{"paper":{"title":"On the commutator of unit quaternions and the numbers 12 and 24","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.DG"],"primary_cat":"math.GT","authors_text":"Thomas Puettmann","submitted_at":"2011-01-26T20:29:19Z","abstract_excerpt":"The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power of the commutator. Subsequently, we construct free S^3-actions on S^7 x S^3 whose quotients are exotic 7-sphere and give a geometric explanation for the order of the stable homotopy groups \\pi_{n+3} (S^n). Intermediate results of perhaps independent interest are a construction o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.5147","kind":"arxiv","version":4},"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"}