\chapter {Eilenberg-Moore spectral sequence I} \section{Introduction} This chapter is devoted to the effective homology version of the spectral sequence of Eilenberg-Moore, in the particular case of loop spaces. More precisely, let $X$ be a {\bf $1$-reduced} simplicial set with effective homology, the software {\tt Kenzo} constructs a $0$-reduced simplicial set $GX$ which is also with effective homology. In particular, if $X$ is $m$-reduced, this process may be iterated $m$ times, producing an effective homology version of $G^kX,\,k\leq m$. So, the software builds an object which contains a complete solution of the Adams problem about the iteration of the cobar construction. The user is advised to consult the paper {\em Stable homotopy and iterated loop spaces, pp. 545} by {\bf Gunnar Carlsson} and {\bf R.James Milgram} in {\em Handbook of Algebraic Topology, pp 545} edited by {\em I.M.James, North-Holland, 1995}. \section{The detailed construction} Let ${\cal C}_*(X)$ be a coalgebra with effective homology. This means that a homotopy equivalence: $$\diagram{ & \hat{C}_* \cr \swarrow\nearrow & & \searrow\nwarrow \cr {\cal C}_*(X) & & EC_* \cr }$$ is provided, where the chain complex $EC_*$ is effective and must be con\-si\-de\-red as describing the homology of ${\cal C}_*(X)$. This scheme includes the case where ${\cal C}_*(X)$ is itself effective; without any other information, the program constructs automatically a trivial homotopy equivalence. \par Now, if we apply the {\em Cobar} functor to this homotopy equivalence we obtain the homotopy equivalence $H_R$: $$\diagram{ &{\widetilde{Cobar}}^{{\hat C}_*}({\Z},{\Z}) \cr \swarrow\nearrow & & \searrow\nwarrow \cr Cobar^{{\cal C}_*(X)}({\Z}, {\Z}) & & {\widetilde{Cobar}}^{EC_*}({\Z}, {\Z}) \cr }$$ in which the $\widetilde{Cobar}$'s are cobars construction with respect to the $A_\infty$-co\-al\-ge\-bras structure on ${\hat C}_*$ and $EC_*$ defined by the initial homotopy equivalence. \par Secondly, Julio Rubio\footnote{{\bf J.J. Rubio-Garcia}. {\em Homologie effective des espaces de lacets it\'er\'es: un logiciel}, Th\`ese, Institut Fourier, 1991.} has shown that it is possible to construct another homotopy equivalence, $H_L$: $$\diagram{ & Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \cr \swarrow\nearrow & & \searrow\nwarrow \cr {\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}({\Z}, {\Z}) \cr }$$ Both reductions of $H_L$ are obtained by the basic perturbation lemma, as explained in the following section. The composition of both homotopy e\-qui\-va\-len\-ces, $H_L$ and $H_R$, makes the link between $GX$, i.e. the loop space of $X$ and the effective right bottom chain complex of $H_R$. \subsection {Obtaining the left reduction $H_L$} The homotopy equivalence $H_L$ is obtained by a sequence of intermediate constructions based mainly upon two applications of the basic perturbation lemma. We are led to consider the two following comodules: \begin{itemize} \item ${\cal C}_*(X)$ comodule on itself, with the canonical coproduct. $${\cal C}_*(X)\stackrel {\Delta}{\longrightarrow} {\cal C}_*(X)\otimes {\cal C}_*(X).$$ \item ${\check{\cal C}}_*(X)$, comodule on ${\cal C}_*(X)$, with a ``trivial'' coproduct $${\check{\cal C}}_*(X) \stackrel {\check{\Delta}}{\longrightarrow} {\cal C}_*(X)\otimes{\check{\cal C}}_*(X),\quad \sigma \rightarrow 1\otimes\sigma,\,\sigma \in {\check{\cal C}}_*(X).$$ \end{itemize} Now, we consider the set of the followings cobars (where $u$ means {\em ``untwisted''} and $t$ {\em ``twisted''}): \begin{eqnarray*} {\rm Hat}_{uu} &=& Cobar^{{\cal C}_*(X)}({\Z}, {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX)),\\ {\rm Hat}_{ut} &=& Cobar^{{\cal C}_*(X)}({\Z}, {\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)),\\ {\rm Hat}_{tu} &=& Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)),\\ {\rm Hat}_{tt} &=& Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)). \end{eqnarray*} One may always say that ${\rm Hat}_{ut}$ is obtained from ${\rm Hat}_{uu}$ by a perturbation $\delta_r$ ($r$: right) induced by the twisted tensor product $\otimes_t$ and that ${\rm Hat}_{tu}$ is obtained from ${\rm Hat}_{uu}$ by a perturbation $\delta_l$ ($l$: left) induced by the discrepancy between the coproducts in ${\check{\cal C}}_*(X)$ and ${\cal C}_*(X)$. After that, ${\rm Hat}_{tt}$ is obtained from ${\rm Hat}_{tu}$ by the perturbation $\delta_r$ as well as, by commutativity, from ${\rm Hat}_{ut}$ by the perturbation $\delta_l$. \par This is shown in the following diagram (here, the arrows are not reductions, but denote the perturbations between the differential morphisms): $$\diagram{ & {\rm Hat}_{uu} \cr \delta_l \swarrow & & \searrow \delta_r \cr {\rm Hat}_{tu} & & {\rm Hat}_{ut} \cr \delta_r \searrow & & \swarrow \delta_l \cr & {\rm Hat}_{tt} \cr }$$ The underlying graded modules ${\rm Hat}_{uu}$, ${\rm Hat}_{ut}$, ${\rm Hat}_{tu}$ and ${\rm Hat}_{tt}$ are the same and the program keeps ${\rm Hat}_{uu}$ as the underlying graded module for all the chain complexes. The differential perturbations are given by the formulas: \begin{eqnarray*} \delta_r({\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes c \otimes g) &=& [{\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n] \otimes [d_{\otimes t}(c \otimes g)-d_\otimes (c \otimes g)], \\ \delta_l({\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes c \otimes g) &=& {\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes {\bar{\Delta}}c \otimes g, \end{eqnarray*} where ${\bar{c}}_i, c \in {\cal C}_*(X),\, g \in {\cal C_*}(GX))$ and $\bar{\Delta}= \Delta - \check{\Delta}$. \par \vskip 0.35cm Now, on one hand, we know that there exists a reduction $${\rm Hat}_{tu}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)) \Longrightarrow {\cal C}_*(GX),$$ so, perturbing this reduction by $\delta_r$, one obtains the Rubio reduction $${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow{\cal C}_*(GX).$$ On the other hand, we know also that there exists a reduction $${\rm Hat}_{ut}=Cobar^{{\cal C}_*(X)}({\Z}, {\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow Cobar^{{\cal C}_*(X)}({\Z}, {\Z}),$$ so, perturbing this reduction by $\delta_l$, one obtains the reduction $${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow Cobar^{{\cal C}_*(X)}({\Z}, {\Z}).$$ Finally, we have obtained the wished left homotopy equivalence $H_L$: $$\diagram{ & Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \cr \swarrow\nearrow & & \searrow\nwarrow \cr {\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}({\Z}, {\Z}) \cr }$$ \subsection {The useful functions} For the applications, the only function that the user must know is the function {\tt loop-space-efhm} which builds the final homotopy equivalence. But for the interested user, we give nevertheless a short description of all the functions involved in the described process. \vskip 0.30cm {\parindent=0mm {\leftskip=5mm {\tt loop-space-efhm} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm From the space $X$ ($1$-reduced) with effective homology (here the argument {\em space}), build a homotopy equivalence giving an effective homology version of the space $GX$. This homotopy equivalence will be used by the homology function to compute the homology groups. In fact, due to the slot-unbound mechanism of CLOS, this function will be automatically called, as soon as the user requires a homology group of a loop-space. If $X$ is $n$-reduced and if the wished loop space is $\Omega^k X, \, k \leq n$, then the process will be automatically iterated. \par} {\leftskip=5mm {\tt ls-hat-u-u} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Return the chain complex $Cobar^{{\cal C}_*(X)}({\Z}, {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX))$. Because of the particular structure of ${\check{\cal C}}_*(X)$, this chain complex is nothing but $Cobar^{{\cal C}_*(X)}({\Z},{\Z})\otimes {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX).$ \par} {\leftskip=5mm {\tt ls-hat-left-perturbation} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Return the morphism corresponding to the differential perturbation $\delta_l$, induced by the discrepancy between the coproducts of ${\check{\cal C}}_*(X)$ and ${\cal C}_*(X)$. \par} {\leftskip=5mm {\tt ls-hat-t-u} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Return the chain complex $Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX))$ by applying the differential perturbation {\tt hat-left-perturbation} upon the chain complex built by the function {\tt hat-u-u}. This is realized by the method {\tt add}. \par} {\leftskip=5mm {\tt ls-hat-u-t} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Return the chain complex $Cobar^{{\cal C}_*(X)}({\Z},{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX))$. Because of the particular structure of ${\check{\cal C}}_*(X)$, this chain complex is nothing but $Cobar^{{\cal C}_*(X)}({\Z},{\Z})\otimes[{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)],$ where the twisted tensor product is obtained by a call to {\tt twisted-tnsr-prdc} (see loop space fibrations chapter). \par} {\leftskip=5mm {\tt ls-hat-right-perturbation} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Return the morphism corresponding to the differential perturbation $\delta_r$ induced by the twisted tensor product. This morphism is nothing but the tensor product of two morphisms: the identity morphism on $Cobar^{{\cal C}_*(X)}({\Z},{\Z})$ and the perturbation morphism induced by the twisted tensor product. This last morphism is a by-product of the function {\tt szczarba}. \par} {\leftskip=5mm {\tt ls-left-hmeq-hat} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Return the chain complex ${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX))$ by perturbing the chain complex ${\rm Hat}_{ut}$ by the perturbation $\delta_l$. \par} {\leftskip=5mm {\tt ls-pre-left-hmeq-left-reduction} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Build the reduction $${\rm Hat}_{tu}=Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)) \Longrightarrow {\cal C}_*(GX).$$ \par} {\leftskip=5mm {\tt ls-pre-left-hmeq-right-reduction} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Build the reduction $${\rm Hat}_{ut}=Cobar^{{\cal C}_*(X)}({\Z},{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow Cobar^{{\cal C}_*(X)}({\Z}, {\Z}).$$ \par} {\leftskip=5mm {\tt ls-left-hmeq-left-reduction} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Build the Rubio reduction $$Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow{\cal C}_*(GX)$$ by perturbing by the perturbation $\delta_r$ the reduction $${\rm Hat}_{tu} \Longrightarrow {\cal C}_*(GX)$$ obtained by the function {\tt pre-left-hmeq-left-reduction}. \par} {\leftskip=5mm {\tt ls-left-hmeq-right-reduction} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Build the reduction $$Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow Cobar^{{\cal C}_*(X)}({\Z}, {\Z})$$ by perturbing by the perturbation $\delta_l$ the reduction $${\rm Hat}_{ut} \Longrightarrow Cobar^{{\cal C}_*(X)}({\Z}, {\Z})$$ obtained by the function {\tt pre-left-hmeq-right-reduction}. \par} {\leftskip=5mm {\tt ls-left-hmeq} {\em space} \hfill {\em [Function]} \par} {\leftskip=15mm Build the homotopy equivalence $$\diagram{ & Cobar^{{\cal C}_*(X)}({\Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \cr \swarrow\nearrow & & \searrow\nwarrow \cr {\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}({\Z}, {\Z}) \cr }$$ from the above reductions. The function {\tt loop-space-efhm} composes this left homotopy equivalence $H_L$ with a homotopy equivalence $H_R$ which is the cobar of a pre-existing homotopy equivalence (possibly the trivial one) between the coalgebra $X$ and an effective version of it, as described at the beginning of this chapter. The Lisp definition is given in the next subsection. \par} } \newpage \subsection {Searching homology for loop spaces} The \index{searching homology!loop spaces} origin list of a loop space object has the form {\tt (LOOP-SPACE {\em space})}. The {\tt search-efhm} method applied to a loop space object consists essentially in a call to the function {\tt loop-space-efhm} described in this chapter. {\footnotesize\begin{verbatim} (defmethod SEARCH-EFHM (loop-space (orgn (eql 'loop-space))) (declare (type simplicial-set loop-space)) (loop-space-efhm (second (orgn loop-space)))) \end{verbatim}} The lisp definition of the function {\tt loop-space-efhm} shows clearly the possible recursivity of the process is {\em space} is itself a loop space. {\footnotesize\begin{verbatim} (defun LOOP-SPACE-EFHM (space) (declare (type simplicial-set space)) (let ((left-hmeq (ls-left-hmeq space)) (right-hmeq (cobar (efhm space)))) (declare (type homotopy-equivalence left-hmeq right-hmeq)) (cmps left-hmeq right-hmeq))) \end{verbatim}} \subsection* {Examples} Let us show first how to get the homology groups of some known loop spaces, $\Omega {\cal S}^2$, $\Omega^3 {\cal S}^4$, $\Omega^2{\rm Moore}({\Z}/{2{\Z}},3)$. {\footnotesize\begin{verbatim} (setf s2 (sphere 2)) ==> [K1 Simplicial-Set] (setf os2 (loop-space s2)) ==> [K6 Simplicial-Group] (dotimes (k 10) (homology os2 k)) ==> Homology in dimension 0 : Component Z Homology in dimension 1 : Component Z Homology in dimension 2 : Component Z Homology in dimension 3 : Component Z Homology in dimension 4 : Component Z Homology in dimension 5 : Component Z Homology in dimension 6 : Component Z Homology in dimension 7 : Component Z Homology in dimension 8 : Component Z Homology in dimension 9 : Component Z ---done--- (cat-init) (setf s4 (sphere 4)) ==> [K1 Simplicial-Set] (setf ooos4 (loop-space s4 3)) ==> [K30 Simplicial-Group] (dotimes (k 6) (homology ooos4 k)) ==> Homology in dimension 0 : Component Z Homology in dimension 1 : Component Z Homology in dimension 2 : Component Z/2Z Homology in dimension 3 : Component Z/2Z Homology in dimension 4 : Component Z/3Z Component Z/2Z Component Z Homology in dimension 5 : Component Z/3Z Component Z/2Z Component Z ---done--- (setf moore-23 (moore 2 3)) ==> [K42 Simplicial-Set] (setf oo-moore-23 (loop-space moore-23 2)) ==> [K59 Simplicial-Group] (dotimes (k 6) (homology oo-moore-23 k)) ==> Homology in dimension 0 : Component Z Homology in dimension 1 : Component Z/2Z Homology in dimension 2 : Component Z/2Z Homology in dimension 3 : Component Z/4Z Component Z/2Z Homology in dimension 4 : Component Z/2Z Component Z/2Z Component Z/2Z Homology in dimension 5 : Component Z/2Z Component Z/2Z Component Z/2Z Component Z/2Z ---done--- \end{verbatim}} \newpage \section {The solution of the Adams and Carlsson problem} Let\index{Adams's problem} $X$ an $n$-reduced simplicial set which is not a suspension. The Adams and Carlsson problem\footnote{{\bf Gunnar Carlsson, R.James Milgram.} {\em Stable homotopy and iterated loop spaces} in {\em Handbook of Algebraic Topology, pp 545} edited by {\em I.M.James, North-Holland, 1995}.} asks for a finite CW-complex modelizing the iterated loop space $\Omega^n X$. We show in this section how the {\tt Kenzo} program builds the relevant solution.\par Let us consider for instance the truncated projective space $X=P^\infty {\R}/ P^3 {\R}$ (a non-suspended space) and let us build $\Omega^3 X$. {\footnotesize\begin{verbatim} (setf p4 (R-proj-space 4)) ==> [K1 Simplicial-Set] (setf o3p4 (loop-space p4 3)) ==> [K30 Simplicial-Group] (setf ecc (rbcc (efhm o3p4))) ==> [K390 Chain-Complex] \end{verbatim}} The cellular homology of $\Omega^3 X$ is now known through the symbol {\tt ecc}. Note that $360$ objects ($390-30$) have been built to get the final complex {\tt ecc}. From the printing of the length of the basis in dimensions $0$ to $5$, {\footnotesize\begin{verbatim} (dotimes (i 6) (print (length (basis ecc i)))) ==> 1 1 2 5 13 33 \end{verbatim}} we see that the required CW-complex can be constructed using $1$ $0$-cell, $1$ $1$-cell, $2$ $2$-cells, $5$ $3$-cells, $13$ $4$-cells and $33$ $5$-cells. Let us print the incidence matrix between the $5$ and the $4$-cells: \newpage {\footnotesize\begin{verbatim} (chcm-mat ecc 5) ==> ========== MATRIX 13 lines + 33 columns ===== L1=[C1=-2] L2=[C1=-1] L3=[C1=-4][C2=1][C3=-1][C4=-2] L4=[C2=1][C3=-1][C6=2] L5=[C1=6][C4=1][C6=1] L6=[C1=4][C4=4][C6=4][C7=3] L7=[C1=4][C12=-2][C14=2] L8=[C1=6][C4=1][C6=1] L9=[C1=4][C4=4][C6=4][C7=3] L10=[C8=4][C10=1][C11=-1][C14=-4][C15=-2][C20=-2] L11=[C1=4][C8=4][C10=1][C11=-1][C16=-4][C18=-1][C19=1][C23=-2] L12=[C12=4][C13=2][C16=-4][C18=-1][C19=1][C27=-2] L13=[C1=-1][C20=4][C21=2][C23=-4][C24=-2][C27=4][C28=2] ========== END-MATRIX ---done--- \end{verbatim}} For instance, we see that the first $5$-cell, {\footnotesize\begin{verbatim} (first (basis ecc 5)) ==> <>]>>]>> \end{verbatim}} is glued to the $4$-cells \#5 and \#8 by attaching maps of degree $6$. The $5$-cell \#4 is also glued to the $4$-cell \#5 by an attaching map of degree $1$, etc. \subsection* {Lisp files concerned in this chapter} {\tt lp-space-efhm.lisp}, {\tt searching-homology.lisp}.