chap4.txt

  
  4 SCO methods and functions
  
  
  4.1 Methods and functions for orbifold triangulations
  
  4.1-1 OrbifoldTriangulation
  
  OrbifoldTriangulation( M[, I, mu_data, info] )  function
  Returns:  OrbifoldTriangulation
  
  The  constructor  for  OrbifoldTriangulations.  Needs  the list M of maximal
  simplices,  the  Isotropy  at  certain  vertices as a record I, and the list
  mu_data  that  encodes the function mu. If only one argument is given, I and
  mu_data  are  supposed  to  be  empty.  In case of two arguments, mu_data is
  supposed  to be empty. If the last argument info 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.
  
    Example  
    gap> M := [ [1,2,3], [1,2,4], [1,3,4], [2,3,4] ];;
    gap> S2 := OrbifoldTriangulation( M, "S^2" );
    <OrbifoldTriangulation "S^2" of dimension 2. 4 simplices on 4 vertices without\
     Isotropy>
    gap> I := rec( 1 := Group( (1,2) ) );;
    gap> mu_data := [
    > [ [2], [1,2], [1,2,3], [1,2,4], x->x*(1,2) ],
    > [ [2], [1,2], [1,2,4], [1,2,3], x->x*(1,2) ]
    > ];;
    gap> Teardrop := OrbifoldTriangulation( M, I, mu_data, "Teardrop" );
    <OrbifoldTriangulation "Teardrop" of dimension 2. 4 simplices on 4 vertices wi\
    th Isotropy on 1 vertex and nontrivial mu-maps>
  
  
  4.1-2 Vertices
  
  Vertices( ot )  method
  Returns:  List V
  
  This returns the list of vertices V of the orbifold triangulation ot. Should
  be preferred to the equivalent ot!.vertices.
  
  4.1-3 Simplices
  
  Simplices( ot )  method
  Returns:  List M
  
  This  returns  the list of maximal simplices M of the orbifold triangulation
  ot. Should be preferred to the equivalent ot!.max_simplices.
  
  4.1-4 Isotropy
  
  Isotropy( ot )  method
  Returns:  Record I
  
  This  returns the isotropy record I of the orbifold triangulation ot. Should
  be preferred to the equivalent ot!.isotropy.
  
  4.1-5 Mu
  
  Mu( ot )  method
  Returns:  Function mu
  
  This  returns  the  function  mu of the orbifold triangulation ot. Should be
  preferred to the equivalent ot!.mu.
  
  4.1-6 MuData
  
  MuData( ot )  method
  Returns:  List mu_data
  
  This  returns  the list mu_data that encodes the function mu of the orbifold
  triangulation ot. Should be preferred to the equivalent ot!.mu_data.
  
  4.1-7 InfoString
  
  InfoString( ot )  method
  Returns:  String info
  
  This  return  the  string  info  of the orbifold triangulation ot. Should be
  preferred to the equivalent ot!.info.
  
  
  4.2 Methods and functions for simplicial sets
  
  4.2-1 SimplicialSet
  
  SimplicialSet( ot )  method
  Returns:  SimplicialSet
  
  The  constructor  for simplicial sets based on an orbifold triangulation ot.
  This  just  sets  up  the  object  without  any  computations.  These can be
  triggered later, either explicitly or by SimplicialSet (4.2-2).
  
    Example  
    gap> Teardrop;
    <OrbifoldTriangulation "Teardrop" of dimension 2. 4 simplices on 4 vertices wi\
    th Isotropy on 1 vertex and nontrivial mu-maps>
    gap> S := SimplicialSet( Teardrop );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 0 with Length vector [ 4 ]>
  
  
  4.2-2 SimplicialSet
  
  SimplicialSet( S, i )  method
  Returns:  List S_i
  
  This  returns  the components of dimension i of the simplicial set S. Should
  be used to access existing data instead of using S!.simplicial_set[ i + 1 ],
  as  it has the additional side effect of computing S up to dimension i, thus
  always returning the desired result.
  
    Example  
    gap> S := SimplicialSet( Teardrop );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 0 with Length vector [ 4 ]>
    gap> S!.simplicial_set[1];
    [ [ [ 1, 2, 3 ] ], [ [ 1, 2, 4 ] ], [ [ 1, 3, 4 ] ], [ [ 2, 3, 4 ] ] ]
    gap> S!.simplicial_set[2];;
    Error, List Element: <list>[2] must have an assigned value
    gap> SimplicialSet( S, 0 );
    [ [ [ 1, 2, 3 ] ], [ [ 1, 2, 4 ] ], [ [ 1, 3, 4 ] ], [ [ 2, 3, 4 ] ] ]
    gap> SimplicialSet( S, 1 );;
    gap> S;
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 1 with Length vector [ 4, 12 ]>
  
  
  4.2-3 ComputeNextDimension
  
  ComputeNextDimension( S )  method
  Returns:  S
  
  This computes the component of the next dimension of the simplicial set S. S
  is extended as a side effect.
  
    Example  
    gap> S;
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 1 with Length vector [ 4, 12 ]>
    gap> ComputeNextDimension( S );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 2 with Length vector [ 4, 12, 22 ]>
  
  
  4.2-4 Extend
  
  Extend( S, i )  method
  Returns:  S
  
  This computes the components of the simplicial set S up to dimension i. S is
  extended   as   a   side  effect.  This  method  is  equivalent  to  calling
  ComputeNextDimension (4.2-3) the appropriate number of times.
  
    Example  
    gap> S;
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 2 with Length vector [ 4, 12, 22 ]>
    gap> Extend( S, 5 );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 5 with Length vector [ 4, 12, 22, 33, 51, 73 ]>
  
  
  
  4.3 Methods and functions for matrix creation and computation
  
  4.3-1 BoundaryOperator
  
  BoundaryOperator( i, L, mu )  function
  Returns:  List B
  
  This  returns  the  ith  boundary  of  L,  which  has  to be an element of a
  simplicial  set. mu is the function μ 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.
  
  4.3-2 CreateBoundaryMatrices
  
  CreateBoundaryMatrices( S, d, R )  method
  Returns:  List M
  
  This  returns  the  list  M  of homalg matrices over the homalg ring R up to
  dimension   d,  corresponding  to  the  boundary  matrices  induced  by  the
  simplicial set S. If d is not given, the current dimension of S is used.
  
    Example  
    gap> S := SimplicialSet( Teardrop );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 0 with Length vector [ 4 ]>
    gap> M := CreateBoundaryMatrices( S, 4, HomalgRingOfIntegers() );;
    gap> S;
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 5 with Length vector [ 4, 12, 22, 33, 51, 73 ]>
  
  
  4.3-3 Homology
  
  Homology( M[, R] )  method
  Returns:  a homalg complex
  
  This  returns  the  homology complex of a list M of homalg matrices over the
  homalg ring R.
  
    Example  
    gap> S := SimplicialSet( Teardrop );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 0 with Length vector [ 4 ]>
    gap> R := HomalgRingOfIntegers();
    Z
    gap> M := CreateBoundaryMatrices( S, 4, R );;
    gap> Homology( M, R );
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  Z/< 2 >
    ----------------------------------------------->>>>  0
    <A graded homology object consisting of 5 left modules at degrees [ 0 .. 4 ]>
  
  
  4.3-4 CreateCoboundaryMatrices
  
  CreateCoboundaryMatrices( S[, d], R )  method
  Returns:  List M
  
  This  returns  the  list  M  of homalg matrices over the homalg ring R up to
  dimension  d,  corresponding  to  the  coboundary  matrices  induced  by the
  simplicial set S. If d is not given, the current dimension of S is used.
  
    Example  
    gap> S := SimplicialSet( Teardrop );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 0 with Length vector [ 4 ]>
    gap> M := CreateCoboundaryMatrices( S, 4, HomalgRingOfIntegers() );;
    gap> S;
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 5 with Length vector [ 4, 12, 22, 33, 51, 73 ]>
  
  
  4.3-5 Cohomology
  
  Cohomology( M[, R] )  method
  Returns:  a homalg complex
  
  This  returns the cohomology complex of a list M of homalg matrices over the
  homalg ring R.
  
    Example  
    gap> S := SimplicialSet( Teardrop );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 0 with Length vector [ 4 ]>
    gap> R := HomalgRingOfIntegers();
    Z
    gap> M := CreateCoboundaryMatrices( S, 4, R );;
    gap> Cohomology( M, R );
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  Z/< 2 >
    <A graded cohomology object consisting of 5 left modules at degrees
    [ 0 .. 4 ]>
  
  
  4.3-6 SCO_Examples
  
  SCO_Examples(  )  function
  Returns:  nothing
  
  This  is just an easy way to call the script examples.g, which is located in
  gap/pkg/SCO/examples/.
  
    Example  
    gap> SCO_Examples();
    @@@@@@@@ SCO @@@@@@@@
    
    Select base ring:
     1) Integers (default)
     2) Rationals
     3) Z/nZ
    :1
    
    Select Computer Algebra System:
     1) GAP (default)
     2) External GAP
     3) MAGMA
     4) Maple
     5) Sage
    :3
    ---------------------------------------------------------------
    Magma V2.14-14    Tue Aug 19 2008 08:36:19 on evariste [Seed = 1054613462]
    Type ? for help.  Type <Ctrl>-D to quit.
    ----------------------------------------------------------------
    
    
    Select Method:
     1) Full syzygy computation (default)
     2) matrix creation and rank computation only
    :1
    
    Select orbifold (default="C2")
    :Torus
      
    Select mode:
     1) Cohomology (default)
     2) Homology
    :1
    
    Select dimension (default = 4)
    :4
    Creating the coboundary matrices ...
    Starting cohomology computation ...
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  Z^(1 x 2)
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  0