From 333df40667b41959ee55cf55425ef824911677cc Mon Sep 17 00:00:00 2001 From: Gerd Heber Date: Thu, 15 Sep 2016 10:49:07 -0500 Subject: [PATCH] Added Francis' changes. --- examples/18_Serre_spectral_sequence.ipynb | 130 +++++++++++----------- 1 file changed, 65 insertions(+), 65 deletions(-) diff --git a/examples/18_Serre_spectral_sequence.ipynb b/examples/18_Serre_spectral_sequence.ipynb index 05bec93..b711471 100644 --- a/examples/18_Serre_spectral_sequence.ipynb +++ b/examples/18_Serre_spectral_sequence.ipynb @@ -12,11 +12,11 @@ "\n", "## The topological problem\n", "\n", - "Let ${\\cal G}$ be a simplicial group (the fiber space), $B$, a 1-reduced simplicial set (the base space) and $\\tau$ (the fibration), a simplicial morphism of degree $-1$, $\\tau: B \\longrightarrow {\\cal G}$. We know, from a previous chapter (fibrations) that the software `Kenzo` knows how to build the total space of the fibration: $B \\times_\\tau {\\cal G}$, which has the same simplices as $B \\times {\\cal G}$ and the same face operators except the last one, given by:\n", + "Let ${\\cal G}$ be a simplicial group (the fiber space), $B$, a 1-reduced simplicial set (the base space) and $\\tau$ (the fibration), a simplicial morphism of degree $-1$, and $\\tau: B \\rightarrow \\mathcal{G}$ a fibration, that is, a twisting operator as defined in Chapter 12. In the Kenzo program, this object $\\tau$ is implemented as an object of class `simplicial-mrph` and \"degree\" $-1$ while a genuine simplicial morphism has degree 0. We know the program Kenzo knows how to build the total space of the fibration: $B \\times_\\tau {\\cal G}$, which has the same simplices as $B \\times {\\cal G}$ and the same face operators except the last one, given by:\n", "$$\n", "\\partial_n(b,g) = (\\partial_n b, \\tau(b).\\partial_n g), \\quad b\\in B, g \\in {\\cal G},\n", "$$\n", - "where the '`.`' denotes the group operation in ${\\cal G}$. This discrepancy between the respective face operators $\\partial_n$ induces a discrepancy between the differential operators. We suppose now, that the base and fiber spaces are objects of effective homology type, i.e. there exists homotopy equivalences\n", + "where the '`.`' denotes the group operation in ${\\cal G}$. This discrepancy between the respective face operators $\\partial_n$ induces a discrepancy between the differential operators. We suppose now, that the base and fiber spaces are objects of effective homology type, i.e. there exist homotopy equivalences\n", "$$\n", "\\begin{aligned}\n", " & \\qquad\\widehat{B} & & & & \\qquad\\widehat{\\cal G} & & \\cr\n", @@ -24,7 +24,7 @@ "B & & EB & & {\\cal G} & & EG \\cr\n", "\\end{aligned}\n", "$$\n", - "where the chain complexes $EB$ and $EG$ are effective. The problem consists in finding a homotopy equivalence between $B \\times_\\tau {\\cal G}$ and an effective chain complex noted $EB \\widetilde{\\otimes} EG$; the underlying graded module of $EB \\widetilde{\\otimes} EG$ is $EB \\otimes EG$ but the differential is twisted in a rather complicated way. In fact, the chain complex $EB \\widetilde{\\otimes} EG$ is nothing but the Hirsh complex (**Guy Hirsh.** `Sur les groupes d'homologie des espaces fibres`, Bulletin de la Societe Mathematique de Belgique, 1954, vol. 6, pp. 79-96.).\n", + "where the chain complexes $EB$ and $EG$ are effective. The problem consists in finding a homotopy equivalence between $B \\times_\\tau {\\cal G}$ and an effective chain complex noted $EB \\widetilde{\\otimes} EG$; the underlying graded module of $EB \\widetilde{\\otimes} EG$ is $EB \\otimes EG$ but the differential is twisted in a rather complicated way. In fact, the chain complex $EB \\widetilde{\\otimes} EG$ is nothing but the Hirsch complex (**Guy Hirsch.** `Sur les groupes d'homologie des espaces fibres`, Bulletin de la Societe Mathematique de Belgique, 1954, vol. 6, pp. 79-96.).\n", "\n", "### The Ronald Brown reduction\n", "\n", @@ -36,7 +36,7 @@ " {{\\cal C}_*(B) \\otimes {\\cal C}_*({\\cal G})} & &\\cr\n", "\\end{aligned}\n", "$$\n", - "But, in fact we are interested in the following one (the Ronald Brown reduction):\n", + "Using the *basic perturbation lemma*, we then obtain a new reduction between twisted objects:\n", "$$\n", "\\begin{aligned}\n", "{{\\cal C}_*(B \\times_\\tau {\\cal G})} & \\qquad\\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(B \\times_\\tau {\\cal G})} \\cr\n", @@ -44,7 +44,7 @@ " {{\\cal C}_*(B) \\otimes_t {\\cal C}_*({\\cal G})} & &\\cr\n", "\\end{aligned}\n", "$$\n", - "where the symbol $\\otimes_t$ represents a *twisted tensor product*, induced by the twisting operator $\\tau$; this twisting operator is nothing but the Shih twisting cochain (**Gugenheim.**, *On the chain complex of a fibration*. Illinois Journal of Mathematics, 1972, vol. 16, pp. 398-414.) determined by Szczarba (**R. H. Szczarba.** *The homology of twisted cartesian products*. Transactions of the American Mathematical Society, 1961, vol. 100, pp. 197-216.). Here, to obtain the Szczarba cochain, we apply the perturbation lemma (Ronnie Brown). In effect, writing \n", + "where the symbol $\\otimes_t$ represents a *twisted tensor product*, induced by the twisting operator $\\tau$; this twisting operator is nothing but the Shih twisting cochain (**Gugenheim.**, *On the chain complex of a fibration*. Illinois Journal of Mathematics, 1972, vol. 16, pp. 398-414.) determined by Szczarba (**R. H. Szczarba.** *The homology of twisted cartesian products*. Transactions of the American Mathematical Society, 1961, vol. 100, pp. 197-216.). In effect, writing \n", "$$ d_\\tau = d + (d_\\tau -d),$$\n", "one may consider that the differential $d_\\tau$ of $B \\times_\\tau {\\cal G}$ is the differential $d$ of $B \\times {\\cal G}$ modified by the perturbation $d_\\tau - d$. The perturbation morphism $d_\\tau -d$ is deduced from the difference between both $\\partial_n$ face operators. This being done, a perturbation may possibly be propagated along the Eilenberg-Zilber reduction (see the method `add` applied to a reduction) and the perturbation lemma gives also, as a bonus, the perturbation to be applied to the differential of $B \\otimes {\\cal G}$ to obtain the differential of $B \\otimes_t {\\cal G}$.\n", "\n", @@ -75,7 +75,7 @@ "B\\otimes_t {\\cal G} & & EB \\widetilde{\\otimes} EG \\cr\n", "\\end{aligned}\n", "$$\n", - "where $\\widehat{B}\\widetilde{\\otimes}\\widehat{\\cal G}$ and $EB \\widetilde{\\otimes} EG$ are appropriate strongly twisted tensor products determined by the perturbation lemma: this last homotopy equivalence is obtained by propagating over the previous one, the perturbation to be applied to the differential of $B \\otimes {\\cal G}$ to obtain the differential of $B \\otimes_t {\\cal G}$. The homotopy equivalence so obtained is called in `Kenzo` the *right Serre homotopy equivalence*.\n", + "where $\\widehat{B}\\widetilde{\\otimes}\\widehat{\\cal G}$ and $EB \\widetilde{\\otimes} EG$ are appropriate twisted tensor products determined by the perturbation lemma: this last homotopy equivalence is obtained by propagating over the previous one, the perturbation to be applied to the differential of $B \\otimes {\\cal G}$ to obtain the differential of $B \\otimes_t {\\cal G}$. The homotopy equivalence so obtained is called in `Kenzo` the *right Serre homotopy equivalence*.\n", "\n", "### Assembling the puzzle\n", "\n", @@ -304,7 +304,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", " \n", "End of computing.\n", @@ -314,7 +314,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", " \n", "End of computing.\n", @@ -330,14 +330,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", " \n", "End of computing.\n", @@ -347,7 +347,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", " \n", "End of computing.\n", @@ -363,14 +363,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", " \n", "End of computing.\n", @@ -380,7 +380,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", " \n", "End of computing.\n", @@ -394,14 +394,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", " \n", "End of computing.\n", @@ -421,7 +421,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", @@ -440,7 +440,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", @@ -459,7 +459,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 6.\n", @@ -478,7 +478,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 7.\n", @@ -497,7 +497,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 8.\n", @@ -516,7 +516,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 9.\n", @@ -535,7 +535,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n" ] }, @@ -628,7 +628,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", " \n", "End of computing.\n", @@ -638,7 +638,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", " \n", "End of computing.\n", @@ -654,14 +654,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", " \n", "End of computing.\n", @@ -671,7 +671,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", " \n", "End of computing.\n", @@ -687,14 +687,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", " \n", "End of computing.\n", @@ -704,7 +704,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", " \n", "End of computing.\n", @@ -718,14 +718,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", " \n", "End of computing.\n", @@ -745,7 +745,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", @@ -764,7 +764,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n" ] }, @@ -850,7 +850,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", " \n", "End of computing.\n", @@ -860,7 +860,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", " \n", "End of computing.\n", @@ -876,14 +876,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 1) :\n", " \n", "End of computing.\n", @@ -893,7 +893,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", " \n", "End of computing.\n", @@ -909,14 +909,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", " \n", "End of computing.\n", @@ -926,7 +926,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", " \n", "End of computing.\n", @@ -940,14 +940,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", " \n", "End of computing.\n", @@ -967,7 +967,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n" ] }, @@ -1527,7 +1527,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 0) :\n", ">> \n", "End of computing.\n", @@ -1547,7 +1547,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 1.\n", @@ -1558,7 +1558,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", ">> \n", "End of computing.\n", @@ -1572,14 +1572,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 2.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 2) :\n", ">> \n", "End of computing.\n", @@ -1589,7 +1589,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", ">> \n", "End of computing.\n", @@ -1605,14 +1605,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 3.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 3) :\n", ">> \n", "End of computing.\n", @@ -1622,7 +1622,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", ">> \n", "End of computing.\n", @@ -1636,14 +1636,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 4.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 4) :\n", ">> \n", "End of computing.\n", @@ -1653,7 +1653,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", ">> \n", "End of computing.\n", @@ -1669,14 +1669,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 5.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 5) :\n", ">> \n", "End of computing.\n", @@ -1686,7 +1686,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 6) :\n", ">> \n", "End of computing.\n", @@ -1700,14 +1700,14 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n", "\n", "Computing boundary-matrix in dimension 6.\n", "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 6) :\n", ">> \n", "End of computing.\n", @@ -1717,7 +1717,7 @@ "Rank of the source-module : 1.\n", "\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "Computing the boundary of the generator 1/1 (dimension 7) :\n", ">> \n", "End of computing.\n", @@ -1733,7 +1733,7 @@ "\n", "---done---\n", "\n", - ";; Clock -> 2016-04-23, 23h 30m 17s.\n", + ";; Clock -> 2016-09-15, 15h 39m 0s.\n", "\n" ] },