70 KiB
70 KiB
In [1]:
(ql:quickload "kenzo") (in-package :cat)
To load "kenzo":
Load 1 ASDF system:
kenzo
; Loading "kenzo"
Out[1]:
#<PACKAGE "CAT">
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 [2]:
(def k1 (k-z-1))
Out[2]:
[K1 Abelian-Simplicial-Group]
In [3]:
(def tw (smgr-fibration k1))
Out[3]:
[K25 Fibration K13 -> K1]
If needed, the classifying space may be found in the slot :sorc of the fibration:
In [4]:
(def k2 (sorc tw))
Out[4]:
[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 [5]:
(def tt (fibration-total tw))
Out[5]:
[K31 Kan-Simplicial-Set]
In [6]:
(def gmsm (crpr 0 (gbar 4 0 '(10 11 12) 0 '(20 21) 0 '(30) 0 '()) 0 '(2 4 6 8)))
Out[6]:
<CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>
In [7]:
(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[7]:
NIL
In [8]:
(? tt 4 gmsm)
Out[8]:
----------------------------------------------------------------------{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 [9]:
(? tt *)
Out[9]:
----------------------------------------------------------------------{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 [10]:
(def br (brown-reduction tw))
Out[10]:
[K59 Reduction K31 => K57]
In [11]:
(def tw-pr (bcc br))
Out[11]:
[K57 Chain-Complex]
In [12]:
(? tw-pr 4 (tnpr 4 (gbar 4 0 '(1 10 100) 0 '(1000 10000) 0 '(100000) 0 '()) 0 '()))
Out[12]:
----------------------------------------------------------------------{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 [13]:
(? tw-pr *)
Out[13]:
----------------------------------------------------------------------{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 [14]:
(def k1 (k-z-1))
Out[14]:
[K1 Abelian-Simplicial-Group]
In [15]:
(def tw (smgr-fibration k1))
Out[15]:
[K25 Fibration K13 -> K1]
In [16]:
(def tt (fibration-total tw))
Out[16]:
[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 [17]:
(bspn tt)
Out[17]:
<CrPr - <<GBar>> - NIL>
In [18]:
(crpr 0 +null-gbar+ 0 nil)
Out[18]:
<CrPr - <<GBar>> - NIL>
In [19]:
(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[19]:
[K62 Cohomology-Class on K31 of degree 0]
In [20]:
(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: undefined function: :TYPE
Out[20]:
[K63 Morphism (degree 0): K60 -> K31]
In [21]:
(? aug 0 (bspn tt))
Out[21]:
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
In [22]:
(? coaug 0 :z-gnrt)
Out[22]:
----------------------------------------------------------------------{CMBN 0}
<1 * <CrPr - <<GBar>> - NIL>>
------------------------------------------------------------------------------
In [23]:
(def chi-x-tau (smgr-crts-contraction (k-z-1)))
Out[23]:
[K64 Morphism (degree 1): K31 -> K31]
Let us apply the contraction morphism upon some simplices:
In [24]:
(setf *tc* (cmbn 0 1 (crpr 0 +null-gbar+ 0 '())))
Out[24]:
----------------------------------------------------------------------{CMBN 0}
<1 * <CrPr - <<GBar>> - NIL>>
------------------------------------------------------------------------------
In [25]:
(? chi-x-tau *)
Out[25]:
----------------------------------------------------------------------{CMBN 1}
------------------------------------------------------------------------------
In [26]:
(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[26]:
----------------------------------------------------------------------{CMBN 4}
<1 * <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>>
------------------------------------------------------------------------------
In [27]:
(? chi-x-tau *)
Out[27]:
----------------------------------------------------------------------{CMBN 5}
<-1 * <CrPr - <<GBar<- (2 4 6 8)><- (10 11 12)><- (20 21)><- (30)><- NIL>>> 4-3-2-1-0 NIL>>
------------------------------------------------------------------------------
In [28]:
(setf *tc* (cmbn 3 1 (crpr 0 (gbar 3 0 '(10 11) 0 '(20) 0 '()) 0 '(2 4 6))))
Out[28]:
----------------------------------------------------------------------{CMBN 3}
<1 * <CrPr - <<GBar<- (10 11)><- (20)><- NIL>>> - (2 4 6)>>
------------------------------------------------------------------------------
In [29]:
(? chi-x-tau *)
Out[29]:
----------------------------------------------------------------------{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 [30]:
(def rdct (build-rdct :f aug :g coaug :h chi-x-tau :orgn '(reduction-tt-z)))
Out[30]:
[K65 Reduction K31 => K60]
In [31]:
(pre-check-rdct rdct)
---done---
Out[31]:
NIL
In [32]:
(setf *tc* (cmbn 0 1 (crpr 0 +null-gbar+ 0 '())))
Out[32]:
----------------------------------------------------------------------{CMBN 0}
<1 * <CrPr - <<GBar>> - NIL>>
------------------------------------------------------------------------------
In [33]:
(setf *bc* (cmbn 0 1 :z-gnrt))
Out[33]:
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
In [34]:
(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[34]:
NIL
In [35]:
(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[35]:
----------------------------------------------------------------------{CMBN 4}
<1 * <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>>
------------------------------------------------------------------------------
In [36]:
(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[36]:
NIL
In [37]:
(setf *tc* (cmbn 3 1 (crpr 0 (gbar 3 0 '(10 11) 0 '(20) 0 '()) 0 '(2 4 6))))
Out[37]:
----------------------------------------------------------------------{CMBN 3}
<1 * <CrPr - <<GBar<- (10 11)><- (20)><- NIL>>> - (2 4 6)>>
------------------------------------------------------------------------------
In [38]:
(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[38]:
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 [39]:
(def chi-t-tau (smgr-tnpr-contraction (k-z-1)))
Out[39]:
[K88 Morphism (degree 1): K57 -> K57]
In [40]:
(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[40]:
[K89 Cohomology-Class on K57 of degree 0]
In [41]:
(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[41]:
[K90 Morphism (degree 0): K60 -> K57]
In [42]:
(def rdct-t (build-rdct :f aug-t :g coaug-t :h chi-t-tau :orgn '(reduction-t-tt-z)))
Out[42]:
[K91 Reduction K57 => K60]
In [43]:
(pre-check-rdct rdct-t)
---done---
Out[43]:
NIL
In [44]:
(setf *tc* (cmbn 0 1 (tnpr 0 +null-gbar+ 0 '())))
Out[44]:
----------------------------------------------------------------------{CMBN 0}
<1 * <TnPr <<GBar>> NIL>>
------------------------------------------------------------------------------
In [45]:
(setf *bc* (cmbn 0 1 :z-gnrt))
Out[45]:
----------------------------------------------------------------------{CMBN 0}
<1 * Z-GNRT>
------------------------------------------------------------------------------
In [46]:
(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[46]:
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 [47]:
(setf *tc* (cmbn 3 1 (tnpr 0 +null-gbar+ 3 '(1 10 100))))
Out[47]:
----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr <<GBar>> (1 10 100)>>
------------------------------------------------------------------------------
In [48]:
(? chi-t-tau *tc*)
Out[48]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<- (1 10 100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [49]:
(setf *tc* (cmbn 3 1 (tnpr 2 (gbar 2 0 '(1) 0 '()) 1 '(10))))
Out[49]:
----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr <<GBar<- (1)><- NIL>>> (10)>>
------------------------------------------------------------------------------
In [50]:
(? chi-t-tau *tc*)
Out[50]:
----------------------------------------------------------------------{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 [51]:
(setf *tc* (cmbn 3 1 (tnpr 3 (gbar 3 0 '(1 10) 0 '(100) 0 '()) 0 '())))
Out[51]:
----------------------------------------------------------------------{CMBN 3}
<1 * <TnPr <<GBar<- (1 10)><- (100)><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [52]:
(? chi-t-tau *tc*)
Out[52]:
----------------------------------------------------------------------{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 [53]:
(setf *tc* (cmbn 4 1 (tnpr 0 +null-gbar+ 4 '(1 10 100 1000))))
Out[53]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar>> (1 10 100 1000)>>
------------------------------------------------------------------------------
In [54]:
(? chi-t-tau *tc*)
Out[54]:
----------------------------------------------------------------------{CMBN 5}
<-1 * <TnPr <<GBar<- (1 10 100 1000)><2-1-0 NIL><1-0 NIL><0 NIL><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [55]:
(setf *tc* (cmbn 4 1 (tnpr 2 (gbar 2 0 '(1) 0 '()) 2 '(10 100))))
Out[55]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<- (1)><- NIL>>> (10 100)>>
------------------------------------------------------------------------------
In [56]:
(? chi-t-tau *tc*)
Out[56]:
----------------------------------------------------------------------{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 [57]:
(setf *tc* (cmbn 4 1 (tnpr 3 (gbar 3 0 '(1 10) 0 '(100) 0 '()) 1 '(1000))))
Out[57]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<- (1 10)><- (100)><- NIL>>> (1000)>>
------------------------------------------------------------------------------
In [58]:
(? chi-t-tau *tc*)
Out[58]:
----------------------------------------------------------------------{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 [59]:
(setf *tc* (cmbn 4 1 (tnpr 4 (gbar 4 0 '(1 10 100) 0 '(1000 10000) 0 '(100000) 0 '()) 0 '())))
Out[59]:
----------------------------------------------------------------------{CMBN 4}
<1 * <TnPr <<GBar<- (1 10 100)><- (1000 10000)><- (100000)><- NIL>>> NIL>>
------------------------------------------------------------------------------
In [60]:
(? chi-t-tau *tc*)
Out[60]:
----------------------------------------------------------------------{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 [61]:
(def wgkz1 (fibration-total (smgr-fibration(k-z-1))))
Out[61]:
[K31 Kan-Simplicial-Set]
In [62]:
(def chi-t-tau (tnpr-contraction wgkz1))
Out[62]:
[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 [63]:
(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[63]:
<TnPr <CrPr - <<GBar>> - NIL> <<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\-3]>>>
In [64]:
(? chi-t-tau 0 gnrt0)
Out[64]:
----------------------------------------------------------------------{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 [65]:
(? chi-t-tau *)
Out[65]:
----------------------------------------------------------------------{CMBN 2}
------------------------------------------------------------------------------
In [66]:
(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[66]:
<TnPr <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)> <<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\2]>>>
In [67]:
(? chi-t-tau 4 gnrt1)
Out[67]:
----------------------------------------------------------------------{CMBN 5}
------------------------------------------------------------------------------
At last we verify that the total space of the fibration of $K(\mathbb{Z},1)$ is contractible.
In [68]:
(homology (fibration-total(smgr-fibration (k-z-1))) 0 10)
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing the boundary of the generator 1/1 (dimension 1) : <TnPr <<Abar>> S1> End of computing. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing the boundary of the generator 1/1 (dimension 2) : <TnPr <<Abar[2 S1]>> *> End of computing. Homology in dimension 1 : ---done--- ;; Clock -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing the boundary of the generator 1/1 (dimension 3) : <TnPr <<Abar[2 S1]>> S1> End of computing. Homology in dimension 2 : ---done--- ;; Clock -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 5. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 7. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 8. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. Computing boundary-matrix in dimension 9. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s. 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 -> 2017-10-08, 12h 31m 26s.
Out[68]:
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 [69]:
(def k22 (classifying-space(classifying-space(k-z2-1))))
Out[69]:
[K327 Abelian-Simplicial-Group]
In [70]:
(def fb (smgr-fibration k22))
Out[70]:
[K351 Fibration K339 -> K327]
In [71]:
(def tt2 (fibration-total fb))
Out[71]:
[K357 Kan-Simplicial-Set]
In [ ]:
(homology tt2 0 10)
Lisp file concerned in this chapter¶
cs-twisted-products.lisp.