Date: Sat, 21 Mar 1998 13:01:53 -0500 (EST)
From: Peter Freyd <pjf@saul.cis.upenn.edu>
To: jiewu@math.upenn.edu
Subject: Re: Generating Hypothesis (fwd)

I'm not aware of much progress. But that doesn't mean too much. There
is stuff going on in stable homotopy and I've not kept in tune. If the
toplist has any news on this subject, please forward. And forward this
there, if you wish.

For infinite CW complexes the situation was in hand long before me.
There are countable CW complexes (realizable as open subsets of
3-space) with homotopically non-trivial maps to spheres that whenever
restricted to finite subcomplexes become homotopically trivial. These
can be constructed to lie entirely in a stable range.

In one of my papers, I think one entitled Homotopy is Not Concrete, in
the Steenrod Festschrift (and I've just noticed for the first time
that it's not in my CV) I give a pretty soft construction of such
"phantom maps" (meaning a little abelian group theory, a little
category nonsense, and essentially no topology.)