# Relative phantom maps and rational homotopy

## TL;DR

This paper extends the theory of phantom maps to the relative case in rational homotopy, developing new techniques due to limitations of existing methods.

## Contribution

It introduces a generalization of phantom map relations to relative phantom maps and develops novel techniques beyond traditional $ ext{lim}^1$ and profinite completion methods.

## Key findings

- Established new relations between relative phantom maps and rational homotopy.
- Developed innovative methods applicable where classical techniques fail.
- Extended the theoretical framework of phantom maps in algebraic topology.

## Abstract

We generalize some results of Gray and McGibbon-Roitberg on relations between phantom maps and rational homotopy to relative phantom maps. Since the $\lim^1$ and the profinite completion techniques do not apply to relative phantom maps, we develop new techniques.