Subject: More on generating hypothesis
From: Mark Hovey <mhovey@wesleyan.edu>
Date: 21 Jul 2005 09:06:56 -0400

It appears that Freyd, whose original paper I still cannot lay my hands
on, knew that the generating hypothesis is equivalent to the statement
that

 pi_*(X) is an injective pi_*(S)-module for all finite torsion spectra
X.  

I am not sure how much of my other results were previously known.  The
most surprising one, to me anyway, is this:

Let I denote the Brown-Comenetz dual of the sphere.  If the generating
hypothesis is true, then there is a cofiber sequence

Sigma^{-1} I --delta--> W --> W --> I

in which W is a wedge of spheres and delta is a phantom map.  

This implies that lots of weird things happen.  For example, this
implies that [E,W]=0 if and only if E=0, so that every spectrum maps
into this wedge of spheres, even though almost no spectra map into a
single sphere nontrivially.  So W is not only not harmonic, it is not
E-local for any E (unless E has Bousfield class equal to <S>). 

This also implies that the p-complete sphere S_p has P = S_p v P for
some product P of suspensions of I.  This seems absolutely crazy, but I
am unable to derive any contradictions from it.
                       Mark