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# boundaryMap(ZZ,CellComplex) -- compute the boundary map of a cell complex from r-faces to (r-1)-faces

- Function: boundaryMap
- Usage:
    `boundaryMap(ZZ,CellComplex)`
- Inputs:
  - r, an integer, the source cell-dimension
  - C, a cell complex, the CellComplex
- Optional inputs:
  - Labels => ..., default value {},
- Outputs:
  - a matrix, the boundary map from r-faces to (r-1)-faces of C

## Description

This function returns the map in the chain complex from the r-th homological degree to the (r-1)-th homological degree.

For example, below we construct the Taylor complex for the monomial ideal $\langle x,y,z\rangle$

    i1 : R = QQ[x,y,z];

    i2 : vx = newSimplexCell({},x);

    i3 : vy = newSimplexCell({},y);

    i4 : vz = newSimplexCell({},z);

    i5 : exy = newSimplexCell {vx,vy};

    i6 : exz = newSimplexCell {vx,vz};

    i7 : eyz = newSimplexCell {vy,vz};

    i8 : f = newSimplexCell {exy,exz,eyz};

    i9 : C = cellComplex(R,{f});

    i10 : d1 = boundaryMap_1 C

    o10 = {1} | y  0  z  |
          {1} | -x z  0  |
          {1} | 0  -y -x |

    o10 : Matrix

    i11 : d2 = boundaryMap_2 C

    o11 = {2} | z  |
          {2} | x  |
          {2} | -y |

    o11 : Matrix

    i12 : assert(d1*d2==0)

## See also

- complex(CellComplex) -- compute the cellular chain complex for a cell complex
- complex(SimplicialComplex) -- create the chain complex associated to a simplicial complex.

## Ways to use this method:

- boundaryMap(ZZ,CellComplex) -- compute the boundary map of a cell complex from r-faces to (r-1)-faces

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CellularResolutions/doc.m2:547:0.