{
 "metadata": {
  "language": "common-lisp",
  "name": "",
  "signature": "sha256:369b292c1ecc74297adacb6883a5c1faf08191eaf887911e6915d0cc592d4b0d"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Overview"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this overview we try to  show, without entering in detailed explanations, various possibilities of $\\texttt{Kenzo}$ in algebraic topology.\n",
      "\n",
      "Load the $\\texttt{Kenzo}$ software as follows:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(ql:quickload \"kenzo\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The main functionality of the $\\texttt{Kenzo}$ software is defined package in the $\\texttt{cat}$ Lisp package. Initialize the $\\texttt{cat}$ package as follows:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:cat-init)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Print the $\\texttt{Kenzo}$ version as follows:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:kenzo-version)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us begin by the space $\\mathrm{Moore}(\\mathbb{Z}/2\\mathbb{Z}, 3)$ described as a simplicial set having only three non--degenerate simplices, namely in dimension $0$, $3$ and $4$. In the  representation created by the software, the $0$--simplex (base point), 15 the $3$--simplex and the $4$--simplex are respectively labelled $\\texttt{\"*\"}$, $\\texttt{M3}$ and $\\texttt{N4}$. Two faces of the $4$-simplex $\\texttt{N4}$ are identified with the $3$-simplex $\\texttt{M3}$, the others being contracted on the base point. To create the simplicial set one types simply: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar m23 (cat:moore 2 3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The system answers that a $\\texttt{Kenzo}$ object has been created, with number $1$ and type $\\texttt{SIMPLICIAL SET}$. This object may be referenced  by the symbol $\\texttt{m23}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m23"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We may compute the homology groups of this space, using the underlying chain complex induced by the simplicial set description. Here we compute the $H_i$ from $0$ to $4$ included. When in the answer the component part is void, it means that the corresponding homology group is null."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology m23 0 5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As $\\texttt{m23}$ is a simplicial set, it is possible to create the cartesian product $\\texttt{m23}\\times\\texttt{m23}$ by the function $\\texttt{crts-prdc}$. This is a new simplicial set."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar m23xm23 (cat:crts-prdc m23 m23))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m23xm23"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Being a simplicial set, $\\texttt{m23xm23}$ is also a chain complex object and we may for instance ask for the basis in dimension $6$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:basis m23xm23 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(length (cat:basis m23xm23 6))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The number of elements of the basis is quite large (230). The user will note that the basis elements are formed by cartesian products of *degenerated simplices*. In the list, an element like $\\texttt{<CrPr 1-0 N4 5-3-2 M3>}$ means $\\eta_1\\eta_0\\mathrm{N4}\\times\\eta_5\\eta_3\\eta_2\\mathrm{M3}$.\n",
      "\n",
      "We may construct also the tensor product $\\mathrm{m23}\\otimes\\mathrm{m23}$ from the underlying chain complex of the simplicial set $\\texttt{m23}$. This tensor product is a new chain complex and we see that the basis in dimension $6$ has only one element:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar t2m23 (cat:tnsr-prdc m23 m23))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t2m23"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:basis t2m23 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The Eilenberg-Zilber theorem is used to compute the homology groups of the cartesian product space: as chain complexes, $\\texttt{m23xm23}$ and $\\texttt{t2m23}$ have the same homology groups, but the computations using the tensor product are considerably faster."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology m23xm23 0 8)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us consider now the space $\\mathrm{K}(\\mathbb{Z}, 1)$. This is an Abelian simplicial group created  in $\\texttt{Kenzo}$ by the function $\\texttt{k-z}$. In this simplicial group, a simplex in dimension $n$  \n",
      "is mathematically represented by a sequence of integers, known as a *bar* object: $$ [a_1 \\mid a_2 \\mid \\ldots \\mid a_n].$$ \n",
      "In $\\texttt{Kenzo}$, a non-degenerate simplex of $\\mathrm{K}(\\mathbb{Z}, 1)$ in dimension $n$ will be simply a list of $n$ non-null integers, for instance: $\\texttt{(2 3 4 5)}$. In dimension $0$, the only \n",
      "simplex is $\\texttt{NIL}$ (the base point)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar kz1 (cat:k-z 1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "kz1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "But this object is also a *coalgebra* and an *algebra*, and we may \n",
      "see the effect of the respective induced *coproduct* and *product*: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:cprd kz1 4 '(2 3 4 5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:aprd kz1 6 (cat:tnpr 2 '(1 2) 4 '(3 4 5 6)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The printed results are the printed representation of *combinations*, i.e., integer linear combinations of generators resulting from the application of the morphisms. The degree of the combination is indicated by the information: $\\texttt{CMBN n}$. \n",
      "\n",
      "In the same way, we may create the Abelian simplicial groups $\\mathrm{K}(\\mathbb{Z}/2\\mathbb{Z}, n)$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar k-z2-2 (cat:k-z2 2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "k-z2-2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology k-z2-2 4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us play now with the sphere $S^3$ and its loop spaces. $S^3$ and $\\Omega^2 S^3$ are created by \n",
      "respective calls to the functions $\\texttt{sphere}$ and $\\texttt{loop-space}$. Then we compute the $H_4$ and $H_5$ of $\\Omega^2 S^3$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar s3 (cat:sphere 3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar o2s3 (cat:loop-space s3 2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "o2s3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology o2s3 4 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us take now the first loop space $\\Omega^1 S^3$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar os3 (cat:loop-space s3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "os3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the following instruction, we locate in the symbol $\\texttt{L1}$ the canonical generator of $\\pi_2 (\\Omega^1S^3)$, that is the $2$--simplex coming from the original sphere. In fact, the object created by the command $\\texttt{(loop3 0 's3 1)}$ is the \"word\" $S3^1$ belonging to the Kan simplicial version $G(S^3)$ (a simplicial group) of the loop space $\\Omega S^3$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar L1 (cat:loop3 0 's3 1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us consider also the $2$--degeneracy of the base point of the loop space. In the printed result, the user will recognize the degeneracy $\\eta_1\\eta_0$ of the null loop, base point of $\\Omega^1 S^3$: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar null-simp (cat:absm 3 cat:+null-loop+))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "null-simp"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We may  build now a new space by pasting a disk $D3$ as indicated by the following call. It means that we \"paste\" to the space $\\texttt{os3}$ a $3$--simplex named $\\texttt{D3}$, the attaching map being described by the list of its faces in dimension $2$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar dos3 (cat:disk-pasting os3 3 '<D3> \n",
      "                               (list L1 null-simp L1 null-simp)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dos3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us compute a few homology groups of the new space $\\texttt{dos3}$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology dos3 2 4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "But more interesting, let us  build the  loop space of the object $\\texttt{dos3}$ and let us compute the homology in dimension $5$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar odos3 (cat:loop-space dos3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "odos3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology odos3 5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us continue with the Kan theory. First, we check that $S^3$ is not of type Kan and that $\\Omega S^3$ is indeed of type Kan and a non--Abelian simplicial group."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(typep s3 'cat:kan)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(typep os3 'cat:kan)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(typep os3 'cat:simplicial-group)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(typep os3 'cat:ab-simplicial-group)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us create the word $L2=(S3)^2$, i.e an object belonging to $\\Omega S^3$ and let us apply the product of the underlying algebra upon $L2 \\otimes L2$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar L2 (cat:loop3 0 's3 2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar square (cat:aprd os3 4 (cat:tnpr 2 L2 2 L2)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "square"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We see that the result is a linear combination of words composed from degeneracies of $\\texttt{L2}$. The following instruction selects the generator part of the \n",
      "second element of the previous combination."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar L4 (cat:gnrt (second (cat:cmbn-list square))))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "L4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us use the lisp function $\\texttt{mapcar}$ (one among the various iteration functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object $\\texttt{L4}$, this list is a \"Kan hat\"."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar hat (mapcar #'(lambda (i) (cat:face os3 i 4 l4)) '(1 2 3 4)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "hat"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The function $\\texttt{kfll}$ tries to find a filling of this \"Kan hat\", and we see that the face $2$ of the resulting simplex (which is very different from $\\texttt{L4}$) is  the same as the face $2$ of $\\texttt{L4}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar kan-simplex (cat:kfll os3 0 4 hat))  "
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "kan-simplex"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:face os3 2 4 kan-simplex)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(second hat)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let ${\\cal G}$ be a simplicial group $0$--reduced. $\\Omega S^3$ is such a group. The software $\\texttt{Kenzo}$ allows the construction of the universal bundle $\\bar{\\cal W}{\\cal G}$, i.e. the classifying space of ${\\cal G}$. In our case, as $\\Omega S^3$ in non--Abelian, the result is not a simplicial group but only a simplicial set. We verify that the $H_4$ is null."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar cls-os3 (cat:classifying-space os3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cls-os3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(typep cls-os3 'cat:simplicial-group)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology cls-os3 4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let us end this short overview with an example of computation of homotopy groups. The method used in $\\texttt{Kenzo}$ is the Whitehead tower. In this current version only the case where the first non--null homology group (in non--null dimension) is $\\mathbb{Z}$ or $\\mathbb{Z}/{2\\mathbb{Z}}$ can be processed; however if this homology group is a direct sum of several copies of $\\mathbb{Z}$ or $\\mathbb{Z}/{2 \\mathbb{Z}}$, then the corresponding stage of the Whitehead tower may also be constructed step by step.\n",
      "\n",
      "We take again $\\mathrm{Moore}(\\mathbb{Z}/2\\mathbb{Z}, 3)$ whose $H_3$ is $\\mathbb{Z}/2\\mathbb{Z}$. First the fundamental cohomology class is constructed:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar ch3 (cat:chml-clss m23 3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ch3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Then the function $\\texttt{z2-whitehead}$ is called to build a fibration over the simplicial set $\\texttt{m23}$ canonically associated to the cohomology class $\\texttt{ch3}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar f3 (cat:z2-whitehead m23 ch3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Then the total space of the fibration is built:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar x4 (cat:fibration-total f3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The $H_4$ of this total space is the $\\pi_4$ of $\\mathrm{Moore}(\\mathbb{Z}/2\\mathbb{Z})$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology x4 3 5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We may now iterate the process, to compute the $\\pi_5$ of $\\mathrm{Moore}(\\mathbb{Z}/2\\mathbb{Z})$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar ch4 (cat:chml-clss x4 4))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ch4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar f4 (cat:z2-whitehead x4 ch4))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f4"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(defvar x5 (cat:fibration-total f4))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x5"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cat:homology x5 4 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So $\\pi_5(\\mathrm{Moore}(\\mathbb{Z}/2\\mathbb{Z}))$ is $\\mathbb{Z}/4\\mathbb{Z}$."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}