Relative phantom maps

TL;DR

This paper introduces the concept of relative phantom maps in algebraic topology, connecting combinatorics and topology, and provides conditions under which these maps are trivial or relatively trivial based on rational homology.

Contribution

It defines relative phantom maps and characterizes their triviality using rational homology, extending classical notions in algebraic topology.

Findings

Identifies conditions for triviality of relative phantom maps

Connects combinatorial and topological concepts through relative phantom maps

Provides criteria for relative triviality based on rational homology

Abstract

The de Bruijn-Erd\H{o}s theorem states that the chromatic number of an infinite graph equals the maximum of the chromatic numbers of finite subgraphs. Such a determinativeness by finite subobjects appears in the definition of a phantom map which is classical in algebraic topology. The topological method in combinatorics connects these two, which leads us to define the relative version of a phantom map: a map $f\colon X\to Y$ is called a relative phantom map to a map $\varphi\colon B\to Y$ if the restriction of $f$ to any finite subcomplex of $X$ lifts to $B$ through $\varphi$, up to homotopy. There are two kinds of maps which are obviously relative phantom maps: (1) the composite of a map $X\to B$ with $\varphi$; (2) a usual phantom map $X\to Y$. A relative phantom map of type (1) is called trivial, and a relative phantom map out of a suspension which is a sum of (1) and (2) is called relatively trivial. We study the (relative) triviality of relative phantom maps and in particular, we give rational homology conditions for the (relative) triviality.