# The Gray filtration on phantom maps

Fundamenta Mathematicae (2001)

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

## Access Full Article

top## Abstract

top## How to cite

topLê 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.