Rational equivalence and phantom map out of a loop space

Abstract

McGibbon asked if for a connected finite complex $X$ there is a rational equivalence from the loop space of $X$ to a product of spheres and loop spaces of spheres. We will show that the answer is yes if it has only a finite number of nonzero rational homotopy groups or if spaces are localised at a prime. We will also give a clear picture of phantom maps out of the iterated loop space of a finite complex.