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

- 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

## See also

- idealsComplex -- creates the Billera-Schenck-Stillman chain complex of ideals
- splineComplex -- creates the Billera-Schenck-Stillman chain complex

## Ways to use cellularComplex:

- `cellularComplex(List)`
- `cellularComplex(List,List)`

## For the programmer

The object cellularComplex is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/AlgebraicSplines.m2:2068:0.