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362a97811f
kenzo/examples/Chapter14.ipynb
gheber 362a97811f First cut of Chapter 14.
2016-03-15 07:15:45 -05:00

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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Disk pasting and suspension\n",
"\n",
"## Introduction\n",
"\n",
"Let us recall the technique of space construction by *attaching maps* (**Marvin Greenberg** in\n",
"*Lectures on Algebraic Topology.* Benjamin Inc, 1967.). Let $A$ be a subspace of a space $X$ and a map $f$ of $A$ into a space $Y$. In the space $X \\coprod Y$, let us identify each $x$ in $A$ with its image $f(x)$ in $Y$. The quotient space $Z=X\\bigcup_f Y$ of the space $X\\coprod Y$ by the equivalence relation that the identification above determines, is called the *adjunction space* of the system $(X \\supset A), (Y\\supset f(A))$. The quotient map $g: X\\coprod Y \\rightarrow Z$ sends $Y$ onto a subspace of $Z$. \n",
"\n",
"In this section, we shall take as pair $(X,A)$ the pair $(D^n, S^{n-1})$, where $D^n$ is the unit disc of $\\mathbb{R}^n$ and $S^{n-1}$ is the sphere of dimension ${n-1}$, boundary of $D^n$. In this case, the space $Z=D^n\\bigcup_f Y$ is said to be obtained from $Y$ by *attaching an n-disc via* $f$, or *disk pasting*.\n",
"\n",
"## The functions for disk pasting\n",
"\n",
"The **Kenzo** program implements the previous technique of space construction, via simplicial sets.\n",
"We know that $D^n$ is homeomorphic to the standard simplex $\\Delta^n$. The method used by the implementor is the following:\n",
"\n",
"Starting from a simplicial set *ss* representing $Y$, build a new one having the same simplices as *ss* and in dimension $n$, a new added simplex described by the user. This description consists of:\n",
"- a name (a symbol) for this new generator,\n",
"- the list of all its $n+1$ faces.\n",
"\n",
"`disk-pasting` *smst dmns new faces* *[Function]*\n",
"> Construct, from the simplicial set smst, a new simplicial set by attaching the unit disc of dimension *dmns*. The argument *faces* is the list of $dmn+1$ simplices corresponding to the faces of the new simplex attached to *smst*. These simplices are, in principle, written as abstract simplices (*absm*), nevertheless it is possible to write them as geometric simplices (*gmsm*). In this case, the corresponding geometric simplices must have the correct dimension. The program will transform the list of *gmsm*'s into a list of *absm*'s. The argument *new* allows the user to name this new added simplex. The program verifies the coherence of the addition of the new simplex by calling internally the function `check-faces`.\n",
"\n",
"`chcm-disk-pasting` *chcm dmns new bndr* *[Function]*\n",
"> Construct, from the chain complex *chcm* (chain groups ${\\cal C}_*$), a new chain complex (chain groups ${\\cal C}_*'$), where:\n",
"$$C'_p=C_p,\\quad \\forall p \\not= dmn,$$\n",
"$$C'_{dmn}= C_{dmn} \\oplus \\mathbb{Z}\\, new.$$\n",
"The symbol `new` is the name given by the user to a new generator in dimension $dmns$ and *bndr* is a combination of degree $dmns-1$ representing the boundary of the generator *new*. Of course, one must have $d_{dmns-1}(bndr)=0.$\n",
"\n",
"`hmeq-disk-pasting` *hmeq dmns new bndr* `&key` *new-lbcc* *[Function]*\n",
"> Construct a homotopy equivalence by attaching the generator *new* of dimension *dmns* with boundary *bndr* to the left bottom chain complex of the homotopy equivalence *hmeq*. This modification of the left bottom chain complex is propagated along the whole new homotopy equivalence.\n",
"The key parameter *new-lbcc* is for internal use.\n",
"\n",
"### Example\n",
"\n",
"Let us begin by a trivial example corresponding to the well known following result: if $f$ maps $S^{n-1}$ onto a point $Y$, then by disk pasting we obtain a space homeomorphic to $S^n$. Here, the point $Y$ will be represented by the trivial simplicial set:"
]
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{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
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"outputs": [
{
"name": "stdout",
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"text": [
"\n",
"The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.\n",
"0. CMPR: #<FUNCTION S-CMPR>\n",
"1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {100507BF4B}>\n",
"2. BSGN: *\n",
"3. DFFR: [K2 Morphism (degree -1): K1 -> K1]\n",
"4. GRMD: [K1 Simplicial-Set]\n",
"5. EFHM: \"unbound\"\n",
"6. IDNM: 1\n",
"7. ORGN: (BUILD-FINITE-SS (*))\n",
"8. CPRD: [K5 Morphism (degree 0): K1 -> K3]\n",
"9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {100507BF6B}>\n",
"\n",
"> \n",
"Checking the 0-simplices..."
]
},
{
"data": {
"text/plain": [
"[K1 Simplicial-Set]"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def s0 (build-finite-ss '(*)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the identification process, the 1-simplices boundaries of the 2-simplex $\\Delta^2$ are applied on the 0--degeneracy of the base point $*$."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K6 Simplicial-Set]"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def s2 (disk-pasting s0 \n",
" 2 \n",
" 's2 \n",
" (list (absm 1 '*) (absm 1 '*) (absm 1 '*)) )) "
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"\n",
"Dimension = 0:\n",
"\n",
" Vertices : (*)\n",
"\n",
"Dimension = 1:\n",
"\n",
"Dimension = 2:\n",
"\n",
" Simplex : S2\n",
"\n",
" Faces : (<AbSm 0 *> <AbSm 0 *> <AbSm 0 *>)"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(show-structure s2 2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"More interesting is the construction of the successive projectives spaces $P^i\\mathbb{R}$. Let us begin by $P^1\\mathbb{R}$. We start from a point (the simplicial set `s0`) and we identify the two boundary points of $\\Delta^1$ to the base point of `s0`. The new simplex to be added to `s0` is of dimension 1 and its 2 faces are the base point itself. It is well known that the resulting corresponding space is homeomorphic to the circle $S^1$."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K11 Simplicial-Set]"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def p1r (disk-pasting s0 1 'd1 (list (absm 0 '*)(absm 0 '*))))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
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"name": "stdout",
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"text": [
"\n",
"\n",
"Dimension = 0:\n",
"\n",
" Vertices : (*)\n",
"\n",
"Dimension = 1:\n",
"\n",
" Simplex : D1\n",
"\n",
" Faces : (<AbSm - *> <AbSm - *>)"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(show-structure p1r 1)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"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 -> 2016-03-15, 11h 44m 29s.\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 -> 2016-03-15, 11h 44m 29s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"D1 \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 -> 2016-03-15, 11h 44m 29s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 44m 29s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"D1 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 0.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 1 :\n",
"\n",
"\n",
"Component Z\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 44m 29s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 0.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 3.\n",
"Rank of the source-module : 0.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 2 :\n",
"\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 44m 29s.\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(dotimes (i 3) (homology p1r i))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The projective spaces $P^n\\mathbb{R}$ of higher degree may be constructed by the following iterative rule. Suppose we have built a simplicial set corresponding to $P^{n-1}\\mathbb{R}$ and that we have named the additional simplex $d_{n-1}$. Suppose also that $d_{n-2}$ has been previously defined ($d_0$ is the base point). Then the simplicial set corresponding to $P^n\\mathbb{R}$ is obtained by pasting $\\Delta^n$ with the following description of the additional simplex: \n",
"\n",
"- the name of the new simplex is $d_n$,\n",
"- the faces of $d_n$ are:\n",
"$$\n",
"\\begin{aligned}\n",
"\\partial_0 d_n & = & d_{n-1},\\\\\n",
"\\partial_1 d_n & = & \\eta_0 d_{n-2},\\\\\n",
"\\partial_2 d_n & = & \\eta_1 d_{n-2},\\\\\n",
"\\cdots & = & \\cdots , \\\\\n",
"\\partial_{n-1} d_n & = & \\eta_{n-2} d_{n-2},\\\\\n",
"\\partial_n d_n & = & d_{n-1}.\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"So, applying this rule, we may construct a model for $P^i\\mathbb{R},\\, i=2,3,4$ and verify the well known results about their respective homology groups."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K31 Simplicial-Set]"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def p2r (disk-pasting p1r 2 'd2\n",
" (list (absm 0 'd1)\n",
" (absm 1 '*)\n",
" (absm 0 'd1)) ))"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"\n",
"Dimension = 0:\n",
"\n",
" Vertices : (*)\n",
"\n",
"Dimension = 1:\n",
"\n",
" Simplex : D1\n",
"\n",
" Faces : (<AbSm - *> <AbSm - *>)\n",
"\n",
"Dimension = 2:\n",
"\n",
" Simplex : D2\n",
"\n",
" Faces : (<AbSm - D1> <AbSm 0 *> <AbSm - D1>)"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(show-structure p2r 2)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"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 -> 2016-03-15, 11h 47m 15s.\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 -> 2016-03-15, 11h 47m 15s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"D1 \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 -> 2016-03-15, 11h 47m 15s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 47m 15s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"D1 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 47m 15s.\n",
"Computing the boundary of the generator 1/1 (dimension 2) :\n",
"D2 \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 -> 2016-03-15, 11h 47m 15s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 47m 15s.\n",
"Computing the boundary of the generator 1/1 (dimension 2) :\n",
"D2 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 3.\n",
"Rank of the source-module : 0.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 2 :\n",
"\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 47m 15s.\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(dotimes (i 3) (homology p2r i))"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K49 Simplicial-Set]"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def p3r (disk-pasting p2r 3 'd3\n",
" (list (absm 0 'd2) \n",
" (absm 1 'd1)\n",
" (absm 2 'd1)\n",
" (absm 0 'd2))))"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"\n",
"Dimension = 0:\n",
"\n",
" Vertices : (*)\n",
"\n",
"Dimension = 1:\n",
"\n",
" Simplex : D1\n",
"\n",
" Faces : (<AbSm - *> <AbSm - *>)\n",
"\n",
"Dimension = 2:\n",
"\n",
" Simplex : D2\n",
"\n",
" Faces : (<AbSm - D1> <AbSm 0 *> <AbSm - D1>)\n",
"\n",
"Dimension = 3:\n",
"\n",
" Simplex : D3\n",
"\n",
" Faces : (<AbSm - D2> <AbSm 0 D1> <AbSm 1 D1> <AbSm - D2>)"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(show-structure p3r 3)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"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 -> 2016-03-15, 11h 48m 3s.\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 -> 2016-03-15, 11h 48m 3s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"D1 \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 -> 2016-03-15, 11h 48m 3s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 3s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"D1 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 3s.\n",
"Computing the boundary of the generator 1/1 (dimension 2) :\n",
"D2 \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 -> 2016-03-15, 11h 48m 3s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 3s.\n",
"Computing the boundary of the generator 1/1 (dimension 2) :\n",
"D2 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 3.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 3s.\n",
"Computing the boundary of the generator 1/1 (dimension 3) :\n",
"D3 \n",
"End of computing.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 2 :\n",
"\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 3s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 3.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 3s.\n",
"Computing the boundary of the generator 1/1 (dimension 3) :\n",
"D3 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 4.\n",
"Rank of the source-module : 0.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 3 :\n",
"\n",
"\n",
"Component Z\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 3s.\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(dotimes (i 4)(homology p3r i))"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K67 Simplicial-Set]"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def p4r (disk-pasting p3r 4 'd4 \n",
" (list (absm 0 'd3)\n",
" (absm 1 'd2)\n",
" (absm 2 'd2) \n",
" (absm 4 'd2)\n",
" (absm 0 'd3))))"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"collapsed": false
},
"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 -> 2016-03-15, 11h 48m 48s.\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 -> 2016-03-15, 11h 48m 48s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"D1 \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 -> 2016-03-15, 11h 48m 48s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"D1 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"Computing the boundary of the generator 1/1 (dimension 2) :\n",
"D2 \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 -> 2016-03-15, 11h 48m 48s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"Computing the boundary of the generator 1/1 (dimension 2) :\n",
"D2 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 3.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"Computing the boundary of the generator 1/1 (dimension 3) :\n",
"D3 \n",
"End of computing.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 2 :\n",
"\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 3.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"Computing the boundary of the generator 1/1 (dimension 3) :\n",
"D3 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 4.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"Computing the boundary of the generator 1/1 (dimension 4) :\n",
"D4 \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 -> 2016-03-15, 11h 48m 48s.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 4.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"Computing the boundary of the generator 1/1 (dimension 4) :\n",
"D4 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 5.\n",
"Rank of the source-module : 0.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 4 :\n",
"\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 48m 48s.\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(dotimes (i 5) (homology p4r i))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us give now some examples with the function `chcm-disk-pasting`. We are going to work with the unit chain complex (obtained by the function `z-chcm`, see chapter 1), having a unique non null component, namely a $\\mathbb{Z}$-module of degree 0 generated by the unique generator `:Z-gnrt`."
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K85 Chain-Complex]"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def *ccz* (z-chcm))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Using the function `chcm-disk-pasting`, let us construct from `*ccz*` a new chain complex having in dimension 1 a unique new generator `gen-1` whose boundary is 5 times the unique generator in dimension 0. In this new chain complex the homology group in dimension 0 is $\\mathbb{Z}/5\\,\\mathbb{Z}$."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K87 Chain-Complex]"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def newcc-0 (chcm-disk-pasting *ccz* 1 'gen-1 (cmbn 0 5 :Z-gnrt)))"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {
"collapsed": false
},
"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 -> 2016-03-15, 11h 51m 43s.\n",
"Computing the boundary of the generator 1/1 (dimension 0) :\n",
":Z-GNRT \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 51m 43s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"GEN-1 \n",
"End of computing.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 0 :\n",
"\n",
"Component Z/5Z\n",
"\n"
]
},
{
"data": {
"text/plain": [
"(\n",
"----------------------------------------------------------------------{CMBN 0}\n",
"<1 * Z-GNRT>\n",
"------------------------------------------------------------------------------\n",
")"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(chcm-homology-gen newcc-0 0)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 51m 58s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"GEN-1 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 0.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 1 :\n",
"\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(chcm-homology-gen newcc-0 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us do the same with a generator whose boundary is 165 times the generator `:Z-gnrt`:"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K91 Chain-Complex]"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def newcc-1 (chcm-disk-pasting *ccz* 1 'gen-1 (cmbn 0 165 :Z-gnrt)))"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"collapsed": false
},
"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 -> 2016-03-15, 11h 52m 59s.\n",
"Computing the boundary of the generator 1/1 (dimension 0) :\n",
":Z-GNRT \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 52m 59s.\n",
"Computing the boundary of the generator 1/1 (dimension 1) :\n",
"GEN-1 \n",
"End of computing.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 0 :\n",
"\n",
"Component Z/165Z\n",
"\n"
]
},
{
"data": {
"text/plain": [
"(\n",
"----------------------------------------------------------------------{CMBN 0}\n",
"<1 * Z-GNRT>\n",
"------------------------------------------------------------------------------\n",
")"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(chcm-homology-gen newcc-1 0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now from the chain complex `newcc-1`, let us build a new one, `newcc-2`, by adding in dimension 1 a second generator whose boundary is 182 times the generator `:Z-gnrt`. The homology group in dimension 0 of this new chain complex is the null group, since 165 and 182 are relatively prime integers."
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K93 Chain-Complex]"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def newcc-2 (chcm-disk-pasting newcc-1 1 'gen-2 (cmbn 0 182 :Z-gnrt)))"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": false
},
"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 -> 2016-03-15, 11h 54m 35s.\n",
"Computing the boundary of the generator 1/1 (dimension 0) :\n",
":Z-GNRT \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 2.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 54m 35s.\n",
"Computing the boundary of the generator 1/2 (dimension 1) :\n",
"GEN-2 \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 54m 35s.\n",
"Computing the boundary of the generator 2/2 (dimension 1) :\n",
"GEN-1 \n",
"End of computing.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 0 :\n",
"\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(chcm-homology-gen newcc-2 0)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"Computing boundary-matrix in dimension 1.\n",
"Rank of the source-module : 2.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 54m 50s.\n",
"Computing the boundary of the generator 1/2 (dimension 1) :\n",
"GEN-2 \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 54m 50s.\n",
"Computing the boundary of the generator 2/2 (dimension 1) :\n",
"GEN-1 \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 0.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 1 :\n",
"\n",
"Component Z\n",
"\n"
]
},
{
"data": {
"text/plain": [
"(\n",
"----------------------------------------------------------------------{CMBN 1}\n",
"<-165 * GEN-2>\n",
"<182 * GEN-1>\n",
"------------------------------------------------------------------------------\n",
")"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(chcm-homology-gen newcc-2 1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As a second example, let us start from $\\Omega^1(S^3)$."
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K95 Simplicial-Set]"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def s3 (sphere 3))"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K100 Simplicial-Group]"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def os3 (loop-space s3))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the following instruction, we locate in the symbol `fund-simp` the canonical generator of $\\pi_2 (\\Omega^1S^3)$, that is the 2-simplex coming from the original sphere:"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop[S3]>>>"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def fund-simp (absm 0 (loop3 0 's3 1)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We need also the 2-degeneracy of the base point of the loop space:"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<AbSm 1-0 <<Loop>>>"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def null-simp (absm 3 +null-loop+))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We may build now a new object by pasting a disk $D^3$ as indicated by the following call:"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K112 Simplicial-Set]"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def dos3 (disk-pasting os3 3 '<D3> \n",
" (list fund-simp null-simp fund-simp null-simp)))"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"Computing boundary-matrix in dimension 2.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 57m 56s.\n",
"Computing the boundary of the generator 1/1 (dimension 2) :\n",
"<<AlLp[2 S3]>> \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 3.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 57m 56s.\n",
"Computing the boundary of the generator 1/1 (dimension 3) :\n",
"<D3> \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 -> 2016-03-15, 11h 57m 56s.\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(homology dos3 2)"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"Computing boundary-matrix in dimension 3.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 58m 8s.\n",
"Computing the boundary of the generator 1/1 (dimension 3) :\n",
"<D3> \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 4.\n",
"Rank of the source-module : 1.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 58m 8s.\n",
"Computing the boundary of the generator 1/1 (dimension 4) :\n",
"<<AlLp[2 S3][2 S3]>> \n",
"End of computing.\n",
"\n",
"\n",
"\n",
"\n",
"Homology in dimension 3 :\n",
"\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 58m 8s.\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(homology dos3 3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then, let us build the loop space of the object `dos3` and show the homology in dimension 5:"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K231 Simplicial-Group]"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def odos3 (loop-space dos3))"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"Computing boundary-matrix in dimension 5.\n",
"Rank of the source-module : 14.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 1/14 (dimension 5) :\n",
"<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 2/14 (dimension 5) :\n",
"<<AlLp[2 <D3>][3 <<AlLp[2 S3][2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 3/14 (dimension 5) :\n",
"<<AlLp[3 <<AlLp[2 S3][2 S3]>>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 4/14 (dimension 5) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 5/14 (dimension 5) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 6/14 (dimension 5) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 7/14 (dimension 5) :\n",
"<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 8/14 (dimension 5) :\n",
"<<AlLp[2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 9/14 (dimension 5) :\n",
"<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 10/14 (dimension 5) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 11/14 (dimension 5) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 12/14 (dimension 5) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 13/14 (dimension 5) :\n",
"<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 14/14 (dimension 5) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
"Computing boundary-matrix in dimension 6.\n",
"Rank of the source-module : 26.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 1/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][5 <<AlLp[2 S3][2 S3][2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 2/26 (dimension 6) :\n",
"<<AlLp[3 <<AlLp[2 S3][2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 3/26 (dimension 6) :\n",
"<<AlLp[5 <<AlLp[2 S3][2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 4/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][3 <<AlLp[2 S3][2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 5/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 6/26 (dimension 6) :\n",
"<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 7/26 (dimension 6) :\n",
"<<AlLp[2 <D3>][2 <D3>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 8/26 (dimension 6) :\n",
"<<AlLp[2 <D3>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 9/26 (dimension 6) :\n",
"<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 10/26 (dimension 6) :\n",
"<<AlLp[3 <<AlLp[2 S3][2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 11/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 12/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 13/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 14/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 15/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 16/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 17/26 (dimension 6) :\n",
"<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 18/26 (dimension 6) :\n",
"<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 19/26 (dimension 6) :\n",
"<<AlLp[2 <D3>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 20/26 (dimension 6) :\n",
"<<AlLp[3 <<AlLp[2 S3][2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 21/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 22/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 23/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 24/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 25/26 (dimension 6) :\n",
"<<AlLp[2 <D3>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \n",
"End of computing.\n",
"\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"Computing the boundary of the generator 26/26 (dimension 6) :\n",
"<<AlLp[1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>][1 <<AlLp[2 S3]>>]>> \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",
"Component Z/2Z\n",
"\n",
"Component Z/2Z\n",
"\n",
"Component Z\n",
"\n",
"\n",
"---done---\n",
"\n",
";; Clock -> 2016-03-15, 11h 59m 11s.\n",
"\n"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(homology odos3 5)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Searching Homology for objects created by disk pasting\n",
"\n",
"The comment list of an object created by disk pasting contains various informations as shown by the simple following example"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.\n",
"0. CMPR: #<FUNCTION S-CMPR>\n",
"1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {1007875F3B}>\n",
"2. BSGN: *\n",
"3. DFFR: [K2 Morphism (degree -1): K1 -> K1]\n",
"4. GRMD: [K1 Simplicial-Set]\n",
"5. EFHM: [K19 Homotopy-Equivalence K1 <= K1 => K1]\n",
"6. IDNM: 1\n",
"7. ORGN: (BUILD-FINITE-SS (*))\n",
"8. CPRD: [K5 Morphism (degree 0): K1 -> K3]\n",
"9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {1007875F7B}>\n",
"\n",
"> \n",
"Checking the 0-simplices..."
]
},
{
"data": {
"text/plain": [
"[K1 Simplicial-Set]"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def s0 (build-finite-ss '(*)))"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[K374 Simplicial-Set]"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(def s2 (disk-pasting s0 \n",
" 2 \n",
" 's2 \n",
" (list (absm 1 '*) (absm 1 '*) (absm 1 '*)) ))"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(DISK-PASTING [K1 Simplicial-Set] 2 S2 (<AbSm 0 *> <AbSm 0 *> <AbSm 0 *>))"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(orgn s2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In fact, the comment list contains the very arguments used to realize the attaching. The method retrieves those arguments, gets the homotopy equivalence value of the `efhm` slot of the old object -creating it if necessary - then build a new homotopy equivalence, by calling the function `hmeq-disk-pasting` *hmeq* which takes into account the attaching.\n",
"\n",
"```commonlisp\n",
"(DEFMETHOD SEARCH-EFHM (smst (orgn (eql 'disk-pasting)))\n",
" (declare (type simplicial-set smst))\n",
" (the homotopy-equivalence\n",
" (destructuring-bind (old-smst dmns new faces) (rest (orgn smst))\n",
" (declare\n",
" (type simplicial-set old-smst)\n",
" (fixnum dmns)\n",
" (symbol new)\n",
" (ignore faces))\n",
" (hmeq-disk-pasting \n",
" (efhm old-smst)\n",
" dmns new (? smst dmns new)\n",
" :new-lbcc smst))))\n",
"```\n",
"\n",
"## The functions for the suspension"
]
}
],
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