Homotopy epimorphisms preserve nilpotency (Q1913930)

Let \(X\) and \(Y\) be pointed connected, CW-complexes. It is shown that if \(X\) is nilpotent, then for any homotopy epimorphism \(f : X \to Y\) (homotopy monomorphism \(f : Y \to X)\) \(Y\) is also nilpotent. Two examples are constructed, which show that the reverse implications in the theorem are false. From the above result follows that if \(X\) is nilpotent, then for any homotopy epimorphism \(f : X \to Y\) (homotopy monomorphism \(f : Y \to X\)) its \(p\)-localization \(f_p : X_p \to Y_p\) \((f_p : Y_p \to X_p)\) is a homotopy epimorphism (monomorphism), where \(p\) is either a prime or 0.