PHANTOM MAPS,COGROUPS AND THE SUSPENSION MAP

ABSTRACT

The set Ph(X, Y) of pointed homotopy classes of phantom maps from X to Y admits a natural group structure if either Y is a grouplike space or X is a cogroup. In the present paper, the group structure on Ph(X,Y) is examined in the second case. (The first case was examined in an earlier paper.) The results in the two cases are similar—for instance, the group structure turns out to be abelian, divisible and independent of the grouplike structure on Y or the cogroup structure on X—but the techniques used to establish the results differ substantially in the two cases.

In addition, a study of the map g*: Ph(X,Y₁) → Ph(X,Y₂) induced by a map g: Y₁ → Y₂ of grouplike spaces is initiated. A particularly interesting special case of this situation is the suspension map Ph(X, Y) → Ph(X, ΩσY) ? Ph(σX, σY) with Y a grouplike space.