kenzo/examples/17_Classifying_Spaces.ipynb

In [1]:

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

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

Out[1]:

#<PACKAGE "CAT">

Classifying Spaces¶

Introduction¶

Let ${\cal G}$ be a simplicial group $0$-reduced. The program Kenzo allows the construction of the universal bundle $\bar{\cal W}{\cal G}$, called the classifying space of ${\cal G}$. It is known that this is the base space of a principal bundle such that the total space is contractible. We know also that if ${\cal G}$ is an Abelian simplicial group then $\bar{\cal W}{\cal G}$ is also an Abelian simplicial group, otherwise if ${\cal G}$ is non-Abelian, then $\bar{\cal W}{\cal G}$ is only a simplicial set. In dimension $n$ the simplices of the classifying space are elements of the product: $${\cal G}_{n-1}\times {\cal G}_{n-2}\times\cdots\times{\cal G}_0,$$ where ${\cal G}_k$ is the set of the $k$-simplices of the given simplicial group.

A simplex of $\bar{\cal W}{\cal G}$ is called gbar (geometric bar) and may be seen as an $n$-tuple $$(g_{n-1}, g_{n-2}, \ldots, g_1, g_0),$$ where the $g_i$'s are elements of ${\cal G}_i$, possibly degenerated. In Kenzo a gbar is represented externally under the form:

<<GBar < $[\eta_{n-1}, g_{n-1}]$ >< $[\eta_{n-2}, g_{n-2}]$ > ... < $[\eta_1, g_1]$ > < - $g_0$> >>

where a term such as $[\eta_i, g_i]$ is a simplex of ${\cal G}_i$. Let us recall that our group ${\cal G}$ is assumed 0-reduced; as a consequence, the $g_0$ compound of a gbar is necessarily the identity element of ${\cal G}$.

In the symbolic expression $[\eta_i, g_i]$, the operator $\eta$ represents a sequence (possibly the void one) of degeneracy operators applied coherently upon some element of ${\cal G}_k$, for $k\leq i$. For instance $[\eta_1, g_1]$ can be the pair $((0),g_0)$, internally implemented as an absm object, i.e. the 0-degeneracy of the base point and $[\eta_2, g_2]$ can be $((1 0), g_0)$, that is, the only possible 2-degeneracy of the base point $g_0$ of $mathcal{G}$. In particular, the dimension of the "geometric" simplex $g_i$ is $i - \ell$ with $\ell$ the length of the integer sequence $\eta_i$. To be more explicit, let us take the Abelian simplicial group $K(\mathbb{Z}_2, 1)$ (see the simplicial groups chapter), then some elements of the basis in dimension 4 of its classifying space are, for instance:

<<GBar<- 3><- 2><- 1><- 0>>>
<<GBar<- 3><1-0 0><0 0><- 0>>>
<<GBar<1 2><- 2><0 0><- 0>>>
<<GBar<1 2><0 1><0 0><- 0>>>
<<GBar<2 2><1 1><0 0><- 0>>>
<<GBar<2-0 1><1 1><0 0><- 0>>>
<<GBar<2-1 1><1 1><- 1><- 0>>>
<<GBar<2-1 1><0 1><0 0><- 0>>>

We recall that the symbol -, in front of an element means "no degeneracy".

Face, degeneracy and group operations¶

Let us denote a gbar under the following generic form: $$ g_{bar}=(g_{n-1}, g_{n-2}, \ldots, g_j, \ldots, g_1, g_0). $$ In this symbolism, $g_0$ is also the base point of the initial simplicial group. The rule for the face operators is the following:

• If $j=n$, $\delta_n g_{bar}= (g_{n-2}, g_{n-3}, \ldots, g_1, g_0).$
• If $j \not=n$, $\delta_j g_{bar}= (\delta_j g_{n-1}, \ldots, \delta_j g_{j+1}, g_{j-1}.\delta_j g_j, g_{j-1}, \ldots, g_1, g_0),$ where in the term $g_{j-1}.\delta_j g_j$, the dot represents the group composition.
• If $j=0$, $\delta_0 g_{bar}= (\delta_0 g_{n-1}, \ldots, \delta_0 g_1).$

The rule for the degeneracy operators is:

$$ \eta_j g_{bar}= (\eta_j g_{n-1}, \ldots, \eta_j g_{j+1}, \eta_{j-1}\eta_{j-2}\ldots\eta_1\eta_0 g_0, g_{j-1}, \ldots, g_1, g_0). $$

If ${\cal G}$ is Abelian then $\bar{\cal W}{\cal G}$ is also Abelian and its composition law is simply:

$$ (g_{n-1}, \ldots, g_1, g_0).(g'_{n-1}, g'_1, g'_0)=(g_{n-1}.g'_{n-1}, \ldots, g_1.g'_1, g_0.g'_0). $$

If the initial group is not Abelian then, in general, the previous composition law is not compatible with the simplicial structure.

The functions for the classifying spaces¶

gbar dmns dop-1 elem-1 ... dop-n elem-n [Macro]

Build a gbar in dimension dmns. The sequence dop-1 elem-1 ... dop-n elem-n, is a sequence of pairs (dop-k, elem-k), where dop-k is a binary coded degeneracy operator and elem-k a simplex of the initial simplicial group. The ordering of the pairs in the sequence must conform to the theoritical definition of a gbar. If the sequence of pairs is void, the special call (gbar 0), creates the base point of the classifying space externally represented by <<GBAR>>. This base point can be reached through the system symbol +null-gbar+.

classifying-space-cmpr cmpr [Function]

From a comparison function cmpr defined for a simplicial group ${\cal G}$, build a comparison function for the classifying space of ${\cal G}$.

classifying-space-basis basis [Function]

From the basis function basis of a simplicial group ${\cal G}$, build a basis function for the classifying space of ${\cal G}$. If ${\cal G}$ is locally effective, this function returns the symbol :locally-effective.

classifying-space-face face sintr-grml [Function]

From the face function face and the lisp function for the group law, sintr-grml, of a simplicial group ${\cal G}$, build a face function for the classifying space of ${\cal G}$.

classifying-space smgr [Method]

Build the simplicial set, classifying space of the simplicial group smgr. The construction uses the three preceding auxiliary functions. The base point of the returned simplicial set is the null gbar +null-gbar+.

classifying-space A-smgr [Method]

Build the Abelian simplicial group, classifying space of the Abelian simplicial group A-smgr. The group law and its inverse (slots :sintr-grml and :sintr-grin) are built by the two auxiliary functions:
classifying-space-grml-sintr and
classifying-space-grin-sintr,
defining the laws of this new group from the corresponding laws of A-smgr.

Eilenberg-Mac Lane spaces $K(\pi,n)$¶

We have already described $K(\mathbb{Z},1)$ and $K(\mathbb{Z}_2,1)$ in the chapter on simplicial groups. This section gives some complements about the construction of $K(\mathbb{Z},n)$ and $K(\mathbb{Z}_2,n)$. The trivial case for $n=0$ is not implemented in Kenzo. For $n > 1$, these simplicial groups are built recursively as the classifying spaces of $K(\mathbb{Z},n-1)$ and $K(\mathbb{Z}_2,n-1)$, respectively, as shown in the following description:

k-z n [Function]

Build $K(\mathbb{Z},n)$ recursively from $K(\mathbb{Z},n-1)$. If $n=1$, return $K(\mathbb{Z},1)$

commonlisp
     (defun K-Z (n)
        (the ab-simplicial-group
           (if (= n 1)
               (k-z-1)
               (classifying-space (k-z (1- n))))))

k-z2 n [Function]

Build $K(\mathbb{Z}_2,n)$ recursively from $K(\mathbb{Z}_2,n-1)$. If $n=1$, return $K(\mathbb{Z}_2,1)$

commonlisp
     (defun K-Z2 (n)
        (the ab-simplicial-group
           (if (= n 1)
               (k-z2-1)
               (classifying-space (k-z2 (1- n))))))

These spaces have an effective homology which is computed via a search-efhm method (described in a next chapter). For the case $n=1$, we may recall here that $K(\mathbb{Z},1)$, though being locally effective, has the homology of $S^1$. The search-efhm method applied to this chain complex does the following: a reduction between $K(\mathbb{Z},1)$ and the chain complex circle (described in the chain complex chapter) is built using the function kz1-rdct and the homotopy equivalence, value of the slot efhm, is built between this reduction and the trivial reduction of $K(\mathbb{Z},1)$ upon itself.

On the other hand, $K(\mathbb{Z},2)$ has a very simple finite basis in every dimension (see $\mathbb{R}$-projective spaces), so its homology is computed directly from its basis. The homotopy equivalence, value of the slot efhm, is the trivial homotopy equivalence of $K(\mathbb{Z},2)$ upon itself.

Examples¶

Let us begin by showing some examples of gbars built from various initial simplicial groups. In the second statement, the initial simplicial group is $K(\mathbb{Z}_2, 1)$, in the third, it is $\Omega({\mathit Moore}(2,2))$. We recall that in the call of the function gbar, the degeneracy operators appear as integers, according to the general rule in Kenzo, but the printed result shows in clear the sequence of $\eta_i$'s.

In [2]:

(gbar 2 1 'a 2 'b)

Out[2]:

<<GBar<0 A><1 B>>>

In [3]:

(gbar 4 0 3 3 0 1 0 0 0)

Out[3]:

<<GBar<- 3><1-0 0><0 0><- 0>>>

In [4]:

(def gbar-m22 (gbar 4 0 (loop3 3 'm2 1 4 'n3 1)
                      0 (loop3 0 'n3 1)
                      0 (loop3 0 'm2 1)
                      0 +null-loop+))

Out[4]:

<<GBar<- <<Loop[1-0 M2][2 N3]>>><- <<Loop[N3]>>><- <<Loop[M2]>>><- <<Loop>>>>>

Let us test the face function upon this simplex. The user will note that the face function returns an abstract simplex.

In [5]:

(def om (loop-space (moore 2 2)))

Out[5]:

[K6 Simplicial-Group]

In [6]:

(def face (classifying-space-face (face om) (sintr (grml om))))

Out[6]:

#<CLOSURE (FLET RSLT :IN CLASSIFYING-SPACE-FACE) {1001CE47FB}>

In [7]:

(dotimes (i 5) (print (funcall face i 4 gbar-m22)))

<AbSm - <<GBar<- <<Loop[0 M2][1 M2]>>><- <<Loop[M2]>>><- <<Loop>>>>>> 
<AbSm 0 <<GBar<- <<Loop[M2]>>><- <<Loop>>>>>> 
<AbSm - <<GBar<- <<Loop[0 M2][N3]>>><- <<Loop[M2\2]>>><- <<Loop>>>>>> 
<AbSm - <<GBar<- <<Loop[N3\2]>>><- <<Loop[M2]>>><- <<Loop>>>>>> 
<AbSm - <<GBar<- <<Loop[N3]>>><- <<Loop[M2]>>><- <<Loop>>>>>> 

Out[7]:

NIL

Let us work now, directly with the classifying space of $K(\mathbb{Z}_2, 1)$.

In [8]:

(def cs-kz21 (classifying-space (k-z2-1)))

Out[8]:

[K30 Abelian-Simplicial-Group]

In [9]:

(orgn cs-kz21)

Out[9]:

(CLASSIFYING-SPACE [K18 Abelian-Simplicial-Group])

In [10]:

(def elem-1 (first (basis cs-kz21 4)))

Out[10]:

<<GBar<- 3><- 2><- 1><- 0>>>

In [11]:

(? cs-kz21 4 elem-1)

Out[11]:

----------------------------------------------------------------------{CMBN 3}
<2 * <<GBar<- 2><- 1><- 0>>>>
------------------------------------------------------------------------------

In [12]:

(? cs-kz21 *)

Out[12]:

----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------

In [13]:

(cprd cs-kz21  4 elem-1)

Out[13]:

----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar>> <<GBar<- 3><- 2><- 1><- 0>>>>>
<1 * <TnPr <<GBar<- 1><- 0>>> <<GBar<- 1><- 0>>>>>
<1 * <TnPr <<GBar<- 3><- 2><- 1><- 0>>> <<GBar>>>>
------------------------------------------------------------------------------

In [14]:

(dotimes (i 5)
    (print (face cs-kz21 i 4 elem-1)))

<AbSm - <<GBar<- 2><- 1><- 0>>>> 
<AbSm 0 <<GBar<- 1><- 0>>>> 
<AbSm 1 <<GBar<- 1><- 0>>>> 
<AbSm 2 <<GBar<- 1><- 0>>>> 
<AbSm - <<GBar<- 2><- 1><- 0>>>> 

Out[14]:

NIL

Let us test the law group with $K(\mathbb{Z}, 1)$. We recall the simplices of this Abelian simplicial group are represented as lists of integers. The functions grml and grin accept also the gbars under the form of abstract simplices (see the last statement).

In [15]:

(def cs-kz1 (classifying-space (k-z-1)))

Out[15]:

[K54 Abelian-Simplicial-Group]

In [16]:

(grml cs-kz1 3 (crpr 0 (gbar 3 0 '(1 2) 0 '(3) 0 '())
                     0 (gbar 3 0 '(-1 -2) 0 '(-3) 0 '())))

Out[16]:

<AbSm 2-1-0 <<GBar>>>

In [17]:

(grml cs-kz1 3 (crpr 0 (gbar 3 0 '(1 2) 0 '(3) 0 '())
                     4 (gbar 2 0 '(-3) 0 '())))

Out[17]:

<AbSm - <<GBar<- (1 2)><0 NIL><- NIL>>>>

In [18]:

(grml cs-kz1 3 (crpr 0 (gbar 3 0 '(1 2) 0 '(3) 0 '())
                     1 (gbar 2 0 '(-3) 0 '())))

Out[18]:

<AbSm - <<GBar<- (1 -1)><- (3)><- NIL>>>>

In [19]:

(grin cs-kz1 3 (gbar 3 0 '(1 2) 1 '() 0 '()))

Out[19]:

<AbSm - <<GBar<- (-1 -2)><0 NIL><- NIL>>>>

In [20]:

(grin cs-kz1 3 *)

Out[20]:

<AbSm - <<GBar<- (1 2)><0 NIL><- NIL>>>>

As $K(\mathbb{Z}, 1)$ and $K(\mathbb{Z}_2, 1)$ are Abelian simplicial groups, we may iterate the classifying space construction and retrieve some known results.

In [21]:

(cat-init)    ;; re-initialization

---done---

Out[21]:

NIL

In [22]:

(def k-z-3 (k-z 3))

Out[22]:

[K25 Abelian-Simplicial-Group]

In [23]:

(homology k-z-3 0 10)

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


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 0) :
<<Abar>> 
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, 12h 28m 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 -> 2017-10-08, 12h 28m 49s.


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, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 3) :
<<Abar[3 <<Abar[2 S1]>>]>> 
End of computing.




Homology in dimension 2 :



---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


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


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 3) :
<<Abar[3 <<Abar[2 S1]>>]>> 
End of computing.


Computing boundary-matrix in dimension 4.
Rank of the source-module : 0.




Homology in dimension 3 :


Component Z


---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


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, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 5) :
<<Abar[5 <<Abar[2 S1][2 S1]>>]>> 
End of computing.




Homology in dimension 4 :



---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 5) :
<<Abar[5 <<Abar[2 S1][2 S1]>>]>> 
End of computing.


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


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 6) :
<<Abar[3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>]>> 
End of computing.




Homology in dimension 5 :


Component Z/2Z


---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


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


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 6) :
<<Abar[3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>]>> 
End of computing.


Computing boundary-matrix in dimension 7.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 7) :
<<Abar[7 <<Abar[2 S1][2 S1][2 S1]>>]>> 
End of computing.




Homology in dimension 6 :



---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


Computing boundary-matrix in dimension 7.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 7) :
<<Abar[7 <<Abar[2 S1][2 S1][2 S1]>>]>> 
End of computing.


Computing boundary-matrix in dimension 8.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/2 (dimension 8) :
<<Abar[3 <<Abar[2 S1]>>][5 <<Abar[2 S1][2 S1]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 2/2 (dimension 8) :
<<Abar[5 <<Abar[2 S1][2 S1]>>][3 <<Abar[2 S1]>>]>> 
End of computing.




Homology in dimension 7 :


Component Z/3Z


---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


Computing boundary-matrix in dimension 8.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/2 (dimension 8) :
<<Abar[3 <<Abar[2 S1]>>][5 <<Abar[2 S1][2 S1]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 2/2 (dimension 8) :
<<Abar[5 <<Abar[2 S1][2 S1]>>][3 <<Abar[2 S1]>>]>> 
End of computing.


Computing boundary-matrix in dimension 9.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/2 (dimension 9) :
<<Abar[9 <<Abar[2 S1][2 S1][2 S1][2 S1]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 2/2 (dimension 9) :
<<Abar[3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>]>> 
End of computing.




Homology in dimension 8 :


Component Z/2Z


---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


Computing boundary-matrix in dimension 9.
Rank of the source-module : 2.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/2 (dimension 9) :
<<Abar[9 <<Abar[2 S1][2 S1][2 S1][2 S1]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 2/2 (dimension 9) :
<<Abar[3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>]>> 
End of computing.


Computing boundary-matrix in dimension 10.
Rank of the source-module : 3.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/3 (dimension 10) :
<<Abar[3 <<Abar[2 S1]>>][7 <<Abar[2 S1][2 S1][2 S1]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 2/3 (dimension 10) :
<<Abar[5 <<Abar[2 S1][2 S1]>>][5 <<Abar[2 S1][2 S1]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 3/3 (dimension 10) :
<<Abar[7 <<Abar[2 S1][2 S1][2 S1]>>][3 <<Abar[2 S1]>>]>> 
End of computing.




Homology in dimension 9 :


Component Z/2Z


---done---

;; Clock -> 2017-10-08, 12h 28m 49s.

Out[23]:

NIL

In [24]:

(def k-z2-5 (k-z2 5))

Out[24]:

[K342 Abelian-Simplicial-Group]

In [25]:

(homology k-z2-5 0 7)

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


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 0) :
<<Abar>> 
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, 12h 28m 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 -> 2017-10-08, 12h 28m 49s.


Computing boundary-matrix in dimension 2.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 0.




Homology in dimension 2 :



---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


Computing boundary-matrix in dimension 3.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 4.
Rank of the source-module : 0.




Homology in dimension 3 :



---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


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, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 5) :
<<Abar[5 <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>>]>> 
End of computing.




Homology in dimension 4 :



---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 1.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 5) :
<<Abar[5 <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>>]>> 
End of computing.


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


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 6) :
<<Abar[6 <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>>]>> 
End of computing.




Homology in dimension 5 :


Component Z/2Z


---done---

;; Clock -> 2017-10-08, 12h 28m 49s.


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


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/1 (dimension 6) :
<<Abar[6 <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>>]>> 
End of computing.


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


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 1/2 (dimension 7) :
<<Abar[7 <<Abar[6 <<Abar[5 <<Abar[4 3]>>]>>]>>]>> 
End of computing.


;; Clock -> 2017-10-08, 12h 28m 49s.
Computing the boundary of the generator 2/2 (dimension 7) :
<<Abar[7 <<Abar[6 <<Abar[5 <<Abar[2 1][2 1]>>]>>]>>]>> 
End of computing.




Homology in dimension 6 :



---done---

;; Clock -> 2017-10-08, 12h 28m 49s.

Out[25]:

NIL

Lisp files concerned in this chapter¶

classifying-space.lisp, k-pi-n.lisp.