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.

• GNU-Clisp for versions Clisp_2.n for n > 37 (does not work with Clisp-2.31: bug in the Lisp function change-class). Clisp is a GNU-free program, a little slow, but this is not very important in most applications.
• Any ACL-n (Franz Allegro-Common-Lisp) for n > 4, in particular with the free ACL81_express; this last version is quite convenient, about two times faster than Clisp.
• LispWorks-5. Still faster: three times faster than Clisp, but the free version is terribly heap-size limited, for example the π₆S³ = Z₁₂ is not reachable with this free version.

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.

• 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 A₄ be the 4-th alternating group. Then π₄(ΣK(A₄,1)) = Z/4.

       The elementary method used by the Kenzo program, known as the Whitehead tower, produces a different result, namely π₄(ΣK(A₄,1)) = 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 PR (resp. PⁿR) 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

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.

• Unix tar package.

• Windows zip 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.