The Mislin genus, phantom maps and weak identities

AbstractFor a nilpotent space X, we prove that the cardinality of the set WI(X), which denotes the homotopy classes of weak identities, depends only on the Mislin genus of X. On the other hand, we prove that when X=BG, the classifying space of a connected finite loop space, or X is a 1-connected finite Postnikov space, or X is a 1-connected infinite loop space, and Y is the iterated loop space of a 1-connected finite CW-complex, the isomorphism class of the group Ph(X,Y) depends only on the Mislin genus of X. These generalize recent results of Roitberg