The Gray filtration on phantom maps
This paper is a study of the Gray index of phantom maps. We give a new, tower theoretic, definition of the Gray index, which allows us to study the naturality properties of the Gray index in some detail.
McGibbon and Roitberg have shown that if $f^*$ is surjective on rational cohomology, then the induced map on phantom sets is also surjective. We show that if $f^*$ is surjective just in dimension $k$, then $f$ induces a surjection on a certain subquotient of the phantom set. If the condition holds for all $k$, we recover McGibbon and Roitberg's theorem. There is a dual result, and a theorem on phantom maps into spheres which holds one dimension at a time as well.Finally, we examine the set of phantom maps whose Gray index is infinite. The main theorem is a partial verification of our conjecture that if $X$ and $Y$ are nilpotent and of finite type, then every phantom map $f:X\to Y$ must have finite index.