Title of paper: Phantom maps, SNT-theory, and natural filtrations on lim^1 sets.
Author: Pierre GHIENNE.
AMS Classification: 55Q05, 55S37, 55P15.
Adress of author: Matematisk Institut, Universitetsparken 5, DK--2100 København.
E-mail adress: ghienne@math.ku.dk
Text of abstract: We study the so-called Gray filtration on the set of
phantom maps between two spaces. Using both its algebraic
characterization and the Sullivan completion approach to phantom
maps, we generalize some of the recent results of Le, McGibbon and
Strom. We particularly emphasize on the set of phantom maps with
infinite Gray index, describing it in an original algebraic way.
We furthermore introduce and study a natural filtration on SNT-sets
(that is sets of homotopy types of spaces having the same $n$-type
for all $n$), which appears to have the same algebraic characterization
of the Gray one on phantom maps. For spaces whose rational homotopy type
is that of an $H$-space or a co-$H$-space, we establish criteria permitting
to determinate those subsets of this filtration which are non trivial,
generalizing work of McGibbon and M\o ller.
We finally describe algebraically the natural connection between phantom
maps and SNT-theory, associating to a phantom map its homotopy fiber or
cofiber. We use this description to show that this connection respect
filtrations, and to find generic examples of spaces for which the filtration
on the corresponding SNT-set consists of infinitely many strict inclusions.