Piecewise-linear ball complexes: calculations with GAP

[korepanov]

Dear all,

Recently, some young people and I started a project that may be called "Piecewise-linear ball complexes: calculations with GAP". As to me, my direct aim is to make calculations in some TQFT's (topological quantum field theories) naturally defined on PL ball complexes of any dimensions. I think, however, that calculations with PL ball complexes may be of broader interest.

So, I invite interested mathematicians to work together. As far as I know, great mathematicians of the past liked calculations, and did not limit themselves to scratching something on themselves and waiting for a flash of genius.

Some first programs/functions are already written. I will give a more detailed account of this if needed. Right now let me just explain how we represent a PL ball complex.

First, we assume that all vertices in the complex are numbered (from 1 to their total number N_0). Hence, in this sense, the 0-skeleton of the complex is described. Next, assuming that the k-skeleton is already given, which implies (in particular) the numeration of all k-cells, we describe the (k+1)-skeleton as the list of all (k+1)-cells, each of which, in its turn, is the set of numbers of k-cells in its boundary. Then we compose the list of length n, where n - is the dimension of the complex, whose elements are lists of 1-, ..., n-cells.

Thus, a three-dimensional ball B^3 can be represented by the following PL ball complex with two vertices 1 and 2:

[
[ [1,2], [1,2] ], # two one-dimensional simplexes, each with
# ends 1 and 2, of which the first is referred to
# in the next line as 1, the second - as 2;
[ [1,2], [1,2] ], # two digons (=bigons) bounded each by
# one-dimensional simplexes 1 and 2;
[ [1,2] ] # the three-ball bounded by digons 1 and 2
]

With the best New Year wishes,

Igor Korepanov

 

[korepanov]

Perhaps it will be of use if I add here what "PL ball complex" means, just a quotation from Nikolai Mnev's paper arXiv:math/0609257v3 :

A PL-ball complex is a pair (X, U), where X is a compact Euclidean polyhedron and U is a covering of X by closed PL-balls such that the following axioms are satisfied:

plbc1: the relative interiors of balls from U form a partition of X.
plbc2: The boundary of each ball from U is a union of balls from U.

A PL-ball complex is defined up to PL-homeomorphism only by the combinatorics of adjunctions of its balls.

Igor