Added Francis' changes.
This commit is contained in:
Gerd Heber
parent
74ffaf0957
commit
333df40667
@ -12,11 +12,11 @@
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"\n",
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"## The topological problem\n",
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"\n",
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"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",
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"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",
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"$$\n",
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"\\partial_n(b,g) = (\\partial_n b, \\tau(b).\\partial_n g), \\quad b\\in B, g \\in {\\cal G},\n",
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"$$\n",
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"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",
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"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",
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"$$\n",
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"\\begin{aligned}\n",
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" & \\qquad\\widehat{B} & & & & \\qquad\\widehat{\\cal G} & & \\cr\n",
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@ -24,7 +24,7 @@
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"B & & EB & & {\\cal G} & & EG \\cr\n",
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"\\end{aligned}\n",
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"$$\n",
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"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",
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"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",
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"\n",
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"### The Ronald Brown reduction\n",
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"\n",
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@ -36,7 +36,7 @@
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" {{\\cal C}_*(B) \\otimes {\\cal C}_*({\\cal G})} & &\\cr\n",
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"\\end{aligned}\n",
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"$$\n",
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"But, in fact we are interested in the following one (the Ronald Brown reduction):\n",
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"Using the *basic perturbation lemma*, we then obtain a new reduction between twisted objects:\n",
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"$$\n",
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"\\begin{aligned}\n",
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"{{\\cal C}_*(B \\times_\\tau {\\cal G})} & \\qquad\\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(B \\times_\\tau {\\cal G})} \\cr\n",
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@ -44,7 +44,7 @@
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" {{\\cal C}_*(B) \\otimes_t {\\cal C}_*({\\cal G})} & &\\cr\n",
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"\\end{aligned}\n",
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"$$\n",
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"where the symbol $\\otimes_t$ represents a *twisted tensor product*, induced by the twisting operator $\\tau$; this twisting operator is nothing but the Shih twisting cochain (**Gugenheim.**, *On the chain complex of a fibration*. Illinois Journal of Mathematics, 1972, vol. 16, pp. 398-414.) determined by Szczarba (**R. H. Szczarba.** *The homology of twisted cartesian products*. Transactions of the American Mathematical Society, 1961, vol. 100, pp. 197-216.). Here, to obtain the Szczarba cochain, we apply the perturbation lemma (Ronnie Brown). In effect, writing \n",
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"where the symbol $\\otimes_t$ represents a *twisted tensor product*, induced by the twisting operator $\\tau$; this twisting operator is nothing but the Shih twisting cochain (**Gugenheim.**, *On the chain complex of a fibration*. Illinois Journal of Mathematics, 1972, vol. 16, pp. 398-414.) determined by Szczarba (**R. H. Szczarba.** *The homology of twisted cartesian products*. Transactions of the American Mathematical Society, 1961, vol. 100, pp. 197-216.). In effect, writing \n",
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"$$ d_\\tau = d + (d_\\tau -d),$$\n",
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"one may consider that the differential $d_\\tau$ of $B \\times_\\tau {\\cal G}$ is the differential $d$ of $B \\times {\\cal G}$ modified by the perturbation $d_\\tau - d$. The perturbation morphism $d_\\tau -d$ is deduced from the difference between both $\\partial_n$ face operators. This being done, a perturbation may possibly be propagated along the Eilenberg-Zilber reduction (see the method `add` applied to a reduction) and the perturbation lemma gives also, as a bonus, the perturbation to be applied to the differential of $B \\otimes {\\cal G}$ to obtain the differential of $B \\otimes_t {\\cal G}$.\n",
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"\n",
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@ -75,7 +75,7 @@
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"B\\otimes_t {\\cal G} & & EB \\widetilde{\\otimes} EG \\cr\n",
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"\\end{aligned}\n",
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"$$\n",
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"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",
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"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",
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"\n",
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"### Assembling the puzzle\n",
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"\n",
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@ -304,7 +304,7 @@
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 0) :\n",
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"<TnPr * *> \n",
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"End of computing.\n",
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@ -314,7 +314,7 @@
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 1) :\n",
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"<TnPr * S1> \n",
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"End of computing.\n",
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@ -330,14 +330,14 @@
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 1.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 1) :\n",
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"<TnPr * S1> \n",
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"End of computing.\n",
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@ -347,7 +347,7 @@
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 2) :\n",
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"<TnPr S2 *> \n",
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"End of computing.\n",
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@ -363,14 +363,14 @@
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 2.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 2) :\n",
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"<TnPr S2 *> \n",
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"End of computing.\n",
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@ -380,7 +380,7 @@
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 3) :\n",
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"<TnPr S2 S1> \n",
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"End of computing.\n",
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@ -394,14 +394,14 @@
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 3.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 3) :\n",
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"<TnPr S2 S1> \n",
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"End of computing.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 4.\n",
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@ -440,7 +440,7 @@
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 5.\n",
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@ -459,7 +459,7 @@
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 6.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 7.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 8.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 9.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n"
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]
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},
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 0) :\n",
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"<TnPr * *> \n",
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"End of computing.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 1) :\n",
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"<TnPr * S1> \n",
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"End of computing.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 1.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 1) :\n",
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"<TnPr * S1> \n",
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"End of computing.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 2) :\n",
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"<TnPr S2 *> \n",
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"End of computing.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 2.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 2) :\n",
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"<TnPr S2 *> \n",
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"End of computing.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 3) :\n",
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"<TnPr S2 S1> \n",
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"End of computing.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 3.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 3) :\n",
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"<TnPr S2 S1> \n",
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"End of computing.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 4.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n"
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]
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},
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 0) :\n",
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"<TnPr * *> \n",
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"End of computing.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 1) :\n",
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"<TnPr * S1> \n",
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"End of computing.\n",
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"\n",
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"---done---\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"\n",
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"\n",
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"Computing boundary-matrix in dimension 1.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 1) :\n",
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"<TnPr * S1> \n",
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"End of computing.\n",
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"Rank of the source-module : 1.\n",
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"\n",
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"\n",
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";; Clock -> 2016-04-23, 23h 30m 17s.\n",
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";; Clock -> 2016-09-15, 15h 39m 0s.\n",
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"Computing the boundary of the generator 1/1 (dimension 2) :\n",
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"<TnPr S2 *> \n",
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"End of computing.\n",
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"\n",
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"---done---\n",
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"\n",
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";; 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",
|
||||
"<TnPr S2 *> \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",
|
||||
"<TnPr S2 S1> \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",
|
||||
"<TnPr S2 S1> \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",
|
||||
"<TnPr * <<AlLp>>> \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",
|
||||
"<TnPr * <<AlLp[2 S3]>>> \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",
|
||||
"<TnPr * <<AlLp[2 S3]>>> \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",
|
||||
"<TnPr S3 <<AlLp>>> \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",
|
||||
"<TnPr S3 <<AlLp>>> \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",
|
||||
"<TnPr * <<AlLp[2 S3][2 S3]>>> \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",
|
||||
"<TnPr * <<AlLp[2 S3][2 S3]>>> \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",
|
||||
"<TnPr S3 <<AlLp[2 S3]>>> \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",
|
||||
"<TnPr S3 <<AlLp[2 S3]>>> \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",
|
||||
"<TnPr * <<AlLp[2 S3][2 S3][2 S3]>>> \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",
|
||||
"<TnPr * <<AlLp[2 S3][2 S3][2 S3]>>> \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",
|
||||
"<TnPr S3 <<AlLp[2 S3][2 S3]>>> \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"
|
||||
]
|
||||
},
|
||||
|
||||
Loading…
Reference in New Issue
Block a user