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