Phantom Maps and the Towers which Determine them

Abstract

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 lim1 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 ΩY(n)← ΩY(n+1)). The groups in this tower are not abelian in general; however they do have some nice algebraic properties.