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 → Y must have finite index.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-3-3,
author = {L\^e Minh H\`a and Jeffrey Strom},
title = {The Gray filtration on phantom maps},
journal = {Fundamenta Mathematicae},
volume = {167},
year = {2001},
pages = {251-268},
zbl = {0982.55015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-3-3}
}
Lê Minh Hà; Jeffrey Strom. The Gray filtration on phantom maps. Fundamenta Mathematicae, Tome 167 (2001) pp. 251-268. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-3-3/