examples.xml

<?xml version="1.0" encoding="UTF-8"?>

<!-- 

  examples.xml            SCO package documentation                  Simon Goertzen
                                                                     

         Copyright (C) 2007-2008, Lehrstuhl B für Mathematik, RWTH-Aachen

This chapter gives examples on the usage of this package.

-->

<Chapter Label="examples"><Heading>Examples</Heading>

Although there are some small examples embedded in chapter <Ref
Chap="ch:MandF"/>, we will give some complete examples in this
chapter. Most of these could easily be called with the example script
mentioned in chapter <Ref Chap="usage"/>, but we will do them step by
step for best reproducability.

<Section><Heading>Example 1: Klein Bottle</Heading>
Suppose we want to calculate the cohomology of the Klein
Bottle. First, we need a triangulation.

It could look like this:

<Alt Only="LaTeX">
\begin{figure}[htbp]
\begin{center}
\includegraphics{files/pgt}
\caption{triangulation}
\end{center}
\end{figure}
</Alt>

<Alt Not="LaTeX">
<Listing>
1--2--3--1
|\ |\ |\ |
| \| \| \|
4--5--6--4
|\ |\ |\ |
| \| \| \|
7--8--9--7
|\ |\ |\ |
| \| \| \|
1--3--2--1
</Listing>
</Alt>

<P/>
This results in the following list of maximum simplices:

<Example>
gap> M := [ [1,2,4], [1,2,7], [1,3,6], [1,3,8], [1,4,6], [1,7,8],
> [2,3,5], [2,3,9], [2,4,5], [2,7,9], [3,5,6], [3,8,9],
> [4,5,7], [4,6,9], [4,7,9], [5,6,8], [5,7,8], [6,8,9] ];;
</Example>

As there is no isotropy and therefore no <M>\mu</M>-map, we can
continue with the orbifold triangulation and simplicial set:

<Example><![CDATA[
gap> ot := OrbifoldTriangulation( M, "Klein Bottle" );
<OrbifoldTriangulation "Klein Bottle" of dimension 2. 18 simplices on 9 vertic\
es without Isotropy>
gap> ss := SimplicialSet( ot );
<The simplicial set of the orbifold triangulation "Klein Bottle", computed up \
to dimension 0 with Length vector [ 18 ]>
]]></Example>

Now we will need a &homalg; ring. As this is a small example we can
compute directly over &ZZ;, so we can use &GAP;. In case you have
&RingsForHomalg; installed you might want to try computing in another
computer algebra system with the command
<C>HomalgRingOfIntegersInCAS()</C>, replacing "CAS" with the
corresponding system.

<Example><![CDATA[
gap> R := HomalgRingOfIntegers();
Z
]]></Example>

We are almost there. Let us create some coboundary matrices and compute
their cohomology:

<Example><![CDATA[
gap> c := CreateCoboundaryMatrices( ss, 4, R );;
gap> C := Cohomology( c, R );
----------------------------------------------->>>>  Z^(1 x 1)
----------------------------------------------->>>>  Z^(1 x 1)
----------------------------------------------->>>>  Z/< 2 >
----------------------------------------------->>>>  0
----------------------------------------------->>>>  0
<A graded cohomology object consisting of 5 left modules at degrees
[ 0 .. 4 ]>
]]></Example>

This is the cohomology of the Klein Bottle.

</Section>

<Section><Heading>Example 2: <M>V_4</M></Heading>

&SCO; can also be used to compute group cohomology, as every group can
be represented as an orbifold with just a single point. For
<M>V_4</M>, it would look like this:

<Example><![CDATA[
gap> M := [ [1] ];;
gap> V4 := Group( (1,2), (3,4) );;
gap> iso := rec( 1 := V4 );;
gap> ot := OrbifoldTriangulation( M, iso, "V4" );
<OrbifoldTriangulation "V4" of dimension 0. 1 simplex on 1 vertex with Isotrop\
y on 1 vertex>
gap> ss := SimplicialSet( ot );
<The simplicial set of the orbifold triangulation "V4", computed up to dimensi\
on 0 with Length vector [ 1 ]>
gap> R := HomalgRingOfIntegers();
Z
gap> c := CreateCoboundaryMatrices( ss, 4, R );;
gap> C := Cohomology( c, R );
----------------------------------------------->>>>  Z^(1 x 1)
----------------------------------------------->>>>  0
----------------------------------------------->>>>  Z/< 2 > + Z/< 2 >
----------------------------------------------->>>>  Z/< 2 >
----------------------------------------------->>>>  Z/< 2 > + Z/< 2 > + Z/< 2\
 >
<A graded cohomology object consisting of 5 left modules at degrees
[ 0 .. 4 ]>
]]></Example>

This is the <M>V_4</M> group cohomology up to degree 4.

</Section>

<Section><Heading>Example 3: The "Teardrop" orbifold</Heading>

You have seen a manifold in example 1, and group cohomology in example
2. Now we will meet our first proper orbifold, the Teardrop. This is
the example Moerdijk and Pronk used in their paper <Cite
Key="MP_SCO"/> on which my work is based. It is an easy example, but
includes both nontrivial isotropy and <M>\mu</M>-maps. We take the
isotropy at the top to be <M>C_2</M>.

The triangulation looks like this, with the gluing being at [1,3].

<Alt Only="LaTeX">
\begin{figure}[htbp]
\begin{center}
\includegraphics{files/Teardrop}
\caption{triangulation}
\end{center}
\end{figure}
</Alt>

<Alt Not="LaTeX">
<Listing>
   3=====1=====3
  / \   / \   / \
 /   \ /   \ /   \
5-----2-----4-----5
       \   /
        \ /
         5
</Listing>
</Alt>

<P/>The source code:

<Example><![CDATA[
gap> M := [ [1,2,3], [1,2,4], [1,3,4], [2,3,5], [2,4,5], [3,4,5] ];;
gap> iso := rec( 1 := Group( (1,2) ) );;
gap> mu := [
>            [ [3], [1,3], [1,2,3], [1,3,4], x -> (1,2) ],
>            [ [3], [1,3], [1,3,4], [1,2,3], x -> (1,2) ]
>          ];;
gap> ot := OrbifoldTriangulation( M, iso, mu, "Teardrop" );
<OrbifoldTriangulation "Teardrop" of dimension 2. 6 simplices on 5 vertices wi\
th Isotropy on 1 vertex and nontrivial mu-maps>
gap> ss := SimplicialSet( ot );
<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
imension 0 with Length vector [ 6 ]>
gap> R := HomalgRingOfIntegers();
Z
gap> c := CreateCoboundaryMatrices( ss, 6, R );;
gap> C := Cohomology( c, R );
----------------------------------------------->>>>  Z^(1 x 1)
----------------------------------------------->>>>  0
----------------------------------------------->>>>  Z^(1 x 1)
----------------------------------------------->>>>  0
----------------------------------------------->>>>  Z/< 2 >
----------------------------------------------->>>>  0
----------------------------------------------->>>>  Z/< 2 >
<A graded cohomology object consisting of 7 left modules at degrees
[ 0 .. 6 ]>
]]></Example>

This is the Teardrop cohomology.

</Section>

</Chapter>