Numerical invariants of phantom maps

Abstract

In this paper two numerical invariants of phantom maps are studied; the Gray index G ( f ) and the essential category weight E ( f ) of a phantom map f . The possible values of these invariants are determined along with bounds on their values given in terms of the domain and range of the phantom map. Examples are given which show that the Gray index can take any positive finite value. Furthermore, if Map* ( X,Ŷ ) ∼ *, then every phantom map f : X → Y has finite Gray index. On the other hand, if X has finite type and H *(Σ X ;[inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]/ p ) is not locally finite as an [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] p module, then essential phantoms out of X with G ( f ) = ∞ are shown to exist. If Ω X is homotopy equivalent to a finite dimensional CW complex, then every phantom map f : X → Y has E ( f ) = ∞. However, if Map* ( X,Ŷ ) ∼ *, then every map f : X → Y has E ( f ) < ∞. An example is given such that this mapping space hypothesis is necessary. Calculations of E ( f ) for f : K ([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /], n ) → S m are given; they demonstrate that E ( f ) can also take any positive finite value. Finally, a new filtration is introduced on certain phantom sets. It is used to sharpen a result of McGibbon and Roitberg. The new result is that if X is nilpotent, of finite type, and Ph ( X ,S n +1 ) = * for each n such that QH n ( X ;[inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]) ≠ 0, then Ph ( X,Y ) = * for each finite type nilpotent space Y .