Subject: Re: two questions
From: Mikael Johansson <mik@math.su.se>
Date: Wed, 27 Jun 2007 15:44:09 +0200 (CEST)

> Subject: Question on cubical cohomology reference
> From: Kevin Iga <kiga@pepperdine.edu>
> Date: Tue, 26 Jun 2007 22:26:23 -0700
>
> I am writing a paper where cubical cohomology (that is, the analog of 
simplicial cohomology but using n-dimensional cubes instead of simplices) 
is the correct language and tool to solve some problems.  Though I 
reconstructed all I need for my purposes, I would like to give proper 
credit.  Does anyone know who first developed cubical cohomology and/or 
what papers are the seminal works on the subject?  I know I am not the 
first to come up with it.
>
> Kevin Iga
> kiga@pepperdine.edu
>

One of the first treatments I saw was the book by Kaczynski, Mischaikow 
and Mrozek, title "Computational Homology" and published by Springer.

This probably isn't in any way original, but it has an extensive 
literature list.

-- 
Mikael Johansson         |  To see the world in a grain of sand
mik@math.su.se           |   And heaven in a wild flower
                         |  To hold infinity in the palm of your hand
                         |   And eternity for an hour
__________________________________________________________________

Subject: Cubical cohomology
From: "Ronnie Brown" <ronnie.profbrown@btinternet.com>
Date: Wed, 27 Jun 2007 15:38:22 +0100

I expect lots can give similar references to the next two.

Federer, \emph{Lectures in algebraic topology}, Lecture notes, Brown 
University, Providence, RI, 1962.

   AUTHOR = {Massey, W. S.},
    TITLE = {Singular homology theory},
   SERIES = {Graduate Texts in Mathematics},
   VOLUME = {70},
PUBLISHER = {Springer-Verlag},
  ADDRESS = {New York},
     YEAR = {1980}

But the first is probably not the earliest. Dan Kan's first papers on 
combinatorial homotopy were cubical.

My work with Higgins used cubical theory in an essential way, but enhanced 
by the extra structure of what we called `connections', a kind of extra 
degeneracy. For a survey of the use, see

132. R.Brown `Crossed complexes and homotopy groupoids as non commutative 
tools for higher dimensional local-to-global problems', Proceedings
of the Fields Institute Workshop on Categorical Structures for
Descent and Galois Theory, Hopf Algebras and Semiabelian Categories,
September 23-28, 2002, Fields Institute Communications 43 (2004) 101-130.
math.AT/0212274

Such cubical complexes with connections  are also used in concurrency 
theory (e.g. work of E. Goubault and of P. Gaucher).

A tricky point is to construct a cubical higher homotopy groupoid of a 
filtered space, and prove its major properties, which then allow 
convenient links between homotopy and homology, e.g. generalising the 
Relative Hurewicz Theorem.

I am thus interested in finding further uses of cubical methods.

Ronnie Brown
www.bangor.ac.uk/r.brown
________________________________________________________________________

Subject: Re: two questions
From: Robert Bruner <rrb@math.wayne.edu>
Date: Wed, 27 Jun 2007 12:16:01 -0400 (EDT)

Re cubical homology:  it is worked out in detail in Hilton and Wylie,
and I would expect proper attributions could be found there as well.
My copy is not at hand, so I cannot verify this.

Bob
___________________________________________________________________________ 

Subject: Cubical cohomology
From: Andrew Ranicki <andrew_ranicki@yahoo.co.uk>
Date: Wed, 27 Jun 2007 18:16:00 +0100

Serre's "Homologie singuliere des espaces fibres" (Annals of Maths. 54, 
425-505 (1951)) is the canonical reference for cubical homology and 
cohomology. It appeared only 7 years after Eilenberg's singular homology, 
and this is the first (and still the best!) reference.
I expect many others will point to this reference!

Andrew Ranicki