From 557080c05ff666ee352a348616aac1741041715c Mon Sep 17 00:00:00 2001 From: gheber Date: Sun, 15 May 2016 18:02:40 -0500 Subject: [PATCH] Added Francis' changes. --- examples/14_Disk_pasting_and_suspension.ipynb | 411 +++++++++--------- 1 file changed, 205 insertions(+), 206 deletions(-) diff --git a/examples/14_Disk_pasting_and_suspension.ipynb b/examples/14_Disk_pasting_and_suspension.ipynb index a287a2d..a3bb9c6 100644 --- a/examples/14_Disk_pasting_and_suspension.ipynb +++ b/examples/14_Disk_pasting_and_suspension.ipynb @@ -9,26 +9,25 @@ "## 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", + "*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 $X \\coprod Y$, the disjoint union of $X$ and $Y$, 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", + "The **Kenzo** program implements the previous technique of space construction, via simplicial sets. 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", + "> 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 these possible *gmsm*s into *absm*s in the *faces* argument. The argument *new* allows the user to name this new added simplex by a symbol. 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", + "$$C'_p=C_p,\\quad \\forall p \\not= dmns,$$\n", + "$$C'_{dmns}= C_{dmns} \\oplus \\mathbb{Z}\\, new.$$\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", @@ -52,9 +51,9 @@ "output_type": "stream", "text": [ "\n", - "The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.\n", + "The object is a STANDARD-OBJECT of type SIMPLICIAL-SET.\n", "0. CMPR: #\n", - "1. BASIS: #\n", + "1. BASIS: #\n", "2. BSGN: *\n", "3. DFFR: [K2 Morphism (degree -1): K1 -> K1]\n", "4. GRMD: [K1 Simplicial-Set]\n", @@ -62,7 +61,7 @@ "6. IDNM: 1\n", "7. ORGN: (BUILD-FINITE-SS (*))\n", "8. CPRD: [K5 Morphism (degree 0): K1 -> K3]\n", - "9. FACE: #\n", + "9. FACE: #\n", "\n", "> \n", "Checking the 0-simplices..." @@ -240,7 +239,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "* \n", "End of computing.\n", @@ -250,7 +249,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "D1 \n", "End of computing.\n", @@ -266,14 +265,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "D1 \n", "End of computing.\n", @@ -293,7 +292,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", @@ -312,7 +311,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -335,7 +334,7 @@ "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", + "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 in the same process ($d_0$ 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", @@ -439,7 +438,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "* \n", "End of computing.\n", @@ -449,7 +448,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "D1 \n", "End of computing.\n", @@ -465,14 +464,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "D1 \n", "End of computing.\n", @@ -482,7 +481,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "D2 \n", "End of computing.\n", @@ -498,14 +497,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "D2 \n", "End of computing.\n", @@ -523,7 +522,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -635,7 +634,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "* \n", "End of computing.\n", @@ -645,7 +644,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "D1 \n", "End of computing.\n", @@ -661,14 +660,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "D1 \n", "End of computing.\n", @@ -678,7 +677,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "D2 \n", "End of computing.\n", @@ -694,14 +693,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "D2 \n", "End of computing.\n", @@ -711,7 +710,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "D3 \n", "End of computing.\n", @@ -725,14 +724,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "D3 \n", "End of computing.\n", @@ -752,7 +751,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -814,7 +813,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "* \n", "End of computing.\n", @@ -824,7 +823,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "D1 \n", "End of computing.\n", @@ -840,14 +839,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "D1 \n", "End of computing.\n", @@ -857,7 +856,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "D2 \n", "End of computing.\n", @@ -873,14 +872,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "D2 \n", "End of computing.\n", @@ -890,7 +889,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "D3 \n", "End of computing.\n", @@ -904,14 +903,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "D3 \n", "End of computing.\n", @@ -921,7 +920,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "D4 \n", "End of computing.\n", @@ -937,14 +936,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "D4 \n", "End of computing.\n", @@ -962,7 +961,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -1055,7 +1054,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", ":Z-GNRT \n", "End of computing.\n", @@ -1065,7 +1064,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "GEN-1 \n", "End of computing.\n", @@ -1114,7 +1113,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "GEN-1 \n", "End of computing.\n", @@ -1191,7 +1190,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", ":Z-GNRT \n", "End of computing.\n", @@ -1201,7 +1200,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "GEN-1 \n", "End of computing.\n", @@ -1279,7 +1278,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", ":Z-GNRT \n", "End of computing.\n", @@ -1289,13 +1288,13 @@ "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\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, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/2 (dimension 1) :\n", "GEN-1 \n", "End of computing.\n", @@ -1339,13 +1338,13 @@ "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\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, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/2 (dimension 1) :\n", "GEN-1 \n", "End of computing.\n", @@ -1538,7 +1537,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<> \n", "End of computing.\n", @@ -1548,7 +1547,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", " \n", "End of computing.\n", @@ -1564,7 +1563,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -1599,7 +1598,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", " \n", "End of computing.\n", @@ -1609,7 +1608,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "<> \n", "End of computing.\n", @@ -1623,7 +1622,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -1687,85 +1686,85 @@ "Rank of the source-module : 14.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/14 (dimension 5) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/14 (dimension 5) :\n", "<][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 3/14 (dimension 5) :\n", "<>][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 4/14 (dimension 5) :\n", "<>][1 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 5/14 (dimension 5) :\n", "<>][2 ][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 6/14 (dimension 5) :\n", "<>][3 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 7/14 (dimension 5) :\n", "<][1 <>][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 8/14 (dimension 5) :\n", "<][2 ][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 9/14 (dimension 5) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 10/14 (dimension 5) :\n", "<>][1 <>][1 <>][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 11/14 (dimension 5) :\n", "<>][1 <>][2 ][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 12/14 (dimension 5) :\n", "<>][2 ][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 13/14 (dimension 5) :\n", "<][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 14/14 (dimension 5) :\n", "<>][1 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", @@ -1775,157 +1774,157 @@ "Rank of the source-module : 26.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/26 (dimension 6) :\n", "<>][5 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/26 (dimension 6) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 3/26 (dimension 6) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 4/26 (dimension 6) :\n", "<>][2 ][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 5/26 (dimension 6) :\n", "<>][3 <>][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 6/26 (dimension 6) :\n", "<][1 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 7/26 (dimension 6) :\n", "<][2 ][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 8/26 (dimension 6) :\n", "<][3 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 9/26 (dimension 6) :\n", "<>][1 <>][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 10/26 (dimension 6) :\n", "<>][2 ][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 11/26 (dimension 6) :\n", "<>][1 <>][1 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 12/26 (dimension 6) :\n", "<>][1 <>][2 ][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 13/26 (dimension 6) :\n", "<>][1 <>][3 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 14/26 (dimension 6) :\n", "<>][2 ][1 <>][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 15/26 (dimension 6) :\n", "<>][2 ][2 ][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 16/26 (dimension 6) :\n", "<>][3 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 17/26 (dimension 6) :\n", "<][1 <>][1 <>][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 18/26 (dimension 6) :\n", "<][1 <>][2 ][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 19/26 (dimension 6) :\n", "<][2 ][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 20/26 (dimension 6) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 21/26 (dimension 6) :\n", "<>][1 <>][1 <>][1 <>][2 ]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 22/26 (dimension 6) :\n", "<>][1 <>][1 <>][2 ][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 23/26 (dimension 6) :\n", "<>][1 <>][2 ][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 24/26 (dimension 6) :\n", "<>][2 ][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 25/26 (dimension 6) :\n", "<][1 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 26/26 (dimension 6) :\n", "<>][1 <>][1 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", @@ -1953,7 +1952,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -1993,9 +1992,9 @@ "output_type": "stream", "text": [ "\n", - "The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.\n", + "The object is a STANDARD-OBJECT of type SIMPLICIAL-SET.\n", "0. CMPR: #\n", - "1. BASIS: #\n", + "1. BASIS: #\n", "2. BSGN: *\n", "3. DFFR: [K2 Morphism (degree -1): K1 -> K1]\n", "4. GRMD: [K1 Simplicial-Set]\n", @@ -2003,7 +2002,7 @@ "6. IDNM: 1\n", "7. ORGN: (BUILD-FINITE-SS (*))\n", "8. CPRD: [K5 Morphism (degree 0): K1 -> K3]\n", - "9. FACE: #\n", + "9. FACE: #\n", "\n", "> \n", "Checking the 0-simplices..." @@ -2614,7 +2613,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Let us apply now the function `suspension` to objects built from $S^2$, namely $\\Omega^1(S^2)$ and $\\Omega^2(S^2)$ and let us have a look upon the homology groups. For $S^2$, we know that the only non-null homology groups are in dimension 0 and 2:" + "Let us apply now the function `suspension` to objects built from $S^2$, namely $\\Omega(S^2)$, $S\\Omega(S^2)$, and finally $\\Omega S \\Omega(S^2)$ and let us have a look upon the homology groups. For $S^2$, we know that the only non-null homology groups are in dimension 0 and 2:" ] }, { @@ -2633,7 +2632,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "* \n", "End of computing.\n", @@ -2653,7 +2652,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", @@ -2664,7 +2663,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "S2 \n", "End of computing.\n", @@ -2678,14 +2677,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "S2 \n", "End of computing.\n", @@ -2705,7 +2704,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", @@ -2724,7 +2723,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", @@ -2743,7 +2742,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -2807,7 +2806,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", ":S-BSGN \n", "End of computing.\n", @@ -2827,7 +2826,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", @@ -2846,7 +2845,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", @@ -2857,7 +2856,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "S2 \n", "End of computing.\n", @@ -2871,14 +2870,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "S2 \n", "End of computing.\n", @@ -2898,7 +2897,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", @@ -2917,7 +2916,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -3003,7 +3002,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "<> \n", "End of computing.\n", @@ -3013,7 +3012,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<> \n", "End of computing.\n", @@ -3029,14 +3028,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<> \n", "End of computing.\n", @@ -3046,7 +3045,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<> \n", "End of computing.\n", @@ -3062,14 +3061,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<> \n", "End of computing.\n", @@ -3079,7 +3078,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "<> \n", "End of computing.\n", @@ -3095,14 +3094,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "<> \n", "End of computing.\n", @@ -3112,7 +3111,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "<> \n", "End of computing.\n", @@ -3128,14 +3127,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "<> \n", "End of computing.\n", @@ -3145,7 +3144,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", "<> \n", "End of computing.\n", @@ -3161,7 +3160,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -3225,7 +3224,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", ":S-BSGN \n", "End of computing.\n", @@ -3245,7 +3244,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", @@ -3256,7 +3255,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<> \n", "End of computing.\n", @@ -3270,14 +3269,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", "<> \n", "End of computing.\n", @@ -3287,7 +3286,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "<> \n", "End of computing.\n", @@ -3303,14 +3302,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "<> \n", "End of computing.\n", @@ -3320,7 +3319,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "<> \n", "End of computing.\n", @@ -3336,14 +3335,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", "<> \n", "End of computing.\n", @@ -3353,7 +3352,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", "<> \n", "End of computing.\n", @@ -3369,7 +3368,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] }, @@ -3433,7 +3432,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "<> \n", "End of computing.\n", @@ -3443,7 +3442,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<>]>> \n", "End of computing.\n", @@ -3459,14 +3458,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", "<>]>> \n", "End of computing.\n", @@ -3476,13 +3475,13 @@ "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/2 (dimension 2) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/2 (dimension 2) :\n", "<>][1 <>]>> \n", "End of computing.\n", @@ -3498,20 +3497,20 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/2 (dimension 2) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/2 (dimension 2) :\n", "<>][1 <>]>> \n", "End of computing.\n", @@ -3521,25 +3520,25 @@ "Rank of the source-module : 4.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/4 (dimension 3) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/4 (dimension 3) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 3/4 (dimension 3) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 4/4 (dimension 3) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", @@ -3557,32 +3556,32 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 4.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/4 (dimension 3) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/4 (dimension 3) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 3/4 (dimension 3) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 4/4 (dimension 3) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", @@ -3592,49 +3591,49 @@ "Rank of the source-module : 8.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/8 (dimension 4) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/8 (dimension 4) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 3/8 (dimension 4) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 4/8 (dimension 4) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 5/8 (dimension 4) :\n", "<>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 6/8 (dimension 4) :\n", "<>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 7/8 (dimension 4) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 8/8 (dimension 4) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", @@ -3656,56 +3655,56 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 8.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/8 (dimension 4) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/8 (dimension 4) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 3/8 (dimension 4) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 4/8 (dimension 4) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 5/8 (dimension 4) :\n", "<>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 6/8 (dimension 4) :\n", "<>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 7/8 (dimension 4) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 8/8 (dimension 4) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", @@ -3715,97 +3714,97 @@ "Rank of the source-module : 16.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 1/16 (dimension 5) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 2/16 (dimension 5) :\n", "<>][4 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 3/16 (dimension 5) :\n", "<>][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 4/16 (dimension 5) :\n", "<>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 5/16 (dimension 5) :\n", "<>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 6/16 (dimension 5) :\n", "<>][1 <>][3 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 7/16 (dimension 5) :\n", "<>][2 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 8/16 (dimension 5) :\n", "<>][3 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 9/16 (dimension 5) :\n", "<>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 10/16 (dimension 5) :\n", "<>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 11/16 (dimension 5) :\n", "<>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 12/16 (dimension 5) :\n", "<>][1 <>][1 <>][2 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 13/16 (dimension 5) :\n", "<>][1 <>][2 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 14/16 (dimension 5) :\n", "<>][2 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 15/16 (dimension 5) :\n", "<>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "Computing the boundary of the generator 16/16 (dimension 5) :\n", "<>][1 <>][1 <>][1 <>][1 <>]>> \n", "End of computing.\n", @@ -3835,7 +3834,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-15, 22h 19m 56s.\n", + ";; Clock -> 2016-05-15, 22h 59m 34s.\n", "\n" ] },