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