140 KiB
140 KiB
Simplicial groups fibrations¶
Introduction¶
This chapter is analogous to the Loop spaces fibrations chapter where algorithms have been specially designed for loop spaces. In the present case, we have at our disposal all the tools developped for the fibrations. Let ${\cal G}$ be a 0-reduced simplicial group and $\bar{\cal W}{\cal G}$, its classifying space. We recall 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. The software Kenzo implements the canonical twisted cartesian product of $\bar{\cal W}{\cal G}$ by ${\cal G}$, denoted ${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$. The canonical fibration
$$ {\cal G}\quad \hookrightarrow\quad {\cal WG} \hskip 1.2em \longrightarrow \hskip -1.2em \longrightarrow
\quad\bar{\cal W}{\cal G},$$
is defined by the following twisting operator:
$$\tau[(g_{n-1}, g_{n-2}, \ldots, g_0)]= g_{n-1}.$$
The total space ${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$ is acyclic.
The function for the canonical fibration¶
Build the simplicial morphism of degree -1 corresponding to the canonical fibration. The source is the classifying space of the simplicial group smgr (built internally by the function), the target is smgr and the internal lisp function value of the slot
:sintrimplements
smgr-fibration smgr [Function]
the twisting operator. The returned object is a fibration.
Examples¶
Let us take the Abelian simplicial group $K(\mathbb{Z},1)$ and build the fibration morphism.
In [1]:
(def k1 (k-z-1))
Out[1]:
[K1 Abelian-Simplicial-Group]
In [2]:
(def tw (smgr-fibration k1))
Out[2]:
[K25 Fibration K13 -> K1]
If needed, the classifying space may be found in the slot :sorc of the fibration:
In [3]:
(def k2 (sorc tw))
Out[3]:
[K13 Abelian-Simplicial-Group]
We may build now the total space of the fibration. We see that Kenzo returns a Kan simplicial set because the base and fiber spaces are also of Kan type. Then we test the face and differential operator upon a simplex of degree 4 of the total space tt.
In [4]:
(def tt (fibration-total tw))
Out[4]:
[K31 Kan-Simplicial-Set]
In [5]:
(def gmsm (crpr 0 (gbar 4 0 '(10 11 12) 0 '(20 21) 0 '(30) 0 '()) 0 '(2 4 6 8)))
Out[5]:
<CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>
In [6]:
(dotimes (i 5) (print (face tt i 4 gmsm)))
<AbSm - <CrPr - <<GBar<- (11 12)><- (21)><- NIL>>> - (4 6 8)>> <AbSm - <CrPr - <<GBar<- (21 12)><- (41)><- NIL>>> - (6 6 8)>> <AbSm - <CrPr - <<GBar<- (10 23)><- (50)><- NIL>>> - (2 10 8)>> <AbSm - <CrPr - <<GBar<- (30 32)><- (30)><- NIL>>> - (2 4 14)>> <AbSm - <CrPr - <<GBar<- (20 21)><- (30)><- NIL>>> - (12 15 18)>>
Out[6]:
NIL
In [7]:
(? tt 4 gmsm)
Out[7]:
----------------------------------------------------------------------{CMBN 3}
<1 * <CrPr - <<GBar<- (10 23)><- (50)><- NIL>>> - (2 10 8)>>
<1 * <CrPr - <<GBar<- (11 12)><- (21)><- NIL>>> - (4 6 8)>>
<1 * <CrPr - <<GBar<- (20 21)><- (30)><- NIL>>> - (12 15 18)>>
<-1 * <CrPr - <<GBar<- (21 12)><- (41)><- NIL>>> - (6 6 8)>>
<-1 * <CrPr - <<GBar<- (30 32)><- (30)><- NIL>>> - (2 4 14)>>
------------------------------------------------------------------------------
In [8]:
(? tt *)
Out[8]:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
We may also build the Ronald Brown reduction and test the differential upon a simplex belonging to the twisted tensor product (slot :bcc of the reduction).
In [9]:
(def br (brown-reduction tw))
Out[9]:
[K59 Reduction K31 => K57]
In [10]:
(def tw-pr (bcc br))
Out[10]:
[K57 Chain-Complex]
In [11]:
(? tw-pr 4 (tnpr 4 (gbar 4 0 '(1 10 100) 0 '(1000 10000) 0 '(100000) 0 '()) 0 '()))
Out[11]:
----------------------------------------------------------------------{CMBN 3}
<-1 * <TnPr <<GBar>> (111 11000 100000)>>
<1 * <TnPr <<GBar>> (111 101000 10000)>>
<1 * <TnPr <<GBar>> (11011 100 100000)>>
<-1 * <TnPr <<GBar>> (11011 100000 100)>>
<-1 * <TnPr <<GBar>> (101001 110 10000)>>
<1 * <TnPr <<GBar>> (101001 10010 100)>>
<1 * <TnPr <<GBar<- (100000)><- NIL>>> (100)>>
<1 * <TnPr <<GBar<- (1 110)><- (101000)><- NIL>>> NIL>>
<1 * <TnPr <<GBar<- (10 100)><- (10000)><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<- (11 100)><- (11000)><- NIL>>> NIL>>
... ...
------------------------------------------------------------------------------
In [12]:
(? tw-pr *)
Out[12]:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
The essential contraction for simplicial groups fibrations¶
It is known that if ${\cal G}$ is a 0-reduced simplicial group, the space ${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$ is contractible. So it is possible to build a reduction of this space over $\mathbb{Z}$. This reduction depends on the important contraction $$\chi_{\times\tau}: {\bar{\cal W}{\cal G} \times_\tau {\cal G}} \longrightarrow {\bar{\cal W}{\cal G} \times_\tau {\cal G}}, $$ defined by: $$\chi_{\times\tau}[(g_{n-1}, \ldots, g_0), g_n] = (-1)^{n+1} (g_n, g_{n-1}, \ldots, g_0, *),$$ where $g_n \in {\cal G}$, $(g_{n-1}, \ldots, g_0)$ is a gbar in $\bar{\cal W}{\cal G}$, and $*$ is the neutral element of ${\cal G}_{n+1}$ (in fact the n-th degeneracy of the base point of ${\cal G}$). We recall also that $\chi_{\times\tau}$ is a homotopy operator (degree: +1) satisfying the relation $$d \circ \chi_{\times\tau} + \chi_{\times\tau} \circ d =1.$$ In addition, if we consider the Brown reduction $$ \begin{aligned} {\bar{\cal W}{\cal G} \times_\tau {\cal G}} & \qquad\stackrel{h}{\longrightarrow} & {}^s{\bar{\cal W}{\cal G} \times_\tau {\cal G}} \cr {\scriptstyle f} \downarrow \uparrow {\scriptstyle g} \cr {\bar{\cal W}{\cal G} \otimes_\tau {\cal G}} \cr \end{aligned} $$ there is an induced contraction $$\chi_{\otimes t}: {\bar{\cal W}{\cal G} \otimes_\tau {\cal G}} \longrightarrow {\bar{\cal W}{\cal G} \otimes_\tau {\cal G}}, $$ defined by $\chi_{\otimes t}= f \circ \chi_{\times\tau} \circ g$. This implies that the twisted tensor product is also contractible over $\mathbb{Z}$.
The two functions of Kenzo for this construction are:
smgr-crts-contraction smgr [Function]
Return the homotopy morphism corresponding to the contraction $\chi_{\times\tau}$. The lisp definition is:
commonlisp
(defun SMGR-CRTS-CONTRACTION (smgr)
(the morphism
(build-mrph
:sorc (fibration-total (smgr-fibration smgr))
:trgt (fibration-total (smgr-fibration smgr))
:degr +1
:intr (smgr-crts-contraction-intr (bspn smgr))
:strt :gnrt
:orgn `(smgr-crts-contraction ,smgr))))
At execution time, the work is essentialy done by the function put in the slot :intr. This function is itself built by the internal function smgr-crts-contraction-intr requiring as argument the base point of the simplicial group.
smgr-tnpr-contraction smgr [Function]
Return the induced morphism $\chi_{\otimes t}= f \circ \chi_{\times\tau} \circ g$, where $f$ and $g$ are built by the Brown reduction. The lisp definition is:
commonlisp
(defun SMGR-TNPR-CONTRACTION (smgr
&aux (fibration (smgr-fibration smgr))
(brown (brown-reduction fibration))
(f (f brown))
(g (g brown))
(chi (smgr-crts-contraction smgr)))
(the morphism
(i-cmps f chi g)))
Examples¶
It is an instructive exercise to build and check the reduction over $\mathbb{Z}$ of the space
${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$ as we did in the loop spaces fibrations chapter.
$$
\begin{aligned}
\bar{\cal W}{\cal G} \times_\tau {\cal G} & \qquad\stackrel{\chi_{\times t}}{\longrightarrow} &
{}^s\bar{\cal W}{\cal G} \times_\tau {\cal G} \cr
{\scriptstyle aug} \downarrow \uparrow {\scriptstyle coaug} \cr
{{\cal C}_*(\mathbb{Z})} \quad& & \cr
\end{aligned}
$$
In this case, the two homomorphisms $f$ and $g$ of the reduction, are respectively the augmentation and coaugmentation morphisms. We define both morphisms according to our previous example based on $K(\mathbb{Z},1)$. The unit chain complex corresponding to $\mathbb{Z}$ is built by the function z-chcm; its unique generator in degree 0 is :z-gnrt.
In [13]:
(def k1 (k-z-1))
Out[13]:
[K1 Abelian-Simplicial-Group]
In [14]:
(def tw (smgr-fibration k1))
Out[14]:
[K25 Fibration K13 -> K1]
In [15]:
(def tt (fibration-total tw))
Out[15]:
[K31 Kan-Simplicial-Set]
The base point of the total space, needed for the coaugmentation, may be obtained by either of the following statements:
In [16]:
(bspn tt)
Out[16]:
<CrPr - <<GBar>> - NIL>
In [17]:
(crpr 0 +null-gbar+ 0 nil)
Out[17]:
<CrPr - <<GBar>> - NIL>
In [18]:
(def aug (build-mrph :sorc (fibration-total(smgr-fibration(k-z-1))) :trgt (z-chcm) :degr 0 :intr #'(lambda (degr gnrt) (declare (ignore gnrt)) (if (zerop degr) (term-cmbn 0 1 :z-gnrt) (zero-cmbn degr))) :strt :gnrt :orgn '(aug-fibr-tot-smgr-fibr-k-z-1) ))
Out[18]:
[K62 Cohomology-Class on K31 of degree 0]
In [19]:
(def coaug (build-mrph :sorc (z-chcm) :trgt (fibration-total(smgr-fibration(k-z-1))) :degr 0 :intr #'(lambda (degr gnrt) (declare (ignore gnrt)) (if (zerop degr) (term-cmbn 0 1 (crpr 0 +null-gbar+ 0 '())) (zero-cmbn degr))) :strt :gnrt :orgn '(coaug-fibr-tot-smgr-fibr-k-z-1) ))
SIMPLE-STYLE-WARNING:
#<SB-INT:SIMPLE-STYLE-WARNING "undefined ~(~A~): ~S" {1005EBEC83}>
Out[19]:
[K63 Morphism (degree 0): K60 -> K31]
In [20]:
(? aug 0 (bspn tt))
Out[20]:
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
In [21]:
(? coaug 0 :z-gnrt)
Out[21]:
----------------------------------------------------------------------{CMBN 0}
<1 * <CrPr - <<GBar>> - NIL>>
------------------------------------------------------------------------------
In [22]:
(def chi-x-tau (smgr-crts-contraction (k-z-1)))
Out[22]:
[K64 Morphism (degree 1): K31 -> K31]
Let us apply the contraction morphism upon some simplices:
In [23]:
(setf *tc* (cmbn 0 1 (crpr 0 +null-gbar+ 0 '())))
Out[23]:
----------------------------------------------------------------------{CMBN 0}
<1 * <CrPr - <<GBar>> - NIL>>
------------------------------------------------------------------------------
In [24]:
(? chi-x-tau *)
Out[24]:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
In [25]:
(setf *tc* (cmbn 4 1 (crpr 0 (gbar 4 0 '(10 11 12) 0 '(20 21) 0 '(30) 0 '()) 0 '(2 4 6 8))))
Out[25]:
----------------------------------------------------------------------{CMBN 4}
<1 * <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>>
------------------------------------------------------------------------------
In [26]:
(? chi-x-tau *)
Out[26]:
----------------------------------------------------------------------{CMBN 5}
<-1 * <CrPr - <<GBar<- (2 4 6 8)><- (10 11 12)><- (20 21)><- (30)><- NIL>>> 4-3-2-1-0 NIL>>
------------------------------------------------------------------------------
In [27]:
(setf *tc* (cmbn 3 1 (crpr 0 (gbar 3 0 '(10 11) 0 '(20) 0 '()) 0 '(2 4 6))))
Out[27]:
----------------------------------------------------------------------{CMBN 3}
<1 * <CrPr - <<GBar<- (10 11)><- (20)><- NIL>>> - (2 4 6)>>
------------------------------------------------------------------------------
In [28]:
(? chi-x-tau *)
Out[28]:
----------------------------------------------------------------------{CMBN 4}
<1 * <CrPr - <<GBar<- (2 4 6)><- (10 11)><- (20)><- NIL>>> 3-2-1-0 NIL>>
------------------------------------------------------------------------------
We may build now the reduction over $\mathbb{Z}$ and test it upon the various simplices above:
In [29]:
(def rdct (build-rdct :f aug :g coaug :h chi-x-tau :orgn '(reduction-tt-z)))
Out[29]:
[K65 Reduction K31 => K60]
In [30]:
(pre-check-rdct rdct)
---done---
Out[30]:
NIL
In [31]:
(setf *tc* (cmbn 0 1 (crpr 0 +null-gbar+ 0 '())))
Out[31]:
----------------------------------------------------------------------{CMBN 0}
<1 * <CrPr - <<GBar>> - NIL>>
------------------------------------------------------------------------------
In [32]:
(setf *bc* (cmbn 0 1 :z-gnrt))
Out[32]:
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
In [33]:
(check-rdct-no-wait)
*TC* =>
----------------------------------------------------------------------{CMBN 0}
<1 * <CrPr - <<GBar>> - NIL>>
------------------------------------------------------------------------------
*BC* =>
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
Checking *TDD* = 0
Result:
----------------------------------------------------------------------{CMBN -2}
------------------------------------------------------------------------------
Checking *BDD* = 0
Result:
----------------------------------------------------------------------{CMBN -2}
------------------------------------------------------------------------------
Checking *DF-FD* = 0
Result:
----------------------------------------------------------------------{CMBN -1}
------------------------------------------------------------------------------
Checking *DG-GD* = 0
Result:
----------------------------------------------------------------------{CMBN -1}
------------------------------------------------------------------------------
Checking *ID-FG* = 0
Result:
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
Checking *ID-GF-DH-HD* = 0
Result:
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
Checking *HH* = 0
Result:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Checking *FH* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
Checking *HG* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
---done---
Out[33]:
NIL
In [34]:
(setf *tc* (cmbn 4 1 (crpr 0 (gbar 4 0 '(10 11 12) 0 '(20 21) 0 '(30) 0 '()) 0 '(2 4 6 8))))
Out[34]:
----------------------------------------------------------------------{CMBN 4}
<1 * <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>>
------------------------------------------------------------------------------
In [35]:
(check-rdct-no-wait)
*TC* =>
----------------------------------------------------------------------{CMBN 4}
<1 * <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>>
------------------------------------------------------------------------------
*BC* =>
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
Checking *TDD* = 0
Result:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Checking *BDD* = 0
Result:
----------------------------------------------------------------------{CMBN -2}
------------------------------------------------------------------------------
Checking *DF-FD* = 0
Result:
----------------------------------------------------------------------{CMBN 3}
------------------------------------------------------------------------------
Checking *DG-GD* = 0
Result:
----------------------------------------------------------------------{CMBN -1}
------------------------------------------------------------------------------
Checking *ID-FG* = 0
Result:
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
Checking *ID-GF-DH-HD* = 0
Result:
----------------------------------------------------------------------{CMBN 4}
------------------------------------------------------------------------------
Checking *HH* = 0
Result:
----------------------------------------------------------------------{CMBN 6}
------------------------------------------------------------------------------
Checking *FH* = 0
Result:
----------------------------------------------------------------------{CMBN 5}
------------------------------------------------------------------------------
Checking *HG* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
---done---
Out[35]:
NIL
In [36]:
(setf *tc* (cmbn 3 1 (crpr 0 (gbar 3 0 '(10 11) 0 '(20) 0 '()) 0 '(2 4 6))))
Out[36]:
----------------------------------------------------------------------{CMBN 3}
<1 * <CrPr - <<GBar<- (10 11)><- (20)><- NIL>>> - (2 4 6)>>
------------------------------------------------------------------------------
In [37]:
(check-rdct-no-wait)
*TC* =>
----------------------------------------------------------------------{CMBN 3}
<1 * <CrPr - <<GBar<- (10 11)><- (20)><- NIL>>> - (2 4 6)>>
------------------------------------------------------------------------------
*BC* =>
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
Checking *TDD* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
Checking *BDD* = 0
Result:
----------------------------------------------------------------------{CMBN -2}
------------------------------------------------------------------------------
Checking *DF-FD* = 0
Result:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Checking *DG-GD* = 0
Result:
----------------------------------------------------------------------{CMBN -1}
------------------------------------------------------------------------------
Checking *ID-FG* = 0
Result:
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
Checking *ID-GF-DH-HD* = 0
Result:
----------------------------------------------------------------------{CMBN 3}
------------------------------------------------------------------------------
Checking *HH* = 0
Result:
----------------------------------------------------------------------{CMBN 5}
------------------------------------------------------------------------------
Checking *FH* = 0
Result:
----------------------------------------------------------------------{CMBN 4}
------------------------------------------------------------------------------
Checking *HG* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
---done---
Out[37]:
NIL
We build now the reduction over $\mathbb{Z}$ of the twisted tensor product. The contraction is obtained by a call to the function smgr-tnpr-contraction.
$$
\begin{aligned}
\bar{\cal W}{\cal G} \otimes_\tau {\cal G} & \qquad\stackrel{\chi_{\otimes t}}{\longrightarrow} &
{}^s\bar{\cal W}{\cal G} \otimes_\tau {\cal G} \cr
{\scriptstyle aug} \downarrow \uparrow {\scriptstyle coaug} \cr
{{\cal C}_*(\mathbb{Z})}\quad \cr
\end{aligned}
$$
Of course, we have to define the augmentation and coaugmentation morphisms adapted to this new example. The user will note that the twisted tensor product chain complex may be obtained from the slot :sorc of the contration $\chi_{\otimes t}$.
In [38]:
(def chi-t-tau (smgr-tnpr-contraction (k-z-1)))
Out[38]:
[K88 Morphism (degree 1): K57 -> K57]
In [39]:
(def aug-t (build-mrph :sorc (sorc chi-t-tau) :trgt (z-chcm) :degr 0 :intr #'(lambda (degr gnrt) (declare (ignore gnrt)) (if (zerop degr) (term-cmbn 0 1 :z-gnrt) (zero-cmbn degr))) :strt :gnrt :orgn '(aug-t-fibr-tot-smgr-fibr-k-z-1) ))
Out[39]:
[K89 Cohomology-Class on K57 of degree 0]
In [40]:
(def coaug-t (build-mrph :sorc (z-chcm) :trgt (sorc chi-t-tau) :degr 0 :intr #'(lambda (degr gnrt) (declare (ignore gnrt)) (if (zerop degr) (term-cmbn 0 1 (tnpr 0 +null-gbar+ 0 '())) (zero-cmbn degr))) :strt :gnrt :orgn '(coaug-t-fibr-tot-smgr-fibr-k-z-1) ))
Out[40]:
[K90 Morphism (degree 0): K60 -> K57]
In [41]:
(def rdct-t (build-rdct :f aug-t :g coaug-t :h chi-t-tau :orgn '(reduction-t-tt-z)))
Out[41]:
[K91 Reduction K57 => K60]
In [42]:
(pre-check-rdct rdct-t)
---done---
Out[42]:
NIL
In [43]:
(setf *tc* (cmbn 0 1 (tnpr 0 +null-gbar+ 0 '())))
Out[43]:
----------------------------------------------------------------------{CMBN 0}
<1 * <TnPr <<GBar>> NIL>>
------------------------------------------------------------------------------
In [44]:
(setf *bc* (cmbn 0 1 :z-gnrt))
Out[44]:
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
In [45]:
(check-rdct-no-wait)
*TC* =>
----------------------------------------------------------------------{CMBN 0}
<1 * <TnPr <<GBar>> NIL>>
------------------------------------------------------------------------------
*BC* =>
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
Checking *TDD* = 0
Result:
----------------------------------------------------------------------{CMBN -2}
------------------------------------------------------------------------------
Checking *BDD* = 0
Result:
----------------------------------------------------------------------{CMBN -2}
------------------------------------------------------------------------------
Checking *DF-FD* = 0
Result:
----------------------------------------------------------------------{CMBN -1}
------------------------------------------------------------------------------
Checking *DG-GD* = 0
Result:
----------------------------------------------------------------------{CMBN -1}
------------------------------------------------------------------------------
Checking *ID-FG* = 0
Result:
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
Checking *ID-GF-DH-HD* = 0
Result:
----------------------------------------------------------------------{CMBN 0}
------------------------------------------------------------------------------
Checking *HH* = 0
Result:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
Checking *FH* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
Checking *HG* = 0
Result:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
---done---
Out[45]:
NIL
The check is validated also for the following simplices. We see also that, if we apply the contraction $\chi_{\otimes t}$ upon various simplices outside the base fiber, the result is in general non--null. The nullity of this contraction outside the base fiber, which we have verified experimentally in many cases in loop spaces fibration is not verified in simplicial group fibrations.
In [46]:
(setf *tc* (cmbn 3 1 (tnpr 0 +null-gbar+ 3 '(1 10 100))))
Out[46]:
----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr <<GBar>> (1 10 100)>>
------------------------------------------------------------------------------
In [47]:
(? chi-t-tau *tc*)
Out[47]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<- (1 10 100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [48]:
(setf *tc* (cmbn 3 1 (tnpr 2 (gbar 2 0 '(1) 0 '()) 1 '(10))))
Out[48]:
----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr <<GBar<- (1)><- NIL>>> (10)>>
------------------------------------------------------------------------------
In [49]:
(? chi-t-tau *tc*)
Out[49]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<1-0 (10)><1-0 NIL><- (1)><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<2-0 (10)><1 (1)><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<2-1 (10)><0 (1)><0 NIL><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [50]:
(setf *tc* (cmbn 3 1 (tnpr 3 (gbar 3 0 '(1 10) 0 '(100) 0 '()) 0 '())))
Out[50]:
----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr <<GBar<- (1 10)><- (100)><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [51]:
(? chi-t-tau *tc*)
Out[51]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<2 (1 10)><1 (100)><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<2-1 (11)><0 (100)><0 NIL><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [52]:
(setf *tc* (cmbn 4 1 (tnpr 0 +null-gbar+ 4 '(1 10 100 1000))))
Out[52]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar>> (1 10 100 1000)>>
------------------------------------------------------------------------------
In [53]:
(? chi-t-tau *tc*)
Out[53]:
----------------------------------------------------------------------{CMBN 5}
<-1 * <TnPr <<GBar<- (1 10 100 1000)><2-1-0 NIL><1-0 NIL><0 NIL><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [54]:
(setf *tc* (cmbn 4 1 (tnpr 2 (gbar 2 0 '(1) 0 '()) 2 '(10 100))))
Out[54]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<- (1)><- NIL>>> (10 100)>>
------------------------------------------------------------------------------
In [55]:
(? chi-t-tau *tc*)
Out[55]:
----------------------------------------------------------------------{CMBN 5}
<-1 * <TnPr <<GBar<1-0 (10 100)><2-1-0 NIL><1-0 NIL><- (1)><- NIL>>> NIL>>
<1 * <TnPr <<GBar<2-0 (10 100)><2-1-0 NIL><1 (1)><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<2-1 (10 100)><2-1-0 NIL><0 (1)><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3-0 (10 100)><2-1 (1)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3-1 (10 100)><2-0 (1)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3-2 (10 100)><1-0 (1)><1-0 NIL><0 NIL><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [56]:
(setf *tc* (cmbn 4 1 (tnpr 3 (gbar 3 0 '(1 10) 0 '(100) 0 '()) 1 '(1000))))
Out[56]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<- (1 10)><- (100)><- NIL>>> (1000)>>
------------------------------------------------------------------------------
In [57]:
(? chi-t-tau *tc*)
Out[57]:
----------------------------------------------------------------------{CMBN 5}
<-1 * <TnPr <<GBar<2 (1 10 1000)><2-1-0 NIL><1 (100)><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<2-1 (11 1000)><2-1-0 NIL><0 (100)><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<2-1-0 (1000)><2-1-0 NIL><- (1 10)><- (100)><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3 (1 10 1000)><2-1 (100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3 (1 1000 10)><2-1 (100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3 (1000 1 10)><2-0 (100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3-1 (11 1000)><2-0 (100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3-1-0 (1000)><2 (1 10)><1-0 NIL><- (100)><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3-2 (11 1000)><1-0 (100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3-2 (1000 11)><1-0 (100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
... ...
------------------------------------------------------------------------------
In [58]:
(setf *tc* (cmbn 4 1 (tnpr 4 (gbar 4 0 '(1 10 100) 0 '(1000 10000) 0 '(100000) 0 '()) 0 '())))
Out[58]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<- (1 10 100)><- (1000 10000)><- (100000)><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [59]:
(? chi-t-tau *tc*)
Out[59]:
----------------------------------------------------------------------{CMBN 5}
<1 * <TnPr <<GBar<2 (1001 10010 100)><2-1-0 NIL><1 (100000)><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<2-1 (11011 100)><2-1-0 NIL><0 (100000)><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3 (1 10 100)><2 (1000 10000)><1-0 NIL><- (100000)><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3 (111 1000 10000)><2-0 (100000)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3 (1001 110 10000)><2-1 (100000)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3 (1001 10010 100)><2-1 (100000)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3-1 (11011 100)><2-0 (100000)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3-2 (1 110)><1 (1000 10000)><1 (100000)><0 NIL><- NIL>>> NIL>>
<1 * <TnPr <<GBar<3-2 (111 11000)><1-0 (100000)><1-0 NIL><0 NIL><- NIL>>> NIL>>
<-1 * <TnPr <<GBar<3-2 (11011 100)><1-0 (100000)><1-0 NIL><0 NIL><- NIL>>> NIL>>
... ...
------------------------------------------------------------------------------
Let us verify once again the unproved conjecture signaled in the loop spaces fibrations chapter. In fact, if the conjecture is true, it is sufficient to prove it with $\bar{\Delta}$. But the following is also a good test of verifying all the involved machinery. If we take ${\cal G}= K(\mathbb{Z},1)$, the space ${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$ is 0-reduced. Let us consider its loop space and the induced contraction $\chi_{\otimes t}$ over ${\cal WG} \otimes_\tau \Omega{\cal WG}$. ${\cal WG}$ and the contraction $\chi_{\otimes t}$ are built by the two following statements:
In [60]:
(def wgkz1 (fibration-total (smgr-fibration(k-z-1))))
Out[60]:
[K31 Kan-Simplicial-Set]
In [61]:
(def chi-t-tau (tnpr-contraction wgkz1))
Out[61]:
[K159 Morphism (degree 1): K154 -> K154]
Let us apply successively the contraction upon a simplex belonging to the base fiber (gnrt0) and a simplex out of the base fiber (gnrt1). We see that in the second case we obtain a null result. We verify also that $\chi_{\otimes t}\circ \chi_{\otimes t}=0$.
In [62]:
(def gnrt0 (tnpr 0 (crpr 0 +null-gbar+ 0 nil) 0 (loop3 0 (crpr 0 (gbar 2 0 '(1) 0 '(0) ) 0 '(1)) -3)))
Out[62]:
<TnPr <CrPr - <<GBar>> - NIL> <<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\-3]>>>
In [63]:
(? chi-t-tau 0 gnrt0)
Out[63]:
----------------------------------------------------------------------{CMBN 1}
<1 * <TnPr <CrPr - <<GBar<- (1)><- (0)>>> - (1)> <<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\-3]>>>>
<1 * <TnPr <CrPr - <<GBar<- (1)><- (0)>>> - (1)> <<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\-2]>>>>
<1 * <TnPr <CrPr - <<GBar<- (1)><- (0)>>> - (1)> <<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\-1]>>>>
------------------------------------------------------------------------------
In [64]:
(? chi-t-tau *)
Out[64]:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
In [65]:
(def gnrt1 (tnpr 4 (crpr 0 (gbar 4 0 '(10 11 12) 0 '(20 21) 0 '(30) 0 '()) 0 '(2 4 6 8)) 0 (loop3 0 (crpr 0 (gbar 2 0 '(1) 0 '(0) ) 0 '(1)) 2)))
Out[65]:
<TnPr <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)> <<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\2]>>>
In [66]:
(? chi-t-tau 4 gnrt1)
Out[66]:
----------------------------------------------------------------------{CMBN 5}
------------------------------------------------------------------------------
At last we verify that the total space of the fibration of $K(\mathbb{Z},1)$ is contractible.
In [67]:
(homology (fibration-total(smgr-fibration (k-z-1))) 0 10)
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 0) : <TnPr <<Abar>> *> End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 1) : <TnPr <<Abar>> S1> End of computing. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2016-03-25, 1h 11m 7s. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 1) : <TnPr <<Abar>> S1> End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 2) : <TnPr <<Abar[2 S1]>> *> End of computing. Homology in dimension 1 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 7s. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 2) : <TnPr <<Abar[2 S1]>> *> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 3) : <TnPr <<Abar[2 S1]>> S1> End of computing. Homology in dimension 2 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 7s. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 3) : <TnPr <<Abar[2 S1]>> S1> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 4) : <TnPr <<Abar[2 S1][2 S1]>> *> End of computing. Homology in dimension 3 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 7s. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 4) : <TnPr <<Abar[2 S1][2 S1]>> *> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 5) : <TnPr <<Abar[2 S1][2 S1]>> S1> End of computing. Homology in dimension 4 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 7s. Computing boundary-matrix in dimension 5. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 5) : <TnPr <<Abar[2 S1][2 S1]>> S1> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 6) : <TnPr <<Abar[2 S1][2 S1][2 S1]>> *> End of computing. Homology in dimension 5 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 7s. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 6) : <TnPr <<Abar[2 S1][2 S1][2 S1]>> *> End of computing. Computing boundary-matrix in dimension 7. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 7) : <TnPr <<Abar[2 S1][2 S1][2 S1]>> S1> End of computing. Homology in dimension 6 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 7s. Computing boundary-matrix in dimension 7. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 7) : <TnPr <<Abar[2 S1][2 S1][2 S1]>> S1> End of computing. Computing boundary-matrix in dimension 8. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 7s. Computing the boundary of the generator 1/1 (dimension 8) : <TnPr <<Abar[2 S1][2 S1][2 S1][2 S1]>> *> End of computing. Homology in dimension 7 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. Computing boundary-matrix in dimension 8. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/1 (dimension 8) : <TnPr <<Abar[2 S1][2 S1][2 S1][2 S1]>> *> End of computing. Computing boundary-matrix in dimension 9. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/1 (dimension 9) : <TnPr <<Abar[2 S1][2 S1][2 S1][2 S1]>> S1> End of computing. Homology in dimension 8 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. Computing boundary-matrix in dimension 9. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/1 (dimension 9) : <TnPr <<Abar[2 S1][2 S1][2 S1][2 S1]>> S1> End of computing. Computing boundary-matrix in dimension 10. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/1 (dimension 10) : <TnPr <<Abar[2 S1][2 S1][2 S1][2 S1][2 S1]>> *> End of computing. Homology in dimension 9 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s.
Out[67]:
NIL
Another example is the following: we build the canonical fibration of $\bar{\cal W}^2K(\mathbb{Z}_2, 1)$ and verify that its fibration total space is contractible:
In [68]:
(def k22 (classifying-space(classifying-space(k-z2-1))))
Out[68]:
[K327 Abelian-Simplicial-Group]
In [69]:
(def fb (smgr-fibration k22))
Out[69]:
[K351 Fibration K339 -> K327]
In [70]:
(def tt2 (fibration-total fb))
Out[70]:
[K357 Kan-Simplicial-Set]
In [71]:
(homology tt2 0 10)
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/1 (dimension 0) : <TnPr <<Abar>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 0. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. 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 -> 2016-03-25, 1h 11m 8s. 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 -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/1 (dimension 3) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. Homology in dimension 2 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/1 (dimension 3) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 2. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/2 (dimension 4) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/2 (dimension 4) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Homology in dimension 3 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. Computing boundary-matrix in dimension 4. Rank of the source-module : 2. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/2 (dimension 4) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/2 (dimension 4) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 3. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/3 (dimension 5) : <TnPr <<Abar>> <<Abar[5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/3 (dimension 5) : <TnPr <<Abar>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/3 (dimension 5) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. Homology in dimension 4 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. Computing boundary-matrix in dimension 5. Rank of the source-module : 3. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/3 (dimension 5) : <TnPr <<Abar>> <<Abar[5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/3 (dimension 5) : <TnPr <<Abar>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/3 (dimension 5) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 6. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/6 (dimension 6) : <TnPr <<Abar>> <<Abar[6 <<Abar[5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/6 (dimension 6) : <TnPr <<Abar>> <<Abar[6 <<Abar[2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/6 (dimension 6) : <TnPr <<Abar>> <<Abar[6 <<Abar[3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 4/6 (dimension 6) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 5/6 (dimension 6) : <TnPr <<Abar[6 <<Abar[5 <<Abar[4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 6/6 (dimension 6) : <TnPr <<Abar[6 <<Abar[5 <<Abar[2 1][2 1]>>]>>]>> <<Abar>>> End of computing. Homology in dimension 5 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. Computing boundary-matrix in dimension 6. Rank of the source-module : 6. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/6 (dimension 6) : <TnPr <<Abar>> <<Abar[6 <<Abar[5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/6 (dimension 6) : <TnPr <<Abar>> <<Abar[6 <<Abar[2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/6 (dimension 6) : <TnPr <<Abar>> <<Abar[6 <<Abar[3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 4/6 (dimension 6) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 5/6 (dimension 6) : <TnPr <<Abar[6 <<Abar[5 <<Abar[4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 6/6 (dimension 6) : <TnPr <<Abar[6 <<Abar[5 <<Abar[2 1][2 1]>>]>>]>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 7. Rank of the source-module : 12. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[6 5]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[2 1][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[3 2][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 4/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[4 3][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 5/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[2 1][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 6/12 (dimension 7) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 7/12 (dimension 7) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 8/12 (dimension 7) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 9/12 (dimension 7) : <TnPr <<Abar[7 <<Abar[6 <<Abar[5 4]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 10/12 (dimension 7) : <TnPr <<Abar[7 <<Abar[6 <<Abar[2 1][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 11/12 (dimension 7) : <TnPr <<Abar[7 <<Abar[6 <<Abar[3 2][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 12/12 (dimension 7) : <TnPr <<Abar[7 <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Homology in dimension 6 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. Computing boundary-matrix in dimension 7. Rank of the source-module : 12. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[6 5]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[2 1][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[3 2][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 4/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[4 3][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 5/12 (dimension 7) : <TnPr <<Abar>> <<Abar[7 <<Abar[2 1][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 6/12 (dimension 7) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 7/12 (dimension 7) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 8/12 (dimension 7) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 9/12 (dimension 7) : <TnPr <<Abar[7 <<Abar[6 <<Abar[5 4]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 10/12 (dimension 7) : <TnPr <<Abar[7 <<Abar[6 <<Abar[2 1][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 11/12 (dimension 7) : <TnPr <<Abar[7 <<Abar[6 <<Abar[3 2][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 12/12 (dimension 7) : <TnPr <<Abar[7 <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 8. Rank of the source-module : 23. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[7 6]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[2 1][5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[3 2][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 4/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[4 3][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 5/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[5 4][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 6/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[2 1][2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 7/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[2 1][3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 8/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[3 2][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 9/23 (dimension 8) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 10/23 (dimension 8) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 11/23 (dimension 8) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 12/23 (dimension 8) : <TnPr <<Abar>> <<Abar[5 <<Abar[4 3]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 13/23 (dimension 8) : <TnPr <<Abar>> <<Abar[5 <<Abar[2 1][2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 14/23 (dimension 8) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 15/23 (dimension 8) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 16/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[6 5]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 17/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[2 1][4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 18/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[3 2][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 19/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[4 3][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 20/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[2 1][2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 21/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[3 <<Abar[2 1]>>][4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 22/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[4 <<Abar[3 2]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 23/23 (dimension 8) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>][4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Homology in dimension 7 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 8s. Computing boundary-matrix in dimension 8. Rank of the source-module : 23. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[7 6]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[2 1][5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[3 2][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 4/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[4 3][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 5/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[5 4][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 6/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[2 1][2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 7/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[2 1][3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 8/23 (dimension 8) : <TnPr <<Abar>> <<Abar[8 <<Abar[3 2][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 9/23 (dimension 8) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 10/23 (dimension 8) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 11/23 (dimension 8) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 12/23 (dimension 8) : <TnPr <<Abar>> <<Abar[5 <<Abar[4 3]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 13/23 (dimension 8) : <TnPr <<Abar>> <<Abar[5 <<Abar[2 1][2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 14/23 (dimension 8) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 15/23 (dimension 8) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 16/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[6 5]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 17/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[2 1][4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 18/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[3 2][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 19/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[4 3][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 20/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[7 <<Abar[2 1][2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 21/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[3 <<Abar[2 1]>>][4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 22/23 (dimension 8) : <TnPr <<Abar[8 <<Abar[4 <<Abar[3 2]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 23/23 (dimension 8) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>][4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 9. Rank of the source-module : 44. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 1/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[8 7]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 2/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][6 5]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 3/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[3 2][5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 4/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[4 3][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 5/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[5 4][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 6/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[6 5][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 7/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][2 1][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 8/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][3 2][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 9/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][4 3][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 8s. Computing the boundary of the generator 10/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[3 2][2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 11/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[3 2][3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 12/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[4 3][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 13/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][2 1][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 14/44 (dimension 9) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][6 <<Abar[5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 15/44 (dimension 9) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][6 <<Abar[2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 16/44 (dimension 9) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][6 <<Abar[3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 17/44 (dimension 9) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 18/44 (dimension 9) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 19/44 (dimension 9) : <TnPr <<Abar>> <<Abar[5 <<Abar[4 3]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 20/44 (dimension 9) : <TnPr <<Abar>> <<Abar[5 <<Abar[2 1][2 1]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 21/44 (dimension 9) : <TnPr <<Abar>> <<Abar[6 <<Abar[5 4]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 9s. Computing the boundary of the generator 22/44 (dimension 9) : <TnPr <<Abar>> <<Abar[6 <<Abar[2 1][3 2]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 23/44 (dimension 9) : <TnPr <<Abar>> <<Abar[6 <<Abar[3 2][2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 24/44 (dimension 9) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 25/44 (dimension 9) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 26/44 (dimension 9) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 27/44 (dimension 9) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar[4 <<Abar[3 2]>>]>>> *CR* -- Cut = 6.635904e-5 -- N = 437 End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 28/44 (dimension 9) : <TnPr <<Abar[6 <<Abar[5 <<Abar[4 3]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 29/44 (dimension 9) : <TnPr <<Abar[6 <<Abar[5 <<Abar[2 1][2 1]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 30/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[7 6]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 31/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[2 1][5 4]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 32/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[3 2][4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 33/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[4 3][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 34/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[5 4][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 35/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[2 1][2 1][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 36/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[2 1][3 2][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 10s. Computing the boundary of the generator 37/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[3 2][2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 38/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[3 <<Abar[2 1]>>][5 <<Abar[4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 39/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[3 <<Abar[2 1]>>][5 <<Abar[2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 40/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[4 <<Abar[3 2]>>][4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 41/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[5 <<Abar[4 3]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 42/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[5 <<Abar[2 1][2 1]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 43/44 (dimension 9) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>][5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 44/44 (dimension 9) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>][4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Homology in dimension 8 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 11s. Computing boundary-matrix in dimension 9. Rank of the source-module : 44. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 1/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[8 7]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 2/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][6 5]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 3/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[3 2][5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 4/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[4 3][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 5/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[5 4][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 6/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[6 5][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 7/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][2 1][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 8/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][3 2][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 9/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][4 3][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 10/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[3 2][2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 11/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[3 2][3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 12/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[4 3][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 13/44 (dimension 9) : <TnPr <<Abar>> <<Abar[9 <<Abar[2 1][2 1][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 14/44 (dimension 9) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][6 <<Abar[5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 11s. Computing the boundary of the generator 15/44 (dimension 9) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][6 <<Abar[2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 16/44 (dimension 9) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][6 <<Abar[3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 17/44 (dimension 9) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 18/44 (dimension 9) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 19/44 (dimension 9) : <TnPr <<Abar>> <<Abar[5 <<Abar[4 3]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 20/44 (dimension 9) : <TnPr <<Abar>> <<Abar[5 <<Abar[2 1][2 1]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 21/44 (dimension 9) : <TnPr <<Abar>> <<Abar[6 <<Abar[5 4]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 22/44 (dimension 9) : <TnPr <<Abar>> <<Abar[6 <<Abar[2 1][3 2]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 23/44 (dimension 9) : <TnPr <<Abar>> <<Abar[6 <<Abar[3 2][2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 24/44 (dimension 9) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 25/44 (dimension 9) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 26/44 (dimension 9) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 27/44 (dimension 9) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 28/44 (dimension 9) : <TnPr <<Abar[6 <<Abar[5 <<Abar[4 3]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 29/44 (dimension 9) : <TnPr <<Abar[6 <<Abar[5 <<Abar[2 1][2 1]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 30/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[7 6]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 31/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[2 1][5 4]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 32/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[3 2][4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 33/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[4 3][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 34/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[5 4][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 35/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[2 1][2 1][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 36/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[2 1][3 2][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 37/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[8 <<Abar[3 2][2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 38/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[3 <<Abar[2 1]>>][5 <<Abar[4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 39/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[3 <<Abar[2 1]>>][5 <<Abar[2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 40/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[4 <<Abar[3 2]>>][4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 41/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[5 <<Abar[4 3]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 42/44 (dimension 9) : <TnPr <<Abar[9 <<Abar[5 <<Abar[2 1][2 1]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 43/44 (dimension 9) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>][5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 44/44 (dimension 9) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>][4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Computing boundary-matrix in dimension 10. Rank of the source-module : 85. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 1/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[9 8]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 2/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[2 1][7 6]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 3/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[3 2][6 5]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 4/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[4 3][5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 5/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[5 4][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 6/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[6 5][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 7/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[7 6][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 12s. Computing the boundary of the generator 8/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[2 1][2 1][5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 13s. Computing the boundary of the generator 9/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[2 1][3 2][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 13s. Computing the boundary of the generator 10/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[2 1][4 3][3 2]>>]>>> *CR* -- Cut = 1.2575794e-4 -- N = 845 End of computing. ;; Clock -> 2016-03-25, 1h 11m 13s. Computing the boundary of the generator 11/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[2 1][5 4][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 13s. Computing the boundary of the generator 12/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[3 2][2 1][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 14s. Computing the boundary of the generator 13/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[3 2][3 2][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 14s. Computing the boundary of the generator 14/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[3 2][4 3][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 14s. Computing the boundary of the generator 15/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[4 3][2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 15s. Computing the boundary of the generator 16/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[4 3][3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 15s. Computing the boundary of the generator 17/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[5 4][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 15s. Computing the boundary of the generator 18/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[2 1][2 1][2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 16s. Computing the boundary of the generator 19/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[2 1][2 1][3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 17s. Computing the boundary of the generator 20/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[2 1][3 2][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 18s. Computing the boundary of the generator 21/85 (dimension 10) : <TnPr <<Abar>> <<Abar[10 <<Abar[3 2][2 1][2 1][2 1]>>]>>> *CR* -- Cut = 2.6375928e-4 -- N = 1705 End of computing. ;; Clock -> 2016-03-25, 1h 11m 18s. Computing the boundary of the generator 22/85 (dimension 10) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][7 <<Abar[6 5]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 18s. Computing the boundary of the generator 23/85 (dimension 10) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][7 <<Abar[2 1][4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 19s. Computing the boundary of the generator 24/85 (dimension 10) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][7 <<Abar[3 2][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 19s. Computing the boundary of the generator 25/85 (dimension 10) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][7 <<Abar[4 3][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 19s. Computing the boundary of the generator 26/85 (dimension 10) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][7 <<Abar[2 1][2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 20s. Computing the boundary of the generator 27/85 (dimension 10) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][6 <<Abar[5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 20s. Computing the boundary of the generator 28/85 (dimension 10) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][6 <<Abar[2 1][3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 21s. Computing the boundary of the generator 29/85 (dimension 10) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][6 <<Abar[3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 21s. Computing the boundary of the generator 30/85 (dimension 10) : <TnPr <<Abar>> <<Abar[5 <<Abar[4 3]>>][5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 21s. Computing the boundary of the generator 31/85 (dimension 10) : <TnPr <<Abar>> <<Abar[5 <<Abar[4 3]>>][5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 22s. Computing the boundary of the generator 32/85 (dimension 10) : <TnPr <<Abar>> <<Abar[5 <<Abar[2 1][2 1]>>][5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 22s. Computing the boundary of the generator 33/85 (dimension 10) : <TnPr <<Abar>> <<Abar[5 <<Abar[2 1][2 1]>>][5 <<Abar[2 1][2 1]>>]>>> *CR* -- Cut = 3.9467352e-4 -- N = 2418 End of computing. ;; Clock -> 2016-03-25, 1h 11m 23s. Computing the boundary of the generator 34/85 (dimension 10) : <TnPr <<Abar>> <<Abar[6 <<Abar[5 4]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 23s. Computing the boundary of the generator 35/85 (dimension 10) : <TnPr <<Abar>> <<Abar[6 <<Abar[2 1][3 2]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 23s. Computing the boundary of the generator 36/85 (dimension 10) : <TnPr <<Abar>> <<Abar[6 <<Abar[3 2][2 1]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 24s. Computing the boundary of the generator 37/85 (dimension 10) : <TnPr <<Abar>> <<Abar[7 <<Abar[6 5]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 24s. Computing the boundary of the generator 38/85 (dimension 10) : <TnPr <<Abar>> <<Abar[7 <<Abar[2 1][4 3]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 24s. Computing the boundary of the generator 39/85 (dimension 10) : <TnPr <<Abar>> <<Abar[7 <<Abar[3 2][3 2]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 24s. Computing the boundary of the generator 40/85 (dimension 10) : <TnPr <<Abar>> <<Abar[7 <<Abar[4 3][2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 24s. Computing the boundary of the generator 41/85 (dimension 10) : <TnPr <<Abar>> <<Abar[7 <<Abar[2 1][2 1][2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 25s. Computing the boundary of the generator 42/85 (dimension 10) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>][4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 25s. Computing the boundary of the generator 43/85 (dimension 10) : <TnPr <<Abar>> <<Abar[3 <<Abar[2 1]>>][4 <<Abar[3 2]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 26s. Computing the boundary of the generator 44/85 (dimension 10) : <TnPr <<Abar>> <<Abar[4 <<Abar[3 2]>>][3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 26s. Computing the boundary of the generator 45/85 (dimension 10) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[6 <<Abar[5 4]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 26s. Computing the boundary of the generator 46/85 (dimension 10) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[6 <<Abar[2 1][3 2]>>]>>> *CR* -- Cut = 5.107059e-4 -- N = 3099 End of computing. ;; Clock -> 2016-03-25, 1h 11m 27s. Computing the boundary of the generator 47/85 (dimension 10) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[6 <<Abar[3 2][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 27s. Computing the boundary of the generator 48/85 (dimension 10) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 28s. Computing the boundary of the generator 49/85 (dimension 10) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar[5 <<Abar[4 3]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 28s. Computing the boundary of the generator 50/85 (dimension 10) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar[5 <<Abar[2 1][2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 28s. Computing the boundary of the generator 51/85 (dimension 10) : <TnPr <<Abar[6 <<Abar[5 <<Abar[4 3]>>]>>]>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 28s. Computing the boundary of the generator 52/85 (dimension 10) : <TnPr <<Abar[6 <<Abar[5 <<Abar[2 1][2 1]>>]>>]>> <<Abar[4 <<Abar[3 2]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 28s. Computing the boundary of the generator 53/85 (dimension 10) : <TnPr <<Abar[7 <<Abar[6 <<Abar[5 4]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 28s. Computing the boundary of the generator 54/85 (dimension 10) : <TnPr <<Abar[7 <<Abar[6 <<Abar[2 1][3 2]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 29s. Computing the boundary of the generator 55/85 (dimension 10) : <TnPr <<Abar[7 <<Abar[6 <<Abar[3 2][2 1]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 29s. Computing the boundary of the generator 56/85 (dimension 10) : <TnPr <<Abar[7 <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>]>> <<Abar[3 <<Abar[2 1]>>]>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 30s. Computing the boundary of the generator 57/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[8 7]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 30s. Computing the boundary of the generator 58/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[2 1][6 5]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 30s. Computing the boundary of the generator 59/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[3 2][5 4]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 30s. Computing the boundary of the generator 60/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[4 3][4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 30s. Computing the boundary of the generator 61/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[5 4][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 30s. Computing the boundary of the generator 62/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[6 5][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 30s. Computing the boundary of the generator 63/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[2 1][2 1][4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 31s. Computing the boundary of the generator 64/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[2 1][3 2][3 2]>>]>>]>> <<Abar>>> *CR* -- Cut = 6.035785e-4 -- N = 3821 End of computing. ;; Clock -> 2016-03-25, 1h 11m 31s. Computing the boundary of the generator 65/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[2 1][4 3][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 31s. Computing the boundary of the generator 66/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[3 2][2 1][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 32s. Computing the boundary of the generator 67/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[3 2][3 2][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 32s. Computing the boundary of the generator 68/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[4 3][2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 32s. Computing the boundary of the generator 69/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[9 <<Abar[2 1][2 1][2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 32s. Computing the boundary of the generator 70/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[3 <<Abar[2 1]>>][6 <<Abar[5 4]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 33s. Computing the boundary of the generator 71/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[3 <<Abar[2 1]>>][6 <<Abar[2 1][3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 33s. Computing the boundary of the generator 72/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[3 <<Abar[2 1]>>][6 <<Abar[3 2][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 34s. Computing the boundary of the generator 73/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[4 <<Abar[3 2]>>][5 <<Abar[4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 34s. Computing the boundary of the generator 74/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[4 <<Abar[3 2]>>][5 <<Abar[2 1][2 1]>>]>>]>> <<Abar>>> *CR* -- Cut = 6.9165323e-4 -- N = 4392 End of computing. ;; Clock -> 2016-03-25, 1h 11m 34s. Computing the boundary of the generator 75/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[5 <<Abar[4 3]>>][4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 34s. Computing the boundary of the generator 76/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[5 <<Abar[2 1][2 1]>>][4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 35s. Computing the boundary of the generator 77/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[6 <<Abar[5 4]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 35s. Computing the boundary of the generator 78/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[6 <<Abar[2 1][3 2]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 35s. Computing the boundary of the generator 79/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[6 <<Abar[3 2][2 1]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 35s. Computing the boundary of the generator 80/85 (dimension 10) : <TnPr <<Abar[10 <<Abar[3 <<Abar[2 1]>>][3 <<Abar[2 1]>>][3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 36s. Computing the boundary of the generator 81/85 (dimension 10) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>][6 <<Abar[5 <<Abar[4 3]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 36s. Computing the boundary of the generator 82/85 (dimension 10) : <TnPr <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>][6 <<Abar[5 <<Abar[2 1][2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 36s. Computing the boundary of the generator 83/85 (dimension 10) : <TnPr <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>][5 <<Abar[4 <<Abar[3 2]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 37s. Computing the boundary of the generator 84/85 (dimension 10) : <TnPr <<Abar[6 <<Abar[5 <<Abar[4 3]>>]>>][4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. ;; Clock -> 2016-03-25, 1h 11m 37s. Computing the boundary of the generator 85/85 (dimension 10) : <TnPr <<Abar[6 <<Abar[5 <<Abar[2 1][2 1]>>]>>][4 <<Abar[3 <<Abar[2 1]>>]>>]>> <<Abar>>> End of computing. Homology in dimension 9 : ---done--- ;; Clock -> 2016-03-25, 1h 11m 37s.
Out[71]:
NIL
Lisp file concerned in this chapter¶
cs-twisted-products.lisp.