Subject: Re: clarification on question
From: Mikael Johansson
Date: Thu, 28 Jun 2007 16:36:17 +0200 (CEST)
> Subject: Still about cubical cohomology reference
> From: Kevin Iga
> Date: Thu, 28 Jun 2007 00:32:01 -0700
>
> Thank you for all the people who responded. I am looking up some of
these references now, but as I have gone through some of them, I should
clarify what I am looking for. For instance, I checked out Serre's thesis
but unfortunately it's not what I need. His cubical cohomology is a
variant of SINGULAR cohomology, not SIMPLICIAL cohomology. For Serre,
chains are generated by maps from a standard cube into the space (a sort
of singular cubical homology results). What I need is a notion of a
cubical complex (by analogy to a simplicial complex), where the chains are
generated by the cells of the complex. This is more combinatorial, and
the chain groups are finite dimensional. In my case, I have a polytope
made up of cubes and I am actually working on the cochain level explicitly
(this turns out to classify certain aspects of representations of a
certain Lie superalgebra).
>
Since this is the case, I will, again urge you to look closely at the
Computational Homology book - it deals with combinatorial complexes built
out of cubes with corners on integer coordinates. This might be more
restricted than what you want, but at least it sounds very close.
Mikael Johansson
_______________________________________________________________________________
Subject: Re: clarification on question
From: Brayton Gray
Date: Thu, 28 Jun 2007 11:25:52 -0500
Regarding the homology of cubical complexes:
In my 1975 book there are characteristic axioms for the chain complex of a
cellular complex. From these axioms the simplicial algorithm is easily
obtained. There are also some cubical examples and one could easily derive
an algorithm for cubical complexes in this way.
Brayton Gray
___________________________________________________________________________
Subject:Re: clarification on question
From:jim stasheff
Date:Thu, 28 Jun 2007 13:32:33 -0400
> Subject: Still about cubical cohomology reference
> From: Kevin Iga
> Date: Thu, 28 Jun 2007 00:32:01 -0700
>
> Thank you for all the people who responded. I am looking up some of
these references now, but as I have gone through some of them, I should
clarify what I am looking for. For instance, I checked out Serre's thesis
but unfortunately it's not what I need. His cubical cohomology is a
variant of SINGULAR cohomology, not SIMPLICIAL cohomology. For Serre,
chains are generated by maps from a standard cube into the space (a sort
of singular cubical homology results). What I need is a notion of a
cubical complex (by analogy to a simplicial complex), where the chains are
generated by the cells of the complex. This is more combinatorial, and
the chain groups are finite dimensional. In my case, I have a polytope
made up of cubes and I am actually working on the cochain level explicitly
(this turns out to classify certain aspects of representations of a
certain Lie superalgebra).
Serre's formulas for the boundary apply as well for your situation. What
sort of results do you need beyond that?
jim