Dear Forum Members, I have only be using GAP for a little while and derived great pleasure from it. Out of curiosity I also had a look at the Programmers Reference, and decided, as an exercise to write some functions supporting the study of homological groups on chain complexes. I would like to present the results of what I actually have already at this moment and obtain remarks and comments from the public on something that maybe could become a package concerning the construction and the study of chain complexes.
It has to be said that this work is in a very embryonic stage, programming is lousy, nomenclature is not always consistent and some problems still have to be resolved. Maybe a few examples could give a good idea what it is all about. In this implementation, chain complexes are constructed by using skeletons recursively, increasing dimension, each skeleton of dimension n serving as a building block on which a skeleton of dimension n+1 is constructed. A 0-skeleton is a mere finite set of vertexes (the type of elements of this set is not important yet, so it can be numbers, polynomials, indeterminates etc.). The only effect is that they are labelled and that this label is used as a positional parameter in higher dimensions. A 1-skeleton is built on a given 0-skeleton by connecting some of the vertexes, in fact one considers a set of formal differences between two vertexes, giving a directed edge. If v is the number of vertexes then these edges can be represented as v-dimensional vectors with only two integer entries 1 and -1 put at the position of the corresponding points. For example the edge connecting the i-th vertex with the j-th vertex can be viewed as the vector (0,..0,1,0,..,0,-1,0,..,0) with the i-th coordinate being 1 and the j-th coordinate being -1. In order not to overflow the forum system I will start with a few examples, but it soon will be clear that the construction always follows the same scheme. First a set of n-cycles is found out, and "promoted" to n+1-cells. Then a set of these cells is found out that form cycles to be promoted to n+2-cells and so on. The examples will appear in separate postings in order not to inundate the forum and to have separate references in case of discussion. The examples will include: The 2-Disk The 2-Sphere The 2-Open ended cylinder The Moebius strip The 2-Projective Space The Klein bottle Still to do : the double torus, the 3-Sphere, general constructions like joining and excision. Thanks for your attention. PS: is it OK if I put some images/programming code as attachments or is this "not done"? Marc Bogaerts _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
