Date: Tue, 17 Mar 1998 09:28:10 -0500 (EST)
From: "Douglas C. Ravenel" <drav@harpo.math.rochester.edu>
Subject: Re: Generating Hypothesis

On Mon, 16 Mar 1998, DONALD M. DAVIS wrote:

> From: Michael Cole <mmcole@math.lsa.umich.edu>
> Date: Sun, 15 Mar 1998 00:18:04 -0500 (EST)
> Subject: generating hypothesis.
>
>
>      Freyd's generating hypothesis conjectures that if a map of finite
> CW spectra induces the 0 homomorphism on homotopy groups, then the map
> is null-homotopic.  Question:  Are there known counterexamples for maps
> of infinite CW spectra?  Is there anything good and up to date to read
> on the generating hypothesis?
>
> Mike Cole
>

One should consult the papers of Ethan Devinatz (including one
with Hopkins) for recent progress on the Generating Hypothesis.

There is a much simpler (than the one suggested by Wu)
counterexample among infinite complexes.  Take any nontrivial
positive dimensional element $a$ in the Steenrod algebra and
represent it by a map between mod p Eilenberg-Mac Lane spectra.
This map is essential (since $a$ is nontrivial) but induces the
trival homomorphism in homotopy since $a$ is positive
dimensional.

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