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

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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

### 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.

    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).

    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.

    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.

    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.

    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.

    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.

    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.

    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.

    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/`.

    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

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