Work in progress by Gerd Heber (HDF Group) to make Kenzo easily loadable from and compatible with any Common Lisp system. Already usable. If interested, see the corresponding Jupyter page, reachable by any GitHub user.

## Overview.

The Kenzo program implements the methods of Constructive Algebraic Topology. The funny corresponding acronym CAT gave the idea to name this program as my beloved cat.

The first version of this program, called EAT for Effective Algebraic Topology, was a joint work with Julio Rubio (1990). The EAT program was totally rewritten with Xavier Dousson in 1998, becoming Kenzo, on the one hand to include many technical improvements, in particular in memory management, improvements deduced from the EAT experience, and on the other hand to implement the Whitehead Tower and compute homotopy groups of arbitrary simply connected simplicial sets, the heart of Xavier's thesis. A detailed Kenzo documentation was simulatneously written by Yvon Siret.

The 1.1.8 version now proposed takes account of the new mathematical technology based on Discrete Vector Fields. Discovered by Robin Forman, it happens this method considerably improves the implementation of the Eilenberg-Zilber theorem, twisted or not, and also the computation of the effective homology of the Eilenberg-MacLane spaces, crucial when using the Postnikov or Whitehead towers.

Examples of results obtained thanks to the Kenzo program.
• In the paper:
Roman Mikhailov, Jie Wu
On homotopy groups of the suspended classifying spaces
Algebraic & Geometric Topology, 2010, vol.10, pp.565–625.
the authors state in Theorem 5.4:
Let $$A_4$$ be the 4-th alternating group. Then $$\pi_4(\Sigma K(A_4,1)) = \mathbf{Z}/4$$, with $$K(A_4,1)$$ being the corresponding Eilenberg-MacLane space, and $$\Sigma$$ the suspension functor.

The elementary method used by the Kenzo program, known as the Whitehead tower, produces a different result, namely $$\pi_4(\Sigma K(A_4,1)) = \mathbf{Z}/12$$. The authors of the quoted paper inadvertently forgot the 3-primary component. This Kenzo computation was done by Ana Romero, using extra-modules devoted to group resolutions written by herself.

• The Kenzo program is now used for an application in Neurology, for automatic counting of synapses in snapshots of neurons. See this webpage.

Other examples of results reachable by Kenzo:

• Let $$P\mathbf{R}$$ (resp. $$P^n\mathbf{R}$$) the infinite real projective space (resp. the $$n$$-dimensional real projective space). Then Kenzo has determined the groups Hi(\Omega2(PR/P2R);Z) for i < 8. This computation has been at the origin of an interesting paper by Vladimir Smirnov determining the whole Z2-homology of the iterated loop spaces of the truncated projective spaces (tex, dvi, ps: only the ps version contains the appendix but in a bad format because of TeX problems).
• The Kenzo program has also determined the homotopy groups πi(PR/P2R) for i < 8. These computations have motivated a nice work (dvi, ps) by Fred Cohen and Ran Levi, who go much further with other related spaces.
• The field where Kenzo seems at this time the most in advance is with the spaces whose TeX notation is Y = \Omegak (X) \cup2 Dp, where X is a simplicial set beginning in dimension n=p+k-1 with a Hn = Z; a p-cell is attached to a loop space of X by a map of degree 2. Kenzo can compute the first homology groups of the first loop spaces of Y and also its first homotopy groups. For example Hi(\Omega(\Omega(S3) \cup2 D3)) is computed by Kenzo for i < 10.

• The longest Kenzo computation. Consists in playing the following game:
1. Take the stunted real projective space P4 = P(R)/P3(R).
2. Construct the loop space OP4 = Ω(P4). It is easy to prove the homotopy group π3(OP4) = Z.
3. Attaching a 4-disk e4 to OP4 by a map S3 → OP4 of degree 4 makes sense and products a new space DOP4.
4. Construct the loop space ODOP4 = Ω(DOP4). It is easy to proof π2(ODOP4) = Z/4Z.
5. Attaching a 3-disk e3 to ODOP4 by a map S2 → ODOP4 of degree 2 makes sense and products a new space X = DODOP4.
6. Construct the loop space OX = ODODOP4 = Ω(DODOP4).
7. Exercise: Compute the first homology groups of OX = ODODOP4.

The Kenzo program spent almost exactly two months to compute Hi(OX) for i ≤ 7. The space OX is quite artificial, not so complicated but designed to accumulate some known difficulties: the space P4 is not a suspension; the influence of attaching a disk by a non-trivial attaching map before looping is a difficult subject, so far without algorithmic solution; the loop functor is applied three times.

The point is that most topologists think a spectral sequence is an algorithm computing the desired homology groups and this example is designed to convince them there is in fact some essential gap; they are invited to propose an algorithm computing these homology groups through the usual Eilenberg-Moore spectral sequence; even a "theoretical" algorithm would be enough, we do not think the exact value of these groups has much interest…

On the contrary, the methods of effective homology allow the user to design an algorithm computing the effective homology of a loop space when the effective homology of the initial space is given. And the ordinary homology is a by-product of effective homology. So that the recipe is: starting from the effective homology of the initial space, here P4, therefore trivial, compute the effective homology of the intermediate spaces, and when the final space is reached, you can deduce the ordinary homology groups.

Results:
• H0(OX) = Z.
• H1(OX) = Z/2Z.
• H2(OX) = (Z/2Z)2 + Z.
• H3(OX) = (Z/2Z)4 + Z/8Z.
• H4(OX) = (Z/2Z)10 + Z/4Z + Z2.
• H5(OX) = (Z/2Z)23 + Z/8Z + Z/16Z.
• H6(OX) = (Z/2Z)52 + (Z/4Z)3 + Z3
• H7(OX) = (Z/2Z)113 + Z/4Z + (Z/8Z)3 + Z/16Z + Z/32Z + Z

#### 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.

#### 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)

Previous various compressed archives (tgz, tar and zip) have been replaced by this unique 7zip version, usable under Linux and Windows as well, much more compact! In particular, the previous Kenzo-doc.pdf component, non-searchable, has been replaced by another equal (!) version available elsewhere, searchable. Thanks to Marek Kaluba for the notification.

## 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.