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.