chap2.txt

  
  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  [Gör08a] and I have also created a
  separate document [Gör08b] to present my results.
  
  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