gheber 4d9aca3412 Added Francis' changes.

2016-05-05 09:15:50 -05:00

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Fibrations¶

In Kenzo, the fibration theory is applied in the simplicial frame and limited to the simplicial principal fiber spaces.

Notion of fibration¶

Let $B$ be a simplicial set, called in this context the base space, and $\mathcal{G}$ a simplicial group, the structural group. A simplicial fibration is defined by a twisting operator, $\tau : B \longrightarrow {\cal G}$, of degree -1. For any dimension $n$ and for any element $b \in B_n$, this twisting operator $\tau$ must satisfy the following relations with respect to the face and degeneracy operators of the base space $B$ and the structural group $\mathcal{G}$: $$ \begin{aligned} \partial_i(\tau b) & = & \tau(\partial_i b), \qquad i < n-1, \\ \partial_{n-1}(\tau b) & = & [\tau(\partial_n b)]^{-1}.\tau(\partial_{n-1} b), \\ \eta_i(\tau b) & = & \tau (\eta_i b), \qquad i \leq n-1, \\ e_n & = & \tau(\eta_n b). \end{aligned} $$ where $e_n$ is the neutral element of ${\cal G}_n$.

Now, the twisting operator $\tau$ defines a fibration of base space $B$, of fiber space $\mathcal{G}$, of total space $B \times_\tau \mathcal{G}$ and of structural group $\mathcal{G}$. The simplicial components $(B \times_\tau \mathcal{G})_n$ of the total space $B \times\tau \mathcal{G}$ are: $$ (B \times_{\tau} {\cal G})_n = (B \times {\cal G})_n = (B_n \times {\cal G}_n), $$ and whose face operators are defined, in any dimension $n$ and for any pair $(b,g),\, b \in B_n,\, g \in {\cal G}_n$ by: $$ \begin{aligned} \partial_i(b,g) & = & (\partial_i b, \partial_i g), \quad i < n, \\ \partial_n(b,g) & = & (\partial_n b, \tau(b).\partial_n g), \\ \eta_i(b,g) & = & (\eta_i b, \eta_i g) \end{aligned} $$ As it is well known, there are 4 natural choices for such a definition. The action of the group can be from the right or from the left and the critical index can be the last one ($n$) or the first one (0). In Kenzo, the implementor has chosen the non-usual one (i.e. action on the right hand and, as critical index, the last one), leading to the Szczarba formulas, the best choice in the case of the loop spaces.

Building a twisting operator¶

To build a twisting operator, one uses the function build-smmr designed as a convenient tool to build a simplicial morphism. The source (keyword :sorc) is the base space, a reduced simplicial set, the target (keyword :trgt) is the fiber space, a simplicial group. The degree is -1 and the morphism (keyword :sintr) is the internal lisp function implementing $\tau$. See the example below, where $\tau$ is a simple twisting operator, applying in particular the only (geometrical) simplex of degree 2 of $S^2$ upon the bar object $[1] \in (K(\mathbb{Z}_2,1))_1$.

In Kenzo, the type FIBRATION is defined. An object of this type must be of type SIMPLICIAL-MRPH, the degree of the corresponding morphism being -1. To apply the morphism, one uses the macro ? or the special macro tw-a-sintr3.

? twop dmns gmsm [Macro]

Apply the twisting operator twop to the geometrical simplex gmsm in dimension dmns. One may also make ? work with a combination of geometric simplices as in (? twop cmbn).

tw-a-sintr3 sintr dmns absm bspn [Macro]

Apply the internal lisp function sintr implementing a twisting operator to the abstract simplex absm in dimension dmns. The base point bspn of the target simplicial set (a simplicial group) must be given, since the function may return the neutral element in dimension $dmns - 1$.

Constructing the total space¶

The total space associated to a twisting operator is constructed by the function fibration-total.

fibration-total fibration [Function]

Build the simplicial set, total space of the fibration. This function extracts from the simplicial morphism fibration (a previously defined twisting operator), the base space (a simplicial set) and the fiber space (a simplicial group) and constructs the corresponding twisted cartesian product of these simplicial sets. The face function is built using the internal function fibration-total-face. The non-degenerate simplices of the total space are coded exactly as those of the non-twisted product ($B \times {\cal G}$) obtained by the lisp call (crts-prdc base fiber). This non-twisted cartesian product is saved into the slot grmd (graded module) of the instance object. If the base space and the fiber space are both of type KAN, then the filling function is computed y the internal function fibration-kfll and the function fibration-total returns an object of type KAN.

Examples¶

In [1]:

(def s2 (sphere 2))

Out[1]:

[K1 Simplicial-Set]

In [2]:

(def kz2 (k-z2-1))

Out[2]:

[K6 Abelian-Simplicial-Group]

In [3]:

(def tw2 (build-smmr
             :sorc s2
             :trgt kz2
             :degr -1
             :sintr #'(lambda (dmns gmsm)
                              (declare (ignore dmns gmsm))
                              (absm 0 1))
             :orgn '(s2-tw-kz2)))

Out[3]:

[K18 Fibration K1 -> K6]

In [4]:

(inspect tw2)

The object is a STANDARD-OBJECT of type SIMPLICIAL-MRPH.
0. SORC: [K1 Simplicial-Set]
1. TRGT: [K6 Abelian-Simplicial-Group]
2. DEGR: -1
3. INTR: NIL
4. STRT: :GNRT
5. ???-CLNM: 0
6. ?-CLNM: 0
7. RSLTS: #(#() #() #() #() #() #() #() #() #() #() ...)
8. IDNM: 18
9. ORGN: (S2-TW-KZ2)
10. SINTR: #<FUNCTION (LAMBDA (DMNS GMSM) :IN "/root/cl-jupyter-master/cl-jupyter.lisp") {10054DC76B}>

>

Out[4]:

NIL

The following statement show how to apply the twisting operator to a geometrical simplex. But if we have an abstract simplex, one must use the special function tw-a-sintr3.

In [5]:

(? tw2 2 's2)

Out[5]:

<AbSm - 1>

In [6]:

(tw-a-sintr3 (sintr tw2) 3 (absm 1 's2) 0)

Out[6]:

<AbSm 0 1>

In [7]:

(tw-a-sintr3 (sintr tw2) 3 (absm 7 '*) 0)

Out[7]:

<AbSm 1-0 0>

In [8]:

(dotimes (i 3) (print (tw-a-sintr3 (sintr tw2) 3 
                                   (absm i 's2) 0)))

<AbSm - 1> 
<AbSm 0 1> 
<AbSm 1 1>

Out[8]:

NIL

Let us build the total space of the fibration (a simplicial set). We show some bases and compute some homology groups. The method to compute the homology groups will be explained in a following chapter.

In [9]:

(def tot-spc2 (fibration-total tw2))

Out[9]:

[K24 Simplicial-Set]

In [10]:

(inspect tot-spc2)

The object is a STANDARD-OBJECT of type SIMPLICIAL-SET.
0. CMPR: #<CLOSURE (FLET CAT::RSLT :IN CRTS-PRDC-CMPR) {1005946DCB}>
1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CRTS-PRDC-BASIS) {100594757B}>
2. BSGN: <CrPr - * - 0>
3. DFFR: [K25 Morphism (degree -1): K24 -> K24]
4. GRMD: [K19 Simplicial-Set]
5. EFHM: "unbound"
6. IDNM: 24
7. ORGN: (FIBRATION-TOTAL [K18 Fibration K1 -> K6])
8. CPRD: [K28 Morphism (degree 0): K24 -> K26]
9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FIBRATION-TOTAL-FACE) {100594DB5B}>

>

Out[10]:

NIL

In [11]:

(basis tot-spc2 0)

Out[11]:

(<CrPr - * - 0>)

In [12]:

(basis tot-spc2 1)

Out[12]:

(<CrPr 0 * - 1>)

In [13]:

(basis tot-spc2 2)

Out[13]:

(<CrPr - S2 - 2> <CrPr - S2 0 1> <CrPr - S2 1 1> <CrPr - S2 1-0 0> <CrPr 1-0 * - 2>)

In [14]:

(basis tot-spc2 3)

Out[14]:

(<CrPr 0 S2 - 3> <CrPr 0 S2 1 2> <CrPr 0 S2 2 2> <CrPr 0 S2 2-1 1> <CrPr 1 S2 - 3> <CrPr 1 S2 0 2> <CrPr 1 S2 2 2> <CrPr 1 S2 2-0 1> <CrPr 2 S2 - 3> <CrPr 2 S2 0 2> ...)

In [15]:

(homology tot-spc2 0 8)

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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 0) :
<TnPr * 0> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * 1> 
End of computing.




Homology in dimension 0 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * 1> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 2) :
<TnPr * 2> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 2) :
<TnPr S2 0> 
End of computing.




Homology in dimension 1 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 2) :
<TnPr * 2> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 2) :
<TnPr S2 0> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 3) :
<TnPr * 3> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 3) :
<TnPr S2 1> 
End of computing.




Homology in dimension 2 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 3) :
<TnPr * 3> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 3) :
<TnPr S2 1> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 4) :
<TnPr * 4> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 4) :
<TnPr S2 2> 
End of computing.




Homology in dimension 3 :


Component Z/4Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 4) :
<TnPr * 4> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 4) :
<TnPr S2 2> 
End of computing.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 2.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 5) :
<TnPr * 5> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 5) :
<TnPr S2 3> 
End of computing.




Homology in dimension 4 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 2.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 5) :
<TnPr * 5> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 5) :
<TnPr S2 3> 
End of computing.


Computing boundary-matrix in dimension 6.
Rank of the source-module : 2.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 6) :
<TnPr * 6> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 6) :
<TnPr S2 4> 
End of computing.




Homology in dimension 5 :


Component Z/4Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 6.
Rank of the source-module : 2.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 6) :
<TnPr * 6> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 6) :
<TnPr S2 4> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 7) :
<TnPr * 7> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 7) :
<TnPr S2 5> 
End of computing.




Homology in dimension 6 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 7) :
<TnPr * 7> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 7) :
<TnPr S2 5> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/2 (dimension 8) :
<TnPr * 8> 
End of computing.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 2/2 (dimension 8) :
<TnPr S2 6> 
End of computing.




Homology in dimension 7 :


Component Z/4Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.

Out[15]:

NIL

In the following example, we take $K(\mathbb{Z},1)$ and build a similar twisting operator. Note that in $(K(\mathbb{Z},1))_1$, the bar object $[1]$ is represented as (1) whereas in $K(\mathbb{Z}_2,1)$ it is 1. In this case, it is known that the total space is a model of the sphere $S^3$ (Hopf fibration - see below).

In [16]:

(def s2 (sphere 2))

Out[16]:

[K1 Simplicial-Set]

In [17]:

(def kz1 (k-z-1))

Out[17]:

[K84 Abelian-Simplicial-Group]

In [18]:

(def tw1 (build-smmr
             :sorc s2
             :trgt kz1
             :degr -1
             :sintr #'(lambda (dmns gmsm)
                              (declare (ignore dmns gmsm))
                              (absm 0 (list 1)))
             :orgn '(s2-tw-kz1)))

Out[18]:

[K96 Fibration K1 -> K84]

In [19]:

(def tot-spc1 (fibration-total tw1))

Out[19]:

[K102 Simplicial-Set]

In [20]:

(homology tot-spc1 0 6)

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


;; Clock -> 2016-05-05, 14h 0m 4s.
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 : 1.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.




Homology in dimension 0 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.




Homology in dimension 1 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 3) :
<TnPr S2 S1> 
End of computing.




Homology in dimension 2 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 3) :
<TnPr S2 S1> 
End of computing.


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




Homology in dimension 3 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


Computing boundary-matrix in dimension 5.
Rank of the source-module : 0.




Homology in dimension 4 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 6.
Rank of the source-module : 0.




Homology in dimension 5 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.

Out[20]:

NIL

We may also obtain the boundary and faces of elements of the total space:

In [21]:

(def elem (crpr 0 's2 0 '(1 2)))

Out[21]:

<CrPr - S2 - (1 2)>

In [22]:

(? tot-spc1 2 elem)

Out[22]:

----------------------------------------------------------------------{CMBN 1}
<2 * <CrPr 0 * - (2)>>
<-1 * <CrPr 0 * - (3)>>
------------------------------------------------------------------------------

In [23]:

(? tot-spc1 *)

Out[23]:

----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------

In [24]:

(dotimes (i 3) (print (face tot-spc1 i 2 elem)))

<AbSm - <CrPr 0 * - (2)>> 
<AbSm - <CrPr 0 * - (3)>> 
<AbSm - <CrPr 0 * - (2)>>

Out[24]:

NIL

The program Kenzo provides the function hopf-fibration to build Hopf fibrations. The lisp definition is the following:

commonlisp
(defun HOPF-FIBRATION-SINTR (n)
  (declare (fixnum n))
  (flet ((rslt (dmns gmsm)
         (declare (ignore dmns gmsm))
         (if (zerop n)
             (absm 1 +empty-list+)
             (absm 0 (list n)))))
         (the sintr #'rslt)))

(defun HOPF-FIBRATION (n)
  (declare (fixnum n))
  (the simplicial-mrph
     (build-smmr
        :sorc (sphere 2)
        :trgt (k-z-1)
        :degr -1
        :sintr (hopf-fibration-sintr n)
        :orgn `(hopf ,n))))

In the definition of the lisp function for the slot sintr, we see that the simplicial morphism applies in particular the only geometric simplex in dimension 2 of $S^2$ onto the bar object $[n] \in (K(\mathbb{Z},1))_1$. If $n=0$, this is $\eta_0 [\ ]$, i.e. the 0-degeneracy of the base point NIL. We give some examples of some fibrations and of the corresponding total spaces in function of the parameter n. First we generate 4 fibrations and apply the twisting operator on the geometrical simplex s2.

In [25]:

(def h1 (hopf-fibration 1))
(def h2 (hopf-fibration 2))
(def h3 (hopf-fibration 3))
(def h10 (hopf-fibration 10))

Out[25]:

[K187 Fibration K1 -> K84]

In [26]:

(mapcar #'(lambda (h) (funcall (sintr (eval h)) 2 's2)) '(h1 h2 h3 h10))

Out[26]:

(<AbSm - (1)> <AbSm - (2)> <AbSm - (3)> <AbSm - (10)>)

Then we generate the corresponding total space and get some homology groups. As already seen, t1 is a strange simplicial model for the 3-sphere $S^3$. Also, t2 is a simplcial model for the real projective space $P^3\mathbb{R} = S^3 / (\mathbb{Z}/2)$.

In [27]:

(def t1 (fibration-total h1))
(def t2  (fibration-total h2))
(def t3 (fibration-total h3))
(def t10 (fibration-total h10))

Out[27]:

[K203 Simplicial-Set]

In [28]:

(homology t1 0 10)

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


;; Clock -> 2016-05-05, 14h 0m 4s.
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 : 1.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.




Homology in dimension 0 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.




Homology in dimension 1 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 3) :
<TnPr S2 S1> 
End of computing.




Homology in dimension 2 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 3) :
<TnPr S2 S1> 
End of computing.


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




Homology in dimension 3 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


Computing boundary-matrix in dimension 5.
Rank of the source-module : 0.




Homology in dimension 4 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 6.
Rank of the source-module : 0.




Homology in dimension 5 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 6.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 7.
Rank of the source-module : 0.




Homology in dimension 6 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 7.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 8.
Rank of the source-module : 0.




Homology in dimension 7 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 8.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 9.
Rank of the source-module : 0.




Homology in dimension 8 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 9.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 10.
Rank of the source-module : 0.




Homology in dimension 9 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.

Out[28]:

NIL

In [29]:

(homology t2 0 10)

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


;; Clock -> 2016-05-05, 14h 0m 4s.
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 : 1.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.




Homology in dimension 0 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.




Homology in dimension 1 :


Component Z/2Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 3) :
<TnPr S2 S1> 
End of computing.




Homology in dimension 2 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 3) :
<TnPr S2 S1> 
End of computing.


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




Homology in dimension 3 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


Computing boundary-matrix in dimension 5.
Rank of the source-module : 0.




Homology in dimension 4 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 5.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 6.
Rank of the source-module : 0.




Homology in dimension 5 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 6.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 7.
Rank of the source-module : 0.




Homology in dimension 6 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 7.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 8.
Rank of the source-module : 0.




Homology in dimension 7 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 8.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 9.
Rank of the source-module : 0.




Homology in dimension 8 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


Computing boundary-matrix in dimension 9.
Rank of the source-module : 0.


Computing boundary-matrix in dimension 10.
Rank of the source-module : 0.




Homology in dimension 9 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.

Out[29]:

NIL

In [30]:

(homology t3 0 4)

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


;; Clock -> 2016-05-05, 14h 0m 4s.
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 : 1.


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.




Homology in dimension 0 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.




Homology in dimension 1 :


Component Z/3Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 3) :
<TnPr S2 S1> 
End of computing.




Homology in dimension 2 :



---done---

;; Clock -> 2016-05-05, 14h 0m 4s.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 3) :
<TnPr S2 S1> 
End of computing.


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




Homology in dimension 3 :


Component Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.

Out[30]:

NIL

In [31]:

(homology t10 1)

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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 1) :
<TnPr * S1> 
End of computing.


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


;; Clock -> 2016-05-05, 14h 0m 4s.
Computing the boundary of the generator 1/1 (dimension 2) :
<TnPr S2 *> 
End of computing.




Homology in dimension 1 :


Component Z/10Z


---done---

;; Clock -> 2016-05-05, 14h 0m 4s.

Out[31]:

NIL

Lisp file concerned in this chapter¶

fibrations.lisp.

51 KiB Raw Blame History