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