usage.xml

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  usage.xml            SCO package documentation                  Simon Goertzen
                                                                     

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

This chapter explains the usage of this package.

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<Chapter Label="usage"><Heading>Usage</Heading>

There are different ways to use &SCO;. Please note that for the actual
computations the &homalg; package is required, and you will need both
the &RingsForHomalg; and the &GaussForHomalg; package to make use of
the full computational capabilities. For your information,
&RingsForHomalg; offers support for external computer algebra systems
and the rings they support, while &GaussForHomalg; extends &GAP;
functionality with regards to sparse matrices and computations over
fields and <M>&ZZ; / \langle p^n \rangle</M>. <P/>

<Section Label="script"><Heading>The Examples Script</Heading>

Regardless of the extend of your installation, you will always be able
to call the example script <F>SCO/examples/examples.g</F>. This script
is not only callable in-&GAP; by <Ref Meth="SCO_Examples"/>, but also
automatically checks which packages you have installed and provides
you with the available options.

The example script is designed to take you through the ring creation
process and then load one of the files of your choice located in the
<F>SCO/examples/orbifolds/</F> directory. In there you will find a lot
of test files with small 0- or 1-dimensional orbifolds, but also the
complete triangulations of the 17 orbifolds corresponding to the
2-dimensional wallpaper groups (these should be exactly the
uncapitalized files, ranging from <F>p1.g</F> to
<F>p6m.g</F>). Computing the cohomology of these orbifolds was
an important part of my diploma thesis <Cite Key="Goe"/> and I have
also created a separate document <Cite Key="WGC"/> to present my
results.<P/>

Please note that the variables <A>M</A>, <A>iso</A>, and <A>mu</A>
in the orbifold files have to keep their name for the example script
to work correctly. Refer to chapter <Ref Label="examples"/> for
concrete examples.

</Section>

<Section Label="manual"><Heading>Working Manually</Heading>

Once you are familiar with the example script and want to try out your
own triangulations, it is best to create your own <F>.g</F> file in
the <F>SCO/examples/orbifolds/</F> directory, then call the script
again. If for any reason you do not want to create a file or work with
the script, you can always do every step by hand. Check <Ref
Label="ch:MandF"/> if you need to know more about specific methods and
functions. The basic steps are:<P/>

<List>
 <Item>Define a list of maximum simplices</Item>
 <Item>If applicable, define an isotropy record</Item>
 <Item>If applicable, define a list encoding the <M>\mu</M>-map</Item>
 <Item>From the above data, create an orbifold triangulation</Item>
 <Item>Define the simplicial set of the orbifold triangulation</Item>
 <Item>Create a &homalg; ring <M>R</M></Item>
 <Item>Create boundary or coboundary matrices over <M>R</M></Item>
 <Item>Calculate their homology or cohomology</Item>
</List>

</Section>

</Chapter>