Phantom Maps and Finiteness Conditions

TL;DR

This paper extends Zabrodsky's results on phantom maps, showing that under weaker finiteness conditions on spaces, all maps are phantom maps, and applies this to classify certain self-homotopy equivalences.

Contribution

It generalizes Zabrodsky's finiteness conditions for phantom maps using resolving classes and the Zabrodsky lemma, broadening the scope of when all maps are phantom.

Findings

Zabrodsky's conditions can be relaxed for phantom map results.

Identification of the self-homotopy equivalence group for specific spaces.

Application of resolving classes to homotopy theory problems.

Abstract

A phantom map is a potentially nontrivial map which induces the zero map on every homology theory and on homotopy groups. Zabrodsky has shown that in the presence of particular finiteness conditions on spaces $X$ and $Y$ every map $X\to Y$ is a phantom map. More specifically, Zabrodsky essentially requires $Y$ to be a finite CW complex and $X$ to be a Postnikov space. We show Zabrodsky's observations hold under less restrictive finiteness conditions on the spaces $X$ and $Y$, making use of the Zabrodsky lemma and the machinery of resolving classes. As an application we identify, up to extension, the group of self-homotopy equivalences of spaces belonging to a particular family.