A counter-example to a conjecture of Felix
If X is a simply connected space of finite type, then the rational homotopy groups of the based loop space of X possess the structure of a graded Lie algebra, denoted L-x. The radical of L-x, which is an important rational homotopy invariant of X, is of finite total dimension if the Lusternik-Schnirelmann category of X is finite. Let X be a simply connected space with finite Lusternik-Schnirelmann category. If dim L-x < infinity, i.e., if X is elliptic, then L-x is its own radical, and therefore the total dimension of the radical of L-x in odd degrees is less than or equal to its total dimension in even degrees (Friedlander and Halperin (1979) ). Felix conjectured that this inequality should hold for all simply connected spaces with finite Lusternik-Schnirelmann category. We prove Felix's conjecture in some interesting special cases, then provide a counter-example to the general case. (C) 2010 Elsevier B.V. All rights reserved.