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# HH SimplicialMap -- Compute the induced map on homology of a simplicial map.

## Synopsis

• Function: homology
• Usage:
homology f
• Inputs:
• Outputs:
• , which is the map on homology induced by $f$

## Description

The graded module map from the homology of the source of $f$ to the homology 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 homology.

 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 homology f1 o7 = -1: 0 <--0-- 0 :-1 0: 0 <--0-- 0 :0 2 1 1: ZZ <--| 1 |-- ZZ :1 | 0 | 1 2: ZZ <--0-- 0 :2 o7 : GradedModuleMap i8 : prune homology f2 o8 = -1: 0 <--0-- 0 :-1 0: 0 <--0-- 0 :0 2 1 1: ZZ <--| 0 |-- ZZ :1 | 1 | 1 2: ZZ <--0-- 0 :2 o8 : GradedModuleMap