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Eilenberg-Moore spectral sequence II¶
Introduction¶
This chapter is devoted to the effective homology version of the spectral sequence of Eilenberg-Moore, in the particular case of classifying spaces. More precisely, let ${\cal G}$ be a simplicial group; then its classifying space $\bar{\cal W}{\cal G}$ is a simplicial set canonically defined (J. Peter May. Simplicial objects in algebraic topology, Van Nostrand, 1967.). Furthermore, if ${\cal G}$ is an Abelian simplicial group, then $\bar{\cal W}{\cal G}$ is again an Abelian simplicial group with natural structure, so that the $\bar{\cal W}$ construction can be iterated. The $\bar{\cal W}$ construction is implemented in Kenzo only if ${\cal G}$ is reduced (${\cal G}_0$ has only one element). In particular if ${\cal G}$ is a simplicial group with effective homology, then Kenzo constructs a version of $\bar{\cal W}{\cal G}$ also with effective homology.
The detailed construction¶
Let ${\cal G}$ be a simplicial group with effective homology. This means that a homotopy equivalence: $$\begin{aligned} & \qquad\hat{C}_* \cr \swarrow\nearrow & & \searrow\nwarrow\qquad \cr {\cal C}_*({\cal G}) & & EG_* \cr \end{aligned}$$ is provided, where the chain complex $EG_*$ is effective and must be considered as describing the homology of ${\cal C}_*({\cal G})$. This scheme includes the case where ${\cal C}_*({\cal G})$ itself is effective; without any other information, the program constructs automatically a trivial homotopy equivalence.
Now, if we apply the Bar functor to this homotopy equivalence we obtain the homotopy equivalence $H_R$: $$\begin{aligned} &\qquad{\widetilde{Bar}}^{\hat{ C}_*} \cr \swarrow\nearrow & & \searrow\nwarrow\qquad \cr Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z}) & & {\widetilde{Bar}}^{EG_*}(\mathbb{Z}, \mathbb{Z}) \cr \end{aligned}$$ in which the $\widetilde{Bar}$'s are Bars constructions with respect to the $A_\infty$ structure on $\hat{C}_*$ and $EG_*$ defined by the initial homotopy equivalence.
By analogy with the result of Julio Rubio (J.J. Rubio-Garcia. Homologie effective des espaces de lacets iteres: un logiciel, These, Institut Fourier, 1991.) one may show that it is possible to construct another homotopy equivalence, $H_L$: $$\begin{aligned} & \qquad Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),\mathbb{Z}) \cr \swarrow\nearrow & & \searrow\nwarrow\qquad \cr {\cal C}_*(\bar{\cal W}{\cal G}) & & Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z}) \cr \end{aligned}$$ Both reductions of $H_L$ are obtained by the basic perturbation lemma, as explained in the following section. The composition of both homotopy equivalences, $H_L$ and $H_R$, makes the link between $\bar{\cal W}{\cal G}$, the classifying space of the simplicial group ${\cal G}$ and the effective right bottom chain complex of $H_R$.
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 modules:
- ${\cal C}_*({\cal G})$ module on itself, with the canonical product. $${\cal C}_*({\cal G})\otimes {\cal C}_*({\cal G})\stackrel {\Pi}{\longrightarrow} {\cal C}_*({\cal G}).$$
- ${\check{\cal C}}_*({\cal G})$, module on ${\cal C}_*({\cal G})$, with a "trivial" product $${\check{\cal C}}_*({\cal G})\otimes{\cal C}_*({\cal G}) \stackrel {\check{\Pi}}{\longrightarrow} {\check{\cal C}}_*({\cal G})$$ defined by $$\sigma \otimes \tau \longmapsto \sigma . \varepsilon(\tau),$$ where $\varepsilon$ is the traditional augmentation of ${\cal C}_*({\cal G})$.
Now, we consider the set of the followings Bars (where $u$ means "untwisted" and $t$ "twisted"): $$\begin{aligned} {\rm Hat}_{uu} &=& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\check{\cal C}}_*({\cal G}),\mathbb{Z}),\\ {\rm Hat}_{ut} &=& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\cal C}_*({\cal G}),\mathbb{Z}),\\ {\rm Hat}_{tu} &=& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t{\check{\cal C}}_*({\cal G}),\mathbb{Z}),\\ {\rm Hat}_{tt} &=& Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),\mathbb{Z}). \end{aligned}$$
One may always say that ${\rm Hat}_{ut}$ is obtained from ${\rm Hat}_{uu}$ by a perturbation $\delta_r$ ($r$: right) induced by the discrepancy between the products in ${\check{\cal C}}_*({\cal G})$ and ${\cal C}_*({\cal G})$ and that ${\rm Hat}_{tu}$ is obtained from ${\rm Hat}_{uu}$ by a perturbation $\delta_l$ ($l$: left) induced by the twisted tensor product $\otimes_t$. 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 $\delta_l$.
This is shown in the following diagram (here, the arrows are not reductions, but denote the perturbations between the differential morphisms): $$\begin{aligned} & \qquad{\rm Hat}_{uu} \cr \delta_l \swarrow & & \searrow \delta_r \cr {\rm Hat}_{tu} & & {\rm Hat}_{ut} \cr \delta_r \searrow & & \swarrow \delta_l \cr & \qquad{\rm Hat}_{tt} \cr \end{aligned}$$
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: $$\begin{aligned} \delta_l[(w\otimes {\check g}_0)\otimes (g_1 \otimes\cdots\otimes g_n)] &=& [d_{\otimes t}(w\otimes {\check g}_0) - d_\otimes (w\otimes {\check g}_0)] \otimes (g_1 \otimes\cdots\otimes g_n), \\ \delta_r[(w\otimes {\check g}_0)\otimes (g_1 \otimes\cdots\otimes g_n)] &=& w \otimes(g_0 . g_1) \otimes g_2 \otimes\cdots\otimes g_n, \end{aligned}$$ where ${\check g}_0 \in {\check {\cal G}}_0,\, g_i \in {\cal G},\, w \in \bar{\cal W}{\cal G}.$
Now, on the other hand, we know that there exists a reduction
$${\rm Hat}_{ut} = Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes
{\cal C}_*({\cal G}),\mathbb{Z}) \Longrightarrow {\cal C}_*(\bar{\cal W}{\cal G}),$$
so, perturbing this reduction by $\delta_l$, one obtains the Rubio reduction
$${\rm Hat}_{tt}=Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t
{\cal C}_*({\cal G}),\mathbb{Z}) \Longrightarrow {\cal C}_*(\bar{\cal W}{\cal G}).$$
On the other hand, we know also that there exists a reduction
$${\rm Hat}_{tu} = Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t
{\check{\cal C}}_*({\cal G}),\mathbb{Z}) \Longrightarrow
Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z}),$$
so, perturbing this reduction by $\delta_r$, one obtains the reduction
$${\rm Hat}_{tt}=Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t
{\cal C}_*({\cal G}),\mathbb{Z}) \Longrightarrow Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z}).$$
Finally, we have obtained the wished left homotopy equivalence $H_L$:
$$\begin{aligned}
& \qquad Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),\mathbb{Z}) \cr
\swarrow\nearrow & & \searrow\nwarrow \qquad\cr
{\cal C}_*(\bar{\cal W}{\cal G}) & & Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z}) \cr
\end{aligned}$$
The useful functions¶
For the applications, the only function that the user must know is the function classifying-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.
classifying-space-efhm smgr [Function]
From the simplicial group ${\cal G}$ (0-reduced) with effective homology (here the argument smgr), build a homotopy equivalence giving an effective homology version of the classifying space $\bar{\cal W}{\cal G}$ of ${\cal G}$. 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 for a classifying space.
cs-hat-u-u smgr [Function]
Return the chain complex $Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\check{\cal C}}_*({\cal G}),\mathbb{Z})$. Because of the particular structure of ${\check{\cal C}}_*({\cal G})$, this chain complex is nothing but $[{\cal C}_*(\bar{\cal W}{\cal G})\otimes {\check{\cal C}}_*({\cal G})]\otimes Bar^{{\cal C}_*({\cal G})}(\mathbb{Z},\mathbb{Z}).$
cs-hat-right-perturbation smgr [Function]
Return the morphism corresponding to the differential perturbation $\delta_r$, induced by the discrepancy between the respective products in ${\check{\cal C}}_*({\cal G})$ and in ${\cal C}_*({\cal G})$.
cs-hat-u-t smgr [Function]
Return the chain complex $Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\cal C}_*({\cal G}),\mathbb{Z})$ by applying the differential perturbation
hat-right-perturbationupon the chain complexhat-u-usmgr. This is realized by the methodadd.
cs-hat-t-u smgr [Function]
Return the chain complex $Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\check{\cal C}}_*({\cal G}),\mathbb{Z})$. Because of the particular structure of ${\check{\cal C}}_*({\cal G})$, this chain complex is nothing but $[{\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\check{\cal C}}_*({\cal G})]\otimes Bar^{{\cal C}_*({\cal G})}(\mathbb{Z},\mathbb{Z}),$ where the twisted tensor product is the botton chain complex of the Brown reduction of the fibration of the simplicial group.
cs-hat-left-perturbation smgr [Function]
Return the morphism corresponding to the differential perturbation $\delta_l$ induced by the twisted tensor product. This morphism is nothing but the tensor product of two morphisms: the perturbation morphism by-product of the Brown reduction of the fibration of the simplicial group and the identity morphism on $Bar^{{\cal C}_*({\cal G})}(\mathbb{Z},\mathbb{Z})$.
cs-left-hmeq-hat smgr [Function]
Return the chain complex ${\rm Hat}_{tt}=Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),\mathbb{Z})$ by perturbing the chain complex ${\rm Hat}_{ut}$ by the perturbation $\delta_l$.
cs-pre-left-hmeq-left-reduction smgr [Function]
Build the reduction $${\rm Hat}_{tu}=Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes {\cal C}_*({\cal G}),\mathbb{Z}) \Longrightarrow {\cal C}_*(\bar{\cal W}{\cal G}).$$
cs-pre-left-hmeq-right-reduction smgr [Function]
Build the reduction $${\rm Hat}_{ut}=Bar^{{\cal C}_*({\cal G})}({\check{\cal C}}_*({\cal G})\otimes_t {\cal C}_*(\bar{\cal W}{\cal G}),\mathbb{Z}) \Longrightarrow Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z}).$$
cs-left-hmeq-left-reduction smgr [Function]
Build the Rubio reduction $$Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),\mathbb{Z}) \Longrightarrow{\cal C}_*(\bar{\cal W}{\cal G})$$ by perturbing by the perturbation $\delta_r$ the reduction $${\rm Hat}_{tu} \Longrightarrow {\cal C}_*(\bar{\cal W}{\cal G})$$ obtained by the function {\tt pre-left-hmeq-left-reduction}.
cs-left-hmeq-right-reduction smgr [Function]
Build the reduction $$Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),\mathbb{Z}) \Longrightarrow Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z})$$ by perturbing by the perturbation $\delta_l$ the reduction $${\rm Hat}_{ut} \Longrightarrow Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z})$$ obtained by the function
cs-pre-left-hmeq-right-reduction.
cs-left-hmeq smgr [Function]
Build the homotopy equivalence $$\begin{aligned} &\qquad Bar^{{\cal C}_*({\cal G})}({\cal C}_*(\bar{\cal W}{\cal G})\otimes_t {\cal C}_*({\cal G}),\mathbb{Z}) \cr \swarrow\nearrow & & \searrow\nwarrow\qquad \cr {\cal C}_*(\bar{\cal W}{\cal G}) & & Bar^{{\cal C}_*({\cal G})}(\mathbb{Z}, \mathbb{Z}) \cr \end{aligned}$$ from the above reductions. The function
classifying-space-efhmcomposes this left homotopy equivalence $H_L$ with the homotopy equivalence $H_R$ which is the Bar of a pre-existing homotopy equivalence (possibly the trivial one) between the module ${\cal G}$ and an effective version of it, as described at the beginning of this chapter.
Searching homology for classifying spaces¶
The origin list of a classifying space object has the form (CLASSIFYING-SPACE smgr). The search-efhm method applied to a classifying space object consists essentially in a call to the function classifying-space-efhm described just above.
commonlisp
(defmethod SEARCH-EFHM (classifying-space (orgn (eql 'classifying-space)))
(declare (type simplicial-set classifying-space))
(classifying-space-efhm (second (orgn classifying-space))))
The following Lisp definition of the function classifying-space-efhm shows that the process may be recursive if it happens that smrg is itself a classifying space:
commonlisp
(defun CLASSIFYING-SPACE-EFHM (smgr)
(declare (type simplicial-group smgr))
(let ((left-hmeq (cs-left-hmeq smgr))
(right-hmeq (bar (efhm smgr))))
(declare (type homotopy-equivalence left-hmeq right-hmeq))
(cmps left-hmeq right-hmeq)))
Examples¶
(def kz1 (k-z-1))
[K1 Abelian-Simplicial-Group]
(homology kz1 0 10)
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 0) : * End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 1) : S1 End of computing. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 1) : S1 End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 0. Homology in dimension 1 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 2. Rank of the source-module : 0. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Computing boundary-matrix in dimension 4. Rank of the source-module : 0. Homology in dimension 3 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 4. Rank of the source-module : 0. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Homology in dimension 4 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Computing boundary-matrix in dimension 6. Rank of the source-module : 0. Homology in dimension 5 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 6. Rank of the source-module : 0. Computing boundary-matrix in dimension 7. Rank of the source-module : 0. Homology in dimension 6 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 7. Rank of the source-module : 0. Computing boundary-matrix in dimension 8. Rank of the source-module : 0. Homology in dimension 7 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 8. Rank of the source-module : 0. Computing boundary-matrix in dimension 9. Rank of the source-module : 0. Homology in dimension 8 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 9. Rank of the source-module : 0. Computing boundary-matrix in dimension 10. Rank of the source-module : 0. Homology in dimension 9 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s.
NIL
(def bkz1 (classifying-space kz1))
[K23 Abelian-Simplicial-Group]
(homology bkz1 0 10)
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 0) : <<Abar>> End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 0. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 1. Rank of the source-module : 0. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 2) : <<Abar[2 S1]>> End of computing. Homology in dimension 1 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 2) : <<Abar[2 S1]>> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Homology in dimension 2 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 3. Rank of the source-module : 0. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 4) : <<Abar[2 S1][2 S1]>> End of computing. Homology in dimension 3 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 4) : <<Abar[2 S1][2 S1]>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Homology in dimension 4 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 5. Rank of the source-module : 0. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 6) : <<Abar[2 S1][2 S1][2 S1]>> End of computing. Homology in dimension 5 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 6) : <<Abar[2 S1][2 S1][2 S1]>> End of computing. Computing boundary-matrix in dimension 7. Rank of the source-module : 0. Homology in dimension 6 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 7. Rank of the source-module : 0. Computing boundary-matrix in dimension 8. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 8) : <<Abar[2 S1][2 S1][2 S1][2 S1]>> End of computing. Homology in dimension 7 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 8. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 8) : <<Abar[2 S1][2 S1][2 S1][2 S1]>> End of computing. Computing boundary-matrix in dimension 9. Rank of the source-module : 0. Homology in dimension 8 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 9. Rank of the source-module : 0. Computing boundary-matrix in dimension 10. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 10) : <<Abar[2 S1][2 S1][2 S1][2 S1][2 S1]>> End of computing. Homology in dimension 9 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s.
NIL
(def obkz1 (loop-space bkz1))
[K154 Simplicial-Group]
(homology obkz1 0 6)
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 0) : <<AlLp>> End of computing. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 1) : <<AlLp[1 <<Abar[2 S1]>>]>> End of computing. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 1) : <<AlLp[1 <<Abar[2 S1]>>]>> End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 2) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Homology in dimension 1 : Component Z ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/1 (dimension 2) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 2. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/2 (dimension 3) : <<AlLp[3 <<Abar[2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 2/2 (dimension 3) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Homology in dimension 2 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 3. Rank of the source-module : 2. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/2 (dimension 3) : <<AlLp[3 <<Abar[2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 2/2 (dimension 3) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 3. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/3 (dimension 4) : <<AlLp[1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 2/3 (dimension 4) : <<AlLp[3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 3/3 (dimension 4) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Homology in dimension 3 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 4. Rank of the source-module : 3. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/3 (dimension 4) : <<AlLp[1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 2/3 (dimension 4) : <<AlLp[3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 3/3 (dimension 4) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 5. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/5 (dimension 5) : <<AlLp[5 <<Abar[2 S1][2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 2/5 (dimension 5) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 3/5 (dimension 5) : <<AlLp[1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 4/5 (dimension 5) : <<AlLp[3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 5/5 (dimension 5) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Homology in dimension 4 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 7s. Computing boundary-matrix in dimension 5. Rank of the source-module : 5. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/5 (dimension 5) : <<AlLp[5 <<Abar[2 S1][2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 2/5 (dimension 5) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 3/5 (dimension 5) : <<AlLp[1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 4/5 (dimension 5) : <<AlLp[3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 5/5 (dimension 5) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 8. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 1/8 (dimension 6) : <<AlLp[1 <<Abar[2 S1]>>][5 <<Abar[2 S1][2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 2/8 (dimension 6) : <<AlLp[3 <<Abar[2 S1][2 S1]>>][3 <<Abar[2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 3/8 (dimension 6) : <<AlLp[5 <<Abar[2 S1][2 S1][2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 4/8 (dimension 6) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 5/8 (dimension 6) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 6/8 (dimension 6) : <<AlLp[1 <<Abar[2 S1]>>][3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 7/8 (dimension 6) : <<AlLp[3 <<Abar[2 S1][2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. ;; Clock -> 2017-10-08, 12h 33m 7s. Computing the boundary of the generator 8/8 (dimension 6) : <<AlLp[1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>][1 <<Abar[2 S1]>>]>> End of computing. Homology in dimension 5 : ---done--- ;; Clock -> 2017-10-08, 12h 33m 8s.
NIL
Lisp files concerned in this chapter¶
cs-space-efhm.lisp, searching-homology.lisp.