Educational Materials

This page is currently under construction. It will contain the thorough description of examples of homology computation (with pictures); most of these examples are already available now in the advanced version of the source code distribution with some notes, but the descriptions on this page will be presented in a more accessible form (some of them as exercises to do on one's own).

The Programs CubTop and ShowCubes

The program cubtop performs set-theoretical and topological operations on representable cubical sets of arbitrary dimension and graphically displays those in 2D. It is designed by Sylavain Bérubé and Anik Trahan for educational purposes and, in particular, it accompanies the book Computational Homology by T. Kaczynski, K. Mischaikow, and M. Mrozek. Students without any previous background in topology can learn to understand topological operations such as closure, interior, or boundary of representable sets via exercises based on this program. Cubtop is not aimed at numerical problem solving, and its graphical display function does not handle huge data files. A brief description of the program can be found in the README file included in the program. If you can read French, you will found an ample guide with examples in GRTC CubTop Documentation. Its translation to English can be yet another instructive exercise, for example, in preparation for the preliminary Ph.D. foreign language exam.

In addition to cubtop, there is a more recent program showcubes by Pawel Pilarczyk, which permits a dynamical 3D visualization of cubical sets. The program cubslice cuts cubical sets to slices, thus it permits searching for holes detected by the homology (homcubes or homchain) program. Sets of dimensions higher than 3 are also handled by showcubes, extra dimensions are simply truncated. The program cub2cells permits changing the format of an input file thus allowing for a greater compatibility of homology, cubtop, and showcubes programs.

Note: The cubtop program, as well as the showcubes program have all been integrated with the Advanced version of the CHomP package on October 12, 2005, and are now available for download together with the remainder of CHomP Software.