GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<?xml version="1.0" encoding="UTF-8"?>
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<!-- 
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  examples.xml            SCO package documentation                  Simon Goertzen
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         Copyright (C) 2007-2008, Lehrstuhl B für Mathematik, RWTH-Aachen
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This chapter gives examples on the usage of this package.
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-->
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<Chapter Label="examples"><Heading>Examples</Heading>
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Although there are some small examples embedded in chapter <Ref
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Chap="ch:MandF"/>, we will give some complete examples in this
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chapter. Most of these could easily be called with the example script
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mentioned in chapter <Ref Chap="usage"/>, but we will do them step by
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step for best reproducability.
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<Section><Heading>Example 1: Klein Bottle</Heading>
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Suppose we want to calculate the cohomology of the Klein
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Bottle. First, we need a triangulation.
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It could look like this:
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<Alt Only="LaTeX">
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\begin{figure}[htbp]
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\begin{center}
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\includegraphics{files/pgt}
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\caption{triangulation}
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\end{center}
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\end{figure}
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</Alt>
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<Alt Not="LaTeX">
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<Listing>
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1--2--3--1
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|\ |\ |\ |
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| \| \| \|
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4--5--6--4
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|\ |\ |\ |
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| \| \| \|
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7--8--9--7
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|\ |\ |\ |
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| \| \| \|
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1--3--2--1
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</Listing>
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</Alt>
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<P/>
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This results in the following list of maximum simplices:
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<Example>
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gap> M := [ [1,2,4], [1,2,7], [1,3,6], [1,3,8], [1,4,6], [1,7,8],
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> [2,3,5], [2,3,9], [2,4,5], [2,7,9], [3,5,6], [3,8,9],
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> [4,5,7], [4,6,9], [4,7,9], [5,6,8], [5,7,8], [6,8,9] ];;
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</Example>
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As there is no isotropy and therefore no <M>\mu</M>-map, we can
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continue with the orbifold triangulation and simplicial set:
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<Example><![CDATA[
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gap> ot := OrbifoldTriangulation( M, "Klein Bottle" );
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<OrbifoldTriangulation "Klein Bottle" of dimension 2. 18 simplices on 9 vertic\
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es without Isotropy>
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gap> ss := SimplicialSet( ot );
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<The simplicial set of the orbifold triangulation "Klein Bottle", computed up \
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to dimension 0 with Length vector [ 18 ]>
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]]></Example>
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Now we will need a &homalg; ring. As this is a small example we can
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compute directly over &ZZ;, so we can use &GAP;. In case you have
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&RingsForHomalg; installed you might want to try computing in another
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computer algebra system with the command
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<C>HomalgRingOfIntegersInCAS()</C>, replacing "CAS" with the
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corresponding system.
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<Example><![CDATA[
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gap> R := HomalgRingOfIntegers();
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Z
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]]></Example>
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We are almost there. Let us create some coboundary matrices and compute
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their cohomology:
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<Example><![CDATA[
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gap> c := CreateCoboundaryMatrices( ss, 4, R );;
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gap> C := Cohomology( c, R );
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----------------------------------------------->>>>  Z^(1 x 1)
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----------------------------------------------->>>>  Z^(1 x 1)
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----------------------------------------------->>>>  Z/< 2 >
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----------------------------------------------->>>>  0
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----------------------------------------------->>>>  0
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<A graded cohomology object consisting of 5 left modules at degrees
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[ 0 .. 4 ]>
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]]></Example>
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This is the cohomology of the Klein Bottle.
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</Section>
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<Section><Heading>Example 2: <M>V_4</M></Heading>
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&SCO; can also be used to compute group cohomology, as every group can
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be represented as an orbifold with just a single point. For
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<M>V_4</M>, it would look like this:
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<Example><![CDATA[
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gap> M := [ [1] ];;
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gap> V4 := Group( (1,2), (3,4) );;
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gap> iso := rec( 1 := V4 );;
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gap> ot := OrbifoldTriangulation( M, iso, "V4" );
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<OrbifoldTriangulation "V4" of dimension 0. 1 simplex on 1 vertex with Isotrop\
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y on 1 vertex>
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gap> ss := SimplicialSet( ot );
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<The simplicial set of the orbifold triangulation "V4", computed up to dimensi\
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on 0 with Length vector [ 1 ]>
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gap> R := HomalgRingOfIntegers();
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Z
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gap> c := CreateCoboundaryMatrices( ss, 4, R );;
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gap> C := Cohomology( c, R );
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----------------------------------------------->>>>  Z^(1 x 1)
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----------------------------------------------->>>>  0
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----------------------------------------------->>>>  Z/< 2 > + Z/< 2 >
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----------------------------------------------->>>>  Z/< 2 >
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----------------------------------------------->>>>  Z/< 2 > + Z/< 2 > + Z/< 2\
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 >
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<A graded cohomology object consisting of 5 left modules at degrees
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[ 0 .. 4 ]>
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]]></Example>
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This is the <M>V_4</M> group cohomology up to degree 4.
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</Section>
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<Section><Heading>Example 3: The "Teardrop" orbifold</Heading>
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You have seen a manifold in example 1, and group cohomology in example
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2. Now we will meet our first proper orbifold, the Teardrop. This is
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the example Moerdijk and Pronk used in their paper <Cite
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Key="MP_SCO"/> on which my work is based. It is an easy example, but
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includes both nontrivial isotropy and <M>\mu</M>-maps. We take the
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isotropy at the top to be <M>C_2</M>.
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The triangulation looks like this, with the gluing being at [1,3].
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<Alt Only="LaTeX">
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\begin{figure}[htbp]
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\begin{center}
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\includegraphics{files/Teardrop}
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\caption{triangulation}
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\end{center}
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\end{figure}
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</Alt>
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<Alt Not="LaTeX">
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<Listing>
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   3=====1=====3
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  / \   / \   / \
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 /   \ /   \ /   \
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5-----2-----4-----5
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       \   /
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        \ /
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         5
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</Listing>
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</Alt>
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<P/>The source code:
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<Example><![CDATA[
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gap> M := [ [1,2,3], [1,2,4], [1,3,4], [2,3,5], [2,4,5], [3,4,5] ];;
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gap> iso := rec( 1 := Group( (1,2) ) );;
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gap> mu := [
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>            [ [3], [1,3], [1,2,3], [1,3,4], x -> (1,2) ],
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>            [ [3], [1,3], [1,3,4], [1,2,3], x -> (1,2) ]
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>          ];;
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gap> ot := OrbifoldTriangulation( M, iso, mu, "Teardrop" );
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<OrbifoldTriangulation "Teardrop" of dimension 2. 6 simplices on 5 vertices wi\
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th Isotropy on 1 vertex and nontrivial mu-maps>
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gap> ss := SimplicialSet( ot );
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<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
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imension 0 with Length vector [ 6 ]>
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gap> R := HomalgRingOfIntegers();
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Z
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gap> c := CreateCoboundaryMatrices( ss, 6, R );;
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gap> C := Cohomology( c, R );
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----------------------------------------------->>>>  Z^(1 x 1)
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----------------------------------------------->>>>  0
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----------------------------------------------->>>>  Z^(1 x 1)
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----------------------------------------------->>>>  0
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----------------------------------------------->>>>  Z/< 2 >
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----------------------------------------------->>>>  0
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----------------------------------------------->>>>  Z/< 2 >
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<A graded cohomology object consisting of 7 left modules at degrees
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[ 0 .. 6 ]>
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]]></Example>
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This is the Teardrop cohomology.
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</Section>
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</Chapter>
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