Subject: Re: three postings
From: Bill Richter
Date: Tue, 25 Jul 2006 13:42:05 -0500
The n-ad connectivity theorem is an interpretation [of the paper
with Loday], for the case when the n-cube comes from an n-ad. Maybe
this was not spelled out in the papers.
Ronnie, if this is really true, please write up the details. Why not
just post a sketch here. I'd be shocked if it were true. I only know
of two proofs of the n-ad conn thm, Tom's & Mike's, and they both seem
like very deep arguments to me. I don't think Barratt & Whitehead
ever had a proof. The first part of Tom's proof is a really nice
transversality argument, which is all I think that Barratt & Whitehead
could've hoped for. There's a nice transversality proof of the triad
conn thm in Brayton Gray's book, so you want to generalize that to
higher n. But Tom's transversality generalization is really nice, not
obvious, and it's not the end. Tom goes on some incredible diagram
chase to finish the argument. Mike's proof (actually of just the most
important special case of the n-ad conn thm, for the Barratt ss) uses
deep results of Bousfield. So I don't think this an easy result. I
don't think Toda's original proof makes any sense either, and that's
very surprising, considering how careful Toda was. Chris Stover used
to say he had an easy proof using his "resolutions", and maybe he was
right. If he was, that would show the power of his resolutions.