
#1
Jan1011, 06:33 AM

P: 3

Dear all,
Recently, some young people and I started a project that may be called "Piecewiselinear 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 0skeleton of the complex is described. Next, assuming that the kskeleton is already given, which implies (in particular) the numeration of all kcells, 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 kcells 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, ..., ncells. Thus, a threedimensional ball B^3 can be represented by the following PL ball complex with two vertices 1 and 2: [ [ [1,2], [1,2] ], # two onedimensional 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 # onedimensional simplexes 1 and 2; [ [1,2] ] # the threeball bounded by digons 1 and 2 ] With the best New Year wishes, Igor Korepanov 



#2
Jan1011, 08:52 AM

P: 3

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 PLball complex is a pair (X, U), where X is a compact Euclidean polyhedron and U is a covering of X by closed PLballs 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 PLball complex is defined up to PLhomeomorphism only by the combinatorics of adjunctions of its balls. Igor 


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