Relative phantom maps

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.