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Another snippet of Chapter 7.

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examples/Chapter7.ipynb
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@ -1498,7 +1498,493 @@
"cell_type": "markdown", "cell_type": "markdown",
"metadata": {}, "metadata": {},
"source": [ "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",
" (<a-b> 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 $S^2$."
]
},
{
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"outputs": [],
"source": [
"(def s2 (sphere 2))"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
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},
"outputs": [],
"source": [
"(bspn s2)"
]
},
{
"cell_type": "markdown",
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"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,
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"collapsed": false
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"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
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"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"
] ]
} }
], ],