# boundary(ZZ,SimplicialComplex) -- the boundary map from i-faces to (i-1)-faces

## Synopsis

• Function: boundary
• Usage:
M = boundary(i,D)
• Inputs:
• Outputs:
• M, , the boundary map from i-faces to (i-1)-faces of D

## Description

The columns of the matrix M are indexed by the i-faces of D, and the rows are indexed by the (i-1)-faces, in the order given by faces. M is defined over the coefficient ring of D.The boundary maps for the standard 3-simplex, defined over ZZ.
 i1 : R = ZZ[a..d]; i2 : D = simplicialComplex {a*b*c*d} o2 = | abcd | o2 : SimplicialComplex i3 : boundary(0,D) o3 = | 1 1 1 1 | 1 4 o3 : Matrix ZZ <--- ZZ i4 : faces(0,D) o4 = | a b c d | 1 4 o4 : Matrix R <--- R i5 : boundary(1,D) o5 = | -1 -1 -1 0 0 0 | | 1 0 0 -1 -1 0 | | 0 1 0 1 0 -1 | | 0 0 1 0 1 1 | 4 6 o5 : Matrix ZZ <--- ZZ i6 : faces(1,D) o6 = | ab ac ad bc bd cd | 1 6 o6 : Matrix R <--- R i7 : boundary(2,D) o7 = | 1 1 0 0 | | -1 0 1 0 | | 0 -1 -1 0 | | 1 0 0 1 | | 0 1 0 -1 | | 0 0 1 1 | 6 4 o7 : Matrix ZZ <--- ZZ i8 : faces(2,D) o8 = | abc abd acd bcd | 1 4 o8 : Matrix R <--- R i9 : boundary(3,D) o9 = | -1 | | 1 | | -1 | | 1 | 4 1 o9 : Matrix ZZ <--- ZZ i10 : faces(3,D) o10 = | abcd | 1 1 o10 : Matrix R <--- R i11 : boundary(4,D) o11 = 0 1 o11 : Matrix ZZ <--- 0
The boundary maps depend on the coefficient ring as the following examples illustrate.
 i12 : R = QQ[a..f]; i13 : D = simplicialComplex monomialIdeal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f,b*c*d,b*d*e,b*e*f,c*d*f,c*e*f); i14 : boundary(1,D) o14 = | -1 -1 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 | | 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 | | 0 1 0 0 0 1 0 0 0 -1 -1 -1 0 0 0 | | 0 0 1 0 0 0 1 0 0 1 0 0 -1 -1 0 | | 0 0 0 1 0 0 0 1 0 0 1 0 1 0 -1 | | 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 | 6 15 o14 : Matrix QQ <--- QQ i15 : R' = ZZ/2[a..f]; i16 : D' = simplicialComplex monomialIdeal(a*b*c,a*b*f,a*c*e,a*d*e,a*d*f,b*c*d,b*d*e,b*e*f,c*d*f,c*e*f); i17 : boundary(1,D') o17 = | 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 | | 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 | | 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 | | 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 | | 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 | | 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 | ZZ 6 ZZ 15 o17 : Matrix (--) <--- (--) 2 2