From: Francis Sergeraert
Newsgroups: sci.math.research
Subject: Effective Algebraic Topology Program.
Date: 27 Nov 1997 09:06:01 +0100
The EAT-program (EAT = Effective Algebraic Topology) has been
written by Julio Rubio and myself in 1989-90 in order to concretely
implement the ideas of my paper (Adv in Math, 1994). It is a 5000
lines Common Lisp program which has been demonstrated in many
places.
The program and a *complete* user-guide (240 pp), due to Yvon Siret,
are now available at the ftp point:
"fourier.ujf-grenoble.fr/~ftp/pub/EAT"
The program obtains already non-trivial results with freeware
versions of Common Lisp, such as the low-graded free Allegro-CL for
Windows 95 (www.franz.com), the free Allegro-CL for Linux at the same
address, the GCL (= GNU-Common-Lisp, ex-AKCL, an element of the
GNU-Linux galaxy). For more significant results, a professional
Common-Lisp is necessary. With the actual Allegro-CL for Windows-95
(~$600), several homology groups of iterated loop spaces, unreachable
otherwise, can be computed. Much more powerful (and more expensive
too...) Common Lisp implementations are also available.
The first point where classical Algebraic Topology does *not* give
an algorithm computing some homology group is the case of iterated
loop spaces. It is the Adams' problem: his famous Cobar construction
cannot be iterated without *new* further tools. Four solutions are now
theoretically available to iterate the Cobar construction:
1) V.A. Smirnov.
On the chain complex of an iterated loop space.
Mathematics of the USSR, Izvestiya, 1990, vol. 35, pp 445-455.
2) Rolf Sch"on.
Effective algebraic topology
Memoirs of the American Mathematical Society, 1991, vol. 451.
3) Justin R. Smith.
Iterating the cobar construction.
Memoirs of the American Mathematical Society, 1994, vol. 524.
4) Julio Rubio + FS.
Constructive Algebraic Topology.
www-fourier.ujf-grenoble.fr/~sergerar
The solutions 1), 2) and 3), quite interesting, have not yet lead to
concrete programming work. Several versions of the paper 4) have been
refused by referees, one time because the paper was considered as
trivial, the other times because considered as impossible to be
understood; see www-fourier.ujf-grenoble.fr/~sergerar for details.
In the last refereeing work, when the paper was submitted to
Discrete *Computational* Geometry (DGC) through the redactor Bill
Thurston, the paper was rejected with in particular this appreciation:
"The algorithmic claim of the paper is a joke". In the documentation
now available, our solution for the iterated Cobar construction is in
particular carefully described, following with a high level of details
what happens on the computer, exactly following also what was
explained in the DGC paper.
A new version of the program is in progress, using in particular the
CLOS organization (CLOS = Common Lisp Object System). It should be
available in 1998. The question is now to use the same methods to
reach the first homotopy groups of a *random* simplicial set.
Francis Sergeraert
E-mail address= Francis.Sergeraert@ujf-grenoble.fr
Web-site= www-fourier.ujf-grenoble.fr/~sergerar
==============================================================================
Newsgroup: sci.math.symbolic
From: Francis Sergeraert
Subject: Machine computations of Homotopy Groups.
Date: Tue Dec 29 10:54:10 CST 1998
The Kenzo computer program is now finished. The aim was mainly to
implement our effective versions of the Serre and Eilenberg-Moore
spectral sequences. This allows us to construct the first stages of
the Postnikov and Whitehead towers, and to compute the first homotopy
groups of an *arbitrary* simplicial set with effective homology.
More explanations at:
http://www-fourier.ujf-grenoble.fr/~sergerar/Kenzo