{"paper":{"title":"The realizability of operations on homotopy groups concentrated in two degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Hans-Joachim Baues, Martin Frankland","submitted_at":"2012-08-10T14:18:02Z","abstract_excerpt":"The homotopy groups of a space are endowed with homotopy operations which define the \\Pi-algebra of the space. An Eilenberg-MacLane space is the realization of a \\Pi-algebra concentrated in one degree. In this paper, we provide necessary and sufficient conditions for the realizability of a \\Pi-algebra concentrated in two degrees. We then specialize to the stable case, and list infinite families of such \\Pi-algebras that are not realizable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2184","kind":"arxiv","version":2},"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"}