next | previous | forward | backward | up | top | index | toc | Macaulay2 website
SimplicialComplexes :: homology(SimplicialComplex,Ring)

homology(SimplicialComplex,Ring) -- Compute the homology of a simplicial complex.

Synopsis

Description

The graded module of reduced homologies of C with coefficients in R.

i1 : R=ZZ[x_0..x_5];
i2 : D=simplicialComplex apply({{x_0, x_1, x_2}, {x_1, x_2, x_3}, {x_0, x_1, x_4}, {x_0, x_3, x_4}, {x_2, x_3, x_4}, {x_0, x_2, x_5}, {x_0, x_3, x_5}, {x_1, x_3, x_5}, {x_1, x_4, x_5}, {x_2, x_4, x_5}},face)

o2 = | x_2x_4x_5 x_1x_4x_5 x_1x_3x_5 x_0x_3x_5 x_0x_2x_5 x_2x_3x_4 x_0x_3x_4 x_0x_1x_4 x_1x_2x_3 x_0x_1x_2 |

o2 : SimplicialComplex
i3 : homology(D)

o3 = -1 : cokernel | -1 -1 -1 -1 -1 -1 |                                     

      0 : subquotient (| 1  0  0  0  0  |, | 1  1  1  1  1  0  0  0  0  0  0 
                       | 0  0  1  0  0  |  | -1 0  0  0  0  1  1  1  1  0  0 
                       | 0  1  0  0  0  |  | 0  -1 0  0  0  -1 0  0  0  1  1 
                       | 0  0  0  1  0  |  | 0  0  -1 0  0  0  -1 0  0  -1 0 
                       | -1 -1 0  0  1  |  | 0  0  0  -1 0  0  0  -1 0  0  -1
                       | 0  0  -1 -1 -1 |  | 0  0  0  0  -1 0  0  0  -1 0  0 
          0  0  0  0  |)
          0  0  0  0  |
          1  0  0  0  |
          0  1  1  0  |
          0  -1 0  1  |
          -1 0  -1 -1 |

      1 : subquotient (| 0  1  0  0  0  0  0  0  0  0  |, | -1 -1 0  0  0  0 
                       | 1  0  0  0  0  0  0  0  0  0  |  | 1  0  -1 0  0  0 
                       | 0  -1 1  0  -1 0  1  0  1  0  |  | 0  0  0  -1 -1 0 
                       | 0  0  0  0  0  1  0  0  0  0  |  | 0  1  0  1  0  0 
                       | -1 0  -1 0  1  -1 -1 0  -1 0  |  | 0  0  1  0  1  0 
                       | 0  0  0  0  1  0  0  0  0  0  |  | -1 0  0  0  0  -1
                       | 0  0  0  1  0  0  0  0  1  1  |  | 0  0  0  0  0  1 
                       | 0  1  -1 0  0  0  0  0  -1 0  |  | 0  -1 0  0  0  0 
                       | 0  0  1  -1 -1 0  0  0  0  -1 |  | 0  0  0  0  0  0 
                       | 0  0  0  0  0  0  0  1  0  0  |  | 0  0  0  0  0  -1
                       | 0  0  0  0  0  0  0  0  0  1  |  | 0  0  0  0  0  0 
                       | 1  0  0  0  1  0  0  -1 0  -1 |  | 0  0  -1 0  0  0 
                       | 0  -1 1  0  -1 0  1  1  2  1  |  | 0  0  0  -1 0  0 
                       | 0  0  0  1  0  0  0  0  0  0  |  | 0  0  0  0  -1 0 
                       | 0  0  0  0  -1 1  1  1  1  2  |  | 0  0  0  0  0  0 
          0  0  0  0  |)
          0  0  0  0  |
          0  0  0  0  |
          0  0  0  0  |
          0  0  0  0  |
          0  0  0  0  |
          -1 0  0  0  |
          0  -1 0  0  |
          1  1  0  0  |
          0  0  -1 0  |
          0  0  1  -1 |
          0  0  0  1  |
          0  0  -1 0  |
          -1 0  0  0  |
          0  -1 0  -1 |

      2 : image 0                                                            

o3 : GradedModule
i4 : homology(D,QQ)

o4 = -1 : cokernel | -1 -1 -1 -1 -1 -1 |                                     

      0 : subquotient (| -1 -1 -1 -1 -1 |, | 1  1  1  1  1  0  0  0  0  0  0 
                       | 1  0  0  0  0  |  | -1 0  0  0  0  1  1  1  1  0  0 
                       | 0  1  0  0  0  |  | 0  -1 0  0  0  -1 0  0  0  1  1 
                       | 0  0  1  0  0  |  | 0  0  -1 0  0  0  -1 0  0  -1 0 
                       | 0  0  0  1  0  |  | 0  0  0  -1 0  0  0  -1 0  0  -1
                       | 0  0  0  0  1  |  | 0  0  0  0  -1 0  0  0  -1 0  0 
          0  0  0  0  |)
          0  0  0  0  |
          1  0  0  0  |
          0  1  1  0  |
          0  -1 0  1  |
          -1 0  -1 -1 |

      1 : subquotient (| 1  1  1  1  0  0  0  0  0  0  |, | -1 -1 0  0  0  0 
                       | -1 0  0  0  1  1  1  0  0  0  |  | 1  0  -1 0  0  0 
                       | 0  -1 0  0  -1 0  0  1  1  0  |  | 0  0  0  -1 -1 0 
                       | 0  0  -1 0  0  -1 0  -1 0  1  |  | 0  1  0  1  0  0 
                       | 0  0  0  -1 0  0  -1 0  -1 -1 |  | 0  0  1  0  1  0 
                       | 1  0  0  0  0  0  0  0  0  0  |  | -1 0  0  0  0  -1
                       | 0  1  0  0  0  0  0  0  0  0  |  | 0  0  0  0  0  1 
                       | 0  0  1  0  0  0  0  0  0  0  |  | 0  -1 0  0  0  0 
                       | 0  0  0  1  0  0  0  0  0  0  |  | 0  0  0  0  0  0 
                       | 0  0  0  0  1  0  0  0  0  0  |  | 0  0  0  0  0  -1
                       | 0  0  0  0  0  1  0  0  0  0  |  | 0  0  0  0  0  0 
                       | 0  0  0  0  0  0  1  0  0  0  |  | 0  0  -1 0  0  0 
                       | 0  0  0  0  0  0  0  1  0  0  |  | 0  0  0  -1 0  0 
                       | 0  0  0  0  0  0  0  0  1  0  |  | 0  0  0  0  -1 0 
                       | 0  0  0  0  0  0  0  0  0  1  |  | 0  0  0  0  0  0 
          0  0  0  0  |)
          0  0  0  0  |
          0  0  0  0  |
          0  0  0  0  |
          0  0  0  0  |
          0  0  0  0  |
          -1 0  0  0  |
          0  -1 0  0  |
          1  1  0  0  |
          0  0  -1 0  |
          0  0  1  -1 |
          0  0  0  1  |
          0  0  -1 0  |
          -1 0  0  0  |
          0  -1 0  -1 |

      2 : image 0                                                            

o4 : GradedModule
i5 : homology(D,ZZ/2)

o5 = -1 : cokernel | 1 1 1 1 1 1 |                                      

      0 : subquotient (| 1 1 1 1 1 |, | 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 |)
                       | 1 0 0 0 0 |  | 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 |
                       | 0 1 0 0 0 |  | 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 |
                       | 0 0 1 0 0 |  | 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 |
                       | 0 0 0 1 0 |  | 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 |
                       | 0 0 0 0 1 |  | 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 |

      1 : subquotient (| 1 1 1 1 0 0 0 0 0 0 |, | 1 1 0 0 0 0 0 0 0 0 |)
                       | 1 0 0 0 1 1 1 0 0 0 |  | 1 0 1 0 0 0 0 0 0 0 |
                       | 0 1 0 0 1 0 0 1 1 0 |  | 0 0 0 1 1 0 0 0 0 0 |
                       | 0 0 1 0 0 1 0 1 0 1 |  | 0 1 0 1 0 0 0 0 0 0 |
                       | 0 0 0 1 0 0 1 0 1 1 |  | 0 0 1 0 1 0 0 0 0 0 |
                       | 1 0 0 0 0 0 0 0 0 0 |  | 1 0 0 0 0 1 0 0 0 0 |
                       | 0 1 0 0 0 0 0 0 0 0 |  | 0 0 0 0 0 1 1 0 0 0 |
                       | 0 0 1 0 0 0 0 0 0 0 |  | 0 1 0 0 0 0 0 1 0 0 |
                       | 0 0 0 1 0 0 0 0 0 0 |  | 0 0 0 0 0 0 1 1 0 0 |
                       | 0 0 0 0 1 0 0 0 0 0 |  | 0 0 0 0 0 1 0 0 1 0 |
                       | 0 0 0 0 0 1 0 0 0 0 |  | 0 0 0 0 0 0 0 0 1 1 |
                       | 0 0 0 0 0 0 1 0 0 0 |  | 0 0 1 0 0 0 0 0 0 1 |
                       | 0 0 0 0 0 0 0 1 0 0 |  | 0 0 0 1 0 0 0 0 1 0 |
                       | 0 0 0 0 0 0 0 0 1 0 |  | 0 0 0 0 1 0 1 0 0 0 |
                       | 0 0 0 0 0 0 0 0 0 1 |  | 0 0 0 0 0 0 0 1 0 1 |

      2 : image | 1 |
                | 1 |
                | 1 |
                | 1 |
                | 1 |
                | 1 |
                | 1 |
                | 1 |
                | 1 |
                | 1 |                                                   

o5 : GradedModule

See also