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

## Synopsis

• Function: homology
• Usage:
homology(C,R)
• Inputs:
• Outputs:

## 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