cellularComplex -- create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary

Synopsis

Usage:

C = cellularComplex(F,InputType=>Simplicial) --for use only if $\Delta$ is simplicial

C = cellularComplex(V,F)
Inputs:
- V, a list, list of coordinates of vertices of $\Delta$
- F, a list, list of facets of $\Delta$ (each facet is recorded as a list of indices of vertices taken from V)
Optional inputs:
- BaseRing (missing documentation) => a ring, default value null,
- Homogenize (missing documentation) => a Boolean value, default value true,
- CoefficientRing (missing documentation) => a ring, default value QQ,
- VariableName (missing documentation) => a symbol, default value t,
- InputType (missing documentation) => a string, default value "Polyhedral",
Outputs:
- C, a chain complex, cellular chain complex of $\Delta$ relative to its boundary

Description

This method returns the cellular chain complex of $\Delta$ relative to its boundary. If $\Delta$ is homeomorphic to a disk, the homologies will vanish except for the top dimension.

i1 : V = {{0,0},{1,0},{0,1},{-1,-1}};

i2 : F = {{0,1,2},{0,2,3},{0,1,3}};

i3 : C = cellularComplex(V,F)

                 1                 3                 3
o3 = (QQ[t ..t ])  <-- (QQ[t ..t ])  <-- (QQ[t ..t ])
          0   2             0   2             0   2
                                          
     0                 1                 2

o3 : ChainComplex

i4 : prune HH C

o4 = 0 : 0            

     1 : 0            

                     1
     2 : (QQ[t ..t ])
              0   2

o4 : GradedModule

If the complex is simplicial, there is no need for a vertex list.

i5 : F = {{0,1,2},{0,1,3},{0,2,3}};

i6 : C = cellularComplex(F,InputType=>"Simplicial")

                 1                 3                 3
o6 = (QQ[t ..t ])  <-- (QQ[t ..t ])  <-- (QQ[t ..t ])
          0   2             0   2             0   2
                                          
     0                 1                 2

o6 : ChainComplex

Arbitrary dimensions and polyhedral input are allowed.

i7 : V = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}, {-2, -2, -2}, {-2, 2, -2}, {2, 2, -2}, {2, -2, -2}, {-2, -2, 2}, {-2, 2, 2}, {2, 2, 2}, {2, -2, 2}};

i8 : F = {{0, 1, 2, 3, 4, 5}, {0, 8, 9, 12, 13}, {1, 6, 7, 10, 11}, {2, 7, 8, 11, 12}, {3, 6, 9, 10, 13}, {4, 10, 11, 12, 13}, {5, 6, 7, 8, 9}, {0, 2, 8, 12}, {0, 3, 9, 13}, {0, 4, 12, 13}, {0, 5, 8, 9}, {1, 2, 7, 11}, {1, 3, 6, 10}, {1, 4, 10, 11}, {1, 5, 6, 7}, {2, 4, 11, 12}, {3, 4, 10, 13}, {3, 5, 6, 9}, {2, 5, 7, 8}, {0, 2, 4, 12}, {0, 2, 5, 8}, {0, 3, 4, 13}, {0, 3, 5, 9}, {1, 2, 4, 11}, {1, 2, 5, 7}, {1, 3, 4, 10}, {1, 3, 5, 6}};

i9 : C = cellularComplex(V,F);

i10 : prune HH C

o10 = 0 : 0            

      1 : 0            

      2 : 0            

                      1
      3 : (QQ[t ..t ])
               0   3

o10 : GradedModule

Ways to use cellularComplex :

cellularComplex(List)
cellularComplex(List,List)

For the programmer

The object cellularComplex is a method function with options.

cellularComplex -- create the cellular chain complex whose homologies are the singular homologies of the complex $\Delta$ relative to its boundary

Synopsis

Description

See also

Ways to use cellularComplex :

For the programmer