Numerical invariants of phantom maps
C. A. McGibbon and  Jeffrey Strom
Wayne State University and Dartmouth College

Abstract:
Two numerical homotopy invariants of phantom maps, the Gray index  
G(f)  and the essential category weight  E(f), are studied.  The 
possible values of these invariants are determined.  In certain 
cases bounds on these values are given in terms of rational 
homotopy data.

Examples are provided showing that the Gray index can take any
positive finite value.  For certain cases it is shown that every 
essential phantom  f: X --> Y  has finite Gray index.  However it 
is also shown that there exist spaces, e. g.  CP^\infty, which are 
the domains of essential phantoms with infinite index.

The same type of analysis is carried out on the essential category 
weight of a phantom map.  If the loop space of  X  is homotopy 
equivalent to a finite complex, then every phantom  f: X --> Y  has 
E(f) = \infty.  However, in certain other cases it is shown that 
E(f)  is strictly less than the rational Lusternik-Schnirelmann 
category of the domain.  A homotopy classification of phantoms  
f: K(Z, n)--> S^m  is given along with the values of  E(f). 

The invariants  G  and  E   provide decreasing filtrations on the set
of homotopy classes of phantoms from  X  to  Y.  A third filtration on  
this set is introduced for certain special targets. When the rational 
cohomology of the domain  X  is finitely generated, this
filtration enables one to reduce the search for essential phantoms 
(into finite type targets) to a finite list of spheres.