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# HH_ZZ CellComplex -- compute the homology modules of a cell complex

- Function: homology
- Usage:
    `homology(r,C)`
- Inputs:
  - r, an integer, an integer
  - C, a cell complex, with labels in ring(C)
- Outputs:
  - a module, the r-th homology module of C with coefficients in ring(C)

## Description

This computes the reduced homology in degree r of the cellular complex arising from the cell complex C. For more details on the labels, see HH CellComplex. As an example, we can compute the 0-th and 1st homology of a wedge of two circles.

    i1 : R = QQ

    o1 = QQ

    o1 : Ring

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

    i3 : b1 = newCell {a,a};

    i4 : b2 = newCell {a,a};

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

    i6 : homology(0,C)

    o6 = image 0

                                   1
    o6 : QQ-module, submodule of QQ

    i7 : homology(1,C)

           2
    o7 = QQ

    o7 : QQ-module, free

    i8 : prune oo

           2
    o8 = QQ

    o8 : QQ-module, free

We can make this example slightly more interesting by changing the label ring and adding a non-unit label. Note in particular that this has a non-zero 0-th homology.

    i9 : R = QQ[x]

    o9 = R

    o9 : PolynomialRing

    i10 : a = newSimplexCell({},x);

    i11 : b1 = newCell {a,a};

    i12 : b2 = newCell {a,a};

    i13 : C = cellComplex(R,{b1,b2});

    i14 : homology(0,C)

    o14 = image 0

                                  1
    o14 : R-module, submodule of R

    i15 : prune oo

    o15 = 0

    o15 : R-module

    i16 : homology(1,C)

           2
    o16 = R

    o16 : R-module, free, degrees {2:1}

    i17 : prune oo

           2
    o17 = R

    o17 : R-module, free, degrees {2:1}

## See also

- HH CellComplex -- compute the homology modules of a cell complex
- HH^ZZ CellComplex -- cohomology of a cell complex
- complex(CellComplex) -- compute the cellular chain complex for a cell complex

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CellularResolutions/doc.m2:716:0.