Subject: Distribution.List Date: Wed, 28 Feb 2001 16:56:32 +0100 (CET) From: Juergen Bokowski > Hello Joanna Ellis-Monaghan, > > I received a copy of your question concerning the Steinitz problem > for 2-manifolds from Dave Rusin: > > > Here is what might interest you: > > A first hint: have a look at my paper > Bokowski, J. and Guedes de Oliveira, A., > On the generation of oriented matroids, > Discrete and Computaional Geometry 24:197-208, (2000). > > There are altogether 59 different triangulations > with 12 vertices, 66 edges and 44 faces, > see Altshuler, A., Bokowski, J., and Schuchert, P., > Journal of Combinatorial Theroy A 75,1,148-162, (1996) > > We have proven that no. 54 in our list cannot be realized even with > one triangle less. This tells us that there are non-realizable examples > of orientable triangulated closed 2-manifolds with all genus g \geq 6. > > > The main idea: we consider 5 points in general position in 3-space, 3 points > forming the vertices of a triangle and two other defining the vertices > of an edge. The set of the orientations of all 5 3-simplices concerning > these 5 points tells us whether the edge pierces the triangle or not. > > This is an easy case analysis. > > When you have 12 points in 3-space in general position, write them > as a 12 x 4 matrix with homogeneous coordinates, the signs of all > 12 \choose 4 determinants of 4 x 4 submatrices tell you the signs of > all 3-simplices. Now with the given list of triangles and edges, > you can check (better write a program) the intersection properties. > > We enumerated all possible sign structures, known as oriented matroids, > compatible with the given triangulated map > in an effective way and after several CPU years the result came out > that there is no such sign structure at all. > The method was applied already in many other cases. Compare e.g. my > Springer Lecture Notes in Mathematics Volume 1355, Computational > Synthetic Geometry from 1989. > > > The topological \underline {invariant} (oriented matroid) can be described > as follows which you might not find in the literature this way: > > It is an invariant with respect to homeomorphic transformations of > the projective (r-1)-space. Consider a cell decomposition of the > projective (r-1)-space by topological hyperplanes and require for this > decomposition that any subset of r+2 topological hyperplanes is the > homeomorphic image of a set of r+2 linear hyperplanes. This local > linearity condition turns out to characterize the so called > reaorientation classes of oriented matroids. > > But there is a much more to say about oriented matroids... > > see also the paper > Bokowski, J. and Eggert, A., > All realizations of M\"obius' torus with 7 vertices, > Structural Topolgy 20. > > Please let me know those references about the 7 vertex torus > that are not cited in this paper. > > > Juergen