Abstract
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 rk π i ( X ) . In the second part, we give conditions on the Lusternik–Schnirelmann category of X and on the depth of the Lie algebra π * ( Ω X ) ⊗ Q that imply the exponential growth in k of the sequence ∑ i = 1 d rk π k + i ( X ) , for some d.
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