Ulrich Brehm

Combinatorial manifolds with few vertices

WE SHOW that a d-manifold M with less than 3⌈d2⌉+3 vertices is a sphere and that a d-manifold with 3d/2+3 vertices is either a sphere or d=2, 4, 8, or 16 and M is a “manifold like a projective plane”. There are such examples for d=2, 4, 8. Furthermore we show that a d-manifold M with 2d+3-i vertices is i-connected [i.e. π0(m)=…=πi(M)=0] where 0≤i<d2. In particular the smallest number of vertices of a non-simply-connected d-manifold is exactly n=2d+3, (d≥3).

The shape invariant of triangles and trigonometry in two-point homogeneous spaces

We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces ℂPn and ℍPn (n≥2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces ℂHn and ℍHn (n≥2).

Smooth approximation of polyhedral surfaces regarding curvatures

In this paper we prove that every closed polyhedral surface in Euclidean three-space can be approximated (uniformly with respect to the Hausdorff metric) by smooth surfaces of the same topological type such that not only the (Gaussian) curvature but also the absolute curvature and the absolute mean curvature converge in the measure sense. This gives a direct connection between the concepts of total absolute curvature for both smooth and polyhedral surfaces which have been worked out by several authors, particularly N. H. Kuiper and T. F. Banchoff.

Equivelar maps on the torus

We give a classification of all equivelar polyhedral maps on the torus. In particular, we classify all triangulations and quadrangulations of the torus admitting a vertex transitive automorphism group. These are precisely the ones which are quotients of the regular tessellations {3,6}, {6,3} or {4,4} by a pure translation group. An explicit formula for the number of combinatorial types of equivelar maps (polyhedral and non-polyhedral) with n vertices is obtained in terms of arithmetic functions in elementary number theory, such as the number of integer divisors of n. The asymptotic behaviour for n->~ is also discussed, and an example is given for n such that the number of distinct equivelar triangulations of the torus with n vertices is larger than n itself. The numbers of regular and chiral maps are determined separately, as well as the ones for all other kinds of symmetry. Furthermore, arithmetic properties of the integers of type p^2+pq+q^2 (or p^2+q^2, resp.) can be interpreted and visualized by the hierarchy of covering maps between regular and chiral equivelar maps or type {3,6} (or {4,4}, resp.).

Maximally symmetric polyhedral realizations of Dyck's regular map

We construct realizations of Dycks regular map of genus three as polyhedra in ℝ 3 . One of these has one axis of symmetry of order three and three axes of symmetry of order two. The other polyhedra have three axes of symmetry. We show that a polyhedron realizing Dycks regular map cannot have a symmetry group of order larger than six. Thus the symmetry groups of our realizations are maximal.

MINIMAL SIMPLICIAL DISSECTIONS AND TRIANGULATIONS OF CONVEX 3-POLYTOPES.

Abstract. This paper addresses three questions related to minimal triangulations of a three-dimensional convex polytope P .• Can the minimal number of tetrahedra in a triangulation be decreased if one allows the use of interior points of P as vertices? • Can a dissection of P use fewer tetrahedra than a triangulation? • Does the size of a minimal triangulation depend on the geometric realization of P ? The main result of this paper is that all these questions have an affirmative answer. Even stronger, the gaps of size produced by allowing interior vertices or by using dissections may be linear in the number of points.

A polyhedron of genus 4 with minimal number of vertices and maximal symmetry

We construct a symmetric polyhedron of genus 4 in R3 with 11 vertices. This shows that for given genus g the minimal numbers of vertices of combinatorial manifolds and of polyhedra coincide in the first previously unknown case g=4 also. We show that our polyhedron has the maximal symmetry for the given genus and minimal number of vertices.

Projective Geometry on Modular Lattices

Publisher Summary This chapter focuses on projective geometry on modular lattices. Incidence and Order are basic concepts for a foundation of modern synthetic geometry. These concepts describe the relative location or containment of geometric objects and have led to different lines of geometry, an incidence-geometric and a lattice-theoretic one. Modularity is one of the fundamental properties of classical projective geometry. It makes projections into join-preserving mappings and yields perspectivities to be (interval) isomorphisms. It is therefore natural that order-theoretic generalizations of projective geometry are based on modular lattices and even more, the theory of modular lattices may be considered as a most general concept of projective geometry. In particular, the partially ordered set of all submodules of a module forms a (complete) modular lattice; even more general, any sublattice of the lattice of all normal subgroups of a group is a modular lattice. It considers that lattice-geometric approaches are complete geometrical structures whose geometrical objects form complete (modular) lattices.

Neighborly maps with few vertices

A neighborly map is a simplicial 2-complex which decomposes a closed 2-manifold without boundary, such that any two vertices are joined by an edge (1-cell) in the complex. We find and describe all the neighborly maps with Euler characteristicX>−10 (i.e., genusg<6, if orientable) or, equivalently, all the neighborly maps withV<12 vertices.

Extensions of distance reducing mappings to piecewise congruent mappings on ℓm

In this note I will show any distance reducing mapping f: M → ℓn, where M is a finite subset of ℓm (m ≤ n), can be extended to a piecewise conqruent mapping f: ℓm → ℓn.