475 lines
14 KiB
TeX
475 lines
14 KiB
TeX
\setcounter{chapter}{-1}
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\chapter {Overview}
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In this overview we try to show, without entering in detailed
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explanations, various possibilities of {\tt Kenzo} in algebraic topology.
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The lines ended by the symbol $==>$ are the commands typed by the user
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(don't type this symbol!).
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They are followed by the answer of the program. We have suppressed some
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extra informations, in particular the ones printed during the computation
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of homology groups.
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\vskip 0.5cm
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Let us begin by the space $Moore({\Z}/2{\Z}, 3)$ described as a simplicial set
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having only three non--degenerate simplices, namely in dimension
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$0$, $3$ and $4$. In the representation created by the software, the $0$--simplex (base point),
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the $3$--simplex and the $4$--simplex are respectively labelled
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``{\tt *}'', {\tt M3} and {\tt N4}. Two faces of the $4$-simplex {\tt N4} are identified with the
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$3$-simplex {\tt M3}, the others being contracted on the base point. To create the simplicial set
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one types simply:
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{\footnotesize\begin{verbatim}
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(setf m23 (moore 2 3)) ==>
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\end{verbatim}}
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The system answers that a {\tt Kenzo} object has been created, with number $1$ and
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type {\tt SIMPLICIAL SET}. This object may be referenced by the symbol {\tt m23}.
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{\footnotesize\begin{verbatim}
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[K1 Simplicial-Set]
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\end{verbatim}}
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We may compute the homology groups of this space, using the underlying chain complex
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induced by the simplicial set description. Here we compute the $ H_i$ from $0$ to $4$ included.
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When in the answer the component part is void, it means that the corresponding homology group is null.
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{\footnotesize\begin{verbatim}
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(homology m23 0 5) ==>
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Homology in dimension 0 :
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Component Z
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--done--
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Homology in dimension 1 :
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--done--
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Homology in dimension 2 :
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--done--
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Homology in dimension 3 :
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Component Z/2Z
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--done--
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Homology in dimension 4 :
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---done---
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\end{verbatim}}
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As {\tt m23} is a simplicial set, it is possible to create the cartesian product
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$m23 \times m23$ by the function {\tt crts-prdc}. This is a new simplicial set.
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{\footnotesize\begin{verbatim}
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(setf m23xm23 (crts-prdc m23 m23)) ==>
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[K10 Simplicial-Set]
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\end{verbatim}}
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Being a simplicial set, {\tt m23xm23} is also a chain complex object and we may for instance ask for
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the basis in dimension $6$
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{\footnotesize\begin{verbatim}
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(basis m23xm23 6) ==>
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(<CrPr 1-0 N4 3-2 N4> <CrPr 1-0 N4 4-2 N4> <CrPr 1-0 N4 4-3 N4>
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<CrPr 1-0 N4 4-3-2 M3> <CrPr 1-0 N4 5-2 N4> <CrPr 1-0 N4 5-3 N4>
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<CrPr 1-0 N4 5-3-2 M3> <CrPr 1-0 N4 5-4 N4>
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.... ....
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<CrPr 4-1 N4 5-3-2 M3> <CrPr 4-1-0 M3 3-2 N4> <CrPr 4-1-0 M3 5-2 N4>
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<CrPr 4-1-0 M3 5-3 N4> <CrPr 4-1-0 M3 5-3-2 M3> ... ... ...)
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(length *) ==>
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230
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\end{verbatim}}
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As shown by the last command (* means the previous result),
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the number of elements of the basis is quite large (230). The user
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will note that the basis elements are formed by cartesian products of {\em degenerated simplices}. In
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the list, an element like {\tt <CrPr 1-0 N4 5-3-2 M3>} means $\eta_1\eta_0 N4 \times \eta_5\eta_3\eta_2 M3$.
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\par
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We may construct also the tensor product $m23 \otimes m23$ from the underlying chain complex
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of the simplicial set {\tt m23}. This tensor product is a new chain complex and we see that the
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basis in dimension $6$ has only one element:
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{\footnotesize\begin{verbatim}
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(setf t2m23 (tnsr-prdc m23 m23)) ==>
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[K3 Chain-Complex]
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(basis t2m23 6) ==>
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(<TnPr M3 M3>)
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\end{verbatim}}
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The Eilenberg-Zilber theorem is used to compute the homology groups of the cartesian
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product space: as chain complexes, {\tt m23xm23} and {\tt t2m23} have the same homology groups,
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but the computations using the tensor product are considerably faster.
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{\footnotesize\begin{verbatim}
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(homology m23xm23 0 8) ==>
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Homology in dimension 0 :
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Component Z
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Homology in dimension 1 : (meaning: Group null)
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Homology in dimension 2 :
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Homology in dimension 3 :
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Component Z/2Z
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Component Z/2Z
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Homology in dimension 4 :
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Homology in dimension 5 :
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Homology in dimension 6 :
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Component Z/2Z
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Homology in dimension 7 :
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Component Z/2Z
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---done---
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\end{verbatim}}
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\newpage
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Let us consider now the space $K({\Z}, 1)$. This is an Abelian simplicial group created in {\tt Kenzo}
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by the function {\tt k-z}.
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In this simplicial group, a simplex in dimension $n$
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is mathematically represented by a sequence of integers, known as a {\em bar} object:
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$$ [a_1 \mid a_2 \mid \ldots \mid a_n]. $$
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In {\tt Kenzo}, a non-degenerate simplex of $K({\Z},1)$ in dimension $n$ will be simply
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a list of $n$ non-null integers, for instance: {\tt (2 3 4 5)}. In dimension $0$, the only
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simplex is {\tt NIL} (the base point).
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{\footnotesize\begin{verbatim}
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(setf kz1 (k-z 1)) ==>
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[K38 Abelian-Simplicial-Group]
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\end{verbatim}}
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But this object is also a {\em coalgebra} and an {\em algebra}, and we may
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see the effect of the respective induced {\em coproduct} and {\em product}:
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{\footnotesize\begin{verbatim}
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(cprd kz1 4 '(2 3 4 5)) ==>
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----------------------------------------------------------------------{CMBN 4}
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<1 * <TnPr NIL (2 3 4 5)>>
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<1 * <TnPr (2) (3 4 5)>>
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<1 * <TnPr (2 3) (4 5)>>
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<1 * <TnPr (2 3 4) (5)>>
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<1 * <TnPr (2 3 4 5) NIL>>
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------------------------------------------------------------------------------
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(aprd kz1 6 (tnpr 2 '(1 2) 4 '(3 4 5 6))) ==>
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----------------------------------------------------------------------{CMBN 6}
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<1 * (1 2 3 4 5 6)>
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<-1 * (1 3 2 4 5 6)>
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<1 * (1 3 4 2 5 6)>
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<-1 * (1 3 4 5 2 6)>
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<1 * (1 3 4 5 6 2)>
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<1 * (3 1 2 4 5 6)>
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<-1 * (3 1 4 2 5 6)>
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<1 * (3 1 4 5 2 6)>
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<-1 * (3 1 4 5 6 2)>
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<1 * (3 4 1 2 5 6)>
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<-1 * (3 4 1 5 2 6)>
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<1 * (3 4 1 5 6 2)>
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<1 * (3 4 5 1 2 6)>
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<-1 * (3 4 5 1 6 2)>
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<1 * (3 4 5 6 1 2)>
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------------------------------------------------------------------------------
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\end{verbatim}}
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The printed results are the printed representation of {\em combinations}, i.e.
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integer linear combinations of generators resulting from the application of the
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morphisms. The degree of the combination is indicated by the information: {\tt CMBN $n$}.
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\par
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On the same way, we may create the Abelian simplicial groups $K({\Z}/2{\Z}, n)$:
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{\footnotesize\begin{verbatim}
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(setf k-z2-2 (k-z2 2)) ==>
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[K488 Abelian-Simplicial-Group]
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(homology k-z2-2 4) ==>
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Homology in dimension 4 :
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Component Z/4Z
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---done---
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\end{verbatim}}
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\begin{center}
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----------
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\end{center}
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Let us play now with the sphere $S^3$ and its loop spaces. $S^3$ and $\Omega^2 S^3$ are created by
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respective calls to the functions {\tt sphere} and {\tt loop-space}. Then we compute the $H_4$ and $H_5$
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of $\Omega^2 S^3$:
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{\footnotesize\begin{verbatim}
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(setf s3 (sphere 3)) ==>
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[K52 Simplicial-Set]
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(setf o2s3 (loop-space s3 2)) ==>
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[K69 Simplicial-Group]
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(homology o2s3 4 6) ==>
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Homology in dimension 4 :
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Component Z/3Z
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Component Z/2Z
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Homology in dimension 5 :
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Component Z/3Z
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Component Z/2Z
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---done---
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\end{verbatim}}
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\newpage
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Let us take now the first loop space $\Omega^1 S^3$
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{\footnotesize\begin{verbatim}
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(setf s3 (sphere 3)) ==>
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[K94 Simplicial-Set]
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(setf os3 (loop-space s3)) ==>
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[K99 Simplicial-Group]
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\end{verbatim}}
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In the following instruction, we locate in the symbol {\tt L1} the canonical generator of
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$\pi_2 (\Omega^1S^3)$, that is the $2$--simplex coming from the original sphere. In fact, the object
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created by the command {\tt (loop3 0 's3 1)} is the ``word'' $S3^1$ belonging to the
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Kan simplicial version $G(S^3)$ (a simplicial group) of the loop space $\Omega S^3$.
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{\footnotesize\begin{verbatim}
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(setf L1 (loop3 0 's3 1)) ==>
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<AbSm - <<Loop[S3]>>>
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\end{verbatim}}
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Let us consider also the $2$--degeneracy of the base point of the loop space. In the printed
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result, the user will recognize the degeneracy $\eta_1\eta_0$ of the null loop, base point of
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$\Omega^1 S^3$:
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{\footnotesize\begin{verbatim}
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(setf null-simp (absm 3 +null-loop+)) ==>
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<AbSm 1-0 <<Loop>>>
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\end{verbatim}}
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We may build now a new space by pasting a disk $D3$ as indicated by the
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following call. It means that we ``paste'' to the space {\tt os3} a $3$--simplex named {\tt D3},
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the attaching map being described by the list of its faces in dimension $2$.
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{\footnotesize\begin{verbatim}
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(setf dos3 (disk-pasting os3 3 '<D3>
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(list L1 null-simp L1 null-simp))) ==>
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[K212 Simplicial-Set]
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\end{verbatim}}
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Let us compute a few homology groups of the new space {\tt dos3}:
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{\footnotesize\begin{verbatim}
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(homology dos3 2 4) ==>
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Homology in dimension 2 :
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Component Z/2Z
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Homology in dimension 3 :
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---done---
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\end{verbatim}}
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But more interesting, let us build the loop space of the object {\tt dos3} and
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let us compute the ho\-mo\-lo\-gy in dimension $5$:
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{\footnotesize\begin{verbatim}
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(setf odos3 (loop-space dos3)) ==>
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[K230 Simplicial-Group]
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(homology odos3 5) ==>
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Homology in dimension 5 :
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Component Z/2Z
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Component Z/2Z
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Component Z/2Z
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Component Z/2Z
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Component Z/2Z
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Component Z/2Z
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Component Z
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--done--
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\end{verbatim}}
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\begin{center}
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--------
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\end{center}
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Let us continue with the Kan theory. First, we check that $S^3$ is not of type Kan and that $\Omega S^3$
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is indeed of type Kan and a non--Abelian simplicial group.
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{\footnotesize\begin{verbatim}
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(typep s3 'kan) ==>
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NIL
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(typep os3 'kan) ==>
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T
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(typep os3 'simplicial-group) ==>
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T
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(typep os3 'ab-simplicial-group) ==>
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NIL
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\end{verbatim}}
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Let us create the word $L2=(S3)^2$, i.e an object belonging to $\Omega S^3$ and let us
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apply the product of the underlying algebra upon $L2 \otimes L2$:
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{\footnotesize\begin{verbatim}
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(setf L2 (loop3 0 's3 2)) ==>
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<<Loop[S3\2]>>
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(setf square (aprd os3 4 (tnpr 2 L2 2 L2))) ==>
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----------------------------------------------------------------------{CMBN 4}
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<1 * <<Loop[1-0 S3\2][3-2 S3\2]>>>
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<-1 * <<Loop[2-0 S3\2][3-1 S3\2]>>> <-------
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<1 * <<Loop[2-1 S3\2][3-0 S3\2]>>>
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<1 * <<Loop[3-0 S3\2][2-1 S3\2]>>>
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<-1 * <<Loop[3-1 S3\2][2-0 S3\2]>>>
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<1 * <<Loop[3-2 S3\2][1-0 S3\2]>>>
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------------------------------------------------------------------------------
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\end{verbatim}}
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We see that the result is a linear combination of words composed from degeneracies
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of {\tt L2}. The following instruction selects the generator part of the
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second element of the previous combination (noted by the reverse arrow).
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{\footnotesize\begin{verbatim}
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(setf L4 (gnrt (second (cmbn-list square)))) ==>
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<<Loop[2-0 S3\2][3-1 S3\2]>>
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\end{verbatim}}
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Let us use the lisp function {\tt mapcar} (one among the various iteration functions of Lisp) to
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create the list of the faces 1, 2, 3 and 4 of the object {\tt L4}, this list is a ``Kan hat''.
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{\footnotesize\begin{verbatim}
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(setf hat (mapcar #'(lambda (i) (face os3 i 4 l4)) '(1 2 3 4))) ==>
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(<AbSm - <<Loop[1 S3\2][2 S3\2]>>> <AbSm - <<Loop[0 S3\2][2 S3\2]>>>
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<AbSm - <<Loop[0 S3\2][1 S3\2]>>> <AbSm 1 <<Loop[S3\2]>>>)
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\end{verbatim}}
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The function {\tt kfll} tries to find a filling of this ``Kan hat'', and we see that
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the face $2$ of the resulting simplex (which is very different from {\tt L4}) is the same
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as the face $2$ of {\tt L4}.
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{\footnotesize\begin{verbatim}
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(setf kan-simplex (kfll os3 0 4 hat)) ==>
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<AbSm - <<Loop[3-1 S3\2][2-1 S3\-2][2-0 S3\2][1-0 S3\-2][2-1 S3\2][3-1 S3\-2]
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[1-0 S3\2][3-1 S3\2][3-0 S3\-2][1-0 S3\-2][3-0 S3\2][2-0 S3\-2]
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[1-0 S3\2][3-0 S3\-2][2-0 S3\2][3-0 S3\2]>>>
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(face os3 2 4 kan-simplex) ==>
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<AbSm - <<Loop[0 S3\2][2 S3\2]>>>
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(second hat) ==>
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<AbSm - <<Loop[0 S3\2][2 S3\2]>>>
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\end{verbatim}}
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\begin{center}
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--------
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\end{center}
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Let ${\cal G}$ be a simplicial group $0$--reduced. $\Omega S^3$ is such a group.
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The software {\tt Kenzo} allows
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the construction of the universal bundle $\bar{\cal W}{\cal G}$, i.e. the
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classifying space of ${\cal G}$. In our case, as $\Omega S^3$ in non--Abelian, the
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result is not a simplicial group but only a simplicial set. We verify that the $H_4$ is null.
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{\footnotesize\begin{verbatim}
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(setf cls-os3 (classifying-space os3)) ==>
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[K598 Simplicial-Set]
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(typep cls-os3 'simplicial-group) ==>
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NIL
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(homology cls-os3 4) ==>
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Homology in dimension 4 :
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---done---
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\end{verbatim}}
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\begin{center}
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--------
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\end{center}
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\vskip 0.40cm
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Let us end this short overview with an example of computation of homotopy groups.
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The method used in {\tt Kenzo} is the Whitehead tower.
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In this current version only the case where the first non--null
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homology group (in non--null dimension) is ${\Z}$ or ${\Z}/{2 {\Z}}$ can be processed; however if this
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homology group is a direct sum of several copies of ${\Z}$ or ${\Z}/{2 {\Z}}$, then the corresponding
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stage of the Whitehead tower may also be constructed step by step.
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\vskip 0.40cm
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We take again $Moore({\Z}/2{\Z}, 3)$ whose $H_3$ is ${\Z}/2{\Z}$. First the fundamental cohomology
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class is constructed:
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{\footnotesize\begin{verbatim}
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(setf ch3 (chml-clss m23 3)) ==>
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[K729 Cohomology-Class on K1 of degree 3]
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\end{verbatim}}
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Then the function {\tt z2-whitehead} is called to build a fibration over the simplicial set {\tt m23}
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canonically associated to the cohomology class {\tt ch3}.
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{\footnotesize\begin{verbatim}
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(setf f3 (z2-whitehead m23 ch3)) ==>
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[K730 Fibration K1 -> K488]
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\end{verbatim}}
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Then the total space of the fibration is built:
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{\footnotesize\begin{verbatim}
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(setf x4 (fibration-total f3)) ==>
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[K736 Simplicial-Set]
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\end{verbatim}}
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The $H_4$ of this total space is the $\pi_4$ of $Moore({\Z}/2{\Z})$:
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{\footnotesize\begin{verbatim}
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(homology x4 3 5) ==>
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Homology in dimension 3 :
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---done---
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Homology in dimension 4 :
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Component Z/2Z
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---done---
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\end{verbatim}}
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We may now iterate the process, to compute the $\pi_5$ of $Moore({\Z}/2{\Z})$:
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{\footnotesize\begin{verbatim}
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(setf ch4 (chml-clss x4 4)) ==>
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[K817 Cohomology-Class on K802 of degree 4]
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(setf f4 (z2-whitehead x4 ch4)) ==>
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[K832 Fibration K736 -> K818]
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(setf x5 (fibration-total f4)) ==>
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[K838 Simplicial-Set]
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(homology x5 4 6) ==>
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Homology in dimension 4 :
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---done---
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Homology in dimension 5 :
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Component Z/4Z
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---done---
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\end{verbatim}}
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So $\pi_5(Moore({\Z}/2{\Z}))$ is ${\Z}/4{\Z}$. |