Numerical invariants of phantom maps (Q2747077)

Let \(X\) be a CW complex and let \(Y\) be any space. In this paper a map \(f: X\to Y\) is a phantom map if its restriction to each \(n\)-skeleton \(X_n\) is null-homotopic. The set of homotopy classes of phantom maps from \(X\) to \(Y\) is denoted \(\text{Ph}(X,Y)\). The object of this paper is to partition \(\text{Ph}(X,Y)\) by using numerical invariants. Two invariants for a phantom map \(f: X\to Y\) are considered: the Gray index is the least upper bound for the integers \(n\) such that \(f\) has a factorization \(X\to X/X_n\to Y\) with \(X/X_n\to Y\) a phantom map; the essential category weight is the least upper bound for the integers \(n\) such that the composite \(fg\) is null-homotopic whenever \(g\) is a map with Lyusternik-Shnirel'man category at most \(n\). It is shown that the possible values of the invariants include infinity and every positive integer, but that there are restrictions for certain pairs of spaces \(X\) and \(Y\).NEWLINENEWLINENEWLINEThe paper concludes with an application of a filtration related to essential category weight. Let \(X\) be nilpotent and of finite type such that \(\text{Ph}(X,S^{n+1})\) is trivial for every sphere such that there are \(n\)-dimensional indecomposables in the rational cohomology ring of \(X\). Then \(\text{Ph}(X,Y)\) is trivial for all nilpotent spaces \(Y\) of finite type.