Title: A finite loop space not rationally equivalent to a compact Lie group
Authors: Kasper K. S. Andersen, Tilman Bauer, Jesper Grodal, Erik K. Pedersen
Subj-class: Algebraic Topology; Geometric Topology
MSC-class: 55P35; 55P15, 55R35
Comments: 8 pages, arXiv : math.AT/0306234

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 $5$.