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

## Synopsis

• Function: boundaryMap
• Usage:
boundaryMap(ZZ,CellComplex)
• Inputs:
• r, an integer, the source cell-dimension
• C, , the CellComplex
• Optional inputs:
• Labels => ..., default value {},
• Outputs:
• , 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} | z y 0 | {1} | 0 -x z | {1} | -x 0 -y | o10 : Matrix i11 : d2 = boundaryMap_2 C o11 = {2} | -y | {2} | z | {2} | x | o11 : Matrix i12 : assert(d1*d2==0)