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# HH^ZZ CellComplex -- cohomology of a cell complex

## Synopsis

• Function: cohomology
• Usage:
cohomology(r,C)
• Inputs:
• r, an integer, a non-negative integer
• C, ,
• Optional inputs:
• Degree => ..., default value 0,
• Outputs:
• , 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