369 lines
15 KiB
TeX
369 lines
15 KiB
TeX
\chapter {Eilenberg-Moore spectral sequence II}
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\section{Introduction}
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This chapter is devoted to the effective homology version of the spectral
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sequence of Eilenberg-Moore, in the particular case of classifying spaces.
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More precisely, let ${\cal G}$ be a simplicial group; then its classifying space
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$\bar{\cal W}{\cal G}$ is a simplicial set canonically defined\footnote
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{{\bf J. Peter May}. {\em Simplicial objects in algebraic topology}, Van Nostrand, 1967.}.
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Furthermore, if ${\cal G}$ is an Abelian simplicial group, then $\bar{\cal W}{\cal G}$
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is again an Abelian simplicial group with natural structure, so that the $\bar{\cal W}$
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construction can be iterated. The $\bar{\cal W}$ construction is implemented in {\tt Kenzo}
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only if ${\cal G}$ is reduced (${\cal G}_0$ has only one element). In particular if
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${\cal G}$ is a simplicial group with effective homology, then {\tt Kenzo} constructs
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a version of $\bar{\cal W}{\cal G}$ also with effective homology.
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\newpage
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\section{The detailed construction}
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Let ${\cal G}$ be a simplicial group with effective homology. This means that
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a homotopy equivalence:
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$$\diagram{
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& \hat{C}_* \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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{\cal C}_*({\cal G}) & & EG_* \cr
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}$$
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is provided, where the chain complex $EG_*$ is effective
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and must be con\-si\-de\-red as describing
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the homology of ${\cal C}_*({\cal G})$. This scheme includes the case where
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${\cal C}_*({\cal G})$ itself is effective; without any other information,
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the program constructs automatically a trivial homotopy equivalence.
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\par
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Now, if we apply the {\em Bar} functor to this homotopy equivalence
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we obtain the homotopy equivalence $H_R$:
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$$\diagram{
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&{\widetilde{Bar}}^{\hat{ C}_*} \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z}) & & {\widetilde{Bar}}^{EG_*}({\Z}, {\Z}) \cr
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}$$
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in which the $\widetilde{Bar}$'s are Bars constructions with respect to
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the $A_\infty$ structure on $\hat{C}_*$ and $EG_*$ defined by
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the initial homotopy equivalence.
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\par
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By analogy with the result of Julio Rubio\footnote{{\bf J.J. Rubio-Garcia}. {\em Homologie
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effective des espaces de lacets it\'er\'es: un logiciel}, Th\`ese, Institut Fourier, 1991.}
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one may show that it is possible to construct another homotopy equivalence, $H_L$:
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$$\diagram{
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& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),{\Z}) \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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{\cal C}_*(\bar{\cal W}{\cal G}) & & Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z}) \cr
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}$$
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Both reductions of $H_L$ are obtained by the basic perturbation lemma, as explained
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in the following section.
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The composition of both homotopy e\-qui\-va\-len\-ces, $H_L$ and $H_R$, makes the link between $\bar{\cal W}{\cal G}$,
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the classifying space of the simplicial group ${\cal G}$
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and the effective right bottom chain complex of $H_R$.
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\subsection {Obtaining the left reduction $H_L$}
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The homotopy equivalence $H_L$ is obtained by a sequence of intermediate constructions
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based mainly upon two applications of the basic perturbation lemma.
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We are led to consider the two following modules:
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\begin{itemize}
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\item ${\cal C}_*({\cal G})$ module on itself, with the canonical product.
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$${\cal C}_*({\cal G})\otimes {\cal C}_*({\cal G})\stackrel {\Pi}{\longrightarrow} {\cal C}_*({\cal G}).$$
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\item ${\check{\cal C}}_*({\cal G})$, module on ${\cal C}_*({\cal G})$, with a ``trivial''
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product
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$${\check{\cal C}}_*({\cal G})\otimes{\cal C}_*({\cal G}) \stackrel {\check{\Pi}}{\longrightarrow}
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{\check{\cal C}}_*({\cal G})$$
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defined by
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$$\sigma \otimes \tau \longmapsto \sigma . \varepsilon(\tau),$$
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where $\varepsilon$ is the traditional augmentation of ${\cal C}_*({\cal G})$.
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\end{itemize}
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Now, we consider the set of the followings Bars
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(where $u$ means {\em ``untwisted''} and $t$ {\em ``twisted''}):
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\begin{eqnarray*}
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{\rm Hat}_{uu} &=& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\check{\cal C}}_*({\cal G}),{\Z}),\\
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{\rm Hat}_{ut} &=& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\cal C}_*({\cal G}),{\Z}),\\
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{\rm Hat}_{tu} &=& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t{\check{\cal C}}_*({\cal G}),{\Z}),\\
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{\rm Hat}_{tt} &=& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),{\Z}).
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\end{eqnarray*}
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One may always say that ${\rm Hat}_{ut}$ is obtained from ${\rm Hat}_{uu}$ by a perturbation $\delta_r$ ($r$: right)
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induced by the discrepancy between the products
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in ${\check{\cal C}}_*({\cal G})$ and ${\cal C}_*({\cal G})$
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and that ${\rm Hat}_{tu}$ is obtained from ${\rm Hat}_{uu}$
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by a perturbation $\delta_l$ ($l$: left) induced by the twisted tensor product $\otimes_t$.
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After that,
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${\rm Hat}_{tt}$ is obtained from ${\rm Hat}_{tu}$ by the perturbation $\delta_r$ as well as, by commutativity,
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from ${\rm Hat}_{ut}$ by the perturbation $\delta_l$. \par
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This is shown in the following diagram (here, the arrows are not reductions, but denote
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the perturbations between the differential morphisms):
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$$\diagram{
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& {\rm Hat}_{uu} \cr
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\delta_l \swarrow & & \searrow \delta_r \cr
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{\rm Hat}_{tu} & & {\rm Hat}_{ut} \cr
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\delta_r \searrow & & \swarrow \delta_l \cr
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& {\rm Hat}_{tt} \cr
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}$$
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The underlying graded modules ${\rm Hat}_{uu}$, ${\rm Hat}_{ut}$, ${\rm Hat}_{tu}$ and ${\rm Hat}_{tt}$ are the same
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and the program keeps ${\rm Hat}_{uu}$ as the underlying graded module for all the chain complexes.
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The differential perturbations are given by the formulas:
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\begin{eqnarray*}
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\delta_l[(w\otimes {\check g}_0)\otimes (g_1 \otimes\cdots\otimes g_n)] &=&
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[d_{\otimes t}(w\otimes {\check g}_0) - d_\otimes (w\otimes {\check g}_0)] \otimes (g_1 \otimes\cdots\otimes g_n), \\
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\delta_r[(w\otimes {\check g}_0)\otimes (g_1 \otimes\cdots\otimes g_n)] &=&
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w \otimes(g_0 . g_1) \otimes g_2 \otimes\cdots\otimes g_n,
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\end{eqnarray*}
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where ${\check g}_0 \in {\check {\cal G}}_0,\, g_i \in {\cal G},\, w \in \bar{\cal W}{\cal G}.$
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\par
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Now, on the other hand, we know that there exists a reduction
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$${\rm Hat}_{ut} = Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes
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{\cal C}_*({\cal G}),{\Z}) \Longrightarrow {\cal C}_*(\bar{\cal W}{\cal G}),$$
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so, perturbing this reduction by $\delta_l$, one obtains the Rubio reduction
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$${\rm Hat}_{tt}=Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t
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{\cal C}_*({\cal G}),{\Z}) \Longrightarrow {\cal C}_*(\bar{\cal W}{\cal G}).$$
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On the other hand, we know also that there exists a reduction
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$${\rm Hat}_{tu} = Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t
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{\check{\cal C}}_*({\cal G}),{\Z}) \Longrightarrow
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Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z}),$$
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so, perturbing this reduction by $\delta_r$, one obtains the reduction
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$${\rm Hat}_{tt}=Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t
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{\cal C}_*({\cal G}),{\Z}) \Longrightarrow Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z}).$$
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Finally, we have obtained the wished left homotopy equivalence $H_L$:
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$$\diagram{
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& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),{\Z}) \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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{\cal C}_*(\bar{\cal W}{\cal G}) & & Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z}) \cr
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}$$
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\subsection {The useful functions}
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For the applications, the only function that the user must know is the function {\tt classifying-space-efhm}
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which builds the final homotopy equivalence. But for the
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interested user, we give nevertheless a short description of all the functions
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involved in the described process.
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\vskip 0.30cm
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{\parindent=0mm
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{\leftskip=5mm
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{\tt classifying-space-efhm} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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From the simplicial group ${\cal G}$ ($0$-reduced) with effective homology (here the argument {\em smgr}), build
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a homotopy equivalence giving an effective homology version of the classifying space $\bar{\cal W}{\cal G}$ of ${\cal G}$.
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This homotopy
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equivalence will be used by the homology function to compute the homology groups. In fact, due to
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the slot-unbound mechanism of CLOS, this function will be automatically called, as soon
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as the user requires a homology group for a classifying space. \par}
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{\leftskip=5mm
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{\tt cs-hat-u-u} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the chain complex
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$Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\check{\cal C}}_*({\cal G}),{\Z})$. Because of
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the particular structure of ${\check{\cal C}}_*({\cal G})$, this chain complex is nothing but
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$[{\cal C}_*(\bar{\cal W}{\cal G})\otimes {\check{\cal C}}_*({\cal G})]\otimes Bar^{{\cal C}_*({\cal G})}({\Z},{\Z}).$
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\par}
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{\leftskip=5mm
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{\tt cs-hat-right-perturbation} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the morphism corresponding to the differential perturbation $\delta_r$, induced by
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the discrepancy between the respective products in ${\check{\cal C}}_*({\cal G})$ and
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in ${\cal C}_*({\cal G})$. \par}
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{\leftskip=5mm
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{\tt cs-hat-u-t} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the chain complex
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$Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\cal C}_*({\cal G}),{\Z})$ by applying
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the differential perturbation {\tt hat-right-perturbation} upon
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the chain complex {\tt hat-u-u {\em smgr}}. This is realized by the method
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{\tt add}. \par}
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{\leftskip=5mm
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{\tt cs-hat-t-u} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the chain complex
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$Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\check{\cal C}}_*({\cal G}),{\Z})$. Because of
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the particular structure of ${\check{\cal C}}_*({\cal G})$, this chain complex is nothing but
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$[{\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\check{\cal C}}_*({\cal G})]\otimes Bar^{{\cal C}_*({\cal G})}({\Z},{\Z}),$
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where the twisted tensor product is the botton chain complex of the Brown reduction
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of the fibration of the simplicial group. \par}
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{\leftskip=5mm
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{\tt cs-hat-left-perturbation} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the morphism corresponding to the differential perturbation $\delta_l$ induced
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by the twisted tensor product. This morphism is nothing but the tensor product of two
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morphisms: the perturbation morphism by-product of the Brown reduction of the fibration of
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the simplicial group and the identity morphism on $Bar^{{\cal C}_*({\cal G})}({\Z},{\Z})$. \par}
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{\leftskip=5mm
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{\tt cs-left-hmeq-hat} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the chain complex
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${\rm Hat}_{tt}=Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),{\Z})$ by
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perturbing the chain complex ${\rm Hat}_{ut}$ by the perturbation $\delta_l$. \par}
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{\leftskip=5mm
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{\tt cs-pre-left-hmeq-left-reduction} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the reduction
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$${\rm Hat}_{tu}=Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\cal C}_*({\cal G}),{\Z}) \Longrightarrow {\cal C}_*(\bar{\cal W}{\cal G}).$$
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\par}
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{\leftskip=5mm
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{\tt cs-pre-left-hmeq-right-reduction} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the reduction
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$${\rm Hat}_{ut}=Bar^{{\cal C}_*({\cal G})}({\check{\cal C}}_*({\cal G})\otimes_t {\cal C}_*(\bar{\cal W}{\cal G}),{\Z}) \Longrightarrow
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Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z}).$$ \par}
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{\leftskip=5mm
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{\tt cs-left-hmeq-left-reduction} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the Rubio reduction
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$$Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),{\Z}) \Longrightarrow{\cal C}_*(\bar{\cal W}{\cal G})$$
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by perturbing by the perturbation $\delta_r$ the reduction
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$${\rm Hat}_{tu} \Longrightarrow {\cal C}_*(\bar{\cal W}{\cal G})$$
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obtained by the function {\tt pre-left-hmeq-left-reduction}. \par}
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{\leftskip=5mm
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{\tt cs-left-hmeq-right-reduction} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the reduction
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$$Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),{\Z}) \Longrightarrow
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Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z})$$
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by perturbing by the perturbation $\delta_l$ the reduction
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$${\rm Hat}_{ut} \Longrightarrow Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z})$$
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obtained by the function {\tt cs-pre-left-hmeq-right-reduction}. \par}
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{\leftskip=5mm
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{\tt cs-left-hmeq} {\em smgr} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the homotopy equivalence
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$$\diagram{
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& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),{\Z}) \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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{\cal C}_*(\bar{\cal W}{\cal G}) & & Bar^{{\cal C}_*({\cal G})}({\Z}, {\Z}) \cr
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}$$
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from the above reductions. The function {\tt classifying-space-efhm} composes this left homotopy
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equivalence $H_L$ with the homotopy equivalence $H_R$ which is the Bar of a pre-existing
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homotopy equivalence (possibly the trivial one) between the module ${\cal G}$ and an effective
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version of it, as described at the beginning of this chapter.
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\par}
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}
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\subsection {Searching homology for classifying spaces}
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The \index{searching homology!classifying spaces} origin list of a classifying space object has the form
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{\tt (CLASSIFYING-SPACE {\em smgr})}.
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The {\tt search-efhm} method applied to a classifying space object
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consists essentially in a call to the function {\tt classifying-space-efhm}
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described just above.
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{\footnotesize\begin{verbatim}
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(defmethod SEARCH-EFHM (classifying-space (orgn (eql 'classifying-space)))
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(declare (type simplicial-set classifying-space))
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(classifying-space-efhm (second (orgn classifying-space))))
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\end{verbatim}}
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The following Lisp definition of the function {\tt classifying-space-efhm} shows that
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the process may be recursif if it happens that {\em smrg} is itself a classifying space:
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{\footnotesize\begin{verbatim}
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(defun CLASSIFYING-SPACE-EFHM (smgr)
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(declare (type simplicial-group smgr))
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(let ((left-hmeq (cs-left-hmeq smgr))
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(right-hmeq (bar (efhm smgr))))
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(declare (type homotopy-equivalence left-hmeq right-hmeq))
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(cmps left-hmeq right-hmeq)))
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\end{verbatim}}
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\subsection* {Examples}
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{\footnotesize\begin{verbatim}
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(setf kz1 (k-z-1)) ==>
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[K1 Abelian-Simplicial-Group]
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(homology kz1 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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Component Z
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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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Homology in dimension 6 :
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Homology in dimension 7 :
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Homology in dimension 8 :
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Homology in dimension 9 :
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---done---
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(setf bkz1 (classifying-space kz1)) ==>
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[K23 Abelian-Simplicial-Group]
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(homology bkz1 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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Component Z
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Homology in dimension 3 :
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Homology in dimension 4 :
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Component Z
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Homology in dimension 5 :
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Homology in dimension 6 :
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Component Z
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Homology in dimension 7 :
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Homology in dimension 8 :
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Component Z
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Homology in dimension 9 :
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---done---
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(setf obkz1 (loop-space bkz1)) ==>
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[K154 Simplicial-Group]
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(homology obkz1 0 6) ==>
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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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Component Z
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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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---done---
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\end{verbatim}}
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\subsection* {Lisp files concerned in this chapter}
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{\tt cs-space-efhm.lisp}, {\tt searching-homology.lisp}.
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