# 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) => , default value true,
• CoefficientRing (missing documentation) => a ring, default value QQ,
• VariableName (missing documentation) => , default value t,
• InputType (missing documentation) => , default value Polyhedral,
• Outputs:
• C, , 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