Abstract

The authors define phantom orders, which are ordinal numbers, for homotopy classes of maps between CW complexes \(X\) and \(Y\). Every map from \(X\) to \(Y\) has phantom order at least \(0\). For \(\alpha>0\), a map from \(X\) to \(Y\) has phantom order at least \(\alpha\) if for every \(n\)-skeleton \(X_n\) and for every ordinal \(\beta<\alpha\) there is a factorisation \[ X\to X/X_n\to Y \] such that the map \(X/X_n\to Y\) has phantom order at least \(\beta\). In particular, a map has phantom order at least \(1\) if and only if it is a phantom map, and a map has phantom order at least \(2\) if and only if it is a phantom map of infinite Gray order. There are universal maps on \(X\) of phantom order at least \(\alpha\); in many cases, all the universal maps on \(X\) are essential, so phantom orders of essential maps on \(X\) can be arbitrarily high. But the phantom orders of essential maps from \(X\) to a given space \(Y\) are bounded. The paper concludes with a \(\lim^1\) formula.