HH^ZZ CellComplex -- cohomology of a cell complex

Synopsis

Function: cohomology
Usage:

cohomology(r,C)
Inputs:
- r, an integer, a non-negative integer
- C, a cell complex,
Optional inputs:
- Degree => ..., default value 0,
Outputs:
- a module, the r-th cohomology module of C

Description

This computes the reduced cohomology in degree r of the labeled cell complex. In particular, it constructs the co-chain complex by dualizing by the label ring, and takes the homology of that chain complex. As an example we can compute the cohomology of the wedge of two circles.

i1 : R = QQ[x]

o1 = R

o1 : PolynomialRing

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

i3 : b1 = newCell {a,a};

i4 : b2 = newCell {a,a};

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

i6 : cohomology(-1,C)

o6 = image 0

                             1
o6 : R-module, submodule of R

i7 : cohomology(0,C)

o7 = cokernel {-1} | -x |

                            1
o7 : R-module, quotient of R

i8 : cohomology(1,C)

      2
o8 = R

o8 : R-module, free, degrees {2:-1}

Or in a more interesting case, we have the cohomology over the integers of $\mathbb{RP}^3$.

i9 : C = cellComplexRPn(ZZ,3);

i10 : cohomology(0,C)

o10 = cokernel | -1 |

                               1
o10 : ZZ-module, quotient of ZZ

i11 : cohomology(1,C)

o11 = image 0

                                1
o11 : ZZ-module, submodule of ZZ

i12 : cohomology(2,C)

o12 = cokernel | -2 |

                               1
o12 : ZZ-module, quotient of ZZ

i13 : cohomology(3,C)

        1
o13 = ZZ

o13 : ZZ-module, free

Ways to use this method: