GAP (SCO) - Chapter 4: SCO methods and functions

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.

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

This returns the list of vertices V of the orbifold triangulation ot. Should be preferred to the equivalent ot!.vertices.

4.1-3 Simplices

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

This returns the isotropy record I of the orbifold triangulation ot. Should be preferred to the equivalent ot!.isotropy.

4.1-5 Mu

This returns the function mu of the orbifold triangulation ot. Should be preferred to the equivalent ot!.mu.

4.1-6 MuData

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

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

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).

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

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.

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

This computes the component of the next dimension of the simplicial set S. S is extended as a side effect.

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

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.

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

This returns the ith boundary of L, which has to be an element of a simplicial set. mu 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.

4.3-2 CreateBoundaryMatrices

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.

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

This returns the homology complex of a list M of homalg matrices over the homalg ring R.

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

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.

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

This returns the cohomology complex of a list M of homalg matrices over the homalg ring R.

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

This is just an easy way to call the script examples.g, which is located in gap/pkg/SCO/examples/.

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

4 SCO methods and functions

4.1 Methods and functions for orbifold triangulations

4.1-1 OrbifoldTriangulation