Computing Reduced Homology with SAGEMath

Question

Good morning to everybody. I downloaded SAGEMath9.3 on my Windows in order to make some computations with abstract simplicial complexes.

I found the following reference manual http://www2.math.ritsumei.ac.jp/doc/static/reference/homology/sage/homology/simplicial_complex.html where numerous subroutine and functions are used as an example.

Well, when computing the reduced homology of a given abstract simplicial complex, as in the following example

sage: S = SimplicialComplex([[0,1], [1,2], [0,2]]) # circle

sage: T = S.product(S) # torus

sage: T

Simplicial complex with 9 vertices and 18 facets

sage: T.homology() # this computes reduced homology

{0: 0, 1: Z x Z, 2: Z}

the software outputs the string {0: 0, 1: Z x Z, 2: Z}. It seems that computations stop at the same order of the cardinality of facets. Is there some general relation?

Answer 1

It's probably better to ask questions like this at ask.sagemath.org or the sage-support Google group, since this is more about computing with Sagemath than it is about mathematics. In this case, Sagemath knows the maximum dimension of any simplex in the simplicial complex, and it returns a dictionary of homology groups from dimension 0 to the top dimension. You can ask for homology in another dimension or a range of dimensions, and Sagemath can handle that:

sage: T.homology(range(10))  # dimensions 0 through 9
{0: 0, 1: Z x Z, 2: Z, 3: 0, 4: 0, 5: 0, 6: 0, 7: 0, 8: 0, 9: 0}

But if you don't specify the dimensions, it does indeed stop at the dimension of the simplicial complex.