kenzo/doc/org/Foreword.org

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The Kenzo program implements the general ideas of the second author about Effective Homology¹, mainly around the Serre and Eilenberg-Moore spectral sequences. The first author (re-)discovered the importance of the Basic Perturbation Lemma in these questions, already noted by Victor Guggenheim² and this program directly implements and directly uses this "lemma" which should be called the Fundamental Theorem of Algebraic Topology. The first version of the program, called EAT, was written in 1989-90 by the first and the second authors. It has been demonstrated in several universities: France: Grenoble and Montpellier, Belgium: Louvain-la-Neuve, Italy: Genoa and Pisa, Sweden: Stockholm, Japan: Sapporo, Morioka, Urawa, Tokyo, Kyoto, Nara, Osaka and Hiroshima.

In this Kenzo version, numerous improvements have been integrated in comparison with EAT.

1. Standard CLOS (Common Lisp Object System) techniques.
2. Use of Zermelo theorem about ordered sets (!): any set can be well ordered and this remark gives important new ideas to improve execution speed.
3. Better memory management.
4. Use of Szczarba's formulas instead of Shih's for implementing the twisted Eilenberg-Zilber theorem.

5. New mathematical objects:

   • Serre spectral sequences.
   • Differential algebras.
   • Simplicial morphisms.
   • Kan simplicial sets.
   • Simplicial groups.
   • Fibrations.
   • Classifying spaces.

   In particular, using these tools, the first homotopy groups of arbitrary simplicial sets with effective homology, are now reachable.

The present documentation is a joint work between the second and third authors.

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¹

Francis Sergeraert. The computability problem in algebraic topology. Advances in Mathematics, 1994, vol. 104, pp 1-29.

²

V.K.A.M. Guggenheim. On a perturbation theory for the homology of the loop space. Journal of Pure and Applied Algebra, 1982, vol. 25, pp 197-205.