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SimplicialComplexes :: boundary(ZZ,SimplicialComplex)

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

Synopsis

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

See also