Subject: Re: conf and response
Date: Wed, 31 Oct 2001 07:12:09 -0500
From: jim stasheff <stasheff@email.unc.edu>
To: Don Davis <dmd1@lehigh.edu>, dubois <patricia@QCD.TH.U-PSUD.FR>,
     henneaux <henneaux@ulb.ac.be>

Oh, n-fold meaning d^n = 0

That topic has received a fair amount of attention in recent math
physics.
cf. Dubois-Violette and Henneaux:  math.QA/0110088

jim

Don Davis wrote:
>
> Subject: Re:
> Date: Mon, 29 Oct 2001 23:56:19 -0500
> From: "Fred Cohen" <cohf@math.rochester.edu>
>
> Can you post the following concerning the notice for the CINVESTAV
> CONFERENCE on your site ?
>
> Registration:
>
> Please send the following registration information to
> Miguel Xicotencatl at xico@Poincare.math.cinvestav.mx.
>
> Name:
> Affiliation:
> E-mail address:
>
> There is no registration fee.
>
> Further information: Additional information concerning housing,
> travel, and the program will be posted
> at http://www.math.cinvestav.mx/~xico/december.html.
>
> Thanks very much.
> __________________________________
>
> Subject: Re: two responses and conf
> Date: Tue, 30 Oct 2001 10:31:53 +0100 (CET)
> From: John Rognes <rognes@math.uio.no>
>
> > From: Clarence Wilkerson <wilker@math.purdue.edu>
> >
> > Re: "n-fold" chain complex
> >
> > This is in the older literature ( Hurewicz???).
>
> There are two papers about this by W. Mayer, titled "A new homology
> theory.  I, II" in the Annals of Maths (2) 43, (1942). 370--380,
> 594--605.
>
> My Master's student Erik Strand did something similar a couple of years
> ago, not knowing of Mayer's paper.  Let z be an n-th root of unity in a
> ring R.  For a simplicial abelian R-module M define a degree -1
> homomorphism by
>
>         d(x) = \sum_{i=0}^q z^i d_i(x)
>
> for x in degree i.  Here d_i(x) is the i-th face map.  Then the n-fold
> composite d^n = 0, so (M, d) is an "n-fold" chain complex.  The
> classical
> case is n=2, z=-1.
>
> If each geometric sum \sum_{i=0}^j z^i is a unit in R for 0 < j < n,
> then
> some of the generalized homology groups ker(d^i)/im(d^{n-i}) recover all
>
> the classical homology of M, while the other generalized groups are
> zero.
> (One can be completely precise.)  This includes the case Mayer
> considered,
> namely n = p a prime, R = Z/p and z = 1.
>
> If these sums are not units in R, more complicated things can happen.
>
> - John Rognes