Groups of homotopy classes of phantom maps

Abstract

We introduce a new approach to phantom maps which largely extends the rational-ization-completion approach developed by Meier and Zabrodsky. Our approach enables us to deal with the set $Ph\left(X,Y\right)$ of homotopy classes of phantom maps and the subset $SPh\left(X,Y\right)$ of homotopy classes of special phantom maps simultaneously. We give a sufficient condition for $Ph\left(X,Y\right)$ and $SPh\left(X,Y\right)$ to have natural group structures, which is much weaker than the conditions obtained by Meier and McGibbon. Previous calculations of $Ph\left(X,Y\right)$ have generally assumed that $\left[X,\Omega Ŷ\right]$ is trivial, in which case generalizations of Miller’s theorem are directly applicable, and calculations of $SPh\left(X,Y\right)$ have rarely been reported. Here, we calculate not only $Ph\left(X,Y\right)$ but also $SPh\left(X,Y\right)$ in many important cases of nontrivial $\left[X,\Omega Ŷ\right]$.