reducedSimplicialChainComplex -- the reduced homological chain complex that is determined by an abstract simplicial complex

Usage:

reducedSimplicialChainComplex(K)
Inputs:
- K, an abstract simplicial complex,
Outputs:
- a complex,

Description

This method returns the reduced homological chain complex (i.e., there is a nonzero term in homological degree $-1$ that corresponds to the empty face) that is associated to an abstract simplicial complex. The chain complex is defined over the integers.

i1 : K = abstractSimplicialComplex({{1,2,3},{2,4,9},{1,2,3,5,7,8},{3,4}})

o1 = AbstractSimplicialComplex{-1 => {{}}                                                                                                                                                                                                                                  }
                               0 => {{1}, {2}, {3}, {4}, {5}, {7}, {8}, {9}}
                               1 => {{1, 2}, {1, 3}, {1, 5}, {1, 7}, {1, 8}, {2, 3}, {2, 4}, {2, 5}, {2, 7}, {2, 8}, {2, 9}, {3, 4}, {3, 5}, {3, 7}, {3, 8}, {4, 9}, {5, 7}, {5, 8}, {7, 8}}
                               2 => {{1, 2, 3}, {1, 2, 5}, {1, 2, 7}, {1, 2, 8}, {1, 3, 5}, {1, 3, 7}, {1, 3, 8}, {1, 5, 7}, {1, 5, 8}, {1, 7, 8}, {2, 3, 5}, {2, 3, 7}, {2, 3, 8}, {2, 4, 9}, {2, 5, 7}, {2, 5, 8}, {2, 7, 8}, {3, 5, 7}, {3, 5, 8}, {3, 7, 8}, {5, 7, 8}}
                               3 => {{1, 2, 3, 5}, {1, 2, 3, 7}, {1, 2, 3, 8}, {1, 2, 5, 7}, {1, 2, 5, 8}, {1, 2, 7, 8}, {1, 3, 5, 7}, {1, 3, 5, 8}, {1, 3, 7, 8}, {1, 5, 7, 8}, {2, 3, 5, 7}, {2, 3, 5, 8}, {2, 3, 7, 8}, {2, 5, 7, 8}, {3, 5, 7, 8}}
                               4 => {{1, 2, 3, 5, 7}, {1, 2, 3, 5, 8}, {1, 2, 3, 7, 8}, {1, 2, 5, 7, 8}, {1, 3, 5, 7, 8}, {2, 3, 5, 7, 8}}
                               5 => {{1, 2, 3, 5, 7, 8}}

o1 : AbstractSimplicialComplex

i2 : reducedSimplicialChainComplex(K)

       1       8       19       21       15       6       1
o2 = ZZ  <-- ZZ  <-- ZZ   <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
                                                         
     -1      0       1        2        3        4       5

o2 : Complex

Ways to use reducedSimplicialChainComplex:

reducedSimplicialChainComplex(AbstractSimplicialComplex)

For the programmer

The object reducedSimplicialChainComplex is a method function.

The source of this document is in AbstractSimplicialComplexes.m2:842:0.