360 lines
13 KiB
TeX
360 lines
13 KiB
TeX
\chapter {Classifying Spaces}
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\section {Introduction}
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Let ${\cal G}$ be a simplicial group $0$--reduced. The software {\tt Kenzo} allows
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the construction of the universal bundle $\bar{\cal W}{\cal G}$, called the
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classifying space\index{classifying space} of ${\cal G}$.
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It is known that this is the base space of a principal bundle
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such that the total space is contractible. We know also 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. In dimension $n$ the simplices of the
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classifying space are elements of the product:
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$${\cal G}_{n-1}\times {\cal G}_{n-2}\times\cdots\times{\cal G}_0,$$
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where ${\cal G}_k$ is the set of the $k$--simplices of the given simplicial group. \par
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A simplex of $\bar{\cal W}{\cal G}$ is called {\bf gbar}\index{gbar} ({\em geometric} bar)
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and may be seen as an $n$--uple
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$$(g_{n-1}, g_{n-2}, \ldots, g_1, g_0),$$
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where the $g_i$'s are elements of ${\cal G}_i$, possibly {\em degenerated}.
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In {\tt Kenzo} a gbar is represented externally under the form:\index{gbar!representation}
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\begin{center}
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{\tt <<GBar < $[\eta g]_{n-1}$><$[\eta g]_{n-2}$ > ... < $[\eta g]_1$ > < - $g_0$> >>}
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\end{center}
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where a term such as $[\eta g]_i$ is a simplex of ${\cal G}_i$. Let us recall that
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our group ${\cal G}$ is assumed $0$--reduced; as a consequence, the $g_0$ compound
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of a gbar is necessarily the identity element of ${\cal G}$. \par
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In the symbolic expression $[\eta g]_i$, the operator
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$\eta$ represents a sequence (possibly the void one) of degeneracy operators
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applied coherently upon some element of ${\cal G}_k$, for $k\leq i$. For instance $[\eta g]_1$
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can be $\eta_0 g_0$, i.e. the $0$--degeneracy of the base point and $[\eta g]_2$ can be
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can be $\eta_1\eta_0 g_0$. To be more explicit, let us take the Abelian simplicial group
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$K({\Z}_2, 1)$ (see the simplicial groups chapter), then some elements of the basis in dimension $4$ of
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its classifying space are, for instance:
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{\footnotesize\begin{verbatim}
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<<GBar<- 3><- 2><- 1><- 0>>>
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<<GBar<- 3><1-0 0><0 0><- 0>>>
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<<GBar<1 2><- 2><0 0><- 0>>>
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<<GBar<1 2><0 1><0 0><- 0>>>
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<<GBar<2 2><1 1><0 0><- 0>>>
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<<GBar<2-0 1><1 1><0 0><- 0>>>
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<<GBar<2-1 1><1 1><- 1><- 0>>>
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<<GBar<2-1 1><0 1><0 0><- 0>>>
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\end{verbatim}}
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We recall that the symbol {\tt -}, in front of an element means ``no degeneracy''.
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\section {Face, degeneracy and group operations}
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Let us denote a gbar under the following generic form:
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$$g_{bar}=(g_{n-1}, g_{n-2}, \ldots, {\bf g_j}, \ldots, g_1, g_0).$$
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In this symbolism, $g_0$ is also the base point of the initial simplicial group.
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The rule for the face operators is the following:
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\begin{itemize}
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\item If $j=n$, $\delta_n g_{bar}= (g_{n-2}, g_{n-3}, \ldots, g_1, g_0).$
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\item If $j \not=n$,
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$\delta_j g_{bar}= (\delta_j g_{n-1}, \ldots, {\bf \delta_j g_{j+1}, g_{j-1}.\delta_j g_j, g_{j-1}}, \ldots, g_1, g_0),$
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where in the term $\bf g_{j-1}.\delta_j g_j$, the dot represents the group composition.
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\item If $j=0$, $\delta_0 g_{bar}= (\delta_0 g_{n-1}, \ldots, \delta_0 g_1).$
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\end{itemize}
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The rule for the degeneracy operators is:
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$$\eta_j g_{bar}=
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(\eta_j g_{n-1}, \ldots, {\bf \eta_j g_{j+1}, \eta_{j-1}\eta_{j-2}\ldots\eta_1\eta_0 g_0, g_{j-1}}, \ldots, g_1, g_0).$$
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If ${\cal G}$ is Abelian then $\bar{\cal W}{\cal G}$ is also Abelian and its
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composition law is simply:
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$$(g_{n-1}, \ldots, g_1, g_0).(g'_{n-1}, g'_1, g'_0)=
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(g_{n-1}.g'_{n-1}, \ldots, g_1.g'_1, g_0.g'_0).$$
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It is well known that if the initial group is not Abelian then, in general, the previous
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composition law is not compatible with the simplicial structure.
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\section {The functions for the classifying spaces}
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{\parindent=0mm
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{\leftskip=5mm
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{\tt gbar} {\em dmns dop-1 elem-1 ... dop-n elem-n} \hfill {\em [Macro]} \par}
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{\leftskip=15mm
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Build a gbar in dimension {\em dmns}. The sequence {\em dop-1 elem-1 ... dop-n elem-n},
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is a sequence of pairs ({\em dop-k, elem-k}), where {\em dop-k} is a {\bf binary coded}
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degeneracy operator and {\em elem-k} a simplex of the initial simplicial group. The ordering of
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the pairs in the sequence must conform to the theoritical definition of a gbar.
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If the sequence of pairs is void, the special call {\tt (gbar 0)},
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creates the base point of the classifying space
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externally represented by {\tt <<GBAR>>}. This base point can be reached through
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the system symbol {\tt +null-gbar+}. \par}
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{\leftskip=5mm
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{\tt classifying-space-cmpr} {\em cmpr}\hfill {\em [Function]} \par}
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{\leftskip=15mm
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From a comparison function {\em cmpr} defined for a simplicial group ${\cal G}$, build a
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comparison function for the classifying space of ${\cal G}$. \par}
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{\leftskip=5mm
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{\tt classifying-space-basis} {\em basis}\hfill {\em [Function]} \par}
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{\leftskip=15mm
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From the basis function {\em basis} of a simplicial group ${\cal G}$, build a
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basis function for the classifying space of ${\cal G}$. If ${\cal G}$ is locally effective,
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this function returns the symbol {\tt :locally-effective}.\par}
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{\leftskip=5mm
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{\tt classifying-space-face} {\em face sintr-grml}\hfill {\em [Function]} \par}
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{\leftskip=15mm
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From the face function {\em face} and the lisp function for the group law, {\em sintr-grml},
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of a simplicial group ${\cal G}$, build a
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face function for the classifying space of ${\cal G}$. \par}
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{\leftskip=5mm
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{\tt classifying-space} {\em smgr}\hfill {\em [Method]} \par}
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{\leftskip=15mm
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Build the simplicial set, classifying space of the simplicial group {\em smgr}.
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The construction uses the three preceding auxiliary functions. The base point
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of the returned simplicial set is the null gbar {\tt +null-gbar+}. \par}
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{\leftskip=5mm
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{\tt classifying-space} {\em A-smgr}\hfill {\em [Method]} \par}
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{\leftskip=15mm
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Build the Abelian simplicial group, classifying space of the Abelian simplicial group {\em A-smgr}.
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The group law and its inverse (slots {\tt :sintr-grml} and {\tt :sintr-grin}) are built
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by the two auxiliary functions: \par}
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{\leftskip=20mm
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{\tt classifying-space-grml-sintr} and \par}
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{\leftskip=20mm
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{\tt classifying-space-grin-sintr}, \par}
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{\leftskip=15mm
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defining the laws of this new group from the corresponding laws of {\em A-smgr}. \par}
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}
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\section {Eilenberg-Mac Lane spaces $K(\pi,n)$}
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We\index{$K(\pi,n)$} have already described $K({\Z},1)$ and $K({\Z}_2,1)$ in the chapter on simplicial groups.
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This section gives some complements about the construction of $K({\Z},n)$ and $K({\Z}_2,n)$. The trivial case
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for $n=0$ is not implemented in {\tt Kenzo}. For $n > 1$, these simplicial groups
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are built recursively as the classifying spaces of $K({\Z},n-1)$ and $K({\Z}_2,n-1)$, respectively, as shown
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in the following description:
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\par
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\vskip 0.40cm
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{\parindent=0mm
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{\leftskip=5mm
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{\tt k-z} {\em n} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build $K({\Z},n)$ recursively from $K({\Z},n-1)$. If $n=1$, return $K({\Z},1)$
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{\footnotesize\begin{verbatim}
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(defun K-Z (n)
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(the ab-simplicial-group
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(if (= n 1)
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(k-z-1)
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(classifying-space (k-z (1- n))))))
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\end{verbatim}}
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\par}
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{\leftskip=5mm
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{\tt k-z2} {\em n} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build $K({\Z}_2,n)$ recursively from $K({\Z}_2,n-1)$. If $n=1$, return $K({\Z}_2,1)$
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{\footnotesize\begin{verbatim}
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(defun K-Z2 (n)
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(the ab-simplicial-group
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(if (= n 1)
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(k-z2-1)
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(classifying-space (k-z2 (1- n))))))
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\end{verbatim}}
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\par}}
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These spaces have an effective homology which is computed via a {\em search-efhm} method (described in a next chapter).
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For the case $n=1$, we may recall here that $K({\Z},1)$, though being locally effective,
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has the homology of $S^1$. The {\em search-efhm} method applied to this chain complex does the following: a reduction
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between $K({\Z},1)$ and the chain complex {\tt circle} (described in the chain complex chapter) is
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built using the function {\tt kz1-rdct} and the homotopy equivalence, value of the slot {\tt efhm}, is
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built between this reduction and the trivial reduction of $K({\Z},1)$ upon itself.\par
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On the other hand, $K({\Z},2)$ has a very simple finite basis in every dimension (see $\R$-projective
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spaces), so its homology is computed directly from its basis. The homotopy
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equivalence, value of the slot {\tt efhm}, is the trivial homotopy equivalence of $K({\Z},2)$ upon
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itself.
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\newpage
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\subsection* {Examples}
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Let us begin by showing some examples of gbars built from various
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initial simplicial groups. In the second statement, the initial simplicial group
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is $K({\Z}_2, 1)$, in the third, it is $\Omega({\mathit Moore}(2,2))$. We recall that
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in the call of the function {\tt gbar}, the degeneracy operators appear as integers,
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according to the general rule in {\tt Kenzo}, but the printed result shows in clear
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the sequence of $\eta_i$'s.
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{\footnotesize\begin{verbatim}
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(gbar 2 1 'a 2 'b) ==>
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<<GBar<0 A><1 B>>>
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(gbar 4 0 3 3 0 1 0 0 0) ==>
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<<GBar<- 3><1-0 0><0 0><- 0>>>
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(setf gbar-mr22 (gbar 4 0 (loop3 3 'm2 1 4 'n3 1)
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0 (loop3 0 'n3 1)
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0 (loop3 0 'm2 1)
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0 +null-loop+)) ==>
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<<GBar<- <<Loop[1-0 M2][2 N3]>>><- <<Loop[N3]>>><- <<Loop[M2]>>>
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<- <<Loop>>>>>
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\end{verbatim}}
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Let us test the face function upon this simplex. The user will note
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that the face function returns an abstract simplex.
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{\footnotesize\begin{verbatim}
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(setf om (loop-space (moore 2 2))) ==>
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[K18 Simplicial-Group]
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(setf face (classifying-space-face (face om) (sintr (grml om)))) ==>
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#<Closure (FLET CLASSIFYING-SPACE-FACE RSLT) @ #x49af32>
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(dotimes (i 5) (print (funcall face i 4 gbar-m22))) ==>
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<AbSm - <<GBar<- <<Loop[0 M2][1 M2]>>><- <<Loop[M2]>>><- <<Loop>>>>>>
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<AbSm 0 <<GBar<- <<Loop[M2]>>><- <<Loop>>>>>>
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<AbSm - <<GBar<- <<Loop[0 M2][N3]>>><- <<Loop[M2\2]>>><- <<Loop>>>>>>
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<AbSm - <<GBar<- <<Loop[N3\2]>>><- <<Loop[M2]>>><- <<Loop>>>>>>
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<AbSm - <<GBar<- <<Loop[N3]>>><- <<Loop[M2]>>><- <<Loop>>>>>>
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\end{verbatim}}
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Let us work now, directly with the classifying space of $K({\Z}_2, 1)$.
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{\footnotesize\begin{verbatim}
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(setf cs-kz21 (classifying-space (k-z2-1))) ==>
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[K13 Abelian-Simplicial-Group]
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\end{verbatim}}
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\newpage
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{\footnotesize\begin{verbatim}
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(orgn cs-kz21) ==>
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(CLASSIFYING-SPACE [K1 Abelian-Simplicial-Group])
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(setf elem-1 (first (basis cs-kz21 4))) ==>
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<<GBar<- 3><- 2><- 1><- 0>>>
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(? cs-kz21 4 elem-1) ==>
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----------------------------------------------------------------------{CMBN 3}
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<2 * <<GBar<- 2><- 1><- 0>>>>
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------------------------------------------------------------------------------
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(? cs-kz21 *) ==>
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----------------------------------------------------------------------{CMBN 2}
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------------------------------------------------------------------------------
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(cprd cs-kz21 4 elem-1) ==>
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----------------------------------------------------------------------{CMBN 4}
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<1 * <TnPr <<GBar>> <<GBar<- 3><- 2><- 1><- 0>>>>>
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<1 * <TnPr <<GBar<- 1><- 0>>> <<GBar<- 1><- 0>>>>>
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<1 * <TnPr <<GBar<- 3><- 2><- 1><- 0>>> <<GBar>>>>
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------------------------------------------------------------------------------
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(dotimes (i 5)
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(print (face cs-kz21 i 4 elem-1))) ==>
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<AbSm - <<GBar<- 2><- 1><- 0>>>>
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<AbSm 0 <<GBar<- 1><- 0>>>>
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<AbSm 1 <<GBar<- 1><- 0>>>>
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<AbSm 2 <<GBar<- 1><- 0>>>>
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<AbSm - <<GBar<- 2><- 1><- 0>>>>
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\end{verbatim}}
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Let us test the law group with $K({\Z}, 1)$. We recall the simplices
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of this Abelian simplicial group are represented as lists of integers.
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The functions {\tt grml} and {\tt grin} accept also the gbars under the form
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of abstract simplices (see the last statement).
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{\footnotesize\begin{verbatim}
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(setf cs-kz1 (classifying-space (k-z-1))) ==>
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[K13 Abelian-Simplicial-Group]
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(grml cs-kz1 3 (crpr 0 (gbar 3 0 '(1 2) 0 '(3) 0 '())
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0 (gbar 3 0 '(-1 -2) 0 '(-3) 0 '()))) ==>
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<AbSm 2-1-0 <<GBar>>>
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(grml cs-kz1 3 (crpr 0 (gbar 3 0 '(1 2) 0 '(3) 0 '())
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4 (gbar 2 0 '(-3) 0 '()))) ==>
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<AbSm - <<GBar<- (1 2)><0 NIL><- NIL>>>>
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(grml cs-kz1 3 (crpr 0 (gbar 3 0 '(1 2) 0 '(3) 0 '())
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1 (gbar 2 0 '(-3) 0 '()))) ==>
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<AbSm - <<GBar<- (1 -1)><- (3)><- NIL>>>>
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(grin cs-kz1 3 (gbar 3 0 '(1 2) 1 '() 0 '())) ==>
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<AbSm - <<GBar<- (-1 -2)><0 NIL><- NIL>>>>
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(grin cs-kz1 3 *) ==>
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<AbSm - <<GBar<- (1 2)><0 NIL><- NIL>>>>
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\end{verbatim}}
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As $K({\Z}, 1)$ and $K({\Z_2}, 1)$ are Abelian simplicial groups, we may iterate the classifying
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space construction and retrieve some known results.
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{\footnotesize\begin{verbatim}
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(cat-init) ;; re-initialization
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(setf k-z-3 (k-z 3)) ==>
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[K25 Abelian-Simplicial-Group]
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(homology k-z-3 0 10) ==>
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Homology in dimension 0 :
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Component Z
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Homology in dimension 1 :
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Homology in dimension 2 :
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Homology in dimension 3 :
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Component Z
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Homology in dimension 4 :
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Homology in dimension 5 :
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Component Z/2Z
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Homology in dimension 6 :
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Homology in dimension 7 :
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Component Z/3Z
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Homology in dimension 8 :
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Component Z/2Z
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Homology in dimension 9 :
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Component Z/2Z
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(setf k-z2-5 (k-z2 5)) ==>
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[K342 Abelian-Simplicial-Group]
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(homology k-z2-5 0 7) ==>
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Homology in dimension 0 :
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Component Z
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Homology in dimension 1 :
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Homology in dimension 2 :
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Homology in dimension 3 :
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Homology in dimension 4 :
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Homology in dimension 5 :
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Component Z/2Z
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Homology in dimension 6 :
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--done--
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\end{verbatim}}
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\subsection* {Lisp files concerned in this chapter}
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{\tt classifying-space.lisp}, {\tt k-pi-n.lisp}.
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