Date: Fri, 20 Mar 1998 14:52:17 +0000 (GMT)
From: Ronnie Brown
Subject: Re: cubical theory?
The book by W S Massey, `Singular homology theory', Springer 1980 uses
the cubical theory. As he explains: `The subdivision of an n-dimensional
cube is very easy to describe explicitly, hence the proof of the excision
property is easier to motivate ...'
In work with Philip Higgins, (see references below) we showed how to use
cubical methods to define higher homotopy groupoids of a filtered space,
and to use subdivision methods to prove a higher order Van Kampen
theorem for relative homotopy groups. This implies for example the
relative Hurewicz theorem in the form of a description of \pi_n(X\cup CA)
if (X,A) is (n-1)-connected, and also other results not otherwise
available, for example non abelian results in dim 2.
Our cubical methods require `degeneracies' additional to those usually
considered, derived from the max and min monoid structures on {0,1}.
Workers in computer science are also considering such structures.
R. BROWN, P.J. HIGGINS, ``On the algebra of cubes'', {\em J.
Pure Appl. Algebra} 21 (1981) 233-260. \par
R.BROWN, P.J. HIGGINS, ``Colimit theorems for relative
homotopy groups'', {\em J. Pure Appl. Algebra} 22 (1981) 11-41.
We did use cubical realisation theory from an unpublished MPhil by S
Hintze (Warwick).
Such cubical and non abelian homotopical methods were developed further in
R.BROWN, J.-L. LODAY, ``Van Kampen theorems for diagrams of
spaces'', {\em Topology} 26 (1987) 311-334.
Ronnie Brown
On Fri, 20 Mar 1998, DONALD M. DAVIS wrote:
> Date: Fri, 20 Mar 1998 07:40:40 -0500 (EST)
> From: James Stasheff
>
> A combinatorial friend writes:
>
> < In joint work with Vic Reiner we need (for some
> arrangement bussiness) define cup products
> for a regular CW-complex all of whse cells
> are (hyper)-cubes. We already found a singular
> cubical theory (in Serre's work) but we could not
> find an actual cubical theory -- like simplicial
> in contrast to singular (simplicial) theory --
> worked out.
> I am sure this has been done ! >>
>
> I dimly recall seeing something like this
> many years ago; any references??
>
>
> ************************************************************
> Until August 10, 1998, I am on leave from UNC
> and am at the University of Pennsylvania
>
> Jim Stasheff jds@math.upenn.edu
>
> 146 Woodland Dr
> Lansdale PA 19446 (215)822-6707
>
>
>
> Jim Stasheff jds@math.unc.edu
> Math-UNC (919)-962-9607
> Chapel Hill NC FAX:(919)-962-2568
> 27599-3250
>
>
Prof R. Brown, School of Mathematics,
University of Wales, Bangor
Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom
Tel. direct:+44 1248 382474|office: 382475
fax: +44 1248 383663
World Wide Web:
home page: http://www.bangor.ac.uk/~mas010/
New article: Higher dimensional group theory
Symbolic Sculpture and Mathematics:
http://www.bangor.ac.uk/SculMath/
Mathematics and Knots:
http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm