# Piecewise-linear ball complexes: calculations with GAP

• korepanov
In summary, Igor Korepanov is inviting interested mathematicians to work together on a project focused on making calculations in topological quantum field theories on PL ball complexes of any dimension. He explains the structure of a PL ball complex and how to represent it, and mentions that some initial programs and functions have already been written. Igor believes that this project may be of interest to mathematicians and is open to collaboration.

#### 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

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