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
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
(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:
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.
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.
Examples of recent 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 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
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
by Fred Cohen and Ran Levi, who go much further with other related
- 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:
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
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.
Take the stunted real projective space P4 =
Construct the loop space OP4 = Ω(P4). It is easy to prove the homotopy
group π3(OP4) = Z.
Attaching a 4-disk e4 to OP4 by a map S3 → OP4 of
degree 4 makes sense and products a new space DOP4.
Construct the loop space ODOP4 = Ω(DOP4). It is easy to proof
π2(ODOP4) = Z/4Z.
Attaching a 3-disk e3 to ODOP4 by a map S2 → ODOP4
of degree 2 makes sense and products a new space X = DODOP4.
Construct the loop space OX = ODODOP4 = Ω(DODOP4).
Exercise: Compute the first homology
groups of OX = ODODOP4.
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
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
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!
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.