491 lines
17 KiB
TeX
491 lines
17 KiB
TeX
\chapter {Eilenberg-Moore spectral sequence I}
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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 loop spaces.
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More precisely, let $X$ be a {\bf $1$-reduced} simplicial set with effective homology,
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the software {\tt Kenzo} constructs a $0$-reduced simplicial set $GX$ which is
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also with effective homology. In particular, if $X$ is $m$-reduced, this
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process may be iterated $m$ times, producing an effective homology
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version of $G^kX,\,k\leq m$. So, the software builds an object which contains
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a complete solution of the Adams problem about the iteration of the
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cobar construction. The user is advised to consult the paper
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{\em Stable homotopy and iterated loop spaces, pp. 545} by {\bf Gunnar Carlsson} and
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{\bf R.James Milgram} in {\em Handbook of Algebraic Topology, pp 545} edited
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by {\em I.M.James, North-Holland, 1995}.
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\section{The detailed construction}
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Let ${\cal C}_*(X)$ be a coalgebra with effective homology. This means that
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a homotopy equivalence:
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$$\diagram{
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& \hat{C}_* \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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{\cal C}_*(X) & & EC_* \cr
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}$$
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is provided, where the chain complex $EC_*$ is effective
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and must be con\-si\-de\-red as describing
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the homology of ${\cal C}_*(X)$. This scheme includes the case where
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${\cal C}_*(X)$ is itself 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 Cobar} functor to this homotopy equivalence
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we obtain the homotopy equivalence $H_R$:
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$$\diagram{
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&{\widetilde{Cobar}}^{{\hat C}_*}({\Z},{\Z}) \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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Cobar^{{\cal C}_*(X)}({\Z}, {\Z}) & & {\widetilde{Cobar}}^{EC_*}({\Z}, {\Z}) \cr
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}$$
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in which the $\widetilde{Cobar}$'s are cobars construction with respect to
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the $A_\infty$-co\-al\-ge\-bras structure on ${\hat C}_*$ and $EC_*$ defined by
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the initial homotopy equivalence.
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\par
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Secondly, 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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has shown that it is possible to construct another homotopy equivalence, $H_L$:
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$$\diagram{
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& Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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{\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}({\Z}, {\Z}) \cr
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}$$
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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 $GX$,
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i.e. the loop space of $X$
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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 comodules:
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\begin{itemize}
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\item ${\cal C}_*(X)$ comodule on itself, with the canonical coproduct.
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$${\cal C}_*(X)\stackrel {\Delta}{\longrightarrow} {\cal C}_*(X)\otimes {\cal C}_*(X).$$
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\item ${\check{\cal C}}_*(X)$, comodule on ${\cal C}_*(X)$, with a ``trivial''
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coproduct
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$${\check{\cal C}}_*(X) \stackrel {\check{\Delta}}{\longrightarrow} {\cal C}_*(X)\otimes{\check{\cal C}}_*(X),\quad
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\sigma \rightarrow 1\otimes\sigma,\,\sigma \in {\check{\cal C}}_*(X).$$
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\end{itemize}
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Now, we consider the set of the followings cobars
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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} &=& Cobar^{{\cal C}_*(X)}({\Z}, {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX)),\\
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{\rm Hat}_{ut} &=& Cobar^{{\cal C}_*(X)}({\Z}, {\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)),\\
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{\rm Hat}_{tu} &=& Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)),\\
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{\rm Hat}_{tt} &=& Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)).
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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 twisted tensor product $\otimes_t$ 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 discrepancy between the coproducts
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in ${\check{\cal C}}_*(X)$ and ${\cal C}_*(X)$. 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_r({\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes c \otimes g) &=&
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[{\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n] \otimes [d_{\otimes t}(c \otimes g)-d_\otimes (c \otimes g)], \\
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\delta_l({\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes c \otimes g) &=&
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{\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes {\bar{\Delta}}c \otimes g,
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\end{eqnarray*}
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where ${\bar{c}}_i, c \in {\cal C}_*(X),\, g \in {\cal C_*}(GX))$ and $\bar{\Delta}= \Delta - \check{\Delta}$.
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\par
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\vskip 0.35cm
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Now, on one hand, we know that there exists a reduction
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$${\rm Hat}_{tu}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)) \Longrightarrow {\cal C}_*(GX),$$
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so, perturbing this reduction by $\delta_r$, one obtains the Rubio reduction
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$${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow{\cal C}_*(GX).$$
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On the other hand, we know also that there exists a reduction
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$${\rm Hat}_{ut}=Cobar^{{\cal C}_*(X)}({\Z}, {\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow
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Cobar^{{\cal C}_*(X)}({\Z}, {\Z}),$$
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so, perturbing this reduction by $\delta_l$, one obtains the reduction
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$${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow
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Cobar^{{\cal C}_*(X)}({\Z}, {\Z}).$$
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Finally, we have obtained the wished left homotopy equivalence $H_L$:
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$$\diagram{
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& Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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{\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}({\Z}, {\Z}) \cr
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}$$
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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 loop-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 loop-space-efhm} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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From the space $X$ ($1$-reduced) with effective homology (here the argument {\em space}), build
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a homotopy equivalence giving an effective homology version of the space $GX$. 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 of a loop-space.
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If $X$ is $n$-reduced and if the wished loop space is $\Omega^k X, \, k \leq n$, then
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the process will be automatically iterated. \par}
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{\leftskip=5mm
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{\tt ls-hat-u-u} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the chain complex
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$Cobar^{{\cal C}_*(X)}({\Z}, {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX))$. Because of
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the particular structure of ${\check{\cal C}}_*(X)$, this chain complex is nothing but
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$Cobar^{{\cal C}_*(X)}({\Z},{\Z})\otimes {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX).$
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\par}
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{\leftskip=5mm
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{\tt ls-hat-left-perturbation} {\em space} \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 by
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the discrepancy between the coproducts of ${\check{\cal C}}_*(X)$ and ${\cal C}_*(X)$. \par}
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{\leftskip=5mm
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{\tt ls-hat-t-u} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the chain complex
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$Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX))$ by applying
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the differential perturbation {\tt hat-left-perturbation} upon
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the chain complex built by the function {\tt hat-u-u}. This is realized by the method
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{\tt add}. \par}
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{\leftskip=5mm
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{\tt ls-hat-u-t} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the chain complex
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$Cobar^{{\cal C}_*(X)}({\Z},{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX))$. Because of
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the particular structure of ${\check{\cal C}}_*(X)$, this chain complex is nothing but
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$Cobar^{{\cal C}_*(X)}({\Z},{\Z})\otimes[{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)],$
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where the twisted tensor product is obtained by a call to {\tt twisted-tnsr-prdc}
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(see loop space fibrations chapter). \par}
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{\leftskip=5mm
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{\tt ls-hat-right-perturbation} {\em space} \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
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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 identity morphism on $Cobar^{{\cal C}_*(X)}({\Z},{\Z})$ and
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the perturbation morphism induced by the twisted tensor product. This last
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morphism is a by-product of the function {\tt szczarba}. \par}
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{\leftskip=5mm
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{\tt ls-left-hmeq-hat} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Return the chain complex
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${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX))$ 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 ls-pre-left-hmeq-left-reduction} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the reduction
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$${\rm Hat}_{tu}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)) \Longrightarrow {\cal C}_*(GX).$$
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\par}
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{\leftskip=5mm
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{\tt ls-pre-left-hmeq-right-reduction} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the reduction
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$${\rm Hat}_{ut}=Cobar^{{\cal C}_*(X)}({\Z},{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow
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Cobar^{{\cal C}_*(X)}({\Z}, {\Z}).$$ \par}
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{\leftskip=5mm
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{\tt ls-left-hmeq-left-reduction} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the Rubio reduction
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$$Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow{\cal C}_*(GX)$$
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by perturbing by the perturbation $\delta_r$ the reduction
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$${\rm Hat}_{tu} \Longrightarrow {\cal C}_*(GX)$$
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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 ls-left-hmeq-right-reduction} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the reduction
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$$Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow
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Cobar^{{\cal C}_*(X)}({\Z}, {\Z})$$
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by perturbing by the perturbation $\delta_l$ the reduction
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$${\rm Hat}_{ut} \Longrightarrow Cobar^{{\cal C}_*(X)}({\Z}, {\Z})$$
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obtained by the function {\tt pre-left-hmeq-right-reduction}. \par}
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{\leftskip=5mm
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{\tt ls-left-hmeq} {\em space} \hfill {\em [Function]} \par}
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{\leftskip=15mm
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Build the homotopy equivalence
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$$\diagram{
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& Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \cr
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\swarrow\nearrow & & \searrow\nwarrow \cr
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{\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}({\Z}, {\Z}) \cr
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}$$
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from the above reductions. The function {\tt loop-space-efhm} composes this left homotopy
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equivalence $H_L$ with a homotopy equivalence $H_R$ which is the cobar of a pre-existing
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homotopy equivalence (possibly the trivial one) between the coalgebra $X$ and an effective
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version of it, as described at the beginning of this chapter. The Lisp definition is given
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in the next subsection.
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\par}
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}
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\newpage
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\subsection {Searching homology for loop spaces}
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The \index{searching homology!loop spaces} origin list of a loop space object has the form {\tt (LOOP-SPACE {\em space})}.
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The {\tt search-efhm} method applied to a loop space object
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consists essentially in a call to the function {\tt loop-space-efhm}
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described in this chapter.
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{\footnotesize\begin{verbatim}
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(defmethod SEARCH-EFHM (loop-space (orgn (eql 'loop-space)))
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(declare (type simplicial-set loop-space))
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(loop-space-efhm (second (orgn loop-space))))
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\end{verbatim}}
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The lisp definition of the function {\tt loop-space-efhm} shows clearly
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the possible recursivity of the process is {\em space} is itself
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a loop space.
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{\footnotesize\begin{verbatim}
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(defun LOOP-SPACE-EFHM (space)
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(declare (type simplicial-set space))
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(let ((left-hmeq (ls-left-hmeq space))
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(right-hmeq (cobar (efhm space))))
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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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Let us show first how to get the homology groups of some known loop spaces,
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$\Omega {\cal S}^2$, $\Omega^3 {\cal S}^4$, $\Omega^2{\rm Moore}({\Z}/{2{\Z}},3)$.
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{\footnotesize\begin{verbatim}
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(setf s2 (sphere 2)) ==>
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[K1 Simplicial-Set]
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(setf os2 (loop-space s2)) ==>
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[K6 Simplicial-Group]
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(dotimes (k 10) (homology os2 k)) ==>
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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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Component Z
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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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Component Z
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Homology in dimension 5 :
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Component Z
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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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Component Z
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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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Component Z
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---done---
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(cat-init)
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(setf s4 (sphere 4)) ==>
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[K1 Simplicial-Set]
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(setf ooos4 (loop-space s4 3)) ==>
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[K30 Simplicial-Group]
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(dotimes (k 6) (homology ooos4 k)) ==>
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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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Component Z/2Z
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Homology in dimension 3 :
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Component Z/2Z
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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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Component Z
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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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Component Z
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---done---
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(setf moore-23 (moore 2 3)) ==>
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[K42 Simplicial-Set]
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(setf oo-moore-23 (loop-space moore-23 2)) ==>
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[K59 Simplicial-Group]
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(dotimes (k 6) (homology oo-moore-23 k)) ==>
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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/2Z
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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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Component Z/4Z
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Component Z/2Z
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Homology in dimension 4 :
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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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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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---done---
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\end{verbatim}}
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\newpage
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\section {The solution of the Adams and Carlsson problem}
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Let\index{Adams's problem} $X$ an $n$-reduced simplicial set which is not a suspension.
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The Adams and Carlsson
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problem\footnote{{\bf Gunnar Carlsson, R.James Milgram.} {\em Stable homotopy and iterated loop spaces} in
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{\em Handbook of Algebraic Topology, pp 545} edited by {\em I.M.James, North-Holland, 1995}.}
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asks for a finite CW-complex modelizing the iterated loop space $\Omega^n X$. We show in this section
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how the {\tt Kenzo} program builds the relevant solution.\par
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Let us consider for instance the truncated projective space $X=P^\infty {\R}/ P^3 {\R}$ (a non-suspended space)
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and let us build $\Omega^3 X$.
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{\footnotesize\begin{verbatim}
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(setf p4 (R-proj-space 4)) ==>
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[K1 Simplicial-Set]
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(setf o3p4 (loop-space p4 3)) ==>
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[K30 Simplicial-Group]
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(setf ecc (rbcc (efhm o3p4))) ==>
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[K390 Chain-Complex]
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\end{verbatim}}
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The cellular homology of $\Omega^3 X$ is now known through the symbol {\tt ecc}. Note that $360$ objects
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($390-30$) have been built to get the final complex {\tt ecc}. From the printing of the length of the
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basis in dimensions $0$ to $5$,
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{\footnotesize\begin{verbatim}
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(dotimes (i 6) (print (length (basis ecc i)))) ==>
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1
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1
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2
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5
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13
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33
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\end{verbatim}}
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we see that the required CW-complex can be constructed using $1$ $0$-cell, $1$ $1$-cell, $2$ $2$-cells,
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$5$ $3$-cells, $13$ $4$-cells and $33$ $5$-cells. Let us print the incidence matrix between
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the $5$ and the $4$-cells:
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\newpage
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{\footnotesize\begin{verbatim}
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(chcm-mat ecc 5) ==>
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========== MATRIX 13 lines + 33 columns =====
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L1=[C1=-2]
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L2=[C1=-1]
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L3=[C1=-4][C2=1][C3=-1][C4=-2]
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L4=[C2=1][C3=-1][C6=2]
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L5=[C1=6][C4=1][C6=1]
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L6=[C1=4][C4=4][C6=4][C7=3]
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L7=[C1=4][C12=-2][C14=2]
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L8=[C1=6][C4=1][C6=1]
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L9=[C1=4][C4=4][C6=4][C7=3]
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L10=[C8=4][C10=1][C11=-1][C14=-4][C15=-2][C20=-2]
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L11=[C1=4][C8=4][C10=1][C11=-1][C16=-4][C18=-1][C19=1][C23=-2]
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L12=[C12=4][C13=2][C16=-4][C18=-1][C19=1][C27=-2]
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L13=[C1=-1][C20=4][C21=2][C23=-4][C24=-2][C27=4][C28=2]
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|
========== END-MATRIX
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---done---
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\end{verbatim}}
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For instance, we see that the first $5$-cell,
|
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{\footnotesize\begin{verbatim}
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(first (basis ecc 5)) ==>
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<<AlLp[5 <<AlLp[6 <<AlLp[7 8]>>]>>]>>
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\end{verbatim}}
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|
is glued to the $4$-cells \#5 and \#8 by attaching maps of degree $6$.
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The $5$-cell \#4 is also glued to the $4$-cell \#5 by an attaching map
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of degree $1$, etc.
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\subsection* {Lisp files concerned in this chapter}
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{\tt lp-space-efhm.lisp}, {\tt searching-homology.lisp}.
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