From 74ffaf09570bf6c979a5dfba2ba64b84e48630bd Mon Sep 17 00:00:00 2001 From: Gerd Heber Date: Thu, 15 Sep 2016 10:29:21 -0500 Subject: [PATCH] Added Francis' changes. --- examples/17_Classifying_Spaces.ipynb | 108 +++++++++++++-------------- 1 file changed, 54 insertions(+), 54 deletions(-) diff --git a/examples/17_Classifying_Spaces.ipynb b/examples/17_Classifying_Spaces.ipynb index e04bf9b..549a07f 100644 --- a/examples/17_Classifying_Spaces.ipynb +++ b/examples/17_Classifying_Spaces.ipynb @@ -8,7 +8,7 @@ "\n", "## Introduction\n", "\n", - "Let ${\\cal G}$ be a simplicial group $0$-reduced. The software `Kenzo` allows the construction of the universal bundle $\\bar{\\cal W}{\\cal G}$, called the classifying space of ${\\cal G}$. It is known that this is the base space of a principal bundle such that the total space is contractible. We know also that if ${\\cal G}$ is an Abelian simplicial group then $\\bar{\\cal W}{\\cal G}$ is also an Abelian simplicial group, otherwise if ${\\cal G}$ is non-Abelian, then $\\bar{\\cal W}{\\cal G}$ is only a simplicial set. In dimension $n$ the simplices of the classifying space are elements of the product:\n", + "Let ${\\cal G}$ be a simplicial group $0$-reduced. The program `Kenzo` allows the construction of the universal bundle $\\bar{\\cal W}{\\cal G}$, called the classifying space of ${\\cal G}$. It is known that this is the base space of a principal bundle such that the total space is contractible. We know also that if ${\\cal G}$ is an Abelian simplicial group then $\\bar{\\cal W}{\\cal G}$ is also an Abelian simplicial group, otherwise if ${\\cal G}$ is non-Abelian, then $\\bar{\\cal W}{\\cal G}$ is only a simplicial set. In dimension $n$ the simplices of the classifying space are elements of the product:\n", "$${\\cal G}_{n-1}\\times {\\cal G}_{n-2}\\times\\cdots\\times{\\cal G}_0,$$\n", "where ${\\cal G}_k$ is the set of the $k$-simplices of the given simplicial group.\n", "\n", @@ -17,11 +17,11 @@ "where the $g_i$'s are elements of ${\\cal G}_i$, possibly *degenerated*.\n", "In `Kenzo` a gbar is represented externally under the form:\n", "\n", - "`<<`$[\\eta g]_{n-2}$ `> ... <` $[\\eta g]_1$ `> < -` $g_0$`> >>`\n", + "`<<` $[\\eta_{n-2}, g_{n-2}]$ `> ... <` $[\\eta_1, g_1]$ `> < -` $g_0$`> >>`\n", "\n", - "where a term such as $[\\eta g]_i$ is a simplex of ${\\cal G}_i$. Let us recall that our group ${\\cal G}$ is assumed 0-reduced; as a consequence, the $g_0$ compound of a gbar is necessarily the identity element of ${\\cal G}$.\n", + "where a term such as $[\\eta_i, g_i]$ is a simplex of ${\\cal G}_i$. Let us recall that our group ${\\cal G}$ is assumed 0-reduced; as a consequence, the $g_0$ compound of a gbar is necessarily the identity element of ${\\cal G}$.\n", "\n", - "In the symbolic expression $[\\eta g]_i$, the operator $\\eta$ represents a sequence (possibly the void one) of degeneracy operators applied coherently upon some element of ${\\cal G}_k$, for $k\\leq i$. For instance $[\\eta g]_1$ can be $\\eta_0 g_0$, i.e. the 0-degeneracy of the base point and $[\\eta g]_2$ can be can be $\\eta_1\\eta_0 g_0$. To be more explicit, let us take the Abelian simplicial group $K(\\mathbb{Z}_2, 1)$ (see the simplicial groups chapter), then some elements of the basis in dimension 4 of its classifying space are, for instance:\n", + "In the symbolic expression $[\\eta_i, g_i]$, the operator $\\eta$ represents a sequence (possibly the void one) of degeneracy operators applied coherently upon some element of ${\\cal G}_k$, for $k\\leq i$. For instance $[\\eta_1, g_1]$ can be the pair $((0),g_0)$, internally implemented as an `absm` object, i.e. the 0-degeneracy of the base point and $[\\eta_2, g_2]$ can be $((1 0), g_0)$, that is, the only possible 2-degeneracy of the base point $g_0$ of $mathcal{G}$. In particular, the dimension of the \"geometric\" simplex $g_i$ is $i - \\ell$ with $\\ell$ the length of the integer sequence $\\eta_i$. To be more explicit, let us take the Abelian simplicial group $K(\\mathbb{Z}_2, 1)$ (see the simplicial groups chapter), then some elements of the basis in dimension 4 of its classifying space are, for instance:\n", "```\n", "<<- 2><- 1><- 0>>>\n", "<<1-0 0><0 0><- 0>>>\n", @@ -38,17 +38,17 @@ "\n", "Let us denote a gbar under the following generic form:\n", "$$\n", - "g_{bar}=(g_{n-1}, g_{n-2}, \\ldots, {\\bf g_j}, \\ldots, g_1, g_0).\n", + "g_{bar}=(g_{n-1}, g_{n-2}, \\ldots, g_j, \\ldots, g_1, g_0).\n", "$$\n", "In this symbolism, $g_0$ is also the base point of the initial simplicial group. The rule for the face operators is the following:\n", "- If $j=n$, $\\delta_n g_{bar}= (g_{n-2}, g_{n-3}, \\ldots, g_1, g_0).$\n", - "- If $j \\not=n$, $\\delta_j g_{bar}= (\\delta_j g_{n-1}, \\ldots, {\\bf \\delta_j g_{j+1}, g_{j-1}.\\delta_j g_j, g_{j-1}}, \\ldots, g_1, g_0),$ where in the term $\\bf g_{j-1}.\\delta_j g_j$, the dot represents the group composition.\n", + "- If $j \\not=n$, $\\delta_j g_{bar}= (\\delta_j g_{n-1}, \\ldots, \\delta_j g_{j+1}, g_{j-1}.\\delta_j g_j, g_{j-1}, \\ldots, g_1, g_0),$ where in the term $g_{j-1}.\\delta_j g_j$, the dot represents the group composition.\n", "- If $j=0$, $\\delta_0 g_{bar}= (\\delta_0 g_{n-1}, \\ldots, \\delta_0 g_1).$\n", "\n", "The rule for the degeneracy operators is:\n", "\n", "$$\n", - "\\eta_j g_{bar}= (\\eta_j g_{n-1}, \\ldots, {\\bf \\eta_j g_{j+1}, \\eta_{j-1}\\eta_{j-2}\\ldots\\eta_1\\eta_0 g_0, g_{j-1}}, \\ldots, g_1, g_0).\n", + "\\eta_j g_{bar}= (\\eta_j g_{n-1}, \\ldots, \\eta_j g_{j+1}, \\eta_{j-1}\\eta_{j-2}\\ldots\\eta_1\\eta_0 g_0, g_{j-1}, \\ldots, g_1, g_0).\n", "$$\n", "\n", "If ${\\cal G}$ is Abelian then $\\bar{\\cal W}{\\cal G}$ is also Abelian and its composition law is simply:\n", @@ -57,7 +57,7 @@ "(g_{n-1}, \\ldots, g_1, g_0).(g'_{n-1}, g'_1, g'_0)=(g_{n-1}.g'_{n-1}, \\ldots, g_1.g'_1, g_0.g'_0).\n", "$$\n", "\n", - "It is well known that if the initial group is not Abelian then, in general, the previous composition law is not compatible with the simplicial structure.\n", + "If the initial group is not Abelian then, in general, the previous composition law is not compatible with the simplicial structure.\n", "\n", "## The functions for the classifying spaces\n", "\n", @@ -223,7 +223,7 @@ { "data": { "text/plain": [ - "#" + "#" ] }, "execution_count": 5, @@ -599,7 +599,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "As $K(\\mathbb{Z}, 1)$ and $K(\\mathbb{Z_2}, 1)$ are Abelian simplicial groups, we may iterate the classifying space construction and retrieve some known results." + "As $K(\\mathbb{Z}, 1)$ and $K(\\mathbb{Z}_2, 1)$ are Abelian simplicial groups, we may iterate the classifying space construction and retrieve some known results." ] }, { @@ -670,7 +670,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "<> \n", "End of computing.\n", @@ -690,7 +690,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", @@ -709,7 +709,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", @@ -720,7 +720,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "<>]>> \n", "End of computing.\n", @@ -734,14 +734,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", "<>]>> \n", "End of computing.\n", @@ -761,7 +761,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", @@ -772,7 +772,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", "<>]>> \n", "End of computing.\n", @@ -786,14 +786,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", "<>]>> \n", "End of computing.\n", @@ -803,7 +803,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 6) :\n", "<>][3 <>]>> \n", "End of computing.\n", @@ -819,14 +819,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 6.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 6) :\n", "<>][3 <>]>> \n", "End of computing.\n", @@ -836,7 +836,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 7) :\n", "<>]>> \n", "End of computing.\n", @@ -850,14 +850,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 7.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 7) :\n", "<>]>> \n", "End of computing.\n", @@ -867,13 +867,13 @@ "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/2 (dimension 8) :\n", "<>][5 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 2/2 (dimension 8) :\n", "<>][3 <>]>> \n", "End of computing.\n", @@ -889,20 +889,20 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 8.\n", "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/2 (dimension 8) :\n", "<>][5 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 2/2 (dimension 8) :\n", "<>][3 <>]>> \n", "End of computing.\n", @@ -912,13 +912,13 @@ "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/2 (dimension 9) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 2/2 (dimension 9) :\n", "<>][3 <>][3 <>]>> \n", "End of computing.\n", @@ -934,20 +934,20 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 9.\n", "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/2 (dimension 9) :\n", "<>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 2/2 (dimension 9) :\n", "<>][3 <>][3 <>]>> \n", "End of computing.\n", @@ -957,19 +957,19 @@ "Rank of the source-module : 3.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/3 (dimension 10) :\n", "<>][7 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 2/3 (dimension 10) :\n", "<>][5 <>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 3/3 (dimension 10) :\n", "<>][3 <>]>> \n", "End of computing.\n", @@ -985,7 +985,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n" ] }, @@ -1042,7 +1042,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", "<> \n", "End of computing.\n", @@ -1062,7 +1062,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", @@ -1081,7 +1081,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", @@ -1100,7 +1100,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", @@ -1119,7 +1119,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", @@ -1130,7 +1130,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", "<>]>>]>>]>> \n", "End of computing.\n", @@ -1144,14 +1144,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", "<>]>>]>>]>> \n", "End of computing.\n", @@ -1161,7 +1161,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 6) :\n", "<>]>>]>>]>> \n", "End of computing.\n", @@ -1177,14 +1177,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n", "\n", "Computing boundary-matrix in dimension 6.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/1 (dimension 6) :\n", "<>]>>]>>]>> \n", "End of computing.\n", @@ -1194,13 +1194,13 @@ "Rank of the source-module : 2.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 1/2 (dimension 7) :\n", "<>]>>]>>]>> \n", "End of computing.\n", "\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "Computing the boundary of the generator 2/2 (dimension 7) :\n", "<>]>>]>>]>> \n", "End of computing.\n", @@ -1214,7 +1214,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-03-19, 17h 57m 16s.\n", + ";; Clock -> 2016-09-15, 15h 27m 10s.\n", "\n" ] },