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.