734 lines
28 KiB
TeX
734 lines
28 KiB
TeX
\chapter {Simplicial groups fibrations}
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\section {Introduction}
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This chapter\index{fibrations!simplicial groups} is analogous to the Loop spaces fibrations chapter
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where algorithms have been specially designed for loop spaces. In the present case, we have at our
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disposal all the tools developped for the fibrations. Let ${\cal G}$
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be a $0$--reduced simplicial group and $\bar{\cal W}{\cal G}$, its
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classifying space. We recall that if ${\cal G}$ is
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an Abelian simplicial group then $\bar{\cal W}{\cal G}$ is also an Abelian
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simplicial group, otherwise if ${\cal G}$ is non--Abelian, then $\bar{\cal W}{\cal G}$
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is only a simplicial set. The software {\tt Kenzo} implements the
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{\bf canonical} twisted cartesian product of $\bar{\cal W}{\cal G}$ by ${\cal G}$,
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denoted ${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$.
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The {\em canonical fibration}
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$$ {\cal G}\quad \hookrightarrow\quad {\cal WG} \hskip 1.2em \longrightarrow \hskip -1.2em \longrightarrow
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\quad\bar{\cal W}{\cal G},$$
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is defined by the following twisting operator:
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$$\tau[(g_{n-1}, g_{n-2}, \ldots, g_0)]= g_{n-1}.$$
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The total space ${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$ is acyclic.
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\section {The function for the canonical fibration}
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{\parindent=0mm
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{\leftskip=5mm
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{\tt smgr-fibration} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the simplicial morphism of degree $-1$ corresponding to the canonical fibration.
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The source is the classifying space of the simplicial group {\em smgr}
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(built internally by the function), the target
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is {\em smgr} and the internal lisp function value of the slot {\tt :sintr} implements
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the twisting operator. The returned object is a fibration. \par}
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}
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\subsection* {Examples}
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Let us take the Abelian simplicial group $K({\Z},1)$ and build the fibration morphism.
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{\footnotesize\begin{verbatim}
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(setf k1 (k-z-1)) ==>
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[K1 Abelian-Simplicial-Group]
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(setf tw (smgr-fibration k1)) ==>
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[K25 Fibration]
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\end{verbatim}}
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If needed, the classifying space may be found in the slot {\tt :sorc} of the fibration:
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{\footnotesize\begin{verbatim}
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(setf k2 (sorc tw)) ==>
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[K13 Abelian-Simplicial-Group]
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\end{verbatim}}
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We may build now the total space of the fibration. We see that {\tt Kenzo} returns
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a Kan simplicial set because the base and fiber spaces are also of
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Kan type. Then we test the face and differential operator upon a simplex of degree $4$ of
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the total space {\tt tt}.
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{\footnotesize\begin{verbatim}
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(setf tt (fibration-total tw)) ==>
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[K31 Kan-Simplicial-Set]
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(setf gmsm (crpr 0 (gbar 4 0 '(10 11 12) 0 '(20 21) 0 '(30) 0 '())
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0 '(2 4 6 8))) ==>
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<CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>
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(dotimes (i 5) (print (face tt i 4 gmsm))) ==>
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<AbSm - <CrPr - <<GBar<- (11 12)><- (21)><- NIL>>> - (4 6 8)>>
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<AbSm - <CrPr - <<GBar<- (21 12)><- (41)><- NIL>>> - (6 6 8)>>
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<AbSm - <CrPr - <<GBar<- (10 23)><- (50)><- NIL>>> - (2 10 8)>>
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<AbSm - <CrPr - <<GBar<- (30 32)><- (30)><- NIL>>> - (2 4 14)>>
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<AbSm - <CrPr - <<GBar<- (20 21)><- (30)><- NIL>>> - (12 15 18)>>
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(? tt 4 gmsm) ==>
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\end{verbatim}}
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\newpage
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{\footnotesize\begin{verbatim}
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----------------------------------------------------------------------{CMBN 3}
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<1 * <CrPr - <<GBar<- (10 23)><- (50)><- NIL>>> - (2 10 8)>>
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<1 * <CrPr - <<GBar<- (11 12)><- (21)><- NIL>>> - (4 6 8)>>
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<1 * <CrPr - <<GBar<- (20 21)><- (30)><- NIL>>> - (12 15 18)>>
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<-1 * <CrPr - <<GBar<- (21 12)><- (41)><- NIL>>> - (6 6 8)>>
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<-1 * <CrPr - <<GBar<- (30 32)><- (30)><- NIL>>> - (2 4 14)>>
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------------------------------------------------------------------------------
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(? tt *) ==>
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----------------------------------------------------------------------{CMBN 2}
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------------------------------------------------------------------------------
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\end{verbatim}}
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We may also build the Ronald Brown reduction and test the differential
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upon a simplex belonging to the twisted tensor product (slot {\tt :bcc} of the
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reduction).
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{\footnotesize\begin{verbatim}
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(setf br (brown-reduction tw)) ==>
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[K59 Reduction]
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(setf tw-pr (bcc br)) ==>
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[K57 Chain-Complex]
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(? tw-pr 4 (tnpr 4 (gbar 4 0 '(1 10 100) 0 '(1000 10000) 0 '(100000) 0 '())
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0 '()))
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----------------------------------------------------------------------{CMBN 3}
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<-1 * <TnPr <<GBar>> (111 11000 100000)>>
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<1 * <TnPr <<GBar>> (111 101000 10000)>>
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<1 * <TnPr <<GBar>> (11011 100 100000)>>
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<-1 * <TnPr <<GBar>> (11011 100000 100)>>
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<-1 * <TnPr <<GBar>> (101001 110 10000)>>
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<1 * <TnPr <<GBar>> (101001 10010 100)>>
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<1 * <TnPr <<GBar<- (100000)><- NIL>>> (100)>>
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<1 * <TnPr <<GBar<- (1 110)><- (101000)><- NIL>>> NIL>>
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<1 * <TnPr <<GBar<- (10 100)><- (10000)><- NIL>>> NIL>>
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<-1 * <TnPr <<GBar<- (11 100)><- (11000)><- NIL>>> NIL>>
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<1 * <TnPr <<GBar<- (1000 10000)><- (100000)><- NIL>>> NIL>>
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<-1 * <TnPr <<GBar<- (1001 10010)><- (100000)><- NIL>>> NIL>>
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------------------------------------------------------------------------------
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(? tw-pr *) ==>
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----------------------------------------------------------------------{CMBN 2}
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------------------------------------------------------------------------------
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\end{verbatim}}
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\section {The essential contraction for simplicial groups fibrations}
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It is known that if ${\cal G}$ is a $0$--reduced simplicial group,
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the space ${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$ is
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contractible. So it is possible to build a reduction of this space over ${\Z}$.
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This reduction depends on the important contraction
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$$\chi_{\times\tau}: {\bar{\cal W}{\cal G} \times_\tau {\cal G}} \longrightarrow
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{\bar{\cal W}{\cal G} \times_\tau {\cal G}}, $$
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defined by:
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$$\chi_{\times\tau}[(g_{n-1}, \ldots, g_0), g_n] = (-1)^{n+1} (g_n, g_{n-1}, \ldots, g_0, *),$$
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where $g_n \in {\cal G}$, $(g_{n-1}, \ldots, g_0)$ is a gbar in $\bar{\cal W}{\cal G}$, and
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$*$ is the neutral element of ${\cal G}_{n+1}$ (in fact the $n$--th degeneracy of the base point of
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${\cal G}$). We recall also that $\chi_{\times\tau}$
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is a homotopy operator (degree: $+1$) satisfying the relation
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$$d \circ \chi_{\times\tau} + \chi_{\times\tau} \circ d =1.$$
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In addition, if we consider the Brown reduction
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$$
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\diagram{
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{\bar{\cal W}{\cal G} \times_\tau {\cal G}} & \stackrel{h}{\longrightarrow} &
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{}^s{\bar{\cal W}{\cal G} \times_\tau {\cal G}} \cr
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{\scriptstyle f} \downarrow \uparrow {\scriptstyle g} \cr
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{\bar{\cal W}{\cal G} \otimes_\tau {\cal G}} \cr
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}
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$$
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there is an induced contraction
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$$\chi_{\otimes t}: {\bar{\cal W}{\cal G} \otimes_\tau {\cal G}} \longrightarrow
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{\bar{\cal W}{\cal G} \otimes_\tau {\cal G}}, $$
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defined by $\chi_{\otimes t}= f \circ \chi_{\times\tau} \circ g$.
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This implies that the twisted tensor product is also contractible over ${\Z}$.
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\par
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The two functions of {\tt Kenzo} for this construction are:
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\vskip 0.35cm
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{\parindent=0mm
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{\leftskip=5mm
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{\tt smgr-crts-contraction} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the homotopy morphism corresponding to the contraction
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$\chi_{\times\tau}$. The lisp definition is:
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{\footnotesize\begin{verbatim}
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(defun SMGR-CRTS-CONTRACTION (smgr)
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(the morphism
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(build-mrph
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:sorc (fibration-total (smgr-fibration smgr))
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:trgt (fibration-total (smgr-fibration smgr))
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:degr +1
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:intr (smgr-crts-contraction-intr (bspn smgr))
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:strt :gnrt
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:orgn `(smgr-crts-contraction ,smgr))))
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\end{verbatim}}
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At execution time, the work is essentialy done by the function put in the
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slot {\tt :intr}. This function is itself built by the internal function {\tt smgr-crts-contraction-intr}
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requiring as argument the base point of the simplicial group. \par}
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{\leftskip=5mm
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{\tt smgr-tnpr-contraction} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the induced morphism $\chi_{\otimes t}= f \circ \chi_{\times\tau} \circ g$,
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where $f$ and $g$ are built by the Brown reduction.
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The lisp definition is:
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{\footnotesize\begin{verbatim}
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(defun SMGR-TNPR-CONTRACTION (smgr
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&aux (fibration (smgr-fibration smgr))
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(brown (brown-reduction fibration))
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(f (f brown))
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(g (g brown))
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(chi (smgr-crts-contraction smgr)))
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(the morphism
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(i-cmps f chi g)))
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\end{verbatim}} \par}
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}
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\subsection* {Examples}
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It is an instructive exercise to build and check the reduction over ${\Z}$ of the space
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${\cal WG}=\bar{\cal W}{\cal G} \times_\tau {\cal G}$ as we did in the
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loop spaces fibrations chapter.
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$$
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\diagram{
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\bar{\cal W}{\cal G} \times_\tau {\cal G} & \stackrel{\chi_{\times t}}{\longrightarrow} &
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{}^s\bar{\cal W}{\cal G} \times_\tau {\cal G} \cr
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{\scriptstyle aug} \downarrow \uparrow {\scriptstyle coaug} \cr
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{{\cal C}_*({\Z})} \cr
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}
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$$
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In this case, the two homomorphisms
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$f$ and $g$ of the reduction, are respectively the {\em augmentation} and {\em coaugmentation}
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morphisms.
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We define both morphisms according to our previous example based on $K({\Z},1)$.
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The unit chain complex corresponding to ${\Z}$ is built by the function
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{\tt z-chcm}; its unique generator in degree $0$ is {\tt :z-gnrt}.
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{\footnotesize\begin{verbatim}
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(setf k1 (k-z-1)) ==>
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[K1 Abelian-Simplicial-Group]
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(setf tw (smgr-fibration k1)) ==>
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[K25 Fibration]
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(setf tt (fibration-total tw)) ==>
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[K31 Kan-Simplicial-Set]
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\end{verbatim}}
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The base point of the total space, needed for the coaugmentation, may be obtained
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by either of the following statements:
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{\footnotesize\begin{verbatim}
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(bspn tt) ==>
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<CrPr - <<GBar>> - NIL>
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(crpr 0 +null-gbar+ 0 nil) ==>
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<CrPr - <<GBar>> - NIL>
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(setf aug (build-mrph
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:sorc (fibration-total(smgr-fibration(k-z-1)))
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:trgt (z-chcm)
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:degr 0
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:intr #'(lambda (degr gnrt)
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(if (zerop degr)
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(term-cmbn 0 1 :z-gnrt)
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(zero-cmbn degr)))
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:strt :gnrt
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:orgn '(aug-fibr-tot-smgr-fibr-k-z-1) )) ==>
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[K38 Cohomology-Class (degree 0)]
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(setf coaug (build-mrph
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:sorc (z-chcm)
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:trgt (fibration-total(smgr-fibration(k-z-1)))
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:degr 0
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:intr #'(lambda (degr gnrt)
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(if (zerop degr)
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(term-cmbn 0 1 (crpr 0 +null-gbar+ 0 '()))
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(zero-cmbn degr)))
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:strt :gnrt
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:orgn '(coaug-fibr-tot-smgr-fibr-k-z-1) )) ==>
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[K39 Morphism (degree 0)]
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(? aug 0 (bspn tt)) ==>
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----------------------------------------------------------------------{CMBN 0}
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<1 * Z-GNRT>
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------------------------------------------------------------------------------
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(? coaug 0 :z-gnrt) ==>
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----------------------------------------------------------------------{CMBN 0}
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<1 * <CrPr - <<GBar>> - NIL>>
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------------------------------------------------------------------------------
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(setf chi-x-tau (smgr-crts-contraction (k-z-1))) ==>
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[K40 Morphism (degree 1)]
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\end{verbatim}}
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Let us apply the contraction morphism upon some simplices:
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{\footnotesize\begin{verbatim}
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(setf *tc* (cmbn 0 1 (crpr 0 +null-gbar+ 0 '()))) ==>
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----------------------------------------------------------------------{CMBN 0}
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<1 * <CrPr - <<GBar>> - NIL>>
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------------------------------------------------------------------------------
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(? chi-x-tau *) ==>
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----------------------------------------------------------------------{CMBN 1}
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------------------------------------------------------------------------------
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(setf *tc* (cmbn 4 1 (crpr 0 (gbar 4 0 '(10 11 12) 0 '(20 21) 0 '(30) 0 '())
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0 '(2 4 6 8)))) ==>
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----------------------------------------------------------------------{CMBN 4}
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<1 * <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>>
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------------------------------------------------------------------------------
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(? chi-x-tau *) ==>
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----------------------------------------------------------------------{CMBN 5}
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<-1 * <CrPr - <<GBar<- (2 4 6 8)><- (10 11 12)><- (20 21)><- (30)><- NIL>>>
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4-3-2-1-0 NIL>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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\newpage
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{\footnotesize\begin{verbatim}
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(setf *tc* (cmbn 3 1 (crpr 0 (gbar 3 0 '(10 11) 0 '(20) 0 '())
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0 '(2 4 6)))) ==>
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----------------------------------------------------------------------{CMBN 3}
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<1 * <CrPr - <<GBar<- (10 11)><- (20)><- NIL>>> - (2 4 6)>>
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------------------------------------------------------------------------------
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(? chi-x-tau *) ==>
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----------------------------------------------------------------------{CMBN 4}
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<1 * <CrPr - <<GBar<- (2 4 6)><- (10 11)><- (20)><- NIL>>> 3-2-1-0 NIL>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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We may build now the reduction over ${\Z}$ and test it upon the various simplices
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above:
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{\footnotesize\begin{verbatim}
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(setf rdct (build-rdct :f aug
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:g coaug
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:h chi-x-tau
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:orgn '(reduction-tt-z))) ==>
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[K41 Reduction]
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(pre-check-rdct rdct) ==>
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---done---
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(setf *tc* (cmbn 0 1 (crpr 0 +null-gbar+ 0 '()))) ==>
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----------------------------------------------------------------------{CMBN 0}
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<1 * <CrPr - <<GBar>> - NIL>>
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------------------------------------------------------------------------------
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(setf *bc* (cmbn 0 1 :z-gnrt)) ==>
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----------------------------------------------------------------------{CMBN 0}
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<1 * Z-GNRT>
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------------------------------------------------------------------------------
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(check-rdct) ==>
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*TC* =>
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----------------------------------------------------------------------{CMBN 0}
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<1 * <CrPr - <<GBar>> - NIL>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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\newpage
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{\footnotesize\begin{verbatim}
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*BC* =>
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----------------------------------------------------------------------{CMBN 0}
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<1 * Z-GNRT>
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------------------------------------------------------------------------------
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....... All results null ........
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---done---
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(setf *tc* (cmbn 4 1 (crpr 0 (gbar 4 0 '(10 11 12) 0 '(20 21) 0 '(30) 0 '())
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0 '(2 4 6 8)))) ==>
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----------------------------------------------------------------------{CMBN 4}
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<1 * <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>>
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------------------------------------------------------------------------------
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(check-rdct) ==>
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....... All results null ........
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---done---
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(setf *tc* (cmbn 3 1 (crpr 0 (gbar 3 0 '(10 11) 0 '(20) 0 '())
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0 '(2 4 6)))) ==>
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----------------------------------------------------------------------{CMBN 3}
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<1 * <CrPr - <<GBar<- (10 11)><- (20)><- NIL>>> - (2 4 6)>>
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------------------------------------------------------------------------------
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(check-rdct) ==>
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....... All results null ........
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---done---
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\end{verbatim}}
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\vskip 0.30cm
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We build now the reduction over ${\Z}$ of the twisted tensor product. The contraction
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is obtained by a call to the function {\tt smgr-tnpr-contraction}.
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$$
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\diagram{
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\bar{\cal W}{\cal G} \otimes_\tau {\cal G} & \stackrel{\chi_{\otimes t}}{\longrightarrow} &
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{}^s\bar{\cal W}{\cal G} \otimes_\tau {\cal G} \cr
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{\scriptstyle aug} \downarrow \uparrow {\scriptstyle coaug} \cr
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{{\cal C}_*({\Z})} \cr
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}
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$$
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Of course, we have to define
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the augmentation and coaugmentation morphisms adapted to this new example. The user will note
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that the twisted tensor product chain complex may be obtained from the slot {\tt :sorc} of the contration
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$\chi_{\otimes t}$.
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{\footnotesize\begin{verbatim}
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(setf chi-t-tau (smgr-tnpr-contraction (k-z-1))) ==>
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[K100 Morphism (degree 1)]
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(setf aug-t (build-mrph
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:sorc (sorc chi-t-tau)
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:trgt (z-chcm)
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:degr 0
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:intr #'(lambda (degr gnrt)
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(if (zerop degr)
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(term-cmbn 0 1 :z-gnrt)
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(zero-cmbn degr)))
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:strt :gnrt
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:orgn '(aug-t-fibr-tot-smgr-fibr-k-z-1) )) ==>
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[K209 Cohomology-Class (degree 0)]
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(setf coaug-t (build-mrph
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:sorc (z-chcm)
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:trgt (sorc chi-t-tau)
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:degr 0
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:intr #'(lambda (degr gnrt)
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(if (zerop degr)
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(term-cmbn 0 1 (tnpr 0 +null-gbar+ 0 '()))
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(zero-cmbn degr)))
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:strt :gnrt
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:orgn '(coaug-t-fibr-tot-smgr-fibr-k-z-1) )) ==>
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[K210 Morphism (degree 0)]
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(setf rdct-t (build-rdct :f aug-t
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:g coaug-t
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:h chi-t-tau
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:orgn '(reduction-t-tt-z))) ==>
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[K211 Reduction]
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|
|
|
(pre-check-rdct rdct-t)
|
|
|
|
---done---
|
|
|
|
(setf *tc* (cmbn 0 1 (tnpr 0 +null-gbar+ 0 '()))) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 0}
|
|
<1 * <TnPr <<GBar>> NIL>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(setf *bc* (cmbn 0 1 :z-gnrt)) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 0}
|
|
<1 * Z-GNRT>
|
|
------------------------------------------------------------------------------
|
|
|
|
(check-rdct) ==>
|
|
|
|
....... All results null .....
|
|
|
|
---done---
|
|
\end{verbatim}}
|
|
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.
|
|
{\footnotesize\begin{verbatim}
|
|
|
|
(setf *tc* (cmbn 3 1 (tnpr 0 +null-gbar+ 3 '(1 10 100)))) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
<1 * <TnPr <<GBar>> (1 10 100)>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(? chi-t-tau *tc*) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 4}
|
|
<1 * <TnPr <<GBar<- (1 10 100)><1-0 NIL><0 NIL><- NIL>>> NIL>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(setf *tc* (cmbn 3 1 (tnpr 2 (gbar 2 0 '(1) 0 '()) 1 '(10)))) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
<1 * <TnPr <<GBar<- (1)><- NIL>>> (10)>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(? chi-t-tau *tc*) ==>
|
|
|
|
----------------------------------------------------------------------{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>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(setf *tc* (cmbn 3 1 (tnpr 3 (gbar 3 0 '(1 10) 0 '(100) 0 '()) 0 '()))) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 3}
|
|
<1 * <TnPr <<GBar<- (1 10)><- (100)><- NIL>>> NIL>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(? chi-t-tau *tc*) ==>
|
|
|
|
----------------------------------------------------------------------{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>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(setf *tc* (cmbn 4 1 (tnpr 0 +null-gbar+ 4 '(1 10 100 1000)))) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 4}
|
|
<1 * <TnPr <<GBar>> (1 10 100 1000)>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(? chi-t-tau *tc*) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 5}
|
|
<-1 * <TnPr <<GBar<- (1 10 100 1000)><2-1-0 NIL><1-0 NIL><0 NIL><- NIL>>> NIL>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(setf *tc* (cmbn 4 1 (tnpr 2 (gbar 2 0 '(1) 0 '()) 2 '(10 100)))) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 4}
|
|
<1 * <TnPr <<GBar<- (1)><- NIL>>> (10 100)>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(? chi-t-tau *tc*) ==>
|
|
|
|
----------------------------------------------------------------------{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>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(setf *tc* (cmbn 4 1 (tnpr 3 (gbar 3 0 '(1 10) 0 '(100) 0 '())
|
|
1 '(1000)))) ==>
|
|
\end{verbatim}}
|
|
\newpage
|
|
{\footnotesize\begin{verbatim}
|
|
----------------------------------------------------------------------{CMBN 4}
|
|
<1 * <TnPr <<GBar<- (1 10)><- (100)><- NIL>>> (1000)>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(? chi-t-tau *tc*) ==>
|
|
|
|
----------------------------------------------------------------------{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>>
|
|
<-1 * <TnPr <<GBar<3-2-0 (1000)><1 (1 10)><1 (100)><0 NIL><- NIL>>> NIL>>
|
|
<1 * <TnPr <<GBar<3-2-1 (1000)><0 (1 10)><0 (100)><0 NIL><- NIL>>> NIL>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(setf *tc* (cmbn 4 1 (tnpr 4 (gbar 4 0 '(1 10 100)
|
|
0 '(1000 10000)
|
|
0 '(100000)
|
|
0 '())
|
|
0 '()))) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 4}
|
|
<1 * <TnPr <<GBar<- (1 10 100)><- (1000 10000)><- (100000)><- NIL>>> NIL>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(? chi-t-tau *tc*) ==>
|
|
|
|
----------------------------------------------------------------------{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>>
|
|
<-1 * <TnPr <<GBar<3-2-1 (111)><0 (1000 10000)><0 (100000)><0 NIL><- NIL>>> NIL>>
|
|
------------------------------------------------------------------------------
|
|
\end{verbatim}}
|
|
\newpage
|
|
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 ${\cal G}= K({\Z},1)$, 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:
|
|
{\footnotesize\begin{verbatim}
|
|
(setf wgkz1 (fibration-total (smgr-fibration(k-z-1)))) ==>
|
|
|
|
[K31 Kan-Simplicial-Set]
|
|
|
|
(setf chi-t-tau (tnpr-contraction wgkz1)) ==>
|
|
|
|
[K84 Morphism (degree 1)]
|
|
\end{verbatim}}
|
|
Let us apply successively the contraction upon a simplex belonging to
|
|
the base fiber ({\tt gnrt0}) and a simplex out of the base fiber ({\tt gnrt1}).
|
|
We see that in the second case we obtain a null result. We verify also that
|
|
$\chi_{\otimes t}\circ \chi_{\otimes t}=0$.
|
|
{\footnotesize\begin{verbatim}
|
|
(setf gnrt0 (tnpr 0 (crpr 0 +null-gbar+ 0 nil)
|
|
0 (loop3 0 (crpr 0 (gbar 2 0 '(1) 0 '(0) ) 0 '(1)) -3))) ==>
|
|
|
|
<TnPr <CrPr - <<GBar>> - NIL> <<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\-3]>>>
|
|
|
|
(? chi-t-tau 0 gnrt0) ==>
|
|
|
|
----------------------------------------------------------------------{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]>>>>
|
|
------------------------------------------------------------------------------
|
|
|
|
(? chi-t-tau *) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 2}
|
|
------------------------------------------------------------------------------
|
|
|
|
(setf 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))) ==>
|
|
|
|
<TnPr <CrPr - <<GBar<- (10 11 12)><- (20 21)><- (30)><- NIL>>> - (2 4 6 8)>
|
|
<<Loop[<CrPr - <<GBar<- (1)><- (0)>>> - (1)>\2]>>>
|
|
\end{verbatim}}
|
|
\newpage
|
|
{\footnotesize\begin{verbatim}
|
|
(? chi-t-tau 4 gnrt1) ==>
|
|
|
|
----------------------------------------------------------------------{CMBN 5}
|
|
------------------------------------------------------------------------------
|
|
\end{verbatim}}
|
|
At last we verify that the total space of the fibration of $K({\Z},1)$ is
|
|
contractible.
|
|
{\footnotesize\begin{verbatim}
|
|
(homology (fibration-total(smgr-fibration (k-z-1))) 0 10) ==>
|
|
|
|
Homology in dimension 0 :
|
|
|
|
Component Z
|
|
|
|
Homology in dimension 1 :
|
|
|
|
Homology in dimension 2 :
|
|
|
|
Homology in dimension 3 :
|
|
|
|
Homology in dimension 4 :
|
|
|
|
Homology in dimension 5 :
|
|
|
|
Homology in dimension 6 :
|
|
|
|
Homology in dimension 7 :
|
|
|
|
Homology in dimension 8 :
|
|
|
|
Homology in dimension 9 :
|
|
|
|
---done---
|
|
\end{verbatim}}
|
|
Another example is the following:
|
|
we build the canonical fibration of $\bar{\cal W}^2K({\Z_2}, 1)$ and verify
|
|
that its fibration total space is contractible:
|
|
{\footnotesize\begin{verbatim}
|
|
(setf k22 (classifying-space(classifying-space(k-z2-1)))) ==>
|
|
|
|
[K306 Abelian-Simplicial-Group]
|
|
|
|
(setf fb (smgr-fibration k22)) ==>
|
|
|
|
[K836 Fibration]
|
|
|
|
(setf tt2 (fibration-total fb)) ==>
|
|
|
|
[K837 Kan-Simplicial-Set]
|
|
|
|
(homology tt2 0 10) ==>
|
|
|
|
Homology in dimension 0 :
|
|
|
|
Component Z
|
|
|
|
Homology in dimension 1 :
|
|
|
|
Homology in dimension 2 :
|
|
|
|
Homology in dimension 3 :
|
|
|
|
Homology in dimension 4 :
|
|
|
|
Homology in dimension 5 :
|
|
|
|
Homology in dimension 6 :
|
|
|
|
Homology in dimension 7 :
|
|
|
|
Homology in dimension 8 :
|
|
|
|
Homology in dimension 9 :
|
|
|
|
--done--
|
|
\end{verbatim}}
|
|
|
|
\subsection* {Lisp file concerned in this chapter}
|
|
|
|
{\tt cs-twisted-products.lisp}.
|
|
|
|
|