Homology Computation - Software and Examples

Throughout my academic career, I have put a lot of work into software for the computation of homology of cubical sets, chain complexes and chain maps. To begin with, I implemented a generalization of the algorithm by Kaczynski, Mrozek and Slusarek for the computation of homology of finitely generated free chain complexes and chain maps on them. This program was part of my master dissertation in computer science written under direction of Professor Marian Mrozek and completed in 1999.

My programs for homology computation were rewritten twice and developed significantly since the first version had been released, and also new algorithms were designed and implemented. As a result, the homology software now has several completely new features, for example:

  • support for cubical sets that are not necessarily unions of cubes of maximal dimension,
  • relative homology computation of cubical sets,
  • the ability of performing the computations over an arbitrary Euclidean domain (such as the ring of integers), so that the torsion is no longer lost,
  • (relative) homology computation of abstract simplicial complexes (not as efficient as for cubes, though),
  • the computation of homomorphisms induced in (relative) homology by cubical maps.

For more information about the software and the source code itself, you are welcome to visit the Web page http://chomp.rutgers.edu/software/. The source code of the CHomP software is available there for download, together with a set of interesting examples. Note that in addition to my code (which is clearly marked in each source file), the CHomP software also contains contributions of other authors; in particular, a selection of files from the CAPD library.