A counter-example to a conjecture of Félix

Abstract

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 < ∞ , 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) [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.