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