24 KiB
Overview¶
In this overview we try to show, without entering in detailed explanations, various possibilities of $\texttt{Kenzo}$ in algebraic topology.
Load the $\texttt{Kenzo}$ software as follows:
(ql:quickload "kenzo")
The main functionality of the $\texttt{Kenzo}$ software is defined package in the $\texttt{cat}$ Lisp package. Initialize the $\texttt{cat}$ package as follows:
(cat:cat-init)
Print the $\texttt{Kenzo}$ version as follows:
(cat:kenzo-version)
Let us begin by the space $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}, 3)$ described as a simplicial set having only three non--degenerate simplices, namely in dimension $0$, $3$ and $4$. In the representation created by the software, the $0$--simplex (base point), 15 the $3$--simplex and the $4$--simplex are respectively labelled $\texttt{"*"}$, $\texttt{M3}$ and $\texttt{N4}$. Two faces of the $4$-simplex $\texttt{N4}$ are identified with the $3$-simplex $\texttt{M3}$, the others being contracted on the base point. To create the simplicial set one types simply:
(defvar m23 (cat:moore 2 3))
The system answers that a $\texttt{Kenzo}$ object has been created, with number $1$ and type $\texttt{SIMPLICIAL SET}$. This object may be referenced by the symbol $\texttt{m23}$.
m23
We may compute the homology groups of this space, using the underlying chain complex induced by the simplicial set description. Here we compute the $H_i$ from $0$ to $4$ included. When in the answer the component part is void, it means that the corresponding homology group is null.
(cat:homology m23 0 5)
As $\texttt{m23}$ is a simplicial set, it is possible to create the cartesian product $\texttt{m23}\times\texttt{m23}$ by the function $\texttt{crts-prdc}$. This is a new simplicial set.
(defvar m23xm23 (cat:crts-prdc m23 m23))
m23xm23
Being a simplicial set, $\texttt{m23xm23}$ is also a chain complex object and we may for instance ask for the basis in dimension $6$
(cat:basis m23xm23 6)
(length (cat:basis m23xm23 6))
The number of elements of the basis is quite large (230). The user will note that the basis elements are formed by cartesian products of degenerated simplices. In the list, an element like $\texttt{<CrPr 1-0 N4 5-3-2 M3>}$ means $\eta_1\eta_0\mathrm{N4}\times\eta_5\eta_3\eta_2\mathrm{M3}$.
We may construct also the tensor product $\mathrm{m23}\otimes\mathrm{m23}$ from the underlying chain complex of the simplicial set $\texttt{m23}$. This tensor product is a new chain complex and we see that the basis in dimension $6$ has only one element:
(defvar t2m23 (cat:tnsr-prdc m23 m23))
t2m23
(cat:basis t2m23 6)
The Eilenberg-Zilber theorem is used to compute the homology groups of the cartesian product space: as chain complexes, $\texttt{m23xm23}$ and $\texttt{t2m23}$ have the same homology groups, but the computations using the tensor product are considerably faster.
(cat:homology m23xm23 0 8)
Let us consider now the space $\mathrm{K}(\mathbb{Z}, 1)$. This is an Abelian simplicial group created in $\texttt{Kenzo}$ by the function $\texttt{k-z}$. In this simplicial group, a simplex in dimension $n$
is mathematically represented by a sequence of integers, known as a bar object: $$ [a_1 \mid a_2 \mid \ldots \mid a_n].$$
In $\texttt{Kenzo}$, a non-degenerate simplex of $\mathrm{K}(\mathbb{Z}, 1)$ in dimension $n$ will be simply a list of $n$ non-null integers, for instance: $\texttt{(2 3 4 5)}$. In dimension $0$, the only
simplex is $\texttt{NIL}$ (the base point).
(defvar kz1 (cat:k-z 1))
kz1
But this object is also a coalgebra and an algebra, and we may see the effect of the respective induced coproduct and product:
(cat:cprd kz1 4 '(2 3 4 5))
(cat:aprd kz1 6 (cat:tnpr 2 '(1 2) 4 '(3 4 5 6)))
The printed results are the printed representation of combinations, i.e., integer linear combinations of generators resulting from the application of the morphisms. The degree of the combination is indicated by the information: $\texttt{CMBN n}$.
In the same way, we may create the Abelian simplicial groups $\mathrm{K}(\mathbb{Z}/2\mathbb{Z}, n)$:
(defvar k-z2-2 (cat:k-z2 2))
k-z2-2
(cat:homology k-z2-2 4)
Let us play now with the sphere $S^3$ and its loop spaces. $S^3$ and $\Omega^2 S^3$ are created by respective calls to the functions $\texttt{sphere}$ and $\texttt{loop-space}$. Then we compute the $H_4$ and $H_5$ of $\Omega^2 S^3$:
(defvar s3 (cat:sphere 3))
s3
(defvar o2s3 (cat:loop-space s3 2))
o2s3
(cat:homology o2s3 4 6)
Let us take now the first loop space $\Omega^1 S^3$
(defvar os3 (cat:loop-space s3))
os3
In the following instruction, we locate in the symbol $\texttt{L1}$ the canonical generator of $\pi_2 (\Omega^1S^3)$, that is the $2$--simplex coming from the original sphere. In fact, the object created by the command $\texttt{(loop3 0 's3 1)}$ is the "word" $S3^1$ belonging to the Kan simplicial version $G(S^3)$ (a simplicial group) of the loop space $\Omega S^3$.
(defvar L1 (cat:loop3 0 's3 1))
L1
Let us consider also the $2$--degeneracy of the base point of the loop space. In the printed result, the user will recognize the degeneracy $\eta_1\eta_0$ of the null loop, base point of $\Omega^1 S^3$:
(defvar null-simp (cat:absm 3 cat:+null-loop+))
null-simp
We may build now a new space by pasting a disk $D3$ as indicated by the following call. It means that we "paste" to the space $\texttt{os3}$ a $3$--simplex named $\texttt{D3}$, the attaching map being described by the list of its faces in dimension $2$.
(defvar dos3 (cat:disk-pasting os3 3 '<D3> (list L1 null-simp L1 null-simp)))
dos3
Let us compute a few homology groups of the new space $\texttt{dos3}$:
(cat:homology dos3 2 4)
But more interesting, let us build the loop space of the object $\texttt{dos3}$ and let us compute the homology in dimension $5$:
(defvar odos3 (cat:loop-space dos3))
odos3
(cat:homology odos3 5)
Let us continue with the Kan theory. First, we check that $S^3$ is not of type Kan and that $\Omega S^3$ is indeed of type Kan and a non--Abelian simplicial group.
(typep s3 'cat:kan)
(typep os3 'cat:kan)
(typep os3 'cat:simplicial-group)
(typep os3 'cat:ab-simplicial-group)
Let us create the word $L2=(S3)^2$, i.e an object belonging to $\Omega S^3$ and let us apply the product of the underlying algebra upon $L2 \otimes L2$:
(defvar L2 (cat:loop3 0 's3 2))
L2
(defvar square (cat:aprd os3 4 (cat:tnpr 2 L2 2 L2)))
square
We see that the result is a linear combination of words composed from degeneracies of $\texttt{L2}$. The following instruction selects the generator part of the second element of the previous combination.
(defvar L4 (cat:gnrt (second (cat:cmbn-list square))))
L4
Let us use the lisp function $\texttt{mapcar}$ (one among the various iteration functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object $\texttt{L4}$, this list is a "Kan hat".
(defvar hat (mapcar #'(lambda (i) (cat:face os3 i 4 l4)) '(1 2 3 4)))
hat
The function $\texttt{kfll}$ tries to find a filling of this "Kan hat", and we see that the face $2$ of the resulting simplex (which is very different from $\texttt{L4}$) is the same as the face $2$ of $\texttt{L4}$.
(defvar kan-simplex (cat:kfll os3 0 4 hat))
kan-simplex
(cat:face os3 2 4 kan-simplex)
(second hat)
Let ${\cal G}$ be a simplicial group $0$--reduced. $\Omega S^3$ is such a group. The software $\texttt{Kenzo}$ allows the construction of the universal bundle $\bar{\cal W}{\cal G}$, i.e. the classifying space of ${\cal G}$. In our case, as $\Omega S^3$ in non--Abelian, the result is not a simplicial group but only a simplicial set. We verify that the $H_4$ is null.
(defvar cls-os3 (cat:classifying-space os3))
cls-os3
(typep cls-os3 'cat:simplicial-group)
(cat:homology cls-os3 4)
Let us end this short overview with an example of computation of homotopy groups. The method used in $\texttt{Kenzo}$ is the Whitehead tower. In this current version only the case where the first non--null homology group (in non--null dimension) is $\mathbb{Z}$ or $\mathbb{Z}/{2\mathbb{Z}}$ can be processed; however if this homology group is a direct sum of several copies of $\mathbb{Z}$ or $\mathbb{Z}/{2 \mathbb{Z}}$, then the corresponding stage of the Whitehead tower may also be constructed step by step.
We take again $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}, 3)$ whose $H_3$ is $\mathbb{Z}/2\mathbb{Z}$. First the fundamental cohomology class is constructed:
(defvar ch3 (cat:chml-clss m23 3))
ch3
Then the function $\texttt{z2-whitehead}$ is called to build a fibration over the simplicial set $\texttt{m23}$ canonically associated to the cohomology class $\texttt{ch3}$.
(defvar f3 (cat:z2-whitehead m23 ch3))
f3
Then the total space of the fibration is built:
(defvar x4 (cat:fibration-total f3))
x4
The $H_4$ of this total space is the $\pi_4$ of $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:
(cat:homology x4 3 5)
We may now iterate the process, to compute the $\pi_5$ of $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:
(defvar ch4 (cat:chml-clss x4 4))
ch4
(defvar f4 (cat:z2-whitehead x4 ch4))
f4
(defvar x5 (cat:fibration-total f4))
x5
(cat:homology x5 4 6)
So $\pi_5(\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}))$ is $\mathbb{Z}/4\mathbb{Z}$.