Title: Relative phantom maps,
Abstract: We define a map $f\colon X\to Y$ to be a phantom map relative to a map
$\varphi\colon B\to Y$ if the restriction of $f$ to any finite dimensional
skeleton 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 from a suspension, and in
particular, we give rational homotopy conditions for the (relative) triviality.
We also give a rational homotopy condition for the triviality of relative
phantom maps from a non-suspension to a finite Postnikov section.