A finite loop space not rationally equivalent to a compact Lie group

2003/06/16 by Kasper K. S. Andersen, Tilman Bauer, Jesper Grodal +2
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #math.AT #math.GT #msc:55P15 #msc:55P35 #msc:55R35

paper · pdf · doi:10.1007/s00222-003-0341-4

published as Invent. Math 157 (2004), no. 1, 1--10. · 8 pages

arxiv created 2003/06/16 · openalex publication_date 2004/03/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5.

Citations

• Genetics of Homotopy Theory and the Adams Conjecture

Related

• Noetherian loop spaces
• Quasi finite loop spaces are manifolds
• On the String Topology Coproduct for Lie Groups
• Finite Chevalley groups and loop groups
• String topology of finite groups of Lie type