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(X,Y) of homotopy classes of phantom maps and the subset SPh(X,Y) of homotopy classes of special phantom maps simultaneously. We give a sufficient condition for Ph(X,Y) and SPh(X,Y) to have natural group structures, which is much weaker than the conditions obtained by Meier and McGibbon. Previous calculations of Ph(X,Y) have generally assumed that [X,ΩŶ] is trivial, in which case generalizations of Miller’s theorem are directly applicable, and calculations of SPh(X,Y) have rarely been reported. Here, we calculate not only Ph(X,Y) but also SPh(X,Y) in many important cases of nontrivial [X,ΩŶ].