GAP (SCO) - Chapter 2: Usage

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2 Usage
2.1 The Examples Script
2.2 Working Manually

### 2 Usage

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 ℤ / ⟨ p^n ⟩.

#### 2.1 The Examples Script

Regardless of the extend of your installation, you will always be able to call the example script `SCO/examples/examples.g`. This script is not only callable in-GAP by `SCO_Examples` (4.3-6), 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 `SCO/examples/orbifolds/` 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 `p1.g` to `p6m.g`). Computing the cohomology of these orbifolds was an important part of my diploma thesis [G\t08].

Please note that the variables M, iso, and mu in the orbifold files have to keep their name for the example script to work correctly. Refer to chapter 3 for concrete examples.

#### 2.2 Working Manually

Once you are familiar with the example script and want to try out your own triangulations, it is best to create your own `.g` file in the `SCO/examples/orbifolds/` 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 4 if you need to know more about specific methods and functions. The basic steps are:

- Define a list of maximum simplices

- If applicable, define an isotropy record

- If applicable, define a list encoding the μ-map

- From the above data, create an orbifold triangulation

- Define the simplicial set of the orbifold triangulation

- Create a homalg ring R

- Create boundary or coboundary matrices over R

- Calculate their homology or cohomology

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