Universal phantom maps out of loop spaces

Abstract

We consider the universal phantom map out of a non-finite loop space. First we obtain a necessary and sufficient condition for the universal phantom map out of ΩG for a simply connected compact Lie group G to be essential. Next we prove that the universal phantom map out of ΩkX is essential for all k ≥ 2 if X is a simply connected non-contractible finite CW-complex. Ingredients in the proof are the Browder's ∞-implication argument and the Eilenberg–Moore spectral sequence.