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

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

           2              1
o7 = 1 : ZZ  <--------- ZZ  : 1
                | 1 |
                | 0 |

o7 : ComplexMap
 
i8 : prune homology f2

           2              1
o8 = 1 : ZZ  <--------- ZZ  : 1
                | 0 |
                | 1 |

o8 : ComplexMap

See also

Ways to use this method:

  • HH SimplicialMap -- Compute the induced map on homology of a simplicial map.
  • homology(Nothing,SimplicialMap)

The source of this document is in SimplicialComplexes/Documentation.m2:4335:0.