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# HH^ZZ SimplicialMap -- Compute the induced map on cohomology of a simplicial map.

## Synopsis

• Function: cohomology
• Usage:
cohomology(i,f)
• Inputs:
• Optional inputs:
• Degree => an integer, default value 0, which is ignored by this particular function
• Outputs:
• , which is the induced map on cohomology

## Description

The map from the $i$-th cohomology of the source of $f$ to the $i$-th cohomology of the target of $f$. As an example, we map a circle into the torus in two ways, and we get two distinct maps in cohomology.

 i1 : S = ZZ[x_0..x_6]; i2 : R = ZZ[y_0..y_2]; i3 : Torus = smallManifold(2,7,6,S); i4 : Circle = simplicialComplex{R_0*R_1, R_0*R_2, R_1*R_2}; i5 : f1 = map(Torus,Circle,matrix{{S_3,S_6,S_5}}); o5 : SimplicialMap simplicialComplex | x_2x_5x_6 x_0x_5x_6 x_1x_4x_6 x_0x_4x_6 x_2x_3x_6 x_1x_3x_6 x_3x_4x_5 x_1x_4x_5 x_0x_3x_5 x_1x_2x_5 x_2x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | <--- simplicialComplex | y_1y_2 y_0y_2 y_0y_1 | i6 : f2 = map(Torus,Circle,matrix{{S_0,S_2,S_3}}); o6 : SimplicialMap simplicialComplex | x_2x_5x_6 x_0x_5x_6 x_1x_4x_6 x_0x_4x_6 x_2x_3x_6 x_1x_3x_6 x_3x_4x_5 x_1x_4x_5 x_0x_3x_5 x_1x_2x_5 x_2x_3x_4 x_0x_2x_4 x_0x_1x_3 x_0x_1x_2 | <--- simplicialComplex | y_1y_2 y_0y_2 y_0y_1 | i7 : prune cohomology(1, f1) o7 = | 1 -2 | 1 2 o7 : Matrix ZZ <-- ZZ i8 : prune cohomology(1, f2) o8 = | 0 1 | 1 2 o8 : Matrix ZZ <-- ZZ