On conjectures of Moore and Serre in the case of torsion-free suspensions

Abstract

Let p be a prime. A space X is said to have a homotopy exponent at p if multiplication by pr annihilates the p-torsion of πn(X) for some non-negative integer r independent of n. X is said to have totally finite rational homotopy if ⊕n πn (X) ⊗ ℚ is a finite vector space. Moore has conjectured that these properties are related for finite simply connected CW complexes.Moore's Conjecture. A space having the homotopy type of a finite simply connected CW complex has a homotopy exponent at p if and only if it has totally finite rational homotopy.For convenience I will divide the conjecture into its two implications so that I can refer to each separately.