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# SCO

## SCO - Simplicial Cohomology of Orbifolds

2022.09-01

7 September 2022

Simon Görtzen
Email: simon.goertzen@rwth-aachen.de
Homepage: https://www.linkedin.com/in/simongoertzen/
Address:
Simon Görtzen
Lehrstuhl B fuer Mathematik, RWTH Aachen
Templergraben 64
52062 Aachen
Germany

### Abstract

This document explains the primary uses of the SCO package. Included in this manual is a documented list of the most important methods and functions you will need. For the theoretical basis of this package please refer to my diploma thesis and the corresponding paper (work in progress; [G\t08]).

### Copyright

© 2007-2011 by Simon Goertzen

This package may be distributed under the terms and conditions of the GNU Public License Version 2 or (at your option) any later version.

### Acknowledgements

The SCO package would not have been possible without the theoretical work by I. Moerdijk and D. A. Pronk concerning simplicial cohomology of orbifolds [MP99]. Many thanks to these two, as well as Mohamed Barakat and the Lehrstuhl B für Mathematik at RWTH Aachen University in general. It should be noted that SCO in its current functionality is dependent on the GAP package homalg by M. Barakat [BLH20], as it relies on homalg to do the actual computations. This manual was created with the help of the GAPDoc package by M. Neunhöffer and F. Lübeck.

### Contents

1 Introduction
1.1 Overview over this manual
1.2 Installation of the SCO Package
2 Usage
2.1 The Examples Script
2.2 Working Manually
3 Examples
3.1 Example 1: Klein Bottle
3.2 Example 2: V_4
3.3 Example 3: The "Teardrop" orbifold
4 SCO methods and functions
4.1 Methods and functions for orbifold triangulations
4.1-1 OrbifoldTriangulation
4.1-2 Vertices
4.1-3 Simplices
4.1-4 Isotropy
4.1-5 Mu
4.1-6 MuData
4.1-7 InfoString
4.2 Methods and functions for simplicial sets
4.2-1 SimplicialSet
4.2-2 SimplicialSet
4.2-3 ComputeNextDimension
4.2-4 Extend
4.3 Methods and functions for matrix creation and computation
4.3-1 BoundaryOperator
4.3-2 CreateBoundaryMatrices
4.3-3 Homology
4.3-4 CreateCoboundaryMatrices
4.3-5 Cohomology
4.3-6 SCO_Examples
A An Overview of the SCO package source code
References
Index
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