HH CellComplex -- compute the homology modules of a cell complex

Synopsis

Function: homology
Usage:

homology(C)
Inputs:
- C, a cell complex, with labels in ring(C)
Outputs:
- a graded module, the graded module, the homology of C with coefficients in ring(C)

Description

This computes the reduced homology of the cellular complex arising from the labeled cell complex C.

If the labels are all 1, then this will be the standard homology of the cell complex over the label ring, as in the following example.

i1 : R = QQ[x]

o1 = R

o1 : PolynomialRing

i2 : a = newSimplexCell({},1);

i3 : b1 = newCell {a,a};

i4 : b2 = newCell {a,a};

i5 : C = cellComplex(R,{b1,b2});

i6 : HH C

                                      2
o6 = cokernel | -1 | <-- image 0 <-- R
                                      
     -1                  0           1

o6 : Complex

i7 : prune oo

      2
o7 = R
      
     1

o7 : Complex

However if the cells instead labeled with monomials (or monomial ideals) from the ring the homology of the corresponding complex of R modules is given.

i8 : R = QQ[x]

o8 = R

o8 : PolynomialRing

i9 : a = newSimplexCell({},x);

i10 : b1 = newCell {a,a};

i11 : b2 = newCell {a,a};

i12 : C = cellComplex(R,{b1,b2});

i13 : HH C

                                       2
o13 = cokernel | -x | <-- image 0 <-- R
                                       
      -1                  0           1

o13 : Complex

i14 : prune oo

                                2
o14 = cokernel | x | <-- 0 <-- R
                                
      -1                 0     1

o14 : Complex

Ways to use this method: