Date: Mon, 16 Mar 1998 22:50:39 -0500 (EST)
From: Jie Wu <jiewu@math.upenn.edu>
Subject: Re: Generating Hypothesis

I am also interested in knowing anything about this conjecture.

Does  the following example  work as a counter example for infinite
spectra?

Let f\colon \vee S^{n_{\alpha} \to X such that \pi_*(f) is onto, i.e, the
map f is from a (big) wedge of spheres to X which kills all elements in
the homotopy groups. Then the canonical map g from X to the homotopy
cofiber of f should satisfies that g_* is zero on homotopy groups. The map
g can not be trivial in general. Otherwise X must be a retract of a wedge
of spheres. This can not happen for most spaces (spectra) because the
integral homology of X must be a projective Z-module (=free Z-module) if
this happed (e.g., mod p Moore spaces).

Jie