\begin{center}
\line(1,0){450}
\end{center}

The *Kenzo* program implements the general ideas of the second author about
/Effective Homology/[fn:1], 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
Guggenheim[fn:2] 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.

\begin{center}
\line(1,0){450}
\end{center}

[fn:1] *Francis Sergeraert.* /The computability problem in algebraic topology./
Advances in Mathematics, 1994, vol. 104, pp 1-29.

[fn: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.