Stable suspension order of universal phantom maps and some stably indecomposable loop spaces (Q873725)

A map out of a CW-complex \(X\) is called a phantom map if its restriction to each \(n\)-skeleton \(X_n\) is null homotopic. The universal phantom map out of \(X\) is a based map \(X\to\bigvee^\infty_{n=1} \Sigma X_n\) through which all other phantom maps out of \(X\) factor. For a map \(f: X\to Y\) the stable suspension order of \(f\) is the order of the class \([f]\in\lim_n[\Sigma^n X,\Sigma^n X]\). The author proves that the stable suspension order of the universal phantom map is infinite when \(X\) is a non-trivial finite Postnikov space, a classifying space of a connected Lie group or a loop space on a connected Lie group with torsion.