On Phantom Maps into co-H-spaces

TL;DR

This paper investigates the existence of essential phantom maps into co-H-spaces, extending previous results to a broader class of spaces and deepening the understanding of their homotopy and coalgebra structures.

Contribution

It extends Iriye's observation about phantom maps to include nilpotent, finite type co-H-spaces, using advanced homotopy and coalgebra decomposition techniques.

Findings

Essential phantom maps exist into certain co-H-spaces.

Homotopy decompositions relate to coalgebra decompositions of tensor algebras.

Extension of Iriye's observation to a broader class of spaces.

Abstract

We study the existence of essential phantom maps into co-H-spaces, motivated by Iriye's observation that every suspension space $Y$ of finite type with $H_i(Y;\QQ)\neq 0$ for some $i>1$ is the target of essential phantom maps. We show that Iriye's observation can be extended to the collection of nilpotent, finite type co-H-spaces. This work hinges on an enhanced understanding of the connections between homotopy decompositions of looped co-H-spaces and coalgebra decompositions of tensor algebras due to Grbi\`{c}, Theriault, and Wu.