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Cell[CellGroupData[{
Cell["Homology Iterated Groups Package - Readme", "Title"],
Cell["\<\
Victor Alvarez Solano, Jose Andres Armario Sampalo, Maria Dolores \
Frau Garcia and Pedro Real Jurado
Departamento de Matematica Aplicada I, University of Seville (Spain).
20 August 2006\
\>", "Subtitle"],
Cell["valvarez@us.es", "Subsubtitle"],
Cell[TextData[{
"Determining the homology of a group at a desired dimension is in general a \
very difficult task, which is actually not feasible from a computational \
point of view. Taking into account the tools that the ",
StyleBox["Homological Perturbation Theory",
FontSlant->"Italic"],
" provides, these calculations may be significantly improved with the use \
of ",
StyleBox["homological models.",
FontSlant->"Italic"],
" This way, if a homological model hG for a group G is known, it suffices \
to compute the homology modules for hG instead of those for G,which is \
meaningfully simpler. This package provides a means of computing the homology \
of any iterated product of central extensions, semidirect products and direct \
products of finite abelian groups. Furthermore, explicit calculations of the \
maps involved in the definition of the homological model may be achieved with \
a slight modification of the notebook."
}], "Text"],
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Cell["Loading the package HomologyIteratedGroups", "Subsection"],
Cell[TextData[{
"The package `HomologyIteratedGroups` may be placed in folder: ",
Cell[BoxData[
\("/Applications/Mathematica 5.2.app/AddOns/Applications"\)],
"Output"],
"/.\nI use the Get request and select initialization as a cell property \
such as:"
}], "Text"],
Cell[BoxData[
\(\(<< HomologyIteratedGroups.m;\)\)], "Input",
InitializationCell->True],
Cell["For questions or comments please email me at valvarez@us.es", "Text"]
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Cell["Examples to use the package and possible tests", "Section"],
Cell["\<\
Let consider the iterated product (Z_t x_chi[2] Z_2) x_chi[1] Z_2, \
for the 2-cocycle chi[2]:Z_2 x Z_2 --\[Rule] Z_t given by chi[2][1,1]= \
Mod[1+ceiling(t/2),t] and 0 otherwise; and an the dihedral action chi[1]:Z_2 \
x (Z_t x_chi[2] Z_2)--\[Rule](Z_t x_chi[2]_2) given by \
chi[1][1,{a,g}]=opposite({a,g}) and {a,g} otherwise.
These maps should be pre-defined by the user, before calling the function \
HomologyIteratedGroup. For instance, for t=5, one could type\
\>", "Text"],
Cell[BoxData[{
\(t = 5; \(chi[2]\)[a_, g_] :=
If[g*a \[Equal] 1, Mod[Ceiling[t/2] + 1, t],
0];\), "\[IndentingNewLine]",
\(\(\(chi[1]\)[g_, {a_, b_}] :=
If[g \[Equal] 1, {Mod[\(-\((a + \(chi[2]\)[b, b])\)\), t], b}, {0,
0}];\)\)}], "Input"],
Cell["\<\
A tree-list representing the iterated group is given by \
{{2},{4,0}}. A list of cardinalities, from left to the right, is {5,2,2}. If \
we are interested in computing the homology groups from 0 to 10, we should \
type\
\>", "Text"],
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Cell[BoxData[
InterpretationBox[\("The homology at degree "\[InvisibleSpace]0\
\[InvisibleSpace]" is Z^"\[InvisibleSpace]1\[InvisibleSpace]" + Z_"\
\[InvisibleSpace]{}\),
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Editable->False]], "Print"],
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Editable->False]], "Print"],
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\[InvisibleSpace]" is Z^"\[InvisibleSpace]0\[InvisibleSpace]" + Z_"\
\[InvisibleSpace]{2, 2, 10}\),
SequenceForm[
"The homology at degree ", 3, " is Z^", 0, " + Z_", {2, 2, 10}],
Editable->False]], "Print"],
Cell[BoxData[
InterpretationBox[\("The homology at degree "\[InvisibleSpace]4\
\[InvisibleSpace]" is Z^"\[InvisibleSpace]0\[InvisibleSpace]" + Z_"\
\[InvisibleSpace]{2, 2}\),
SequenceForm[
"The homology at degree ", 4, " is Z^", 0, " + Z_", {2, 2}],
Editable->False]], "Print"]
}, Open ]],
Cell["\<\
and introduce 1 each time we are asked to going on computing the \
next homology group.\
\>", "Text"]
}, Open ]]
}, Open ]]
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