# Relative phantom maps

A phantom map is a map between pointed CW-complexes such that its restriction to any finite dimensional subcomplex is null-homotopic. In this talk, we introduce the following generalization of phantom maps: Let X be a pointed CW-complex and $\varphi \colon B \to Y$ a map between spaces. A phantom map relative to $\varphi$ or a relative phantom map from $X$ to $\varphi$ is a map $f \to Y$ such that a restriction to any finite dimensional subcomplex of $X$ has a lift with respect to $\varphi$, up to homotopy. We study when there is (no) non-trivial phantom maps from $X$ to $\varphi$ and give some examples. For example, there is a non-trivial relative phantom map from some space $X$ to the inclusion $j_n \colon \mathbb{R} P ^n \to \mathbb{R} P^\infty$ for $n \ge 3$, although there is no non-trivial phantom map from a suspension space to $j_n$.

This is joint work with Kouyemon Iriye and Daisuke Kishitmoto.