user_guide:tutorials:latest:chain_complex_homology

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—user_guide:tutorials:latest:chain_complex_homology [2023/11/06 10:57] (current) – created - external edit 127.0.0.1
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 +====== 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>