Gerd Heber adcd323909 Updated Dockerfile & Jupyter notebooks.

2017-10-08 20:11:31 -05:00

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In [1]:

(ql:quickload "kenzo")
(in-package :cat)

To load "kenzo":
  Load 1 ASDF system:
    kenzo
; Loading "kenzo"

Out[1]:

#<PACKAGE "CAT">

Overview¶

In this overview we try to show, without entering in detailed explanations, various possibilities of Kenzo in algebraic topology.

The material is arranged in a sequence of cells. Cells marked with In [*] represent (editable) input expressions, which in a traditional REPL (Read-Evaluate-Print Loop) would be typed by the user in a Lisp session. Only the In [*] mark is visible if the Lisp statement is not yet run; if run, another Out [*] mark is displayed with the corresponding object returned by the execution. Finally, there might be additional, non-editable, diagnostic output printed before the output component, most often a side-effect of the input form evaluation.

Other cells are the text cells, giving various explanations about the Lisp statements and their evaluations. Not editable by the user.

`Kenzo` in Action¶

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), the 3-simplex and the 4-simplex are respectively labelled *, M3 and N4. The faces 0 and 2 of the 4-simplex N4 are identified with (pasted to) the 3-simplex M3, the others being contracted on the base point. To create the simplicial set one types simply:

In [2]:

(def m23 (moore 2 3))

Out[2]:

[K1 Simplicial-Set]

The system answers that a Kenzo object has been created, with number 1 and type SIMPLICIAL SET. This object may be referenced by the symbol m23.

In [3]:

m23

Out[3]:

[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 [4]:

(homology m23 0 5)

Computing boundary-matrix in dimension 0.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.


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 -> 2017-10-08, 0h 48m 17s.


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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/1 (dimension 3) :
M3 
End of computing.




Homology in dimension 2 :



---done---

;; Clock -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/1 (dimension 4) :
N4 
End of computing.




Homology in dimension 3 :


Component Z/2Z


---done---

;; Clock -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 4.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.

Out[4]:

NIL

The homology of m23 is $(Z,0,0,Z/2,0,\ldots)$; the reduced homology is only $Z/2$ in dimension 3, defining the Moore space up to homotopy.

As m23 is a simplicial set, it is possible to create the cartesian product $\texttt{m23}\times\texttt{m23}$ by the function crts-prdc (= CaRTeSian PRoDuCt). This is a new simplicial set.

In [5]:

(def m23xm23 (crts-prdc m23 m23))

Out[5]:

[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 [6]:

(basis m23xm23 6)

Out[6]:

(<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 [7]:

(length *)

Out[7]:

As shown by the last command, the number of elements of the basis is quite large (230). The user will note that the basis elements are in this particular case formed by cartesian products of degenerated simplices. In the list, an element like <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 m23. The tensor product of two simplicial sets do not make sense, and Kenzo understands to work on the underlying chain complex m23. This tensor product is a new chain complex and we see that the basis in dimension 6 has only one element:

In [8]:

(def t2m23 (tnsr-prdc m23 m23))

Out[8]:

[K3 Chain-Complex]

In [9]:

(basis t2m23 6)

Out[9]:

(<TnPr M3 M3>)

The Eilenberg-Zilber theorem is used to compute the homology groups of the cartesian product space: as chain complexes, m23xm23 and t2m23 have the same homology groups, but the computations using the tensor product are considerably faster. The user can check Künneth's theorem.

In [10]:

(homology m23xm23 0 8)

Computing boundary-matrix in dimension 0.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.


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 -> 2017-10-08, 0h 48m 17s.


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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/2 (dimension 3) :
<TnPr * M3> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/2 (dimension 3) :
<TnPr M3 *> 
End of computing.




Homology in dimension 2 :



---done---

;; Clock -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/2 (dimension 3) :
<TnPr * M3> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/2 (dimension 4) :
<TnPr * N4> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 4.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/2 (dimension 4) :
<TnPr * N4> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.


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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/1 (dimension 6) :
<TnPr M3 M3> 
End of computing.




Homology in dimension 5 :



---done---

;; Clock -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 6.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/2 (dimension 7) :
<TnPr M3 N4> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 7.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/2 (dimension 7) :
<TnPr M3 N4> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.

Out[10]:

NIL

Let us consider now the space $\mathrm{K}(\mathbb{Z}, 1)$. This is an Abelian simplicial group created in Kenzo by the function 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 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: (2 3 4 5). In dimension 0, the only simplex is NIL (the base point).

In [11]:

(def kz1 (k-z 1))

Out[11]:

[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 [12]:

(cprd kz1 4 '(2 3 4 5))

Out[12]:

----------------------------------------------------------------------{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 [13]:

(aprd kz1 6 (tnpr 2 '(1 2) 4 '(3 4 5 6)))

Out[13]:

----------------------------------------------------------------------{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: CMBN n.

In the same way, we may create the Abelian simplicial groups $\mathrm{K}(\mathbb{Z}/2\mathbb{Z}, n)$:

In [14]:

(def k-z2-2 (k-z2 2))

Out[14]:

[K64 Abelian-Simplicial-Group]

In [15]:

(homology k-z2-2 4)

Computing boundary-matrix in dimension 4.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/2 (dimension 4) :
<<Abar[4 3]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/3 (dimension 5) :
<<Abar[5 4]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/3 (dimension 5) :
<<Abar[2 1][3 2]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.

Out[15]:

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 sphere and loop-space. Then we compute the $H_4$ and $H_5$ of $\Omega^2 S^3$:

In [16]:

(def s3 (sphere 3))

Out[16]:

[K174 Simplicial-Set]

In [17]:

(def o2s3 (loop-space s3 2))

Out[17]:

[K191 Simplicial-Group]

In [18]:

(homology o2s3 4 6)

Computing boundary-matrix in dimension 4.
Rank of the source-module : 3.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/5 (dimension 5) :
<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 5.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/5 (dimension 5) :
<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.

Out[18]:

NIL

Let us take now the first loop space $\Omega^1 S^3$

In [19]:

(def os3 (loop-space s3))

Out[19]:

[K179 Simplicial-Group]

In the following instruction, we locate in the symbol 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 [20]:

(def L1 (loop3 0 's3 1))

Out[20]:

<<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 [21]:

(def null-simp (absm 3 +null-loop+))

Out[21]:

<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 os3 a 3-simplex named D3, the attaching map being described by the list of its faces in dimension 2. The faces 0 and 2 of D3 are pasted to L1, the faces 1 and 3 are collapsed on the base point; in this way the attaching map $(S^2 \rightarrow \Omega S^3)$ has degree 2.

In [22]:

(def dos3 (disk-pasting os3 3 '<D3>
                        (list L1 null-simp L1 null-simp)))

Out[22]:

[K435 Simplicial-Set]

Let us compute a few homology groups of the new space dos3:

In [23]:

(homology dos3 2 4)

Computing boundary-matrix in dimension 2.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/1 (dimension 3) :
<D3> 
End of computing.




Homology in dimension 2 :


Component Z/2Z


---done---

;; Clock -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/1 (dimension 4) :
<<AlLp[2 S3][2 S3]>> 
End of computing.




Homology in dimension 3 :



---done---

;; Clock -> 2017-10-08, 0h 48m 17s.

Out[23]:

NIL

But more interesting, let us build the loop space of the object dos3 and let us compute the homology in dimension 5:

In [24]:

(def odos3 (loop-space dos3))

Out[24]:

[K453 Simplicial-Group]

In [25]:

(homology odos3 5)

Computing boundary-matrix in dimension 5.
Rank of the source-module : 14.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/14 (dimension 5) :
<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/14 (dimension 5) :
<<AlLp[2 <D3>][3 <<AlLp[2 S3][2 S3]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 3/14 (dimension 5) :
<<AlLp[3 <<AlLp[2 S3][2 S3]>>][2 <D3>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 5/14 (dimension 5) :
<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 7/14 (dimension 5) :
<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 8/14 (dimension 5) :
<<AlLp[2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 7/26 (dimension 6) :
<<AlLp[2 <D3>][2 <D3>][2 <D3>]>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.

Out[25]:

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 [26]:

(typep s3 'kan)

Out[26]:

NIL

In [27]:

(typep os3 'kan)

Out[27]:

In [28]:

(typep os3 'simplicial-group)

Out[28]:

In [29]:

(typep os3 'ab-simplicial-group)

Out[29]:

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 [30]:

(def L2 (loop3 0 's3 2))

Out[30]:

<<Loop[S3\2]>>

In [31]:

(def square (aprd os3 4 (tnpr 2 L2 2 L2)))

Out[31]:

----------------------------------------------------------------------{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 L2. The following instruction selects the generator part of the second element of the previous combination.

In [32]:

(def L4 (gnrt (second (cmbn-list square))))

Out[32]:

<<Loop[2-0 S3\2][3-1 S3\2]>>

Let us use the lisp function mapcar (one among the various iteration functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object L4, this list is a "Kan horn".

In [33]:

(def horn (mapcar #'(lambda (i) (face os3 i 4 l4)) '(1 2 3 4)))

Out[33]:

(<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 kfll tries to find a filling of this "Kan horn", and we see that the face 2 of the resulting simplex (which is very different from L4) is the same as the face 2 of L4.

In [34]:

(def kan-simplex (kfll os3 0 4 horn))

Out[34]:

<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 [35]:

(face os3 2 4 kan-simplex)

Out[35]:

<AbSm - <<Loop[0 S3\2][2 S3\2]>>>

In [36]:

(second horn)

Out[36]:

<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 program Kenzo allows the construction of the universal bundle $\mathcal{WG}$ and in particular of its base space $\overline{\mathcal{W}} \mathcal{G}$, 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 $H_4$, which in principle should be isomorphic to $H_4 S^3 = 0$, is really null.

In [37]:

(def cls-os3 (classifying-space os3))

Out[37]:

[K598 Simplicial-Set]

In [38]:

(typep cls-os3 'simplicial-group)

Out[38]:

NIL

In [39]:

(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 -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.

Out[39]:

NIL

Let us end this short overview with an example of computation of homotopy groups. The method used in Kenzo is the Whitehead tower. An extended version of the Kenzo program due to Ana Romero is more general, but in this present 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 [40]:

(def ch3 (chml-clss m23 3))

Out[40]:

[K729 Cohomology-Class on K1 of degree 3]

Then the function z2-whitehead is called to build a fibration over the simplicial set m23 canonically associated to the cohomology class ch3.

In [41]:

(def f3 (z2-whitehead m23 ch3))

Out[41]:

[K730 Fibration K1 -> K64]

Then the total space of the fibration is built:

In [42]:

(def x4 (fibration-total f3))

Out[42]:

[K736 Simplicial-Set]

The $H_4$ of this total space is the $\pi_4$ of $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:

In [43]:

(homology x4 3 5)

Computing boundary-matrix in dimension 3.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/2 (dimension 3) :
<TnPr * <<Abar[3 2]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/3 (dimension 4) :
<TnPr * <<Abar[4 3]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/3 (dimension 4) :
<TnPr * <<Abar[2 1][2 1]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 3/3 (dimension 4) :
<TnPr N4 <<Abar>>> 
End of computing.




Homology in dimension 3 :



---done---

;; Clock -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 4.
Rank of the source-module : 3.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/3 (dimension 4) :
<TnPr * <<Abar[4 3]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/3 (dimension 4) :
<TnPr * <<Abar[2 1][2 1]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/4 (dimension 5) :
<TnPr * <<Abar[5 4]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/4 (dimension 5) :
<TnPr * <<Abar[2 1][3 2]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 3/4 (dimension 5) :
<TnPr * <<Abar[3 2][2 1]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.

Out[43]:

NIL

We may now iterate the process, to compute the $\pi_5$ of $\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z})$:

In [44]:

(def ch4 (chml-clss x4 4))

Out[44]:

[K817 Cohomology-Class on K802 of degree 4]

In [45]:

(def f4 (z2-whitehead x4 ch4))

Out[45]:

[K832 Fibration K736 -> K818]

In [46]:

(def x5 (fibration-total f4))

Out[46]:

[K838 Simplicial-Set]

In [47]:

(homology x5 4 6)

Computing boundary-matrix in dimension 4.
Rank of the source-module : 4.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/4 (dimension 4) :
<TnPr <TnPr * <<Abar>>> <<Abar[4 <<Abar[3 2]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/4 (dimension 4) :
<TnPr <TnPr * <<Abar[4 3]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 3/4 (dimension 4) :
<TnPr <TnPr * <<Abar[2 1][2 1]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/7 (dimension 5) :
<TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[4 3]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/7 (dimension 5) :
<TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 3/7 (dimension 5) :
<TnPr <TnPr * <<Abar[2 1]>>> <<Abar[3 <<Abar[2 1]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 4/7 (dimension 5) :
<TnPr <TnPr * <<Abar[5 4]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 5/7 (dimension 5) :
<TnPr <TnPr * <<Abar[2 1][3 2]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 6/7 (dimension 5) :
<TnPr <TnPr * <<Abar[3 2][2 1]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 7.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/7 (dimension 5) :
<TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[4 3]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/7 (dimension 5) :
<TnPr <TnPr * <<Abar>>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 3/7 (dimension 5) :
<TnPr <TnPr * <<Abar[2 1]>>> <<Abar[3 <<Abar[2 1]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 4/7 (dimension 5) :
<TnPr <TnPr * <<Abar[5 4]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 5/7 (dimension 5) :
<TnPr <TnPr * <<Abar[2 1][3 2]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 6/7 (dimension 5) :
<TnPr <TnPr * <<Abar[3 2][2 1]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 1/14 (dimension 6) :
<TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[5 4]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 2/14 (dimension 6) :
<TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[2 1][3 2]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 3/14 (dimension 6) :
<TnPr <TnPr * <<Abar>>> <<Abar[6 <<Abar[3 2][2 1]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 5/14 (dimension 6) :
<TnPr <TnPr * <<Abar[2 1]>>> <<Abar[4 <<Abar[3 2]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 6/14 (dimension 6) :
<TnPr <TnPr * <<Abar[3 2]>>> <<Abar[3 <<Abar[2 1]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 7/14 (dimension 6) :
<TnPr <TnPr M3 <<Abar>>> <<Abar[3 <<Abar[2 1]>>]>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 8/14 (dimension 6) :
<TnPr <TnPr * <<Abar[6 5]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 9/14 (dimension 6) :
<TnPr <TnPr * <<Abar[2 1][4 3]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 10/14 (dimension 6) :
<TnPr <TnPr * <<Abar[3 2][3 2]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 11/14 (dimension 6) :
<TnPr <TnPr * <<Abar[4 3][2 1]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 12/14 (dimension 6) :
<TnPr <TnPr * <<Abar[2 1][2 1][2 1]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
Computing the boundary of the generator 13/14 (dimension 6) :
<TnPr <TnPr M3 <<Abar[3 2]>>> <<Abar>>> 
End of computing.


;; Clock -> 2017-10-08, 0h 48m 17s.
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 -> 2017-10-08, 0h 48m 17s.

Out[47]:

NIL

So $\pi_5(\mathrm{Moore}(\mathbb{Z}/2\mathbb{Z}))$ is $\mathbb{Z}/4\mathbb{Z}$.

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71 KiB

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`Kenzo` in Action¶