From 7102d88cc12b1585952212d6b33dd5e06eb6b760 Mon Sep 17 00:00:00 2001 From: gheber Date: Fri, 4 Mar 2016 06:44:59 -0600 Subject: [PATCH] Another snippet of Chapter 7. --- examples/Chapter7.ipynb | 488 +++++++++++++++++++++++++++++++++++++++- 1 file changed, 487 insertions(+), 1 deletion(-) diff --git a/examples/Chapter7.ipynb b/examples/Chapter7.ipynb index f10cb8a..cc33aab 100644 --- a/examples/Chapter7.ipynb +++ b/examples/Chapter7.ipynb @@ -1498,7 +1498,493 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "### Spheres, Moore spaces and projectives spaces" + "### Spheres, Moore spaces and projectives spaces\n", + "\n", + "``sphere`` *n* *[Function]*\n", + "> Create a simplicial set, a model for the sphere of dimension $n$, ($n \\geq 0$). This is a typical example where the differential is known a priori to be null, so the ``:intr-bndr`` keyword parameter is set to function ``zero-pure-dffr``. This function generates a name ``S``*n* for the **unique** simplex of dimension $n$ whose faces are the degeneracies of the base point labelled \"``*``\".\n", + "\n", + "``sphere-wedge`` *dmns1 ... dmnsn* *[Function]*\n", + "> Create a simplicial set for the wedge of spheres. Here, the $dmns_i$ are integers, namely the dimensions of the spheres to be wedged. The differential is null. In the representation created by the software, the 0-simplex (base point) is labelled \"``*``\" and in dimension $p$ the simplexes are labelled ``S``*p*``-``*1*, ..., ``S``*p*``-``*s*, where *s* is the number of spheres of dimension $p$ in the wedge. (See the example).\n", + "\n", + "``moore`` *p n* *[Function]*\n", + "> Construct a simplicial set, a model for ${\\rm Moore}(\\mathbb{Z}/{p\\mathbb{Z}},n)$. The integers $p$ and $n$ must satisfy the conditions $p>1,\\ n> 2p-4$, otherwise the result is undefined. ${\\rm Moore}(\\mathbb{Z}/{p\\mathbb{Z}},n)$ is a space, whose the only non-null homology groups are $H_0=\\mathbb{Z}$ and $H_n=\\mathbb{Z}/{p\\mathbb{Z}}$. A Moore space has only three non-degenerate simplices, namely in dimension $0$, $n$ and $n+1$. A number $p$ of faces of the $(n+1)$-simplex are identified with the $n$-simplex, the others faces being contracted on the base point. In the representation created by the software, the 0-simplex (base point), the $n$-simplex and the $(n+1)$-simplex are respectively labelled \"``*``\", ``M``$n$ and ``N``$n'$, where $n'=n+1$.\n", + "\n", + "``R-proj-space &optional`` *k l* *[Function]*\n", + "> If $k=1$ or omitted, build a simplicial set model of $K(\\mathbb{Z}_2,1)= P^\\infty\\mathbb{R}$. In dimension $n$, this simplicial set has only one non-degenerate simplex, namely the integer $n$. \n", + "The faces of this non-degenerate simplex $n$ are given by the following formulas: $\\partial_0 n = \\partial_n n = n-1$ and for $i \\not= 0$ and $i \\not=n$, $\\partial_i n= \\eta_{i-1} (n-2)$. If $k >1$, build an analogous simplicial set but with no simplices in dimensions $1 \\leq m < k$. If in addition to $k$ the argument $l$, ($l \\geq k$), is provided, build an analogous simplicial set but with no simplices in dimensions $m \\geq l$.\n", + "\n", + "### Examples\n", + "\n", + "Let us define first, an auxiliary function ``show-structure`` with 2 arguments *ss* and *dmn*, to show the structure (i.e. generators and faces) of the simplicial set *ss*, from the dimension 0 up to the dimension *dmn* included.\n", + "\n", + "```lisp\n", + "(defun show-structure (ss dmn)\n", + " (dotimes (i (1+ dmn))\n", + " (format t \"~2%Dimension = ~D :\" i)\n", + " (case i\n", + " (0 (format t \"~2%~8TVertices : ~8T~A\" (basis ss 0)))\n", + " (otherwise\n", + " (dolist (s (basis ss i))\n", + " (format t \"~2%~8TSimplex : ~A~2%~16TFaces : ~A\"\n", + " s (mapcar #'(lambda (j) (face ss j i s))\n", + " ( 0 i)))))\n", + " )))\n", + "\n", + "```\n", + "\n", + "In these elementary examples, we show the structure of some simplicial sets built\n", + "from the functions above. Let us begin with $S^2$." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def s2 (sphere 2))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(bspn s2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Applying the function ``basis``, we see that the only non-null simplices are in dimension 0 and 2." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(dotimes (i 4) (print (basis s2 i)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In dimension 2, the 3 faces of the simplicial set are all the degeneracy $\\eta_0$ of the base point." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(mapcar #'(lambda(i)(face s2 i 2 's2)) '(0 1 2))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(face s2 5 7 (absm (dgop-ext-int '(5 3 1 0)) 's2))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(face s2 2 7 (absm (dgop-ext-int '(5 3 1 0)) 's2))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The differential of the simplex ``s2`` in dimension 2 is of course the null combination of degree 1:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(? s2 2 's2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now, let us see the stucture of $S^3$, using our auxiliary function ``show-structure``." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def s3 (sphere 3))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure s3 3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The space ${\\rm Moore}(2,1)$ is the projective plane:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def p2 (moore 2 1)) " + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure p2 2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "In the following example, note the identification of $p$ faces of the $(n+1)$-simplex with the $n$-simplex." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def sp2r (moore 2 2))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure sp2r 3)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let us see an example of a wedge of spheres:" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def w (sphere-wedge 3 2 3))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure w 5)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(cmpr w 's3-1 's3-2)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(face w 2 3 's3-1)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(? w 3 's3-2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let us show now some examples with the simplicial sets generated by the function ``R-proj-space``." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def p1 (R-proj-space))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(dotimes (i 7)(print(basis p1 i)))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure p1 5)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(dotimes (i 5)(chcm-homology p1 i))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def p2 (R-proj-space 2))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure p2 5)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(dotimes (i 5)(chcm-homology p2 i))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def pr3 (R-proj-space 3))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure pr3 4)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(dotimes (i 7) (print (? pr3 i i)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Let us use now the parameter *l*, allowing the truncation in the upper dimensions." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def p12 (R-proj-space 1 2))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure p12 2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Setting $k=l$ does not creates something amazing!" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def p22 (R-proj-space 2 2))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure p22 2)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(def p47 (R-proj-space 4 7))" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "(show-structure p47 8)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Cartesian product of simplicials sets" ] } ],