  
  [1X4 [5XSCO[1X methods and functions[0X
  
  
  [1X4.1 Methods and functions for orbifold triangulations[0X
  
  [1X4.1-1 OrbifoldTriangulation[0m
  
  [2X> OrbifoldTriangulation( [0X[3XM[, I, mu_data, info][0X[2X ) ___________________[0Xfunction
  [6XReturns:[0X  OrbifoldTriangulation
  
  The  constructor  for  OrbifoldTriangulations.  Needs  the list [3XM[0m of maximal
  simplices,  the  Isotropy  at  certain  vertices as a record [3XI[0m, and the list
  [3Xmu_data[0m  that  encodes the function mu. If only one argument is given, [3XI[0m and
  [3Xmu_data[0m  are  supposed  to  be  empty.  In case of two arguments, [3Xmu_data[0m is
  supposed  to be empty. If the last argument [3Xinfo[0m is given as a string, it is
  stored  in  the  info  component  of the orbifold triangulation and does not
  count towards the total number of arguments.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> M := [ [1,2,3], [1,2,4], [1,3,4], [2,3,4] ];;[0X
    [4Xgap> S2 := OrbifoldTriangulation( M, "S^2" );[0X
    [4X<OrbifoldTriangulation "S^2" of dimension 2. 4 simplices on 4 vertices without\[0X
    [4X Isotropy>[0X
    [4Xgap> I := rec( 1 := Group( (1,2) ) );;[0X
    [4Xgap> mu_data := [[0X
    [4X> [ [2], [1,2], [1,2,3], [1,2,4], x->x*(1,2) ],[0X
    [4X> [ [2], [1,2], [1,2,4], [1,2,3], x->x*(1,2) ][0X
    [4X> ];;[0X
    [4Xgap> Teardrop := OrbifoldTriangulation( M, I, mu_data, "Teardrop" );[0X
    [4X<OrbifoldTriangulation "Teardrop" of dimension 2. 4 simplices on 4 vertices wi\[0X
    [4Xth Isotropy on 1 vertex and nontrivial mu-maps>[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.1-2 Vertices[0m
  
  [2X> Vertices( [0X[3Xot[0X[2X ) _____________________________________________________[0Xmethod
  [6XReturns:[0X  List [3XV[0m
  
  This returns the list of vertices [3XV[0m of the orbifold triangulation [3Xot[0m. Should
  be preferred to the equivalent [10Xot!.vertices[0m.
  
  [1X4.1-3 Simplices[0m
  
  [2X> Simplices( [0X[3Xot[0X[2X ) ____________________________________________________[0Xmethod
  [6XReturns:[0X  List [3XM[0m
  
  This  returns  the list of maximal simplices [3XM[0m of the orbifold triangulation
  [3Xot[0m. Should be preferred to the equivalent [10Xot!.max_simplices[0m.
  
  [1X4.1-4 Isotropy[0m
  
  [2X> Isotropy( [0X[3Xot[0X[2X ) _____________________________________________________[0Xmethod
  [6XReturns:[0X  Record [3XI[0m
  
  This  returns the isotropy record [3XI[0m of the orbifold triangulation [3Xot[0m. Should
  be preferred to the equivalent [10Xot!.isotropy[0m.
  
  [1X4.1-5 Mu[0m
  
  [2X> Mu( [0X[3Xot[0X[2X ) ___________________________________________________________[0Xmethod
  [6XReturns:[0X  Function [3Xmu[0m
  
  This  returns  the  function  [3Xmu[0m of the orbifold triangulation [3Xot[0m. Should be
  preferred to the equivalent [10Xot!.mu[0m.
  
  [1X4.1-6 MuData[0m
  
  [2X> MuData( [0X[3Xot[0X[2X ) _______________________________________________________[0Xmethod
  [6XReturns:[0X  List [3Xmu_data[0m
  
  This  returns  the list [3Xmu_data[0m that encodes the function mu of the orbifold
  triangulation [3Xot[0m. Should be preferred to the equivalent [10Xot!.mu_data[0m.
  
  [1X4.1-7 InfoString[0m
  
  [2X> InfoString( [0X[3Xot[0X[2X ) ___________________________________________________[0Xmethod
  [6XReturns:[0X  String [3Xinfo[0m
  
  This  return  the  string  [3Xinfo[0m  of the orbifold triangulation [3Xot[0m. Should be
  preferred to the equivalent [10Xot!.info[0m.
  
  
  [1X4.2 Methods and functions for simplicial sets[0X
  
  [1X4.2-1 SimplicialSet[0m
  
  [2X> SimplicialSet( [0X[3Xot[0X[2X ) ________________________________________________[0Xmethod
  [6XReturns:[0X  SimplicialSet
  
  The  constructor  for simplicial sets based on an orbifold triangulation [3Xot[0m.
  This  just  sets  up  the  object  without  any  computations.  These can be
  triggered later, either explicitly or by [2XSimplicialSet[0m ([14X4.2-2[0m).
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> Teardrop;[0X
    [4X<OrbifoldTriangulation "Teardrop" of dimension 2. 4 simplices on 4 vertices wi\[0X
    [4Xth Isotropy on 1 vertex and nontrivial mu-maps>[0X
    [4Xgap> S := SimplicialSet( Teardrop );[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 0 with Length vector [ 4 ]>[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.2-2 SimplicialSet[0m
  
  [2X> SimplicialSet( [0X[3XS, i[0X[2X ) ______________________________________________[0Xmethod
  [6XReturns:[0X  List [3XS[0m_[3Xi[0m
  
  This  returns  the components of dimension [3Xi[0m of the simplicial set [3XS[0m. Should
  be used to access existing data instead of using [10XS!.simplicial_set[ i + 1 ][0m,
  as  it has the additional side effect of computing [3XS[0m up to dimension [3Xi[0m, thus
  always returning the desired result.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> S := SimplicialSet( Teardrop );[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 0 with Length vector [ 4 ]>[0X
    [4Xgap> S!.simplicial_set[1];[0X
    [4X[ [ [ 1, 2, 3 ] ], [ [ 1, 2, 4 ] ], [ [ 1, 3, 4 ] ], [ [ 2, 3, 4 ] ] ][0X
    [4Xgap> S!.simplicial_set[2];;[0X
    [4XList Element: <list>[2] must have an assigned value[0X
    [4Xnot in any function[0X
    [4XEntering break read-eval-print loop ...[0X
    [4Xyou can 'quit;' to quit to outer loop, or[0X
    [4Xyou can 'return;' after assigning a value to continue[0X
    [4Xbrk> [0X
    [4Xgap> SimplicialSet( S, 0 );[0X
    [4X[ [ [ 1, 2, 3 ] ], [ [ 1, 2, 4 ] ], [ [ 1, 3, 4 ] ], [ [ 2, 3, 4 ] ] ][0X
    [4Xgap> SimplicialSet( S, 1 );;[0X
    [4Xgap> S;[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 1 with Length vector [ 4, 12 ]>[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.2-3 ComputeNextDimension[0m
  
  [2X> ComputeNextDimension( [0X[3XS[0X[2X ) __________________________________________[0Xmethod
  [6XReturns:[0X  [3XS[0m
  
  This computes the component of the next dimension of the simplicial set [3XS[0m. [3XS[0m
  is extended as a side effect.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> S;[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 1 with Length vector [ 4, 12 ]>[0X
    [4Xgap> ComputeNextDimension( S );[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 2 with Length vector [ 4, 12, 22 ]>[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.2-4 Extend[0m
  
  [2X> Extend( [0X[3XS, i[0X[2X ) _____________________________________________________[0Xmethod
  [6XReturns:[0X  [3XS[0m
  
  This computes the components of the simplicial set [3XS[0m up to dimension [3Xi[0m. [3XS[0m is
  extended   as   a   side  effect.  This  method  is  equivalent  to  calling
  [2XComputeNextDimension[0m ([14X4.2-3[0m) the appropriate number of times.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> S;[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 2 with Length vector [ 4, 12, 22 ]>[0X
    [4Xgap> Extend( S, 5 );[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 5 with Length vector [ 4, 12, 22, 33, 51, 73 ]>[0X
  [4X------------------------------------------------------------------[0X
  
  
  [1X4.3 Methods and functions for matrix creation and computation[0X
  
  [1X4.3-1 BoundaryOperator[0m
  
  [2X> BoundaryOperator( [0X[3Xi, L, mu[0X[2X ) _____________________________________[0Xfunction
  [6XReturns:[0X  List B
  
  This  returns  the  [3Xi[0mth  boundary  of  [3XL[0m,  which  has  to be an element of a
  simplicial set. [3Xmu[0m is the function mu that has to be taken into account when
  computing  orbifold  boundaries.  This function is used for matrix creation,
  there should not be much reason for calling it independently.
  
  [1X4.3-2 CreateBoundaryMatrices[0m
  
  [2X> CreateBoundaryMatrices( [0X[3XS, d, R[0X[2X ) __________________________________[0Xmethod
  [6XReturns:[0X  List [3XM[0m
  
  This  returns  the  list  [3XM[0m  of homalg matrices over the homalg ring [3XR[0m up to
  dimension   [3Xd[0m,  corresponding  to  the  boundary  matrices  induced  by  the
  simplicial set [3XS[0m. If [3Xd[0m is not given, the current dimension of [3XS[0m is used.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> S := SimplicialSet( Teardrop );[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 0 with Length vector [ 4 ]>[0X
    [4Xgap> M := CreateBoundaryMatrices( S, 4, HomalgRingOfIntegers() );;[0X
    [4Xgap> S;[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 5 with Length vector [ 4, 12, 22, 33, 51, 73 ]>[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.3-3 Homology[0m
  
  [2X> Homology( [0X[3XM[, R][0X[2X ) _________________________________________________[0Xmethod
  [6XReturns:[0X  a [5Xhomalg[0m complex
  
  This  returns  the  homology complex of a list [3XM[0m of [5Xhomalg[0m matrices over the
  [5Xhomalg[0m ring [3XR[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> S := SimplicialSet( Teardrop );[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 0 with Length vector [ 4 ]>[0X
    [4Xgap> R := HomalgRingOfIntegers();[0X
    [4X<A homalg internal ring>[0X
    [4Xgap> M := CreateBoundaryMatrices( S, 4, R );;[0X
    [4Xgap> Homology( M, R );[0X
    [4X----------------------------------------------->>>>  Z^(1 x 1)[0X
    [4X----------------------------------------------->>>>  0[0X
    [4X----------------------------------------------->>>>  Z^(1 x 1)[0X
    [4X----------------------------------------------->>>>  Z/< 2 >[0X
    [4X----------------------------------------------->>>>  0[0X
    [4X<A graded homology object consisting of 5 left modules at degrees [ 0 .. 4 ]>[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.3-4 CreateCoboundaryMatrices[0m
  
  [2X> CreateCoboundaryMatrices( [0X[3XS[, d], R[0X[2X ) ______________________________[0Xmethod
  [6XReturns:[0X  List [3XM[0m
  
  This  returns  the  list  [3XM[0m  of homalg matrices over the homalg ring [3XR[0m up to
  dimension  [3Xd[0m,  corresponding  to  the  coboundary  matrices  induced  by the
  simplicial set [3XS[0m. If [3Xd[0m is not given, the current dimension of [3XS[0m is used.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> S := SimplicialSet( Teardrop );[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 0 with Length vector [ 4 ]>[0X
    [4Xgap> M := CreateCoboundaryMatrices( S, 4, HomalgRingOfIntegers() );;[0X
    [4Xgap> S;[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 5 with Length vector [ 4, 12, 22, 33, 51, 73 ]>[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.3-5 Cohomology[0m
  
  [2X> Cohomology( [0X[3XM[, R][0X[2X ) _______________________________________________[0Xmethod
  [6XReturns:[0X  a [5Xhomalg[0m complex
  
  This  returns the cohomology complex of a list [3XM[0m of [5Xhomalg[0m matrices over the
  [5Xhomalg[0m ring [3XR[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> S := SimplicialSet( Teardrop );[0X
    [4X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[0X
    [4Ximension 0 with Length vector [ 4 ]>[0X
    [4Xgap> R := HomalgRingOfIntegers();[0X
    [4X<A homalg internal ring>[0X
    [4Xgap> M := CreateCoboundaryMatrices( S, 4, R );;[0X
    [4Xgap> Cohomology( M, R );[0X
    [4X----------------------------------------------->>>>  Z^(1 x 1)[0X
    [4X----------------------------------------------->>>>  0[0X
    [4X----------------------------------------------->>>>  Z^(1 x 1)[0X
    [4X----------------------------------------------->>>>  0[0X
    [4X----------------------------------------------->>>>  Z/< 2 >[0X
    [4X<A graded cohomology object consisting of 5 left modules at degrees[0X
    [4X[ 0 .. 4 ]>[0X
  [4X------------------------------------------------------------------[0X
  
  [1X4.3-6 SCO_Examples[0m
  
  [2X> SCO_Examples( [0X[3X[0X[2X ) _________________________________________________[0Xfunction
  [6XReturns:[0X  nothing
  
  This  is just an easy way to call the script [11Xexamples.g[0m, which is located in
  [11Xgap/pkg/SCO/examples/[0m.
  
  [4X---------------------------  Example  ----------------------------[0X
    [4Xgap> SCO_Examples();[0X
    [4X@@@@@@@@ SCO @@@@@@@@[0X
    [4X[0X
    [4XSelect base ring:[0X
    [4X 1) Integers (default)[0X
    [4X 2) Rationals[0X
    [4X 3) Z/nZ[0X
    [4X:1[0X
    [4X[0X
    [4XSelect Computer Algebra System:[0X
    [4X 1) GAP (default)[0X
    [4X 2) External GAP[0X
    [4X 3) MAGMA[0X
    [4X 4) Maple[0X
    [4X 5) Sage[0X
    [4X:3[0X
    [4X---------------------------------------------------------------[0X
    [4XMagma V2.14-14    Tue Aug 19 2008 08:36:19 on evariste [Seed = 1054613462][0X
    [4XType ? for help.  Type <Ctrl>-D to quit.[0X
    [4X----------------------------------------------------------------[0X
    [4X[0X
    [4X[0X
    [4XSelect Method:[0X
    [4X 1) Full syzygy computation (default)[0X
    [4X 2) matrix creation and rank computation only[0X
    [4X:1[0X
    [4X[0X
    [4XSelect orbifold (default="C2")[0X
    [4X:Torus[0X
    [4X  [0X
    [4XSelect mode:[0X
    [4X 1) Cohomology (default)[0X
    [4X 2) Homology[0X
    [4X:1[0X
    [4X[0X
    [4XSelect dimension (default = 4)[0X
    [4X:4[0X
    [4XCreating the coboundary matrices ...[0X
    [4XStarting cohomology computation ...[0X
    [4X----------------------------------------------->>>>  Z^(1 x 1)[0X
    [4X----------------------------------------------->>>>  Z^(1 x 2)[0X
    [4X----------------------------------------------->>>>  Z^(1 x 1)[0X
    [4X----------------------------------------------->>>>  0[0X
    [4X----------------------------------------------->>>>  0    [0X
  [4X------------------------------------------------------------------[0X
  
