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.

• 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 H_i(\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 π_i(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 = \Omega^k (X) \cup₂ D^p, where X is a simplicial set beginning in dimension n=p+k-1 with a H_n = 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 H_i(\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 H_i(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
• [December 2020] The Kenzo program constructs and uses an extra garbage collector for the specific results computed by the program. Changing the parameters of this garbage collector dramatically improved the computing speed. For example the last result H₇(OX) is now obtained in 40 minutes.
• [July 2022] Using the new machine IFNODE3 of the Fourier Institute, the next homology group of the same space has been obtained:
  • H₈(OX) = (Z/2Z)²⁵³ + (Z/4Z)⁹ + Z/8Z + Z⁵
  Kenzo was first able to compute the boundary matrices between the dimensions 9-8-7 (about a week) but failed to Smith reduce the boundary matrix d₉ between the dimensions 9 and 8; this matrix is 1995x5796 with 24374 non-zero terms. This last step was done by Jose Divason (Logroņo) in 17 minutes using Pari via Sagemath on his laptop. Also done by Jean-Guillaume Dumas (Grenoble) in 5m48s using his own program Livebox. This failure of Kenzo led us to reconsider the algorithm of Smith reduction of Kenzo. A new improvement of the Kannan-Bachem algorithm was discovered allowing us to write a much simpler algorithm, using the same kernel for the Hermite reduction, but a totally different process for Smith reducing the obtained Hermite matrix, different and much simpler. The Smith reduction of d₉ is now obtained by Kenzo in 4m15s with IFNODE3.

Kenzo extensions.

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)

• Unix tgz package.

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

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