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HH^ZZ CellComplex -- cohomology of a cell complex

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

See also

Ways to use this method:

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CellularResolutions/doc.m2:760:0.