====== General chain complexes in topaz ======

Apart from being capable of computing integer homology of simplicial complexes (see this [[apps_topaz|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:

<code perl>
> application 'topaz';
</code>
==== Constructing a ChainComplex ====

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:

<code perl>
> $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);
</code>
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:

<code perl>
> 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)
</code>
==== Computing integer homology ====

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.

<code perl>
> print homology($cc,0);
({} 1)
({(2 1)} 0)
({} 0)
</code>
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.

<code perl>
> 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)
<>
)
</code>
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.

==== Computing Betti numbers ====

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:

<code perl>
> print betti_numbers($cc);
1 0 0
</code>