{"paper":{"title":"Relations in the 24-th homotopy groups of spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Juno Mukai, Toshiyuki Miyauchi","submitted_at":"2026-06-10T13:12:23Z","abstract_excerpt":"The main purpose of this note is to give a proof of the fact that the Toda brackets \\ $\\langle\\bar{\\nu},\\sigma,\\bar{\\nu}\\rangle$ and $\\langle\\nu,\\eta, \\bar{\\sigma}\\rangle$ are not trivial. This is an affirmative answer of M.~Mahowald's Conjecture (J. Mukai, Determination of the $P$-image by Toda brackets, Geometry and Topology Monographs \\textbf{13}(2008), 355--383). The second purpose is to determine the relations including $\\bar{\\nu}_6\\omega_{14}$ in $\\pi^6_{30}$ and $\\bar{\\nu}_7\\omega_{15}$ in $\\pi^7_{31}$. To this end, we provide relations between the Toda bracket and the $J$-homomorphism,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.12046","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/2606.12046/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"}