Phantom Maps and Monoids of Endomorphisms of $K(\mathbf{Z},m)\times S^n$

Abstract

The compositions $hf$, $hg$ of phantom pair $f,g$ and a phantom map $h$ are homotopic. The compositions $hf$, $kf$ of phantom pair $h,k$ and a phantom map $f$ are homotopic. We determine the homotopy set $[K(\mathbf{Z},m)\times S_{,}^{n}K(\mathbf{Z},m)\times S^n]$ and its monoid structure given by the composition of maps for all $m,n\ge 1$.