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 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 was 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, you are welcome to visit the website http://chomp.rutgers.edu/. Please, select Software from the main menu. The source code of my software is available there for download, together with a set of interesting examples.

The development of this software and algorithms it is based on research which was partially supported by the Polish Committee for Scientific Research (KBN), grants no. 2 P03A 029 12, no. 2 P03A 011 18, no. 5 P03A 029 35, and no. 2 P03A 041 24.