Two postings in response to this morning's question about cubical
approximation. Lots of people have sent url's with Dunwoody's
preprint on the Poincare conjecture. Some of these, and comments,
appear below..............DMD
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Subject: Re: 2 postings & rumor
Date: Thu, 11 Apr 2002 16:39:12 +0100
From: "Prof. T.Porter"
I am sure that Ronnie brown will reply more fully but to start the
discussion it is worth asking for more precision on what type of cubical
complexes are being considered. There are several that occur in the
literature some with the basic cubical structure only. Some with
connections have been used extensively by Ronnie and his collaborators
and have been explored in the abstract by Andy Tonks, Heiner Kamps and
others.
Might I also recommend a GOOGLE search on cubical complex. There are
lots of researchers (non-topologists) who are trying to use algebraic
topology in a multitude of contexts some of which look very interesting!
(e.g. image analysis, modelling physical phenomena etc)
Tim Porter
______________________
Subject: cubical sets in algebraic topology
Date: Thu, 11 Apr 2002 16:08:28 +0100
From: "Ronald Brown"
reply to r.brown@bangor.ac.uk
> Subject: question for the list: cubical approximation
> Date: Wed, 10 Apr 2002 12:35:36 -0700
> From: John H Palmieri
>
> Here's a question for the list: a colleague of mine wants to know if
> there is a cubical analogue of simplicial approximation. Also, what's
> a good reference for cubical chains and related things?
There are old notes of Federer from Brown University and also Massey's book
on Singular Homology. In a series of papers starting with
``On the algebra of cubes'', {\em J. Pure Appl. Algebra} 21 (1981) 233-260.
``Colimit theorems for relative homotopy groups'', {\em J. Pure Appl.
Algebra} 22 (1981) 11-41.
Philip Higgins and I gave a somewhat non abelian version of some basic
concepts in algebraic topology using cubical sets with connections to
define and apply higher homotopy groupoids of a filtered space. Because
these higher homotopy groupoid methods are convenient for `algebraic
inverses to subdivision' we were able to prove a Generalised Van Kampen
Theorem and so *within the space of the above papers* got as far as (a
generalisation of) the relative Hurewicz theorem, the latter regarded as a
theorem on \pi_n(U \cup CW,x) when the pair (U,W) is (n-1)-connected;
knowledge of homology theory is not assumed! Also included is the case n=2.
(This work does not use a cubical approximation theorem.) There is recent
work in concurrency using cubical multiple categories with connections (do a
web search on GETCO).
NOTE: The above papers were the first to give cubical sets the extra
structure of `connections' (a new set of `degeneracy' operators) and which
are essential for the above applications. Note that cubical abelian groups
are not equivalent to chain complexes (M. Golasinski) but cubical abelian
groups with connection are so equivalent.
References to later work, including work with Loday on n-cubes of spaces,
can be found on links from my home page. In particular, there is an n-adic
Hurewicz theorem, which again is an application of a GVKT.
Ronnie Brown
School of Informatics, Mathematics Division,
University of Wales, Bangor
Dean St., Bangor, Gwynedd LL57 1UT,
United Kingdom
Tel. direct:+44 1248 382474|office: 382681
fax: +44 1248 361429
World Wide Web: home page:
http://www.bangor.ac.uk/~mas010/
(Links to survey articles: Higher dimensional group theory
Groupoids and crossed objects in algebraic topology)
Raising Public Awareness of Mathematics CDRom Version 1.1
http://www.bangor.ac.uk/~mas010/CDadvert.html
Symbolic Sculpture and Mathematics:
http://www.cpm.informatics.bangor.ac.uk/sculmath/
___________________________________________________
(I doubt that this pdf file will work when transmitted this
way............DMD)
Subject: RE: 2 postings & rumor
Date: Thu, 11 Apr 2002 10:56:42 -0400
From: Ron Umble
Dunwoody's paper is attached if anyone wants to look.
Ron
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Poin.pdf Type: Portable Document Format (application/pdf)
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Subject: Re: rumor
Date: Thu, 11 Apr 2002 20:13:28 +0100
From: "Brian Sanderson"
You can pick up the paper from
http://www.maths.soton.ac.uk/pure/preprints.phtml
I believe today you will be getting version 7. It first was first posted
recently.
Brian Sanderson
email: bjs@maths.warwick.ac.uk
web : http://www.maths.warwick.ac.uk/~bjs
Tel : 01788 890092
Mobile: 07957 891668
Fax : 01788 891554
________________________________________________
One skeptic, who wishes to remain anonymous, writes:
"Notice, it is his 6th draft. I know someone who knows one of
the people who is thanked in the preprint. He said that no one
has stepped up to say that this draft has fixed the problems that
appeared in the earlier drafts."
_________________________________________________
Subject: Re: 2 postings & rumor
Date: Thu, 11 Apr 2002 16:21:44 -0400 (EDT)
From: Dev Sinha
This is a serious attempt. But, someone I know and whose
opinion I trust claims that there is an error on page two
(but he's not sure how the erroneous result is used later
in the paper). Apparantly errors have appeared in a few
versions of this paper - the author thanks a number of
people who have already pointed out errors to him.
So, the 3-mflds people I know consider the paper to be
an outline of a program which seems likely to work but
which probably doesn't have all of the details right yet.
-Dev