Subject: Re: n-ad connectivity and critical groups From: "Ronnie Brown" Date: Wed, 26 Jul 2006 22:29:46 +0100 Bill and all, Sorry I failed to convince you in 1992! Do you think the nonabelian tensor product has the real homotopical applications we claim? (www.bangor.ac.uk/r.brown/nonabtens.html) It is certainly a new feature we claim to add to the triad case. I do not see how other methods could reach that. If you want nonabelian results you had better use nonabelian methods, analogous to those for the fundamental group(oid). I invite you to consider the results in 124. (with C.D.WENSLEY), `Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72. and see if you can do better with standard homotopical techniques. Note that we there compute not just some 2nd homotopy groups but some homotopy 2-types (as crossed modules). This is intended to indicate possible new directions, and new tools, to follow through an idea, and to show how to use a computational methods to get some explicit homotopical answers. I would have thought you, Bill, would be interested in that, and the new ideas on rewriting coming from Bangor, leading to new computations, such as of double cosets. I outlined the n=4 case in my last email. Why not take the connectivity result I stated (never mind the proof) and interpret it in terms of ordinary r-ad connectivity of the various cubes, including the degenerate 3-cubes, that arise, and then consider special cases? That should be easy. There is a lot of other algebraic detail needing perhaps a fuller exposition; for example, the relation with Whitehead products. Nick Gilbert made a start on one part of the theory, in a paper in a Barcelona proceedings. At the moment I have other expository problems on my mind, since I have some papers and a planned book to complete on crossed complexes. Of course crossed complexes represent only a `linear ' approximation to homotopy theory, but they still have strong value, since they incorporate the fundamental group(oid), and its presentations. This theory has all sorts of interactions needing exploration. A key tool is the relation between crossed complexes and a cubical notion of homotopy omega-groupoid of a filtered space. If you are not interested in the idea of using a crossed module, or a double groupoid, then too bad! (Necessary inputs!) There are people who are interested in, and enthusiastic about, taking the ideas further, or in clarifying them. I have heard before now the following type of argument: if these ideas really worked, we would all know about them, therefore..... History shows that is a common way of avoiding the consideration of new ideas! all the best Ronnie ___________________________________________________________ Subject: Re: 3 on Blakers-Massey From: Bill Richter Date: Fri, 28 Jul 2006 00:47:23 -0500 Tom, thanks for the long interesting post. One story about B-W: B-W stated both connectivity and computation of critical group. It stayed away from \pi_1 (and \pi_0) difficulties in the same way as the original Blakers-Massey result. I suppose it used homology, but I have not looked at the paper in a long time. I had trouble reading that paper. (I have since heard others say that they couldn't figure it out either.) So I reproved the connectivity result myself, in the form stated above as HBM. Barratt & Whitehead didn't attempt a proof of the n-ad conn thm. Toda has the only proof-attempt of that time, and nobody can could figure it out. Barratt told me once that the short B-W paper had a proof of the n-ad conn thm, and so I showed it to him, and he stared at it for a while, and said, in Michael's inimitable style, "That bostid Henry took it out!" But as I said, I don't think they ever had one. And Ronnie, I'm sorry I misquoted you: I outlined the n=4 case in my last email. But as I said, I can do the 3-ad conn thm EHP sequence, and that's a lot of the n=4 case. It gets harder, I think. I have heard before now the following type of argument: if these ideas really worked, we would all know about them, That's a good point, don't I know it all too well :-D I don't know anything about your crossed module, or double groupoids. I certainly don't mean to disparage your work. But If you want to sell me, write up a comprehensible proof of the n-ad conn thm. __________________________________________________________________ Subject: Re: 3 on Blakers-Massey From: "Ronnie Brown" Date: Fri, 28 Jul 2006 10:07:24 +0100 Bill and all, Tom's post was very interesting. I agree about \pi_0. This is used in discussion of `connectivity' of a filtered space, as called `homotopy full' in 32. (with P.J. HIGGINS), ``Colimit theorems for relative homotopy groups'', {\em J. Pure Appl. Algebra} 22 (1981) 11-41. We require \pi_0 X_0 \to \pi_0 X_n to be surjective for all n > 0 (and higher conditions of course). Note also that the proof here uses lots of (then) new ideas. But if someone wants to avoid multiple groupoids for proofs in this area, I can't help! In the Brown-Loday paper, we stuck to the base pointed case, since we did not, and still do not, have a successful analogue of cat^n-groups for the many pointed case. In fact (n+1)-fold groupoids are in general somewhat mysterious. I can't help Bill more than pointing to the Brown-Loday papers referred to, which are applied in Ellis-Steiner, and to Tom's comments on the relation of connectivity for various homotopy fibres of a cube of spaces to r-ad connectivity in the case the cube is derived from a set of subspaces of a space. The inductive proof of the GvKT uses intrinsically the algebra of cat^n-groups. What fascinated Jean-Louis and I was the algebraic structures which arose easily from pushouts of crossed squares ( ~ cat2-groups). Considering pushouts of crossed squares, and using previous experience of work with Philip Higgins, the nonabelian tensor product leapt out of the page. Michel Zisman remarked that for a SK(G,1) it should also in principle come from a nonabelian chain complex involving repeated free products of G with itself, but Michel could not see it. It is probably related to notions of higher Peiffer rules in simplicial groups, discussed by Arvasi and Porter. As explained in 74. ``Computing homotopy types using crossed $n$-cubes of groups'', {\em Adams Memorial Symposium on Algebraic Topology}, Vol 1, edited N. Ray and G. Walker, Cambridge University Press, 1992, 187-210. one can obtain, for example, precise computation of everything to do with the 3-type of say SK(D_{2n},1) where D_{2n} is the dihedral group of order 2n, including Whitehead products and composition with the Hopf map. I know Jie Wu has results on homotopy groups of SK(G,1)'s, but as far as I am aware that does not lead to such computations. That is probably all that needs to be said by me. Ronnie