{"paper":{"title":"Exponential growth of Lie algebras of finite global dimension","license":"","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Jean-Claude Thomas (LAREMA), Steve Halperin, Yves F\\'elix","submitted_at":"2005-09-23T10:54:10Z","abstract_excerpt":"Let $X$ be a finite simply connected CW complex of dimension $n$. The loop space homology $H\\_*(\\Omega X;\\mathbb Q)$ is the universal enveloping algebra of a graded Lie algebra $L\\_X$ isomorphic with $ pi\\_{*-1} (X)\\otimes \\mathbb Q$. Let $Q\\_X \\subset L\\_X$ be a minimal generating subspace, and set $\\alpha = \\limsup\\_i \\frac{\\log{\\scriptsize rk} \\pi\\_i(X)}{i}$. Theorem: If ${dim} L\\_X = \\infty$ and $\\limsup ({dim} (Q\\_X)\\_k)^{1/k} < \\limsup ({dim} (L\\_X)\\_k)^{1/k}$ then $$\\sum\\_{i=1}^{n-1} {rk} \\pi\\_{k+i}(X) = e^{(\\alpha + \\epsilon\\_k)k} \\hspace{1cm} {where} \\epsilon\\_k \\to 0 {as} k\\to \\infty"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0509546","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"}