A counter-example to a conjecture of Félix

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 LX. The radical of LX, 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 dimLX<∞, i.e., if X is elliptic, then LX is its own radical, and therefore the total dimension of the radical of LX in odd degrees is less than or equal to its total dimension in even degrees (Friedlander and Halperin (1979) [8]). Félix conjectured that this inequality should hold for all simply connected spaces with finite Lusternik–Schnirelmann category.We prove Félix’s conjecture in some interesting special cases, then provide a counter-example to the general case.