# The Gray filtration on phantom maps

Fundamenta Mathematicae (2001)

• Volume: 167, Issue: 3, page 251-268
• ISSN: 0016-2736

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## Abstract

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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.

## How to cite

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Lê Minh Hà, and Jeffrey Strom. "The Gray filtration on phantom maps." Fundamenta Mathematicae 167.3 (2001): 251-268. <http://eudml.org/doc/281959>.

@article{LêMinhHà2001,
abstract = { 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. },
author = {Lê Minh Hà, Jeffrey Strom},
journal = {Fundamenta Mathematicae},
keywords = {phantom maps; Gray index; inverse limit; },
language = {eng},
number = {3},
pages = {251-268},
title = {The Gray filtration on phantom maps},
url = {http://eudml.org/doc/281959},
volume = {167},
year = {2001},
}

TY - JOUR
AU - Lê Minh Hà
AU - Jeffrey Strom
TI - The Gray filtration on phantom maps
JO - Fundamenta Mathematicae
PY - 2001
VL - 167
IS - 3
SP - 251
EP - 268
AB - 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.
LA - eng
KW - phantom maps; Gray index; inverse limit;
UR - http://eudml.org/doc/281959
ER -

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