that $\\beta_1^5 \\neq 0$, whilst $\\beta_1^6 = 0$. Shimomura generalised this to the $144$-periodic family generated by $\\beta_1$, written as $\\{\\beta_{1+9s}\\}_{s\\geq 0}$, and showed that any $5$-fold product $\\prod_5 \\beta_{1+9s} \\neq 0$, whilst all $6$-fold products $\\prod_6 \\beta_{1+9s} = 0$. In this article, we give a simple proof of these results as well as some generalisations to other $144$-periodic families. Our tools include BP-synthetic spectra, and the well-known Adams--No"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.02324","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/2511.02324/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"}