Phantom maps and fibrations

Abstract

Given pointed $CW$-complexes $X$ and $Y$, $\rmph(X, Y)$ denotes the set of homotopy classes of phantom maps from $X$ to $Y$ and $\rmsph(X, Y)$ denotes the subset of $\rmph(X, Y)$ consisting of homotopy classes of special phantom maps. In a preceding paper, we gave a sufficient condition such that $\rmph(X, Y)$ and $\rmsph(X, Y)$ have natural group structures and established a formula for calculating the groups $\rmph(X, Y)$ and $\rmsph(X, Y)$ in many cases where the groups $[X,\Omega \widehat{Y}]$ are nontrivial. In this paper, we establish a dual version of the formula, in which the target is the total space of a fibration, to calculate the groups $\rmph(X, Y)$ and $\rmsph(X, Y)$ for pairs $(X,Y)$ to which the formula or existing methods do not apply. In particular, we calculate the groups $\rmph(X,Y)$ and $\rmsph(X,Y)$ for pairs $(X,Y)$ such that $X$ is the classifying space $BG$ of a compact Lie group $G$ and $Y$ is a highly connected cover $Y' \langle n \rangle$ of a nilpotent finite complex $Y'$ or the quotient $\gbb / H$ of $\gbb = U, O$ by a compact Lie group $H$.