Francis' updates to Overview.ipynb.
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@ -13,11 +13,13 @@
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"source": [
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"In this overview we try to show, without entering in detailed explanations, various possibilities of ``Kenzo`` in algebraic topology.\n",
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"\n",
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"The material is arranged in a sequence of cells. Cells marked with ``In [*]`` represent (editable) input expressions, which in a traditional REPL (Read-Evaluate-Print Loop) would be typed by the user. Cells marked with ``Out [*]`` represent the value of the correpsponding input expression. Note that if an input expression was not evaluated previously, the ouput cell might be missing. Finally, there might be additional, non-editable, diagnostic output printed between an input and output cell. This is a side-effect of the input form evaluation and can be ignored on first reading.\n",
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"The material is arranged in a sequence of cells. Cells marked with ``In [*]`` represent (editable) input expressions, which in a traditional REPL (Read-Evaluate-Print Loop) would be typed by the user in a Lisp session. Only the ``In [*]`` mark is visible if the Lisp statement is not yet run; if run, another ``Out [*]`` mark is displayed with the corresponding object returned by the execution. Finally, there might be additional, non-editable, diagnostic output printed before the output component, most often a side-effect of the input form evaluation.\n",
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"\n",
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"### Getting off the Ground\n",
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"Other cells are the ``text`` cells, giving various explanations about the Lisp statements and their evaluations. Not editable by the user.\n",
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"\n",
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"Load the ``Kenzo`` software by evaluating the follwing expression. When loaded successfully, the value of this form is a single string atom list ``\"kenzo\"``."
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"## Getting off the Ground\n",
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"\n",
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"Load the ``Kenzo`` software by evaluating the following expression. When loaded successfully, the value of this form is a single string atom list ``\"kenzo\"``."
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]
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},
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{
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@ -82,9 +84,9 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### ``Kenzo`` in Action\n",
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"## ``Kenzo`` in Action\n",
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"\n",
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"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), the 3-simplex and the 4-simplex are respectively labelled ``\"*\"``, ``M3`` and ``N4``. Two faces of the 4-simplex ``N4`` are identified with the 3-simplex ``M3``, the others being contracted on the base point. To create the simplicial set one types simply: "
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"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), the 3-simplex and the 4-simplex are respectively labelled ``*``, ``M3`` and ``N4``. The faces 0 and 2 of the 4-simplex ``N4`` are identified with (pasted to) the 3-simplex ``M3``, the others being contracted on the base point. To create the simplicial set one types simply:"
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]
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},
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{
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@ -138,7 +140,9 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"As ``m23`` is a simplicial set, it is possible to create the cartesian product $\\texttt{m23}\\times\\texttt{m23}$ by the function ``crts-prdc``. This is a new simplicial set."
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"The homology of ``m23`` is $(Z,0,0,Z/2,0,\\ldots)$; the reduced homology is only $Z/2$ in dimension 3, defining the Moore space up to homotopy.\n",
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"\n",
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"As ``m23`` is a simplicial set, it is possible to create the cartesian product $\\texttt{m23}\\times\\texttt{m23}$ by the function ``crts-prdc`` (= CaRTeSian PRoDuCt). This is a new simplicial set."
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]
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},
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{
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@ -156,7 +160,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Being a simplicial set, $\\texttt{m23xm23}$ is also a chain complex object and we may for instance ask for the basis in dimension $6$"
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"Being a simplicial set, $\\texttt{m23xm23}$ is also a chain complex object and we may for instance ask for the basis in dimension 6."
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]
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},
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{
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@ -185,9 +189,9 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"As shown by the last command, 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 ``<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",
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"As shown by the last command, the number of elements of the basis is quite large (230). The user will note that the basis elements are in this particular case formed by cartesian products of *degenerated simplices*. In the list, an element like ``<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",
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"\n",
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"We may construct also the tensor product $\\mathrm{m23}\\otimes\\mathrm{m23}$ from the underlying chain complex of the simplicial set ``m23``. This tensor product is a new chain complex and we see that the basis in dimension 6 has only one element:"
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"We may construct also the tensor product $\\mathrm{m23}\\otimes\\mathrm{m23}$ from the underlying chain complex of the simplicial set ``m23``. The tensor product of two simplicial sets do not make sense, and Kenzo understands to work on the underlying chain complex ``m23``. This tensor product is a new chain complex and we see that the basis in dimension 6 has only one element:"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The Eilenberg-Zilber theorem is used to compute the homology groups of the cartesian product space: as chain complexes, $\\texttt{m23xm23}$ and ``t2m23`` have the same homology groups, but the computations using the tensor product are considerably faster."
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"The Eilenberg-Zilber theorem is used to compute the homology groups of the cartesian product space: as chain complexes, ``m23xm23`` and ``t2m23`` have the same homology groups, but the computations using the tensor product are considerably faster. The user can check Künneth's theorem."
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"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 ``os3`` a 3-simplex named ``D3``, the attaching map being described by the list of its faces in dimension 2."
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"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 ``os3`` a 3-simplex named ``D3``, the attaching map being described by the list of its faces in dimension 2. The faces 0 and 2 of D3 are pasted to L1, the faces 1 and 3 are collapsed on the base point; in this way the attaching map $(S^2 \\rightarrow \\Omega S^3)$ has degree 2."
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let us use the lisp function ``mapcar`` (one among the various iteration functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object ``L4``, this list is a \"Kan hat\"."
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"Let us use the lisp function ``mapcar`` (one among the various iteration functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object ``L4``, this list is a \"Kan horn\"."
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]
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{
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"outputs": [],
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"source": [
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"(def hat (mapcar #'(lambda (i) (face os3 i 4 l4)) '(1 2 3 4)))"
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"(def horn (mapcar #'(lambda (i) (face os3 i 4 l4)) '(1 2 3 4)))"
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The function ``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 ``L4``) is the same as the face 2 of ``L4``."
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"The function ``kfll`` tries to find a filling of this \"Kan horn\", and we see that the face 2 of the resulting simplex (which is very different from ``L4``) is the same as the face 2 of ``L4``."
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]
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},
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{
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@ -603,7 +607,7 @@
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},
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"outputs": [],
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"source": [
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"(def kan-simplex (kfll os3 0 4 hat))"
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"(def kan-simplex (kfll os3 0 4 horn))"
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]
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{
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@ -625,14 +629,14 @@
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},
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"outputs": [],
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"source": [
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"(second hat)"
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"(second horn)"
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let ${\\cal G}$ be a simplicial group 0-reduced. $\\Omega S^3$ is such a group. The software ``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."
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"Let ${\\cal G}$ be a simplicial group 0-reduced. $\\Omega S^3$ is such a group. The program ``Kenzo`` allows the construction of the universal bundle $\\mathcal{WG}$ and in particular of its base space $\\overline{\\mathcal{W}} \\mathcal{G}$, 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 $H_4$, which in principle should be isomorphic to $H_4 S^3 = 0$, is really null."
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let us end this short overview with an example of computation of homotopy groups. The method used in ``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",
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"Let us end this short overview with an example of computation of homotopy groups. The method used in ``Kenzo`` is the Whitehead tower. An extended version of the Kenzo program due to Ana Romero is more general, but in this present 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",
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"\n",
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"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:"
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]
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