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 ___________________________________ 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 --------------------------------------------------------------------- Name: Poin.pdf Poin.pdf Type: Portable Document Format (application/pdf) Encoding: base64 Download Status: Not downloaded with message _____________________________________________ 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