News of November 2011.

     A new version of the Kenzo program is in preparation, using the powerful notion of Discrete Vector Field.

     See this Arxiv preprint to understand the root of this technology, new in this area. Observe also the application to classifying spaces and loop spaces is not yet written down in this preprint. We preferred first concretely implement this technology inside the Kenzo program to better estimate which can be hoped, and also have a better understanding. The vector field technology is now implemented for the classifying spaces, giving excellent results.

     For example the computation of π5(Ω(S3) ∪2 D3) (= ?) is now obtained in three minutes instead of one hour on an ordinary laptop. Thanks to the discrete vector field computing very efficiently the Eilenberg-MacLane reduction C(K(π,n)) ⇒ A(π,n). Proving also simultaneously and easily a very old conjecture of Eilenberg and MacLane about this reduction. See Annals of Mathematics, On the groups H(π,n), I. 1953, vol.58, pp.55-106, Section 20.

     For the loop spaces, the situation is much more complex; the appropriate discrete vector field is now understood, which has a terribly complex and fascinating structure.

Overview.

The Kenzo program is the last version (16000 Lisp lines, July 1998) of the CAT (= Constructive Algebraic Topology) computer program. Kenzo is also the name of my beloved cat. The Kenzo program is a joint work with Xavier Dousson. The previous version EAT (May 1990) was a joint work with Julio Rubio.The Kenzo documentation was entirely written by Yvon Siret.

An updated version (1-1-7, October 11, 2008) of the program works with:

The Kenzo program is significantly more powerful than EAT, from several points of view.

On one hand, for the computations which could be done with the EAT program, the computing times are divided by a factor generally between 10 and 100. The reasons are multiple and it is not obvious to decide what the most important are. Some are strictly technical; for example the numerous multi-degeneracy operators are now coded with a unique integer, using an amusing binary trick: various tests show much progress has been obtained in this way. Other reasons are strictly mathematical; for example another choice for the Eilenberg-Zilber homotopy operator leads in the Kenzo program to Szczarba's universal twisting cochain; in the EAT program we used Shih's universal twisting cochain; experience shows that Szczarba's cochain is considerably more efficient than Shih's one. It is a major mathematical problem to understand why a so enormous difference exists between both natural twisting cochains.

On the other hand the scope of the Kenzo program is much larger. The EAT program was mainly devoted to the homology of iterated loop spaces. This field is covered by Kenzo, but Kenzo contains also an effective version of the Serre spectral sequence, and an effective version of the second Eilenberg-Moore spectral sequence, the one allowing to compute the homology of a classifying space, when the homology of the initial simplicial group is given. This allows us in particular to construct the first stages of the Whitehead and Postnikov towers, and to compute some homotopy groups.


Examples of recent results obtained thanks to the Kenzo program.

Other examples of results reachable by Kenzo:


Kenzo extensions.

A Kenzo extension deserves to be signalled: written by Ana Romero, it gives a complete description of the Serre and Eilenberg-Moore spectral sequences when versions with homology effective are known for the initial spaces. See Ana Romero's paper, now published in JSC.

Ana Romero also wrote various new Kenzo-modules devoted to some methods allowing her to compute the effective homology of some groups and so to obtain for example the first homotopy groups πkSBG for some simple non-commutative groups G. Work in progress.

You can be interested by this small Kenzo-demonstration file.

See also the Barcelona demonstration given in the 3rd European Congress of Mathematics.


The detailed Kenzo documentation (340 p.) was written by Yvon Siret in 1998-9. Yvon Siret was not a topologist, he was "only" (?!) a (very good) computer scientist, who learned Algebraic Topology when writing this document. His advices were also often crucial when writing down the source code. Many thanks to him!


Updated Kenzo-Source (Version 1-1-7, October 11, 2008)


EAT (= Effective Algebraic Topology), the previous program.

Before the Kenzo program (1998), the EAT program was written in 1990 by Julio Rubio and FS. It was the first program ever written implementing spectral sequences, in fact only some particular cases of the Eilenberg-Moore spectral sequence. The goal was the computation of the first homology groups of some loop spaces, for which no algorithm was previously concretely available. An implementation rather primitive, just designed to illustrate our methods of Effective Homology by a concrete experimental program.

The EAT program is also studied by logicians and computer scientists. Those possibly interested can download the EAT-program and its documentation.