General chain complexes in topaz

Apart from being capable of computing integer homology of simplicial complexes (see this tutorial for an introduction), polymake is able to handle general chain complexes and compute homology for coefficients from different domains. When experimenting in the interactive shell, switch to the topology application first:

> application 'topaz';

You can construct a chain complex via its differential matrices. For example purposes, we use the sparse boundary matrices of a triangulation of the real projective plane. You can then construct a general chain complex from it like this:

> $bd1 = real_projective_plane()->boundary_matrix(1);
> $bd2 = real_projective_plane()->boundary_matrix(2);
> $a = new Array<SparseMatrix<Integer>>($bd1,$bd2);   # omit the trivial zeroth differential
> $cc = new ChainComplex<SparseMatrix<Integer>>($a,1);

The template parameter of ChainComplex denotes the type of the boundary matrices. It defaults to SparseMatrix<Integer>, as this allows computation of integer homology. The second parameter of the chain complex constructor defaults to 0, indicating whether to perform a sanity check on the matrices (i.e. whether matrix dimensions match and successive maps compose to the zero map).

You can access the data stored in the object like this:

> print $cc->boundary_matrix(2);
(15) (0 1) (1 -1) (2 1)
(15) (0 1) (3 -1) (4 1)
(15) (5 1) (6 -1) (7 1)
(15) (1 -1) (5 1) (8 1)
(15) (3 -1) (6 1) (9 1)
(15) (7 1) (10 1) (11 -1)
(15) (4 -1) (10 1) (12 1)
(15) (2 -1) (11 1) (13 1)
(15) (8 1) (12 -1) (14 1)
(15) (9 -1) (13 1) (14 1)

There is a user function to compute integer homology of your complex. You can access the documentation by typing the name of the function in the interactive shell and then pressing F1.

> print homology($cc,0);
({} 1)
({(2 1)} 0)
({} 0)

The output rows correspond to the dimensions of your homology modules, containing the torsion coefficients in curly brackets, and the betti number. Note that this is non-reduced homology, unlike what gets computed when using the HOMOLOGY property of a simplicial complex.

There is an extra function for computing the generators of the homology modules as well.

> print homology_and_cycles($cc,0);
(({} 1)
<(6) (0 1)
>
)
(({(2 1)} 0)
<(15) (10 1) (11 -1) (12 1) (13 -1) (14 -1)
>
)
(({} 0)
<>
)

The output pairs the homology module representation with a representation of the cycles generating the respective modules, where the indices correspond to the indices in your input matrices.

If your complex' differentials do not have Integer coefficients, computing integer homology is not possible. You can still (and very efficiently!) compute the Betti numbers by using the corresponding user function:

> print betti_numbers($cc);
1 0 0