From: sergerar@isis.imag.fr (Francis Sergeraert) Subject: Calculation of higher homotopy groups. Newsgroups: sci.math.research Date: Mon, 24 Apr 1995 08:15:43 GMT I must add to the recent discussion about calculation of homotopy groups various interesting results recently obtained. Rolf Scho"n published (Memoirs AMS, #451, 1991) a general solution of computability problems in (simply connected) algebraic topology. His solution is a tricky, well organized, systematization of Ed Brown's work on homotopy groups. Rolf Scho"n himself declares not to be interested by concrete implementation. It seems extremely hard: you need know advanced functional programming to implement complicated inductive systems of chain complexes, simplicial sets and so on. This essential difficulty led some referees not to hesitate to declare Scho"n's work incomprehensible, but in fact it is right. I also published a little later a general solution of the same problem (Adv. in Math., 1994, vol. 104, pp 1-29). It is much closer to traditional algebraic topology, certainly the optimal solution from this point of view, but needs still much more sophisticated functional programming techniques. The situation with the referees is then quite amusing. If they have some programming knowledge, they are unable to understand how it could be possible to actually program so complex algorithms, even from a theoretical point of view. Considering standard current programming knowledge, this is normal. A recent referreing work led by B. Thurston gave such a result. But for the people not at all aware of actual programming, the proposed solution is so close to traditional algebraic topology that they think there is no theorem at all. The main examples of this sort of comments are due to Jean-Pierre Serre and Peter May. To overcome both obstacles, I undertook a first actual programming work based on my ideas. It led to the EAT-program (Effective Algebraic Topology) available under anonymous ftp at imag.fr:~ftp/pub/EAT. You can find there a program computing homology groups of iterated loop spaces. Because of unavoidable complexity (cf. Hovey's posting above), please be reasonable about what is possible with such a program. Nevertheless, this program succeeded to compute some so far unreachable homology groups. Typical example : H_* (\Omega (\Omega S^3 \cup_2 D^3)) = ? where D^3 is glued to Omega S^3 by a degree two map S^2 -> \Omega~S^3. Our program computed these groups up to *=7. Try to do it by hand. From several points of view our program is quite primitive and could be significanly improved. A student of mine recently wrote down the entire organigram allowing one to compute higher homotopy groups according to this method, of course in the simply connected case. Please E-mail for preprints. A third quite interesting solution of the same problem is due to Smirnov and Justin Smith, around the general operad techniques, originally due to Peter May. An operad structure installed on a chain complex is a relatively complicated structure which in some cases (symmetric group operads) contains a homotopy type. You get then again a general solution of the computability problem in simply connected algebraic topology which this time does not need functional programming, but which is extremely complicated and not yet actually implemented. See Memoirs AMS #524, 1994, for Smith solution and other references. The actual relationship between the three solutions is a major problem. ============================================================================== From: sergerar@isis.imag.fr (Francis Sergeraert) Newsgroups: sci.math.research Subject: Re: Calculation of higher homotopy groups. Date: 2 May 1995 08:32:52 GMT In article <3nii3o\$pt0@lyra.csx.cam.ac.uk>, John Baez writes: |> I'm curious about what programming language you used to write |> your own program. I was just last week in Bangor, Wales, |> where some people working with Ronnie Brown are writing programs |> in Axiom (a language apparently developed, or half-developed, |> at IBM, and then abandoned) to compute homotopy groups using |> Brown's crossed complex ideas. They are not trying to handle |> *general* homotopy types (or general simply-connected ones), but |> they *are* taking advantage of Axiom's ability to handle abstract |> data types like simplicial sets, crossed complexes, etc.. In our case we _must_ use Common Lisp which in fact is (or at least was) the "Assembler Language" under Axiom. Because the final calculations are frequently rather lengthy (several machine days) the concrete time computing efficiency is an important question. Using a general purpose intermediary language like Axiom is catastrophic from this point of view. The data handled by our programs are extremely complicated and we must be very careful about their implementation and handling if a computing time of two days is preferred to two years. Furthermore Axiom is an old-fashioned (Pascal-like style) language : I've seen the first demonstration twenty years ago (it was then called Scratchpad). The ANSI definition of Common Lisp was finished off last year ; of course Axiom will never have one (?). The assembler-like style of Common-Lisp programs could frighten the developper but in fact Common Lisp allows him to easily define his own high-level language. So that we can directly handle with our program chain complexes, morphisms between them, simplicial sets, cobar construction, loop space construction and so on. The implementation of the so-called homological perturbation lemma (more powerful than spectral sequences) is so short (25 lines including four type declaration lines) in our implementation that a recent anonymous (but easily identifiable) referee did not want to believe it is possible to compile it; yes it is. From a programming point of view, the nice challenge is to use CLOS (Common Lisp Object System), a part of the ANSI definition of Common Lisp, to implement our programs. It is quite easy to rewrite Axiom in CLOS (is it done ?), precisely to define Axiom as an intermediary language betwwen CLOS and the user. Normally the _language_ Axiom should not exist ; the good solution is to have a package of CLOS-written Lisp functions implementing Axiom. In this way you could furthermore use the much more precise and efficient typing mechanism of Common Lisp, the so elegant and powerful macro-generation process of Common Lisp, and so on. The language nature of Axiom is a severe handicap. ============================================================================== 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 ============================================================================== [See also Rolf Schon, "Effective Algebraic Topology", in Memoirs of the AMS, 1991 --djr]