(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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"], Cell[CellGroupData[{ 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"] }, Open ]], Cell[CellGroupData[{ 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"], Cell[CellGroupData[{ Cell[BoxData[ \(HomologyIteratedGroups[{{2}, {4, 0}}, {5, 2, 2}, 0, chi]\)], "Input"], Cell[BoxData[ InterpretationBox[\("The homology at degree "\[InvisibleSpace]0\ \[InvisibleSpace]" is Z^"\[InvisibleSpace]1\[InvisibleSpace]" + Z_"\ \[InvisibleSpace]{}\), SequenceForm[ "The homology at degree ", 0, " is Z^", 1, " + Z_", {}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("The homology at degree "\[InvisibleSpace]1\ \[InvisibleSpace]" is Z^"\[InvisibleSpace]0\[InvisibleSpace]" + Z_"\ \[InvisibleSpace]{2, 2}\), SequenceForm[ "The homology at degree ", 1, " is Z^", 0, " + Z_", {2, 2}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("The homology at degree "\[InvisibleSpace]2\ \[InvisibleSpace]" is Z^"\[InvisibleSpace]0\[InvisibleSpace]" + Z_"\ \[InvisibleSpace]{2}\), SequenceForm[ "The homology at degree ", 2, " is Z^", 0, " + Z_", {2}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\("The homology at degree "\[InvisibleSpace]3\ \[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 ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 727}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{1272, 700}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, ShowSelection->True ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 58, 0, 105, "Title"], Cell[1800, 53, 214, 5, 122, "Subtitle"], Cell[2017, 60, 37, 0, 53, "Subsubtitle"], Cell[2057, 62, 965, 17, 90, "Text"], Cell[CellGroupData[{ Cell[3047, 83, 64, 0, 47, "Subsection"], Cell[3114, 85, 282, 7, 52, "Text"], Cell[3399, 94, 93, 2, 30, "Input", InitializationCell->True], Cell[3495, 98, 75, 0, 33, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[3607, 103, 65, 0, 53, "Section"], Cell[3675, 105, 492, 9, 90, "Text"], Cell[4170, 116, 287, 6, 50, "Input"], Cell[4460, 124, 242, 5, 33, "Text"], Cell[CellGroupData[{ Cell[4727, 133, 89, 1, 30, "Input"], Cell[4819, 136, 285, 5, 25, "Print"], Cell[5107, 143, 300, 6, 25, "Print"], Cell[5410, 151, 287, 5, 25, "Print"], Cell[5700, 158, 308, 6, 25, "Print"], Cell[6011, 166, 300, 6, 25, "Print"] }, Open ]], Cell[6326, 175, 111, 3, 33, "Text"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)