The ranks of the homotopy groups of a space of finite LS category

Let X be a simply connected CW complex with finitely many cells in each degree. The first part of the paper is a report on the different conjectures for the behavior of the sequence rkπi(X). In the second part, we give conditions on the Lusternik-Schnirelmann category of X and on the depth of the Lie algebra π*(ΩX)⊗Q that imply the exponential growth in k of the sequence ∑i=1drkπk+i(X), for some d.