The Mislin genus, phantom maps and classifying spaces (Q1916441)

The author shows that the homotopy classes of phantom maps \(\text{Ph}(X,Y)\) and \(\text{Ph}(X',Y)\) are equivalent as sets if \(X\) and \(X'\) are finite type nilpotent spaces with the same genus and \(Y\) has countable higher homotopy groups. He also shows that if \(X\) and \(X'\) have the same genus where \(X\) is simply connected and \(\pi_2X\) is finite, then \(X_\tau\simeq X_\tau'\), where \(X_\tau\) is the fibre of the rationalization map \(X\to X_{(0)}\), and similarly for \(X_\tau'\). This latter result can be used to compute \(\text{Ph}(X,Y)\), where \(X\) has the genus of \(BG\), \(G\) being a simply connected Lie group and \(Y\simeq\Omega^kZ\), \(Z\) being a simply connected finite CW complex.