Abstract

A phantom map is a map of a CW complex \(X\) whose restriction to every skeleton is null homotopic. This implies that every phantom map factors through the quotient of \(X\) given by collapsing the \(k\)th skeleton for all \(k\). The Gray index is the smallest \(k\) such that the induced map from the quotient to the range \(Y\) cannot be chosen to be a phantom map. Then the authors define a dual index by considering the fact that the phantom map \(f\) must lift to a \(k\) connected covering of the range \(Y\). The dual index is the smallest \(k\) so that the lifting of \(f\) cannot be chosen to be a phantom map. Then the authors show that the dual index is equal to the Gray index plus \(1\). The authors make essential use of this fact in establishing their results.