78 KiB
Eilenberg-Moore spectral sequence I¶
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 1-reduced simplicial set with effective homology, the software 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 Stable homotopy and iterated loop spaces, pp. 545 by Gunnar Carlsson and R.James Milgram in Handbook of Algebraic Topology, pp 545 edited by I.M.James, North-Holland, 1995.
The detailed construction¶
Let ${\cal C}_*(X)$ be a coalgebra with effective homology. This means that a homotopy equivalence: $$ \begin{aligned} & \qquad\hat{C}_* & \cr \swarrow\nearrow & & \searrow\nwarrow \quad\cr {\cal C}_*(X) & & EC_* \cr \end{aligned} $$ is provided, where the chain complex $EC_*$ is effective and must be considered 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.
Now, if we apply the Cobar functor to this homotopy equivalence we obtain the homotopy equivalence $H_R$: $$ \begin{aligned} &\qquad{\widetilde{Cobar}}^{{\hat C}_*}(\mathbb{Z},\mathbb{Z}) & \cr \swarrow\nearrow & & \searrow\nwarrow \qquad\cr Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z}) & & {\widetilde{Cobar}}^{EC_*}(\mathbb{Z}, \mathbb{Z}) \cr \end{aligned} $$ in which the $\widetilde{Cobar}$'s are cobars construction with respect to the $A_\infty$-coalgebras structure on ${\hat C}_*$ and $EC_*$ defined by the initial homotopy equivalence.
Secondly, Julio Rubio (J.J. Rubio-Garcia. Homologie effective des espaces de lacets iteres: un logiciel, These, Institut Fourier, 1991.} has shown that it is possible to construct another homotopy equivalence, $H_L$: $$ \begin{aligned} & \qquad Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \cr \swarrow\nearrow & & \searrow\nwarrow\qquad \cr {\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z}) \cr \end{aligned} $$ 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 $GX$, i.e. the loop space of $X$ 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 comodules:
- ${\cal C}_*(X)$ comodule on itself, with the canonical coproduct. $${\cal C}_*(X)\stackrel {\Delta}{\longrightarrow} {\cal C}_*(X)\otimes {\cal C}_*(X).$$
- ${\check{\cal C}}_*(X)$, comodule on ${\cal C}_*(X)$, with a "trivial" coproduct $${\check{\cal C}}_*(X) \stackrel {\check{\Delta}}{\longrightarrow} {\cal C}_*(X)\otimes{\check{\cal C}}_*(X),\quad \sigma \rightarrow 1\otimes\sigma,\,\sigma \in {\check{\cal C}}_*(X).$$
Now, we consider the set of the followings cobars (where $u$ means "untwisted" and $t$ "twisted"): $$ \begin{aligned} {\rm Hat}_{uu} &=& Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX)),\\ {\rm Hat}_{ut} &=& Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)),\\ {\rm Hat}_{tu} &=& Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)),\\ {\rm Hat}_{tt} &=& Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)). \end{aligned} $$
One may always say that ${\rm Hat}_{ut}$ is obtained from ${\rm Hat}_{uu}$ by a perturbation $\delta_r$ ($r$: right) induced by the 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 $\delta_l$.
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: $$ \begin{aligned} \delta_r({\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes c \otimes g) &=& [{\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n] \otimes [d_{\otimes t}(c \otimes g)-d_\otimes (c \otimes g)], \\ \delta_l({\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes c \otimes g) &=& {\bar{c}}_1 \otimes \cdots \otimes {\bar{c}}_n \otimes {\bar{\Delta}}c \otimes g, \end{aligned} $$ where ${\bar{c}}_i, c \in {\cal C}_*(X),\, g \in {\cal C_*}(GX))$ and $\bar{\Delta}= \Delta - \check{\Delta}$.
Now, on one hand, we know that there exists a reduction
$${\rm Hat}_{tu}=Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)) \Longrightarrow {\cal C}_*(GX),$$
so, perturbing this reduction by $\delta_r$, one obtains the Rubio reduction
$${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow{\cal C}_*(GX).$$
On the other hand, we know also that there exists a reduction
$${\rm Hat}_{ut}=Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow
Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z}),$$
so, perturbing this reduction by $\delta_l$, one obtains the reduction
$${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow
Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z}).$$
Finally, we have obtained the wished left homotopy equivalence $H_L$:
$$
\begin{aligned}
& \qquad Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) & \cr
\swarrow\nearrow & & \searrow\nwarrow\qquad\qquad \cr
{\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z}) \cr
\end{aligned}
$$
The useful functions¶
For the applications, the only function that the user must know is the function 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.
loop-space-efhm space [Function]
From the space $X$ (1-reduced) with effective homology (here the argument 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.
ls-hat-u-u space [Function]
Return the chain complex $Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX))$. Because of the particular structure of ${\check{\cal C}}_*(X)$, this chain complex is nothing but $Cobar^{{\cal C}_*(X)}(\mathbb{Z},\mathbb{Z})\otimes {\check{\cal C}}_*(X)\otimes {\cal C_*}(GX).$
ls-hat-left-perturbation space [Function]
Return the morphism corresponding to the differential perturbation $\delta_l$, induced by the discrepancy between the coproducts of ${\check{\cal C}}_*(X)$ and ${\cal C}_*(X)$.
ls-hat-t-u space [Function]
Return the chain complex $Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX))$ by applying the differential perturbation
hat-left-perturbationupon the chain complex built by the functionhat-u-u. This is realized by the methodadd.
ls-hat-u-t space [Function]
Return the chain complex $Cobar^{{\cal C}_*(X)}(\mathbb{Z},{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX))$. Because of the particular structure of ${\check{\cal C}}_*(X)$, this chain complex is nothing but $Cobar^{{\cal C}_*(X)}(\mathbb{Z},\mathbb{Z})\otimes[{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)],$ where the twisted tensor product is obtained by a call to
twisted-tnsr-prdc(see loop space fibrations chapter).
ls-hat-right-perturbation space [Function]
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 $Cobar^{{\cal C}_*(X)}(\mathbb{Z},\mathbb{Z})$ and the perturbation morphism induced by the twisted tensor product. This last morphism is a by-product of the function
szczarba.
ls-left-hmeq-hat space [Function]
Return the chain complex ${\rm Hat}_{tt}=Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX))$ by perturbing the chain complex ${\rm Hat}_{ut}$ by the perturbation $\delta_l$.
ls-pre-left-hmeq-left-reduction space [Function]
Build the reduction
$${\rm Hat}_{tu}=Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes {\cal C_*}(GX)) \Longrightarrow {\cal C}_*(GX).$$
ls-pre-left-hmeq-right-reduction space [Function]
Build the reduction
$${\rm Hat}_{ut}=Cobar^{{\cal C}_*(X)}(\mathbb{Z},{\check{\cal C}}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow
Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z}).$$
ls-left-hmeq-left-reduction space [Function]
Build the Rubio reduction
$$Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow{\cal C}_*(GX)$$
by perturbing by the perturbation $\delta_r$ the reduction
$${\rm Hat}_{tu} \Longrightarrow {\cal C}_*(GX)$$
obtained by the function pre-left-hmeq-left-reduction.
ls-left-hmeq-right-reduction space [Function]
Build the reduction
$$Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) \Longrightarrow
Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z})$$
by perturbing by the perturbation $\delta_l$ the reduction
$${\rm Hat}_{ut} \Longrightarrow Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z})$$
obtained by the function pre-left-hmeq-right-reduction.
ls-left-hmeq space [Function]
Build the homotopy equivalence
$$
\begin{aligned}
& \qquad Cobar^{{\cal C}_*(X)}(\mathbb{Z}, {\cal C}_*(X)\otimes_t {\cal C_*}(GX)) & \cr
\swarrow\nearrow & & \searrow\nwarrow \qquad\qquad\cr
{\cal C_*}(GX) & & Cobar^{{\cal C}_*(X)}(\mathbb{Z}, \mathbb{Z}) \cr
\end{aligned}
$$
from the above reductions. The function 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.
Searching homology for loop spaces¶
The origin list of a loop space object has the form (LOOP-SPACE space). The search-efhm method applied to a loop space object consists essentially in a call to the function loop-space-efhm described in this chapter.
commonlisp
(defmethod SEARCH-EFHM (loop-space (orgn (eql 'loop-space)))
(declare (type simplicial-set loop-space))
(loop-space-efhm (second (orgn loop-space))))
The lisp definition of the function loop-space-efhm shows clearly the possible recursivity of the process is space is itself a loop space.
commonlisp
(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)))
Examples¶
Let us show first how to get the homology groups of some known loop spaces, $\Omega {\cal S}^2$, $\Omega^3 {\cal S}^4$, $\Omega^2{\rm Moore}(\mathbb{Z}/{2\mathbb{Z}},3)$.
(def s2 (sphere 2))
[K1 Simplicial-Set]
(def os2 (loop-space s2))
[K6 Simplicial-Group]
(dotimes (k 10) (homology os2 k))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. 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 -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 1) : <<AlLp[1 S2]>> End of computing. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 1) : <<AlLp[1 S2]>> End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 2) : <<AlLp[1 S2][1 S2]>> End of computing. Homology in dimension 1 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 2) : <<AlLp[1 S2][1 S2]>> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 3) : <<AlLp[1 S2][1 S2][1 S2]>> End of computing. Homology in dimension 2 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 3. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 3) : <<AlLp[1 S2][1 S2][1 S2]>> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 4) : <<AlLp[1 S2][1 S2][1 S2][1 S2]>> End of computing. Homology in dimension 3 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 4. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 4) : <<AlLp[1 S2][1 S2][1 S2][1 S2]>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 5) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Homology in dimension 4 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 5. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 5) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 6) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Homology in dimension 5 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 6. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 6) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Computing boundary-matrix in dimension 7. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 7) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Homology in dimension 6 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 7. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 7) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Computing boundary-matrix in dimension 8. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 8) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Homology in dimension 7 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 8. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 8) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Computing boundary-matrix in dimension 9. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 9) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Homology in dimension 8 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 9. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 9) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Computing boundary-matrix in dimension 10. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 10) : <<AlLp[1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2][1 S2]>> End of computing. Homology in dimension 9 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s.
NIL
(cat-init)
---done---
NIL
(def s4 (sphere 4))
[K1 Simplicial-Set]
(def ooos4 (loop-space s4 3))
[K30 Simplicial-Group]
(dotimes (k 6) (homology ooos4 k))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. 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 -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 1) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 1) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 2) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Homology in dimension 1 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 2. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/1 (dimension 2) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 2. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/2 (dimension 3) : <<AlLp[3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 2/2 (dimension 3) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Homology in dimension 2 : Component Z/2Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 3. Rank of the source-module : 2. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/2 (dimension 3) : <<AlLp[3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 2/2 (dimension 3) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 4. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/4 (dimension 4) : <<AlLp[4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 2/4 (dimension 4) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 3/4 (dimension 4) : <<AlLp[3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 4/4 (dimension 4) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Homology in dimension 3 : Component Z/2Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 4. Rank of the source-module : 4. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/4 (dimension 4) : <<AlLp[4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 2/4 (dimension 4) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 3/4 (dimension 4) : <<AlLp[3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 4/4 (dimension 4) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 7. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/7 (dimension 5) : <<AlLp[5 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 2/7 (dimension 5) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 3/7 (dimension 5) : <<AlLp[4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 4/7 (dimension 5) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 5/7 (dimension 5) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 6/7 (dimension 5) : <<AlLp[3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 7/7 (dimension 5) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Homology in dimension 4 : Component Z/3Z Component Z/2Z Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 39s. Computing boundary-matrix in dimension 5. Rank of the source-module : 7. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/7 (dimension 5) : <<AlLp[5 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 2/7 (dimension 5) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 3/7 (dimension 5) : <<AlLp[4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 4/7 (dimension 5) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 5/7 (dimension 5) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 6/7 (dimension 5) : <<AlLp[3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 7/7 (dimension 5) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 13. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 1/13 (dimension 6) : <<AlLp[6 <<AlLp[2 <<AlLp[3 S4]>>][5 <<AlLp[3 S4][3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 2/13 (dimension 6) : <<AlLp[6 <<AlLp[5 <<AlLp[3 S4][3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 3/13 (dimension 6) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][5 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 4/13 (dimension 6) : <<AlLp[3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 5/13 (dimension 6) : <<AlLp[5 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 6/13 (dimension 6) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 7/13 (dimension 6) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 8/13 (dimension 6) : <<AlLp[4 <<AlLp[5 <<AlLp[3 S4][3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 9/13 (dimension 6) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 39s. Computing the boundary of the generator 10/13 (dimension 6) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 11/13 (dimension 6) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 12/13 (dimension 6) : <<AlLp[3 <<AlLp[2 <<AlLp[3 S4]>>][2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 13/13 (dimension 6) : <<AlLp[1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>][1 <<AlLp[2 <<AlLp[3 S4]>>]>>]>> End of computing. Homology in dimension 5 : Component Z/3Z Component Z/2Z Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 40s.
NIL
(def moore-23 (moore 2 3))
[K405 Simplicial-Set]
(def oo-moore-23 (loop-space moore-23 2))
[K422 Simplicial-Group]
(dotimes (k 6) (homology oo-moore-23 k))
Computing boundary-matrix in dimension 0. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 40s. 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 -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/1 (dimension 1) : <<AlLp[1 <<AlLp[2 M3]>>]>> End of computing. Homology in dimension 0 : Component Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 40s. Computing boundary-matrix in dimension 1. Rank of the source-module : 1. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/1 (dimension 1) : <<AlLp[1 <<AlLp[2 M3]>>]>> End of computing. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/2 (dimension 2) : <<AlLp[2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/2 (dimension 2) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Homology in dimension 1 : Component Z/2Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 40s. Computing boundary-matrix in dimension 2. Rank of the source-module : 2. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/2 (dimension 2) : <<AlLp[2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/2 (dimension 2) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Computing boundary-matrix in dimension 3. Rank of the source-module : 4. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/4 (dimension 3) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/4 (dimension 3) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 3/4 (dimension 3) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 4/4 (dimension 3) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Homology in dimension 2 : Component Z/2Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 40s. Computing boundary-matrix in dimension 3. Rank of the source-module : 4. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/4 (dimension 3) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/4 (dimension 3) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 3/4 (dimension 3) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 4/4 (dimension 3) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Computing boundary-matrix in dimension 4. Rank of the source-module : 9. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/9 (dimension 4) : <<AlLp[4 <<AlLp[2 M3][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/9 (dimension 4) : <<AlLp[4 <<AlLp[3 N4][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 3/9 (dimension 4) : <<AlLp[1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 4/9 (dimension 4) : <<AlLp[2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 5/9 (dimension 4) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 6/9 (dimension 4) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 7/9 (dimension 4) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 8/9 (dimension 4) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 9/9 (dimension 4) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Homology in dimension 3 : Component Z/4Z Component Z/2Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 40s. Computing boundary-matrix in dimension 4. Rank of the source-module : 9. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/9 (dimension 4) : <<AlLp[4 <<AlLp[2 M3][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/9 (dimension 4) : <<AlLp[4 <<AlLp[3 N4][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 3/9 (dimension 4) : <<AlLp[1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 4/9 (dimension 4) : <<AlLp[2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 5/9 (dimension 4) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 6/9 (dimension 4) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 7/9 (dimension 4) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 8/9 (dimension 4) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 9/9 (dimension 4) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Computing boundary-matrix in dimension 5. Rank of the source-module : 19. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/19 (dimension 5) : <<AlLp[5 <<AlLp[3 N4][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/19 (dimension 5) : <<AlLp[5 <<AlLp[2 M3][2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 3/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][4 <<AlLp[2 M3][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 4/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][4 <<AlLp[3 N4][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 5/19 (dimension 5) : <<AlLp[2 <<AlLp[3 N4]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 6/19 (dimension 5) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 7/19 (dimension 5) : <<AlLp[4 <<AlLp[2 M3][3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 8/19 (dimension 5) : <<AlLp[4 <<AlLp[3 N4][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 9/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 10/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 11/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 12/19 (dimension 5) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 13/19 (dimension 5) : <<AlLp[2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 14/19 (dimension 5) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 15/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 16/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 17/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 18/19 (dimension 5) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 19/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Homology in dimension 4 : Component Z/2Z Component Z/2Z Component Z/2Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 40s. Computing boundary-matrix in dimension 5. Rank of the source-module : 19. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/19 (dimension 5) : <<AlLp[5 <<AlLp[3 N4][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/19 (dimension 5) : <<AlLp[5 <<AlLp[2 M3][2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 3/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][4 <<AlLp[2 M3][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 4/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][4 <<AlLp[3 N4][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 5/19 (dimension 5) : <<AlLp[2 <<AlLp[3 N4]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 6/19 (dimension 5) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 7/19 (dimension 5) : <<AlLp[4 <<AlLp[2 M3][3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 8/19 (dimension 5) : <<AlLp[4 <<AlLp[3 N4][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 9/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 10/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 11/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 12/19 (dimension 5) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 13/19 (dimension 5) : <<AlLp[2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 14/19 (dimension 5) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 15/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 16/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 17/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 18/19 (dimension 5) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 19/19 (dimension 5) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Computing boundary-matrix in dimension 6. Rank of the source-module : 41. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/41 (dimension 6) : <<AlLp[6 <<AlLp[2 M3][2 M3][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 2/41 (dimension 6) : <<AlLp[6 <<AlLp[2 M3][3 N4][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 3/41 (dimension 6) : <<AlLp[6 <<AlLp[3 N4][2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 4/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][5 <<AlLp[3 N4][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 5/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][5 <<AlLp[2 M3][2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 6/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][4 <<AlLp[2 M3][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 7/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][4 <<AlLp[3 N4][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 8/41 (dimension 6) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 9/41 (dimension 6) : <<AlLp[4 <<AlLp[2 M3][3 N4]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 10/41 (dimension 6) : <<AlLp[4 <<AlLp[3 N4][2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 11/41 (dimension 6) : <<AlLp[5 <<AlLp[3 N4][3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 12/41 (dimension 6) : <<AlLp[5 <<AlLp[2 M3][2 M3][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 13/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][4 <<AlLp[2 M3][3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 14/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][4 <<AlLp[3 N4][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 15/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 16/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 17/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][4 <<AlLp[2 M3][3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 18/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][4 <<AlLp[3 N4][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 19/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 20/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 21/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 22/41 (dimension 6) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 23/41 (dimension 6) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 24/41 (dimension 6) : <<AlLp[4 <<AlLp[2 M3][3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 25/41 (dimension 6) : <<AlLp[4 <<AlLp[3 N4][2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 26/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 27/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 28/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 29/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 30/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 31/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 32/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 33/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 34/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 35/41 (dimension 6) : <<AlLp[3 <<AlLp[2 M3][2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 36/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 37/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 38/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 39/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 40/41 (dimension 6) : <<AlLp[2 <<AlLp[3 N4]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 41/41 (dimension 6) : <<AlLp[1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>][1 <<AlLp[2 M3]>>]>> End of computing. Homology in dimension 5 : Component Z/2Z Component Z/2Z Component Z/2Z Component Z/2Z ---done--- ;; Clock -> 2016-03-19, 0h 15m 40s.
NIL
The solution of the Adams and Carlsson problem¶
Let $X$ an $n$-reduced simplicial set which is not a suspension. The Adams and Carlsson problem (Gunnar Carlsson, R.James Milgram. Stable homotopy and iterated loop spaces in Handbook of Algebraic Topology, pp 545 edited by 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 Kenzo program builds the relevant solution.
Let us consider for instance the truncated projective space $X=P^\infty \mathbb{R}/ P^3 \mathbb{R}$ (a non-suspended space) and let us build $\Omega^3 X$.
(cat-init)
---done---
NIL
(def p4 (R-proj-space 4))
[K1 Simplicial-Set]
(def o3p4 (loop-space p4 3))
[K30 Simplicial-Group]
(def ecc (rbcc (efhm o3p4)))
[K390 Chain-Complex]
The cellular homology of $\Omega^3 X$ is now known through the symbol ecc. Note that 360 objects ($390-30$) have been built to get the final complex ecc. From the printing of the length of the basis in dimensions 0 to 5,
(dotimes (i 6) (print (length (basis ecc i))))
1 7 74 782 8264 87332
NIL
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:
(chcm-mat ecc 5)
Computing boundary-matrix in dimension 5. Rank of the source-module : 87332. ;; Clock -> 2016-03-19, 0h 15m 40s. Computing the boundary of the generator 1/87332 (dimension 5) : <<AlLp[5 <<AlLp[6 <<AlLp[1 2][2 3][2 3][2 3]>>]>>]>>
SIMPLE-ERROR: In the combination located by *WRONG-CMBN*, the generators: <<AlLp[1 2][0 1]>> and <<AlLp[1 2][0 1]>> are in a wrong order.
NIL
For instance, we see that the first $5$-cell,
(first (basis ecc 5))
<<AlLp[5 <<AlLp[6 <<AlLp[1 2][2 3][2 3][2 3]>>]>>]>>
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.
Lisp files concerned in this chapter¶
lp-space-efhm.lisp, searching-homology.lisp.