The Gray filtration on phantom maps (Q2717649)

In this well written and interesting paper, the authors study the Gray index of phantom maps. They introduce a new definition of the Gray index which immediately shows its homotopy invariance and allows them to study naturality properties of the index. This allows them to generalize a theorem of McGibbon and Roitberg on surjectivity of induced maps on sets of phantom maps. They also conjecture that if the domain and range of a phanton map is of finite type and nilpotent, then the map must have finite Gray index. The authors prove a special case of this conjecture.NEWLINENEWLINENEWLINEA 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.