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

Synopsis

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}

Ways to use this method:

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

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

Synopsis

Description

See also

Ways to use this method: