{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "To load \"kenzo\":\n", " Load 1 ASDF system:\n", " kenzo\n", "; Loading \"kenzo\"\n", "\n" ] }, { "data": { "text/plain": [ "#" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(ql:quickload \"kenzo\")\n", "(in-package :cat)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Eilenberg-Moore spectral sequence I\n", "\n", "## Introduction\n", "\n", "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 program `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 program builds an object which contains a complete solution of the Adams problem about the iteration of the\n", "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 lovely underlying algorithm is due to Julio Rubio, see *A program computing the homology groups of loop spaces*, SIGSAM Bulletin, 1991, vol. 25, pp. 20-24.\n", "\n", "## The detailed construction\n", "\n", "Let ${\\cal C}_*(X)$ be a coalgebra with effective homology. This means that a homotopy equivalence:\n", "$$\n", "\\begin{aligned}\n", " & \\qquad\\hat{C}_* & \\cr\n", " \\swarrow\\nearrow & & \\searrow\\nwarrow \\quad\\cr\n", "{\\cal C}_*(X) & & EC_* \\cr\n", "\\end{aligned}\n", "$$\n", "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; in this case, without any other information, the program constructs automatically a trivial homotopy equivalence.\n", "\n", "Now, if we apply the *Cobar* functor to this homotopy equivalence we obtain the homotopy equivalence $H_R$:\n", "$$\n", "\\begin{aligned}\n", " &\\qquad{\\widetilde{Cobar}}^{{\\hat C}_*}(\\mathbb{Z},\\mathbb{Z}) & \\cr\n", " \\swarrow\\nearrow & & \\searrow\\nwarrow \\qquad\\cr\n", "Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z}) & & {\\widetilde{Cobar}}^{EC_*}(\\mathbb{Z}, \\mathbb{Z}) \\cr\n", "\\end{aligned}\n", "$$\n", "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.\n", "\n", "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, $H_L$:\n", "$$\n", "\\begin{aligned}\n", " & \\qquad Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes_t {\\cal C_*}(GX)) \\cr\n", " \\swarrow\\nearrow & & \\searrow\\nwarrow\\qquad \\cr\n", " {\\cal C_*}(GX) & & Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z}) \\cr\n", "\\end{aligned}\n", "$$\n", "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 $H_R$.\n", "\n", "### Obtaining the left reduction $H_L$\n", "\n", "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:\n", "\n", "* ${\\cal C}_*(X)$ comodule on itself, with the canonical coproduct.\n", "$${\\cal C}_*(X)\\stackrel {\\Delta}{\\longrightarrow} {\\cal C}_*(X)\\otimes {\\cal C}_*(X).$$\n", "* ${\\check{\\cal C}}_*(X)$, comodule on ${\\cal C}_*(X)$, with a \"trivial\" coproduct \n", "$${\\check{\\cal C}}_*(X) \\stackrel {\\check{\\Delta}}{\\longrightarrow} {\\cal C}_*(X)\\otimes{\\check{\\cal C}}_*(X),\\quad\n", "\\sigma \\rightarrow 1\\otimes\\sigma,\\,\\sigma \\in {\\check{\\cal C}}_*(X).$$\n", "\n", "Now, we consider the set of the followings cobars (where $u$ means *\"untwisted\"* and $t$ *\"twisted\"*):\n", "$$\n", "\\begin{aligned}\n", "{\\rm Hat}_{uu} &=& Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\check{\\cal C}}_*(X)\\otimes {\\cal C_*}(GX)),\\\\\n", "{\\rm Hat}_{ut} &=& Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\check{\\cal C}}_*(X)\\otimes_t {\\cal C_*}(GX)),\\\\\n", "{\\rm Hat}_{tu} &=& Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes {\\cal C_*}(GX)),\\\\\n", "{\\rm Hat}_{tt} &=& Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes_t {\\cal C_*}(GX)).\n", "\\end{aligned}\n", "$$\n", "\n", "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 $\\delta_l$.\n", "\n", "This is shown in the following diagram (here, the arrows are not reductions, but denote the perturbations between the differential morphisms):\n", "$$\n", "\\begin{aligned} \n", " & \\qquad{\\rm Hat}_{uu} & \\cr\n", " \\delta_l \\swarrow & & \\searrow \\delta_r \\cr\n", "{\\rm Hat}_{tu} & & {\\rm Hat}_{ut} \\cr\n", " \\delta_r \\searrow & & \\swarrow \\delta_l \\cr\n", " & \\qquad{\\rm Hat}_{tt} & \\cr\n", "\\end{aligned}\n", "$$\n", "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:\n", "$$\n", "\\begin{aligned}\n", "\\delta_r({\\bar{c}}_1 \\otimes \\cdots \\otimes {\\bar{c}}_n \\otimes c \\otimes g) &=& \n", " [{\\bar{c}}_1 \\otimes \\cdots \\otimes {\\bar{c}}_n] \\otimes [d_{\\otimes t}(c \\otimes g)-d_\\otimes (c \\otimes g)], \\\\\n", "\\delta_l({\\bar{c}}_1 \\otimes \\cdots \\otimes {\\bar{c}}_n \\otimes c \\otimes g) &=&\n", "{\\bar{c}}_1 \\otimes \\cdots \\otimes {\\bar{c}}_n \\otimes {\\bar{\\Delta}}c \\otimes g, \n", "\\end{aligned}\n", "$$\n", "where ${\\bar{c}}_i, c \\in {\\cal C}_*(X),\\, g \\in {\\cal C_*}(GX))$ and $\\bar{\\Delta}= \\Delta - \\check{\\Delta}$.\n", "\n", "Now, on one hand, we know that there exists a reduction \n", "$${\\rm Hat}_{tu}=Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes {\\cal C_*}(GX)) \\Longrightarrow {\\cal C}_*(GX),$$ \n", "so, perturbing this reduction by $\\delta_r$, one obtains the Rubio reduction \n", "$${\\rm Hat}_{tt}=Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes_t {\\cal C_*}(GX)) \\Longrightarrow{\\cal C}_*(GX).$$\n", "On the other hand, we know also that there exists a reduction \n", "$${\\rm Hat}_{ut}=Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\check{\\cal C}}_*(X)\\otimes_t {\\cal C_*}(GX)) \\Longrightarrow\n", "Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z}),$$ \n", "so, perturbing this reduction by $\\delta_l$, one obtains the reduction \n", "$${\\rm Hat}_{tt}=Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes_t {\\cal C_*}(GX)) \\Longrightarrow \n", "Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z}).$$\n", "Finally, we have obtained the wished left homotopy equivalence $H_L$:\n", "$$\n", "\\begin{aligned}\n", " & \\qquad Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes_t {\\cal C_*}(GX)) & \\cr\n", " \\swarrow\\nearrow & & \\searrow\\nwarrow\\qquad\\qquad \\cr\n", " {\\cal C_*}(GX) & & Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z}) \\cr\n", "\\end{aligned}\n", "$$\n", "\n", "### The useful functions\n", "\n", "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.\n", "\n", "`loop-space-efhm` *space* *[Function]*\n", "> 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.\n", "\n", "`ls-hat-u-u` *space* *[Function]*\n", "> 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 $Cobar^{{\\cal C}_*(X)}(\\mathbb{Z},\\mathbb{Z})\\otimes {\\check{\\cal C}}_*(X)\\otimes {\\cal C_*}(GX).$\n", "\n", "`ls-hat-left-perturbation` *space* *[Function]*\n", "> Return the morphism corresponding to the differential perturbation $\\delta_l$, induced by the discrepancy between the coproducts of ${\\check{\\cal C}}_*(X)$ and ${\\cal C}_*(X)$.\n", "\n", "`ls-hat-t-u` *space* *[Function]*\n", "> Return the chain complex $Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes {\\cal C_*}(GX))$ by applying the differential perturbation `hat-left-perturbation` upon the chain complex built by the function `hat-u-u`. This is realized by the method `add`.\n", "\n", "`ls-hat-u-t` *space* *[Function]*\n", "> 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).\n", "\n", "`ls-hat-right-perturbation` *space* *[Function]*\n", "> 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`.\n", "\n", "`ls-left-hmeq-hat` *space* *[Function]*\n", "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 $\\delta_l$.\n", "\n", "`ls-pre-left-hmeq-left-reduction` *space* *[Function]*\n", "Build the reduction\n", "$${\\rm Hat}_{tu}=Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes {\\cal C_*}(GX)) \\Longrightarrow {\\cal C}_*(GX).$$ \n", "\n", "`ls-pre-left-hmeq-right-reduction` *space* *[Function]*\n", "Build the reduction\n", "$${\\rm Hat}_{ut}=Cobar^{{\\cal C}_*(X)}(\\mathbb{Z},{\\check{\\cal C}}_*(X)\\otimes_t {\\cal C_*}(GX)) \\Longrightarrow\n", "Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z}).$$\n", "\n", "`ls-left-hmeq-left-reduction` *space* *[Function]*\n", "Build the Rubio reduction \n", "$$Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes_t {\\cal C_*}(GX)) \\Longrightarrow{\\cal C}_*(GX)$$\n", "by perturbing by the perturbation $\\delta_r$ the reduction\n", "$${\\rm Hat}_{tu} \\Longrightarrow {\\cal C}_*(GX)$$ \n", "obtained by the function `pre-left-hmeq-left-reduction`.\n", "\n", "`ls-left-hmeq-right-reduction` *space* *[Function]*\n", "Build the reduction \n", "$$Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes_t {\\cal C_*}(GX)) \\Longrightarrow \n", "Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z})$$\n", "by perturbing by the perturbation $\\delta_l$ the reduction \n", "$${\\rm Hat}_{ut} \\Longrightarrow Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z})$$\n", "obtained by the function `pre-left-hmeq-right-reduction`.\n", "\n", "`ls-left-hmeq` *space* *[Function]*\n", "Build the homotopy equivalence\n", "$$\n", "\\begin{aligned}\n", " & \\qquad Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, {\\cal C}_*(X)\\otimes_t {\\cal C_*}(GX)) & \\cr\n", " \\swarrow\\nearrow & & \\searrow\\nwarrow \\qquad\\qquad\\cr\n", " {\\cal C_*}(GX) & & Cobar^{{\\cal C}_*(X)}(\\mathbb{Z}, \\mathbb{Z}) \\cr\n", "\\end{aligned}\n", "$$\n", "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.\n", "\n", "### Searching homology for loop spaces\n", "\n", "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.\n", "```commonlisp\n", "(defmethod SEARCH-EFHM (loop-space (orgn (eql 'loop-space)))\n", " (declare (type simplicial-set loop-space))\n", " (loop-space-efhm (second (orgn loop-space))))\n", "```\n", "The lisp definition of the function `loop-space-efhm` shows clearly the possible recursivity of the process if *space* is itself a loop space.\n", "```commonlisp\n", "(defun LOOP-SPACE-EFHM (space)\n", " (declare (type simplicial-set space))\n", " (let ((left-hmeq (ls-left-hmeq space))\n", " (right-hmeq (cobar (efhm space))))\n", " (declare (type homotopy-equivalence left-hmeq right-hmeq))\n", " (cmps left-hmeq right-hmeq)))\n", "```\n", "\n", "### Examples\n", "\n", "Let us show first how to get the homology groups of some known loop spaces, $\\Omega {\\cal S}^2$, $\\Omega^3 {\\cal S}^4$, $\\Omega^2{\\rm Moore}(\\mathbb{Z}/{2\\mathbb{Z}},3)$." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K1 Simplicial-Set]" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def s2 (sphere 2))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K6 Simplicial-Group]" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def os2 (loop-space s2))" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Computing boundary-matrix in dimension 0.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 0 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 1 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 2 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 3 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 4 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 6.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 6) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 5 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 6.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 6) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 7.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 7) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 6 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 7.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 7) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 8.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 8) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 7 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 8.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 8) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 9.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 9) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 8 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 9.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 9) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 10.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 10) :\n", "<> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 9 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n" ] }, { "data": { "text/plain": [ "NIL" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(dotimes (k 10) (homology os2 k))" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "---done---" ] }, { "data": { "text/plain": [ "NIL" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(cat-init)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K1 Simplicial-Set]" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def s4 (sphere 4)) " ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K30 Simplicial-Group]" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def ooos4 (loop-space s4 3))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Computing boundary-matrix in dimension 0.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<>]>>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 0 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<>]>>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 1 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 2.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/2 (dimension 3) :\n", "<>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 2/2 (dimension 3) :\n", "<>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 2 :\n", "\n", "\n", "Component Z/2Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 2.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/2 (dimension 3) :\n", "<>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 2/2 (dimension 3) :\n", "<>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 4.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/4 (dimension 4) :\n", "<>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 2/4 (dimension 4) :\n", "<>]>>][3 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 3/4 (dimension 4) :\n", "<>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 4/4 (dimension 4) :\n", "<>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 3 :\n", "\n", "\n", "Component Z/2Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 4.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/4 (dimension 4) :\n", "<>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 2/4 (dimension 4) :\n", "<>]>>][3 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 3/4 (dimension 4) :\n", "<>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 4/4 (dimension 4) :\n", "<>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 7.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/7 (dimension 5) :\n", "<>][2 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 2/7 (dimension 5) :\n", "<>]>>][4 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 3/7 (dimension 5) :\n", "<>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 4/7 (dimension 5) :\n", "<>]>>][1 <>]>>][3 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 5/7 (dimension 5) :\n", "<>]>>][3 <>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 6/7 (dimension 5) :\n", "<>][2 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 7/7 (dimension 5) :\n", "<>]>>][1 <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 4 :\n", "\n", "\n", "Component Z/3Z\n", "\n", "Component Z/2Z\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 7.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/7 (dimension 5) :\n", "<>][2 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 2/7 (dimension 5) :\n", "<>]>>][4 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 3/7 (dimension 5) :\n", "<>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 4/7 (dimension 5) :\n", "<>]>>][1 <>]>>][3 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 5/7 (dimension 5) :\n", "<>]>>][3 <>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 6/7 (dimension 5) :\n", "<>][2 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 7/7 (dimension 5) :\n", "<>]>>][1 <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 6.\n", "Rank of the source-module : 13.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 1/13 (dimension 6) :\n", "<>][5 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 2/13 (dimension 6) :\n", "<>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 3/13 (dimension 6) :\n", "<>]>>][5 <>][2 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 4/13 (dimension 6) :\n", "<>][2 <>]>>][3 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 5/13 (dimension 6) :\n", "<>][2 <>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 6/13 (dimension 6) :\n", "<>]>>][1 <>]>>][4 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 7/13 (dimension 6) :\n", "<>]>>][4 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 8/13 (dimension 6) :\n", "<>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 9/13 (dimension 6) :\n", "<>]>>][1 <>]>>][1 <>]>>][3 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 54s.\n", "Computing the boundary of the generator 10/13 (dimension 6) :\n", "<>]>>][1 <>]>>][3 <>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 11/13 (dimension 6) :\n", "<>]>>][3 <>][2 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 12/13 (dimension 6) :\n", "<>][2 <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 13/13 (dimension 6) :\n", "<>]>>][1 <>]>>][1 <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 5 :\n", "\n", "\n", "Component Z/3Z\n", "\n", "Component Z/2Z\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "\n" ] }, { "data": { "text/plain": [ "NIL" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(dotimes (k 6) (homology ooos4 k))" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K405 Simplicial-Set]" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def moore-23 (moore 2 3))" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K422 Simplicial-Group]" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def oo-moore-23 (loop-space moore-23 2))" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Computing boundary-matrix in dimension 0.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "<> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 0 :\n", "\n", "\n", "Component Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 2.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/2 (dimension 2) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/2 (dimension 2) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 1 :\n", "\n", "\n", "Component Z/2Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 2.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/2 (dimension 2) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/2 (dimension 2) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 4.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/4 (dimension 3) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/4 (dimension 3) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 3/4 (dimension 3) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 4/4 (dimension 3) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 2 :\n", "\n", "\n", "Component Z/2Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 4.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/4 (dimension 3) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/4 (dimension 3) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 3/4 (dimension 3) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 4/4 (dimension 3) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 9.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/9 (dimension 4) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/9 (dimension 4) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 3/9 (dimension 4) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 4/9 (dimension 4) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 5/9 (dimension 4) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 6/9 (dimension 4) :\n", "<>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 7/9 (dimension 4) :\n", "<>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 8/9 (dimension 4) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 9/9 (dimension 4) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 3 :\n", "\n", "\n", "Component Z/4Z\n", "\n", "Component Z/2Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 9.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/9 (dimension 4) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/9 (dimension 4) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 3/9 (dimension 4) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 4/9 (dimension 4) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 5/9 (dimension 4) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 6/9 (dimension 4) :\n", "<>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 7/9 (dimension 4) :\n", "<>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 8/9 (dimension 4) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 9/9 (dimension 4) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 19.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/19 (dimension 5) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/19 (dimension 5) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 3/19 (dimension 5) :\n", "<>][4 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 4/19 (dimension 5) :\n", "<>][4 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 5/19 (dimension 5) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 6/19 (dimension 5) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 7/19 (dimension 5) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 8/19 (dimension 5) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 9/19 (dimension 5) :\n", "<>][1 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 10/19 (dimension 5) :\n", "<>][2 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 11/19 (dimension 5) :\n", "<>][3 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 12/19 (dimension 5) :\n", "<>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 13/19 (dimension 5) :\n", "<>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 14/19 (dimension 5) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 15/19 (dimension 5) :\n", "<>][1 <>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 16/19 (dimension 5) :\n", "<>][1 <>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 17/19 (dimension 5) :\n", "<>][2 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 18/19 (dimension 5) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 19/19 (dimension 5) :\n", "<>][1 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 4 :\n", "\n", "\n", "Component Z/2Z\n", "\n", "Component Z/2Z\n", "\n", "Component Z/2Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 19.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/19 (dimension 5) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/19 (dimension 5) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 3/19 (dimension 5) :\n", "<>][4 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 4/19 (dimension 5) :\n", "<>][4 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 5/19 (dimension 5) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 6/19 (dimension 5) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 7/19 (dimension 5) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 8/19 (dimension 5) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 9/19 (dimension 5) :\n", "<>][1 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 10/19 (dimension 5) :\n", "<>][2 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 11/19 (dimension 5) :\n", "<>][3 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 12/19 (dimension 5) :\n", "<>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 13/19 (dimension 5) :\n", "<>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 14/19 (dimension 5) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 15/19 (dimension 5) :\n", "<>][1 <>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 16/19 (dimension 5) :\n", "<>][1 <>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 17/19 (dimension 5) :\n", "<>][2 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 18/19 (dimension 5) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 19/19 (dimension 5) :\n", "<>][1 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "Computing boundary-matrix in dimension 6.\n", "Rank of the source-module : 41.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/41 (dimension 6) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/41 (dimension 6) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 3/41 (dimension 6) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 4/41 (dimension 6) :\n", "<>][5 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 5/41 (dimension 6) :\n", "<>][5 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 6/41 (dimension 6) :\n", "<>][4 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 7/41 (dimension 6) :\n", "<>][4 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 8/41 (dimension 6) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 9/41 (dimension 6) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 10/41 (dimension 6) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 11/41 (dimension 6) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 12/41 (dimension 6) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 13/41 (dimension 6) :\n", "<>][1 <>][4 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 14/41 (dimension 6) :\n", "<>][1 <>][4 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 15/41 (dimension 6) :\n", "<>][2 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 16/41 (dimension 6) :\n", "<>][3 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 17/41 (dimension 6) :\n", "<>][4 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 18/41 (dimension 6) :\n", "<>][4 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 19/41 (dimension 6) :\n", "<>][1 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 20/41 (dimension 6) :\n", "<>][2 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 21/41 (dimension 6) :\n", "<>][3 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 22/41 (dimension 6) :\n", "<>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 23/41 (dimension 6) :\n", "<>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 24/41 (dimension 6) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 25/41 (dimension 6) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 26/41 (dimension 6) :\n", "<>][1 <>][1 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 27/41 (dimension 6) :\n", "<>][1 <>][2 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 28/41 (dimension 6) :\n", "<>][1 <>][3 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 29/41 (dimension 6) :\n", "<>][2 <>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 30/41 (dimension 6) :\n", "<>][2 <>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 31/41 (dimension 6) :\n", "<>][3 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 32/41 (dimension 6) :\n", "<>][1 <>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 33/41 (dimension 6) :\n", "<>][1 <>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 34/41 (dimension 6) :\n", "<>][2 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 35/41 (dimension 6) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 36/41 (dimension 6) :\n", "<>][1 <>][1 <>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 37/41 (dimension 6) :\n", "<>][1 <>][1 <>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 38/41 (dimension 6) :\n", "<>][1 <>][2 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 39/41 (dimension 6) :\n", "<>][2 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 40/41 (dimension 6) :\n", "<>][1 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 41/41 (dimension 6) :\n", "<>][1 <>][1 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", "\n", "\n", "Homology in dimension 5 :\n", "\n", "\n", "Component Z/2Z\n", "\n", "Component Z/2Z\n", "\n", "Component Z/2Z\n", "\n", "Component Z/2Z\n", "\n", "\n", "---done---\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "\n" ] }, { "data": { "text/plain": [ "NIL" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(dotimes (k 6) (homology oo-moore-23 k))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The solution of the Adams and Carlsson problem\n", "\n", "Let $X$ be 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.\n", "\n", "Let us consider for instance the stunted projective space $X=P^\\infty \\mathbb{R}/ P^3 \\mathbb{R}$ (a non-suspended space) and let us build $\\Omega^3 X$." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "---done---" ] }, { "data": { "text/plain": [ "NIL" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(cat-init)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K1 Simplicial-Set]" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def p4 (R-proj-space :infinity 4))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K30 Simplicial-Group]" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def o3p4 (loop-space p4 3))" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[K390 Chain-Complex]" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def ecc (rbcc (efhm o3p4)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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 chain complex `ecc`. From the printing of the length of the basis in dimensions 0 to 5," ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "1 \n", "1 \n", "2 \n", "5 \n", "13 \n", "33 " ] }, { "data": { "text/plain": [ "NIL" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(dotimes (i 6) (print (length (basis ecc i))))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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:" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 33.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 1/33 (dimension 5) :\n", "<>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 2/33 (dimension 5) :\n", "<>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 3/33 (dimension 5) :\n", "<>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 4/33 (dimension 5) :\n", "<>][4 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 5/33 (dimension 5) :\n", "<>][3 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 6/33 (dimension 5) :\n", "<>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 7/33 (dimension 5) :\n", "<>][2 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 8/33 (dimension 5) :\n", "<>]>>][4 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 9/33 (dimension 5) :\n", "<>]>>][4 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 10/33 (dimension 5) :\n", "<>]>>][4 <>][3 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 11/33 (dimension 5) :\n", "<>]>>][4 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 12/33 (dimension 5) :\n", "<>]>>][3 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 13/33 (dimension 5) :\n", "<>]>>][3 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 14/33 (dimension 5) :\n", "<>]>>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 15/33 (dimension 5) :\n", "<>][2 <>]>>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 55s.\n", "Computing the boundary of the generator 16/33 (dimension 5) :\n", "<>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 17/33 (dimension 5) :\n", "<>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 18/33 (dimension 5) :\n", "<>][3 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 19/33 (dimension 5) :\n", "<>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 20/33 (dimension 5) :\n", "<>]>>][1 <>]>>][3 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 21/33 (dimension 5) :\n", "<>]>>][1 <>]>>][3 <>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 22/33 (dimension 5) :\n", "<>]>>][2 <>]>>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 23/33 (dimension 5) :\n", "<>]>>][3 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 24/33 (dimension 5) :\n", "<>]>>][3 <>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 25/33 (dimension 5) :\n", "<>]>>][1 <>]>>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 26/33 (dimension 5) :\n", "<>]>>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 27/33 (dimension 5) :\n", "<>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 28/33 (dimension 5) :\n", "<>][2 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 29/33 (dimension 5) :\n", "<>]>>][1 <>]>>][1 <>]>>][2 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 30/33 (dimension 5) :\n", "<>]>>][1 <>]>>][2 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 31/33 (dimension 5) :\n", "<>]>>][2 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 32/33 (dimension 5) :\n", "<>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n", "\n", ";; Clock -> 2017-10-08, 12h 27m 56s.\n", "Computing the boundary of the generator 33/33 (dimension 5) :\n", "<>]>>][1 <>]>>][1 <>]>>][1 <>]>>][1 <>]>>]>> \n", "End of computing.\n", "\n" ] }, { "data": { "text/plain": [ "\n", "========== MATRIX 13 lines + 33 columns =====\n", "L1=[C1=-2]\n", "L2=[C1=-1]\n", "L3=[C1=-4][C2=1][C3=-1][C4=-2]\n", "L4=[C2=1][C3=-1][C6=2]\n", "L5=[C1=6][C4=1][C6=1]\n", "L6=[C1=4][C4=4][C6=4][C7=3]\n", "L7=[C1=4][C12=-2][C14=2]\n", "L8=[C1=6][C4=1][C6=1]\n", "L9=[C1=4][C4=4][C6=4][C7=3]\n", "L10=[C8=4][C10=1][C11=-1][C14=-4][C15=-2][C20=-2]\n", "L11=[C1=4][C8=4][C10=1][C11=-1][C16=-4][C18=-1][C19=1][C23=-2]\n", "L12=[C12=4][C13=2][C16=-4][C18=-1][C19=1][C27=-2]\n", "L13=[C1=-1][C20=4][C21=2][C23=-4][C24=-2][C27=4][C28=2]\n", "========== END-MATRIX" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(chcm-mat ecc 5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For instance, we see that the first $5$-cell," ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "<>]>>]>>" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(first (basis ecc 5)) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "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.\n", "\n", "### Lisp files concerned in this chapter\n", "\n", "`lp-space-efhm.lisp`, `searching-homology.lisp`." ] } ], "metadata": { "kernelspec": { "display_name": "SBCL Lisp", "language": "lisp", "name": "lisp" }, "language_info": { "codemirror_mode": "text/x-common-lisp", "mimetype": "text/x-common-lisp", "name": "common-lisp", "pygments_lexer": "common-lisp", "version": "X3J13" } }, "nbformat": 4, "nbformat_minor": 1 }