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Gerd Heber 2016-09-15 10:29:21 -05:00
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examples/17_Classifying_Spaces.ipynb
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@ -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",
"`<<GBar <` $[\\eta g]_{n-1}$`><`$[\\eta g]_{n-2}$ `> ... <` $[\\eta g]_1$ `> < -` $g_0$`> >>`\n",
"`<<GBar <` $[\\eta_{n-1}, g_{n-1}]$ `><` $[\\eta_{n-2}, g_{n-2}]$ `> ... <` $[\\eta_1, g_1]$ `> < -` $g_0$`> >>`\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",
"<<GBar<- 3><- 2><- 1><- 0>>>\n",
"<<GBar<- 3><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": [
"#<CLOSURE (FLET CAT::RSLT :IN CLASSIFYING-SPACE-FACE) {10054A53CB}>"
"#<CLOSURE (FLET CAT::RSLT :IN CLASSIFYING-SPACE-FACE) {1002FB5DBB}>"
]
},
"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",
"<<Abar>> \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",
"<<Abar[3 <<Abar[2 S1]>>]>> \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",
"<<Abar[3 <<Abar[2 S1]>>]>> \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",
"<<Abar[5 <<Abar[2 S1][2 S1]>>]>> \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",
"<<Abar[5 <<Abar[2 S1][2 S1]>>]>> \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",
"<<Abar[3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>]>> \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",
"<<Abar[3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>]>> \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",
"<<Abar[7 <<Abar[2 S1][2 S1][2 S1]>>]>> \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",
"<<Abar[7 <<Abar[2 S1][2 S1][2 S1]>>]>> \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",
"<<Abar[3 <<Abar[2 S1]>>][5 <<Abar[2 S1][2 S1]>>]>> \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",
"<<Abar[5 <<Abar[2 S1][2 S1]>>][3 <<Abar[2 S1]>>]>> \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",
"<<Abar[3 <<Abar[2 S1]>>][5 <<Abar[2 S1][2 S1]>>]>> \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",
"<<Abar[5 <<Abar[2 S1][2 S1]>>][3 <<Abar[2 S1]>>]>> \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",
"<<Abar[9 <<Abar[2 S1][2 S1][2 S1][2 S1]>>]>> \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",
"<<Abar[3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>]>> \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",
"<<Abar[9 <<Abar[2 S1][2 S1][2 S1][2 S1]>>]>> \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",
"<<Abar[3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>][3 <<Abar[2 S1]>>]>> \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",
"<<Abar[3 <<Abar[2 S1]>>][7 <<Abar[2 S1][2 S1][2 S1]>>]>> \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",
"<<Abar[5 <<Abar[2 S1][2 S1]>>][5 <<Abar[2 S1][2 S1]>>]>> \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",
"<<Abar[7 <<Abar[2 S1][2 S1][2 S1]>>][3 <<Abar[2 S1]>>]>> \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",
"<<Abar>> \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",
"<<Abar[5 <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>>]>> \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",
"<<Abar[5 <<Abar[4 <<Abar[3 <<Abar[2 1]>>]>>]>>]>> \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",
"<<Abar[6 <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>>]>> \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",
"<<Abar[6 <<Abar[5 <<Abar[4 <<Abar[3 2]>>]>>]>>]>> \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",
"<<Abar[7 <<Abar[6 <<Abar[5 <<Abar[4 3]>>]>>]>>]>> \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",
"<<Abar[7 <<Abar[6 <<Abar[5 <<Abar[2 1][2 1]>>]>>]>>]>> \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"
]
},