76 KiB
76 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:
In [1]:
(ql:quickload "kenzo")
Out[1]:
("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:
In [2]:
(cat:cat-init)
---done---
Out[2]:
NIL
Print the $\texttt{Kenzo}$ version as follows:
In [3]:
(cat:kenzo-version)
*** Kenzo-Version 1.1.7 ***
Out[3]:
NIL
Let us begin with 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:
In [4]:
(cat:def m23 (cat:moore 2 3))
Out[4]:
[K1 Simplicial-Set]
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 $\texttt{m23}$.
In [5]:
m23
Out[5]:
[K1 Simplicial-Set]
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.
In [6]:
(cat:homology m23 0 5)
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 32s. Computing the boundary of the generator 1/1 (dimension 0) : * End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 0. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2015-08-08, 10h 8m 32s. Computing boundary-matrix in dimension 1. Rank of the source-module : 0. Computing boundary-matrix in dimension 2. Rank of the source-module : 0. Homology in dimension 1 : ---done--- ;; Clock -> 2015-08-08, 10h 8m 32s. Computing boundary-matrix in dimension 2. Rank of the source-module : 0. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 32s. Computing the boundary of the generator 1/1 (dimension 3) : M3 End of computing. Homology in dimension 2 : ---done--- ;; Clock -> 2015-08-08, 10h 8m 32s. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 32s. Computing the boundary of the generator 1/1 (dimension 3) : M3 End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 32s. Computing the boundary of the generator 1/1 (dimension 4) : N4 End of computing. Homology in dimension 3 : Component Z/2Z ---done--- ;; Clock -> 2015-08-08, 10h 8m 32s. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 32s. Computing the boundary of the generator 1/1 (dimension 4) : N4 End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Homology in dimension 4 : ---done--- ;; Clock -> 2015-08-08, 10h 8m 32s.
Out[6]:
NIL
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.
In [7]:
(cat:def m23xm23 (cat:crts-prdc m23 m23))
Out[7]:
[K10 Simplicial-Set]
Being a simplicial set, $\texttt{m23xm23}$ is also a chain complex object and we may for instance ask for the basis in dimension $6$
In [8]:
(cat:basis m23xm23 6)
Out[8]:
(<CrPr 1-0 N4 3-2 N4> <CrPr 1-0 N4 4-2 N4> <CrPr 1-0 N4 4-3 N4> <CrPr 1-0 N4 4-3-2 M3> <CrPr 1-0 N4 5-2 N4> <CrPr 1-0 N4 5-3 N4> <CrPr 1-0 N4 5-3-2 M3> <CrPr 1-0 N4 5-4 N4> <CrPr 1-0 N4 5-4-2 M3> <CrPr 1-0 N4 5-4-3 M3> ...)
In [9]:
(length (cat:basis m23xm23 6))
Out[9]:
230
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:
In [10]:
(cat:def t2m23 (cat:tnsr-prdc m23 m23))
Out[10]:
[K3 Chain-Complex]
In [11]:
(cat:basis t2m23 6)
Out[11]:
(<TnPr M3 M3>)
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.
In [12]:
(cat:homology m23xm23 0 8)
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/1 (dimension 0) : <TnPr * *> End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 0. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2015-08-08, 10h 8m 49s. Computing boundary-matrix in dimension 1. Rank of the source-module : 0. Computing boundary-matrix in dimension 2. Rank of the source-module : 0. Homology in dimension 1 : ---done--- ;; Clock -> 2015-08-08, 10h 8m 49s. Computing boundary-matrix in dimension 2. Rank of the source-module : 0. Computing boundary-matrix in dimension 3. Rank of the source-module : 2. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/2 (dimension 3) : <TnPr * M3> End of computing. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 2/2 (dimension 3) : <TnPr M3 *> End of computing. Homology in dimension 2 : ---done--- ;; Clock -> 2015-08-08, 10h 8m 49s. Computing boundary-matrix in dimension 3. Rank of the source-module : 2. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/2 (dimension 3) : <TnPr * M3> End of computing. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 2/2 (dimension 3) : <TnPr M3 *> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 2. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/2 (dimension 4) : <TnPr * N4> End of computing. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 2/2 (dimension 4) : <TnPr N4 *> End of computing. Homology in dimension 3 : Component Z/2Z Component Z/2Z ---done--- ;; Clock -> 2015-08-08, 10h 8m 49s. Computing boundary-matrix in dimension 4. Rank of the source-module : 2. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/2 (dimension 4) : <TnPr * N4> End of computing. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 2/2 (dimension 4) : <TnPr N4 *> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Homology in dimension 4 : ---done--- ;; Clock -> 2015-08-08, 10h 8m 49s. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/1 (dimension 6) : <TnPr M3 M3> End of computing. Homology in dimension 5 : ---done--- ;; Clock -> 2015-08-08, 10h 8m 49s. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/1 (dimension 6) : <TnPr M3 M3> End of computing. Computing boundary-matrix in dimension 7. Rank of the source-module : 2. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/2 (dimension 7) : <TnPr M3 N4> End of computing. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 2/2 (dimension 7) : <TnPr N4 M3> End of computing. Homology in dimension 6 : Component Z/2Z ---done--- ;; Clock -> 2015-08-08, 10h 8m 49s. Computing boundary-matrix in dimension 7. Rank of the source-module : 2. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/2 (dimension 7) : <TnPr M3 N4> End of computing. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 2/2 (dimension 7) : <TnPr N4 M3> End of computing. Computing boundary-matrix in dimension 8. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 8m 49s. Computing the boundary of the generator 1/1 (dimension 8) : <TnPr N4 N4> End of computing. Homology in dimension 7 : Component Z/2Z ---done--- ;; Clock -> 2015-08-08, 10h 8m 49s.
Out[12]:
NIL
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).
In [13]:
(cat:def kz1 (cat:k-z 1))
Out[13]:
[K38 Abelian-Simplicial-Group]
But this object is also a coalgebra and an algebra, and we may see the effect of the respective induced coproduct and product:
In [14]:
(cat:cprd kz1 4 '(2 3 4 5))
Out[14]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr NIL (2 3 4 5)>>
<1 * <TnPr (2) (3 4 5)>>
<1 * <TnPr (2 3) (4 5)>>
<1 * <TnPr (2 3 4) (5)>>
<1 * <TnPr (2 3 4 5) NIL>>
------------------------------------------------------------------------------
In [15]:
(cat:aprd kz1 6 (cat:tnpr 2 '(1 2) 4 '(3 4 5 6)))
Out[15]:
----------------------------------------------------------------------{CMBN 6}
<1 * (1 2 3 4 5 6)>
<-1 * (1 3 2 4 5 6)>
<1 * (1 3 4 2 5 6)>
<-1 * (1 3 4 5 2 6)>
<1 * (1 3 4 5 6 2)>
<1 * (3 1 2 4 5 6)>
<-1 * (3 1 4 2 5 6)>
<1 * (3 1 4 5 2 6)>
<-1 * (3 1 4 5 6 2)>
<1 * (3 4 1 2 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 $\mathrm{K}(\mathbb{Z}/2\mathbb{Z}, n)$:
In [16]:
(cat:def k-z2-2 (cat:k-z2 2))
Out[16]:
[K64 Abelian-Simplicial-Group]
In [17]:
(cat:homology k-z2-2 4)
Computing boundary-matrix in dimension 4. Rank of the source-module : 2. ;; Clock -> 2015-08-08, 10h 9m 9s. Computing the boundary of the generator 1/2 (dimension 4) : <<Abar[4 3]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 9s. Computing the boundary of the generator 2/2 (dimension 4) : <<Abar[2 1][2 1]>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 3. ;; Clock -> 2015-08-08, 10h 9m 9s. Computing the boundary of the generator 1/3 (dimension 5) : <<Abar[5 4]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 9s. Computing the boundary of the generator 2/3 (dimension 5) : <<Abar[2 1][3 2]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 9s. Computing the boundary of the generator 3/3 (dimension 5) : <<Abar[3 2][2 1]>> End of computing. Homology in dimension 4 : Component Z/4Z ---done--- ;; Clock -> 2015-08-08, 10h 9m 9s.
Out[17]:
NIL
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 $\Omega^2 S^3$:
In [18]:
(cat:def s3 (cat:sphere 3))
Out[18]:
[K174 Simplicial-Set]
In [19]:
(cat:def o2s3 (cat:loop-space s3 2))
Out[19]:
[K191 Simplicial-Group]
In [20]:
(cat:homology o2s3 4 6)
Computing boundary-matrix in dimension 4. Rank of the source-module : 3. ;; Clock -> 2015-08-08, 10h 9m 18s. Computing the boundary of the generator 1/3 (dimension 4) : <<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 18s. Computing the boundary of the generator 2/3 (dimension 4) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 18s. Computing the boundary of the generator 3/3 (dimension 4) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 5. ;; Clock -> 2015-08-08, 10h 9m 18s. Computing the boundary of the generator 1/5 (dimension 5) : <<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 18s. Computing the boundary of the generator 2/5 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 3/5 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 4/5 (dimension 5) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 5/5 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. Homology in dimension 4 : Component Z/3Z Component Z/2Z ---done--- ;; Clock -> 2015-08-08, 10h 9m 19s. Computing boundary-matrix in dimension 5. Rank of the source-module : 5. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 1/5 (dimension 5) : <<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 2/5 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 3/5 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 4/5 (dimension 5) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 5/5 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 8. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 1/8 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][5 <<AlLp[2 S3][2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 19s. Computing the boundary of the generator 2/8 (dimension 6) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 20s. Computing the boundary of the generator 3/8 (dimension 6) : <<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 20s. Computing the boundary of the generator 4/8 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 20s. Computing the boundary of the generator 5/8 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 21s. Computing the boundary of the generator 6/8 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 21s. Computing the boundary of the generator 7/8 (dimension 6) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 21s. Computing the boundary of the generator 8/8 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. Homology in dimension 5 : Component Z/3Z Component Z/2Z ---done--- ;; Clock -> 2015-08-08, 10h 9m 22s.
Out[20]:
NIL
Let us take now the first loop space $\Omega^1 S^3$
In [21]:
(cat:def os3 (cat:loop-space s3))
Out[21]:
[K179 Simplicial-Group]
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 $\Omega S^3$.
In [22]:
(cat:def L1 (cat:loop3 0 's3 1))
Out[22]:
<<Loop[S3]>>
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 $\Omega^1 S^3$:
In [23]:
(cat:def null-simp (cat:absm 3 cat:+null-loop+))
Out[23]:
<AbSm 1-0 <<Loop>>>
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 $2$.
In [24]:
(cat:def dos3 (cat:disk-pasting os3 3 '<D3> (list L1 null-simp L1 null-simp)))
Out[24]:
[K435 Simplicial-Set]
Let us compute a few homology groups of the new space $\texttt{dos3}$:
In [25]:
(cat:homology dos3 2 4)
Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 9m 44s. Computing the boundary of the generator 1/1 (dimension 2) : <<AlLp[2 S3]>> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 9m 44s. Computing the boundary of the generator 1/1 (dimension 3) : <D3> End of computing. Homology in dimension 2 : Component Z/2Z ---done--- ;; Clock -> 2015-08-08, 10h 9m 44s. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 9m 44s. Computing the boundary of the generator 1/1 (dimension 3) : <D3> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 9m 44s. Computing the boundary of the generator 1/1 (dimension 4) : <<AlLp[2 S3][2 S3]>> End of computing. Homology in dimension 3 : ---done--- ;; Clock -> 2015-08-08, 10h 9m 44s.
Out[25]:
NIL
But more interesting, let us build the loop space of the object $\texttt{dos3}$ and let us compute the homology in dimension $5$:
In [26]:
(cat:def odos3 (cat:loop-space dos3))
Out[26]:
[K453 Simplicial-Group]
In [27]:
(cat:homology odos3 5)
Computing boundary-matrix in dimension 5. Rank of the source-module : 14. ;; Clock -> 2015-08-08, 10h 9m 51s. Computing the boundary of the generator 1/14 (dimension 5) : <<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 51s. Computing the boundary of the generator 2/14 (dimension 5) : <<AlLp[2 <D3>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 51s. Computing the boundary of the generator 3/14 (dimension 5) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 51s. Computing the boundary of the generator 4/14 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 5/14 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 6/14 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 7/14 (dimension 5) : <<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 8/14 (dimension 5) : <<AlLp[2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 9/14 (dimension 5) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 10/14 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 11/14 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 12/14 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 13/14 (dimension 5) : <<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 14/14 (dimension 5) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 26. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 1/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][5 <<AlLp[2 S3][2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 52s. Computing the boundary of the generator 2/26 (dimension 6) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 53s. Computing the boundary of the generator 3/26 (dimension 6) : <<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 53s. Computing the boundary of the generator 4/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 53s. Computing the boundary of the generator 5/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 53s. Computing the boundary of the generator 6/26 (dimension 6) : <<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 53s. Computing the boundary of the generator 7/26 (dimension 6) : <<AlLp[2 <D3>][2 <D3>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 53s. Computing the boundary of the generator 8/26 (dimension 6) : <<AlLp[2 <D3>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 53s. Computing the boundary of the generator 9/26 (dimension 6) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 53s. Computing the boundary of the generator 10/26 (dimension 6) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 54s. Computing the boundary of the generator 11/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 54s. Computing the boundary of the generator 12/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 54s. Computing the boundary of the generator 13/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 54s. Computing the boundary of the generator 14/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 54s. Computing the boundary of the generator 15/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 54s. Computing the boundary of the generator 16/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 55s. Computing the boundary of the generator 17/26 (dimension 6) : <<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 55s. Computing the boundary of the generator 18/26 (dimension 6) : <<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 55s. Computing the boundary of the generator 19/26 (dimension 6) : <<AlLp[2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 55s. Computing the boundary of the generator 20/26 (dimension 6) : <<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 55s. Computing the boundary of the generator 21/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 55s. Computing the boundary of the generator 22/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 56s. Computing the boundary of the generator 23/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 56s. Computing the boundary of the generator 24/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 56s. Computing the boundary of the generator 25/26 (dimension 6) : <<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. ;; Clock -> 2015-08-08, 10h 9m 56s. Computing the boundary of the generator 26/26 (dimension 6) : <<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> End of computing. Homology in dimension 5 : Component Z/2Z Component Z/2Z Component Z/2Z Component Z/2Z Component Z/2Z Component Z/2Z Component Z ---done--- ;; Clock -> 2015-08-08, 10h 9m 56s.
Out[27]:
NIL
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.
In [28]:
(typep s3 'cat:kan)
Out[28]:
NIL
In [29]:
(typep os3 'cat:kan)
Out[29]:
T
In [30]:
(typep os3 'cat:simplicial-group)
Out[30]:
T
In [31]:
(typep os3 'cat:ab-simplicial-group)
Out[31]:
NIL
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 $L2 \otimes L2$:
In [32]:
(cat:def L2 (cat:loop3 0 's3 2))
Out[32]:
<<Loop[S3\2]>>
In [33]:
(cat:def square (cat:aprd os3 4 (cat:tnpr 2 L2 2 L2)))
Out[33]:
----------------------------------------------------------------------{CMBN 4}
<1 * <<Loop[1-0 S3\2][3-2 S3\2]>>>
<-1 * <<Loop[2-0 S3\2][3-1 S3\2]>>>
<1 * <<Loop[2-1 S3\2][3-0 S3\2]>>>
<1 * <<Loop[3-0 S3\2][2-1 S3\2]>>>
<-1 * <<Loop[3-1 S3\2][2-0 S3\2]>>>
<1 * <<Loop[3-2 S3\2][1-0 S3\2]>>>
------------------------------------------------------------------------------
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.
In [34]:
(cat:def L4 (cat:gnrt (second (cat:cmbn-list square))))
Out[34]:
<<Loop[2-0 S3\2][3-1 S3\2]>>
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".
In [35]:
(cat:def hat (mapcar #'(lambda (i) (cat:face os3 i 4 l4)) '(1 2 3 4)))
Out[35]:
(<AbSm - <<Loop[1 S3\2][2 S3\2]>>> <AbSm - <<Loop[0 S3\2][2 S3\2]>>> <AbSm - <<Loop[0 S3\2][1 S3\2]>>> <AbSm 1 <<Loop[S3\2]>>>)
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 $\texttt{L4}$.
In [36]:
(cat:def kan-simplex (cat:kfll os3 0 4 hat))
Out[36]:
<AbSm - <<Loop[3-1 S3\2][2-1 S3\-2][2-0 S3\2][1-0 S3\-2][2-1 S3\2][3-1 S3\-2][1-0 S3\2][3-1 S3\2][3-0 S3\-2][1-0 S3\-2][3-0 S3\2][2-0 S3\-2][1-0 S3\2][3-0 S3\-2][2-0 S3\2][3-0 S3\2]>>>
In [37]:
(cat:face os3 2 4 kan-simplex)
Out[37]:
<AbSm - <<Loop[0 S3\2][2 S3\2]>>>
In [38]:
(second hat)
Out[38]:
<AbSm - <<Loop[0 S3\2][2 S3\2]>>>
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.
In [39]:
(cat:def cls-os3 (cat:classifying-space os3))
Out[39]:
[K598 Simplicial-Set]
In [40]:
(typep cls-os3 'cat:simplicial-group)
Out[40]:
NIL
In [41]:
(cat:homology cls-os3 4)
Computing boundary-matrix in dimension 4. Rank of the source-module : 0. Computing boundary-matrix in dimension 5. Rank of the source-module : 1. ;; Clock -> 2015-08-08, 10h 10m 43s. Computing the boundary of the generator 1/1 (dimension 5) : <<Abar[5 <<AlLp[2 S3][2 S3]>>]>> End of computing. Homology in dimension 4 : ---done--- ;; Clock -> 2015-08-08, 10h 10m 43s.
Out[41]:
NIL
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:
In [42]:
(cat:def ch3 (cat:chml-clss m23 3))
Out[42]:
[K729 Cohomology-Class on K1 of degree 3]
Then the function $\texttt{z2-whitehead}$ is called to build a fibration over the simplicial set $\texttt{m23}$ canonically associated to the cohomology class $\texttt{ch3}$.
In [43]:
(cat:def f3 (cat:z2-whitehead m23 ch3))
Out[43]:
[K730 Fibration K1 -> K64]
Then the total space of the fibration is built:
In [44]:
(cat:def x4 (cat:fibration-total f3))
Out[44]:
[K736 Simplicial-Set]
The $H_4$ of this total space is the $\pi_4$ of $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:
In [45]:
(cat:homology x4 3 5)
Computing boundary-matrix in dimension 3. Rank of the source-module : 2. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 1/2 (dimension 3) : <TnPr * <<Abar[3 2]>>> End of computing. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 2/2 (dimension 3) : <TnPr M3 <<Abar>>> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 3. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 1/3 (dimension 4) : <TnPr * <<Abar[4 3]>>> End of computing. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 2/3 (dimension 4) : <TnPr * <<Abar[2 1][2 1]>>> End of computing. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 3/3 (dimension 4) : <TnPr N4 <<Abar>>> End of computing. Homology in dimension 3 : ---done--- ;; Clock -> 2015-08-08, 10h 10m 55s. Computing boundary-matrix in dimension 4. Rank of the source-module : 3. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 1/3 (dimension 4) : <TnPr * <<Abar[4 3]>>> End of computing. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 2/3 (dimension 4) : <TnPr * <<Abar[2 1][2 1]>>> End of computing. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 3/3 (dimension 4) : <TnPr N4 <<Abar>>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 4. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 1/4 (dimension 5) : <TnPr * <<Abar[5 4]>>> End of computing. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 2/4 (dimension 5) : <TnPr * <<Abar[2 1][3 2]>>> End of computing. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 3/4 (dimension 5) : <TnPr * <<Abar[3 2][2 1]>>> End of computing. ;; Clock -> 2015-08-08, 10h 10m 55s. Computing the boundary of the generator 4/4 (dimension 5) : <TnPr M3 <<Abar[2 1]>>> End of computing. Homology in dimension 4 : Component Z/2Z ---done--- ;; Clock -> 2015-08-08, 10h 10m 55s.
Out[45]:
NIL
We may now iterate the process, to compute the $\pi_5$ of $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:
In [46]:
(cat:def ch4 (cat:chml-clss x4 4))
Out[46]:
[K817 Cohomology-Class on K802 of degree 4]
In [47]:
(cat:def f4 (cat:z2-whitehead x4 ch4))
Out[47]:
[K832 Fibration K736 -> K818]
In [48]:
(cat:def x5 (cat:fibration-total f4))
Out[48]:
[K838 Simplicial-Set]
In [49]:
(cat:homology x5 4 6)
Computing boundary-matrix in dimension 4. Rank of the source-module : 4. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 1/4 (dimension 4) : <TnPr <TnPr * <<Abar>>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 2/4 (dimension 4) : <TnPr <TnPr * <<Abar[4 3]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 3/4 (dimension 4) : <TnPr <TnPr * <<Abar[2 1][2 1]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 4/4 (dimension 4) : <TnPr <TnPr N4 <<Abar>>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 7. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 1/7 (dimension 5) : <TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 2/7 (dimension 5) : <TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 3/7 (dimension 5) : <TnPr <TnPr * <<Abar[2 1]>>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 4/7 (dimension 5) : <TnPr <TnPr * <<Abar[5 4]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 5/7 (dimension 5) : <TnPr <TnPr * <<Abar[2 1][3 2]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 6/7 (dimension 5) : <TnPr <TnPr * <<Abar[3 2][2 1]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 9s. Computing the boundary of the generator 7/7 (dimension 5) : <TnPr <TnPr M3 <<Abar[2 1]>>> <<Abar>>> End of computing. Homology in dimension 4 : ---done--- ;; Clock -> 2015-08-08, 10h 11m 10s. Computing boundary-matrix in dimension 5. Rank of the source-module : 7. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 1/7 (dimension 5) : <TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 2/7 (dimension 5) : <TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 3/7 (dimension 5) : <TnPr <TnPr * <<Abar[2 1]>>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 4/7 (dimension 5) : <TnPr <TnPr * <<Abar[5 4]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 5/7 (dimension 5) : <TnPr <TnPr * <<Abar[2 1][3 2]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 6/7 (dimension 5) : <TnPr <TnPr * <<Abar[3 2][2 1]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 7/7 (dimension 5) : <TnPr <TnPr M3 <<Abar[2 1]>>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 14. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 1/14 (dimension 6) : <TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[5 4]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 2/14 (dimension 6) : <TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[2 1][3 2]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 3/14 (dimension 6) : <TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[3 2][2 1]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 4/14 (dimension 6) : <TnPr <TnPr * <<Abar>>> <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 5/14 (dimension 6) : <TnPr <TnPr * <<Abar[2 1]>>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 6/14 (dimension 6) : <TnPr <TnPr * <<Abar[3 2]>>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 7/14 (dimension 6) : <TnPr <TnPr M3 <<Abar>>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 8/14 (dimension 6) : <TnPr <TnPr * <<Abar[6 5]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 10s. Computing the boundary of the generator 9/14 (dimension 6) : <TnPr <TnPr * <<Abar[2 1][4 3]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 11s. Computing the boundary of the generator 10/14 (dimension 6) : <TnPr <TnPr * <<Abar[3 2][3 2]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 11s. Computing the boundary of the generator 11/14 (dimension 6) : <TnPr <TnPr * <<Abar[4 3][2 1]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 12s. Computing the boundary of the generator 12/14 (dimension 6) : <TnPr <TnPr * <<Abar[2 1][2 1][2 1]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 13s. Computing the boundary of the generator 13/14 (dimension 6) : <TnPr <TnPr M3 <<Abar[3 2]>>> <<Abar>>> End of computing. ;; Clock -> 2015-08-08, 10h 11m 14s. Computing the boundary of the generator 14/14 (dimension 6) : <TnPr <TnPr N4 <<Abar[2 1]>>> <<Abar>>> End of computing. Homology in dimension 5 : Component Z/4Z ---done--- ;; Clock -> 2015-08-08, 10h 11m 15s.
Out[49]:
NIL
So $\pi_5(\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}))$ is $\mathbb{Z}/4\mathbb{Z}$.