The Gray filtration on phantom maps

Lê Minh Hà ¹   ; Jeffrey Strom ²

¹ Department of Mathematics University of Toledo Toledo, OH 43606, U.S.A.
² Department of Mathematics Dartmouth College Hanover, NH 03755, U.S.A.

Fundamenta Mathematicae, Tome 167 (2001) no. 3, pp. 251-268

Résumé

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.

Détail

DOI : 10.4064/fm167-3-3

Keywords: paper study gray index phantom maps tower theoretic definition gray index which allows study naturality properties gray index detail mcgibbon roitberg have shown * surjective rational cohomology induced map phantom sets surjective * surjective just dimension induces surjection certain subquotient phantom set condition holds recover mcgibbon roitbergs theorem there dual result theorem phantom maps spheres which holds dimension time finally examine set phantom maps whose gray index infinite main theorem partial verification conjecture nilpotent finite type every phantom map have finite index