user_guide:tutorials:latest:chain_complex_homology

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 — user_guide:tutorials:latest:chain_complex_homology [2020/01/22 09:02] (current) Line 1: Line 1: + ====== 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: + + + > application '​topaz';​ + ​ + ==== 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: + + + > $bd1 = real_projective_plane()->​boundary_matrix(1);​ + >$bd2 = real_projective_plane()->​boundary_matrix(2);​ + > $a = new Array<​SparseMatrix<​Integer>>​($bd1,​$bd2); ​ # omit the trivial zeroth differential + >$cc = new ChainComplex<​SparseMatrix<​Integer>>​($a,​1);​ + ​ + The template parameter of ''​ChainComplex''​ denotes the type of the boundary matrices. It defaults to `SparseMatrix<​html><​Integer>,​ as this allows computation of integer homology. The second parameter of the chain complex constructor defaults to 0, indicating whether to perform a sanity check on the matrices (i.e. whether matrix dimensions match and successive maps compose to the zero map). + + You can access the data stored in the object like this: + + + > print$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) + ​ + ==== 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. + + + > 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. + + ==== 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>​ + print betti_numbers(\$cc);​ + 1 0 0 +
• user_guide/tutorials/latest/chain_complex_homology.txt