Phantom maps and monoids of endomorphisms of \(K(\mathbf Z, m)\times S^ n\) (Q1311659)

A phantom map is a map \(f: X \to Y\) between CW-complexes such that \(f\mid X_ n\) is null-homotopic for each skeleton \(X_ n\) of \(X\). More generally, a phantom pair is a pair of maps \(f,g: X\to Y\) between CW- complexes such that \(f\mid X_ n \simeq g\mid X_ n\) fo reach \(X_ n\). The author proves two dual results for maps between nilpotent CW- complexes of finite type: in the situation \[ X \overset {f,g} \rightrightarrows Y @>h>> Z, \] if \(f,g\) is a phantom pair, \(h\) is a phantom map, and \(\pi_ 1(Y)\) is finite, then \(hf\simeq hg\); in the situation \[ X@>f>> Y \overset {g,h} \rightrightarrows Z, \] if \(f\) is a phantom map, \(g,h\) is a phantom pair, and \(\pi_ 1(Y)\) and \(\pi_ 1(Z)\) are finite, then \(gf \simeq hf\). The author also computes the monoids of homotopy classes of self-maps of \(K({\mathbf Z},m) \times S^ n\) and \(S^ 1 \times S^ n\), where \(K({\mathbf Z},m)\) is an Eilenberg-MacLane space and \(S^ 1\), \(S^ n\) are spheres.