Phantom Maps and the Towers which Determine them

Let X and Y be pointed spaces. A phantom map from X to Y is a map whose restriction to any finite skeleton of X is null‐homotopic. Let Ph ( X , Y ) denote the set of homotopy classes of phantom maps from X to Y . As a pointed set it is isomorphic to the lim 1 term of the tower of groups[ X , Ω Y ( 1 )] ← [ X , Ω Y ( 2 )] ← ··· ← [ X , Ω Y ( n )] ← ··· ,where Y ( n ) denotes the Postnikov approximation of Y through dimension n . The homomorphisms in this tower are induced by the projections Ω Y ( n ) ← Ω Y ( n +1) ). The groups in this tower are not abelian in general; however they do have some nice algebraic properties.