Topological Affine Research Articles

A new discrete theory of pseudoconvexity

Recently geometric hypergraphs that can be defined by intersections of pseudohalfplanes with a finite point set were defined in a purely combinatorial way. This led to extensions of earlier results about points and halfplanes to pseudohalfplanes, including polychromatic colorings and discrete Helly-type theorems about pseudohalfplanes. Here we continue this line of research and introduce the notion of convex sets of such pseudohalfplane hypergraphs. In this context we prove several results corresponding to classical results about convexity, namely Helly's Theorem, Carath\'eodory's Theorem, Kirchberger's Theorem, Separation Theorem, Radon's Theorem and the Cup-Cap Theorem. These results imply the respective results about pseudoconvex sets in the plane defined using pseudohalfplanes. It turns out that most of our results can be also proved using oriented matroids and topological affine planes (TAPs) but our approach is different from both of them. Compared to oriented matroids, our theory is based on a linear ordering of the vertex set which makes our definitions and proofs quite different and perhaps more elementary. Compared to TAPs, which are continuous objects, our proofs are purely combinatorial and again quite different in flavor. Altogether, we believe that our new approach can further our understanding of these fundamental convexity results.

THE ERDŐS–SZEKERES PROBLEM FOR NON‐CROSSING CONVEX SETS

We show an equivalence between a conjecture of Bisztriczky and Fejes T{\'o}th about arrangements of planar convex bodies and a conjecture of Goodman and Pollack about point sets in topological affine planes. As a corollary of this equivalence we improve the upper bound of Pach and T\'{o}th on the Erd\H{o}s-Szekeres theorem for disjoint convex bodies, as well as the recent upper bound obtained by Fox, Pach, Sudakov and Suk, on the Erd\H{o}s-Szekeres theorem for non-crossing convex bodies. Our methods also imply improvements on the positive fraction Erd\H{os}-Szekeres theorem for disjoint (and non-crossing) convex bodies, as well as a generalization of the partitioned Erd\H{o}s-Szekeres theorem of P\'{o}r and Valtr to arrangements of non-crossing convex bodies.

Representation of relatively uniform and order convergence topologies by an inductive limit

Using the concept of a topological affine space, it is proved that a partially ordered topological linear spaces associated with relatively uniform and order convergence can be represented by an inductive limit of its subspaces.

Convexity in Topological Affine Planes

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here.

Topological affine planes with affine connections

Abstract We consider the question whether the system of lines of a two-dimensional topological plane can be described as the system of geodesics of a Riemannian metric or an affine connection. In [4] we have shown that non-classical compact projective planes do not admit Riemannian metrics. Here we use different methods that are suited to two-dimensional planes in general. We apply them to three families of topological affine planes with large collineation groups. It turns out that for the standard models of the skew-parabola planes and the planes over cartesian fields no affine connection, and hence no Riemannian metric exist. However, for the Moulton planes affine connections do exist and we determine all of them. Among them is at least one Riemannian connection. We also give an example where no Riemannian metric exists. Moreover, we derive a characterization of the classical affine plane in the case that ℝ2 acts by vector space translations as a subgroup of the collineation group.

Unitals in Topological Affine Translation Planes Need Not Be Strictly Convex

We study the relationship of two incidence geometric convexity notions, namely, ovoids in real affine spaces and compact unitals of codimension 1 in topological affine translation planes. In [3] we showed that every ovoid in a translation plane is a unital, and we asked if the converse is true. Here we introduce the notion of a shell, which is distinctly weaker than that of an ovoid and still implies the unital property if the translation plane is properly chosen (and the shell is not too degenerate). We give an explicit example of a shell that is not an ovoid. The question remains whether or not conversely, every compact unital of codimension 1 in a translation plane is a shell.

Planar-topologische Fastkörper

Topological planar nearfields (F,T) are suitable for coordinatizing topological affine and projective planes only if the solutions of equations of the type\( ax - bx = c, a \neq b \), depend continuously on\( a, b, c \in F \). In this case we call T a p-topology and deal with the problem, under which conditions a nearfield topology even is a p-topology. Satisfactory answers are given in each of the following three situations:¶¶1. T is induced by a valuation.¶2. T is locally compact.¶3. T is derived from a coupling \( \kappa \) on a topological skewfield with an open kernel and with continuous images \( \kappa(a) \).¶¶Furthermore, it is shown that the interval topology of an ordered nearfield always is a p‐topology.

Smooth Stable Planes and the Moduli Spaces of Locally Compact Translation Planes

Smooth stable planes have been introduced in [3]. At every point p of a smooth stable plane \(\) the tangent spaces of the lines through p form a compact spread (see the definition in Section 2) on the tangent space \(\) thus defining a locally compact topological affine translation plane \(\). We introduce the moduli space \(\) of isomorphism classes of compact spreads, \(\). We show that for \(\) the topology of \(\) is not \(\) by constructing a sequence of non-classical spreads in \(\) that converges to the classical spread in \(\), where \(\). Moreover, we prove that the isomorphism type of \(\) varies continuously with the point p. Finally, we give examples of smooth affine planes which have both classical and non-classical tangent translation planes.

Continuous planar functions

This paper deals with continuous planar functions and their associated topological affine and projective planes. These associated (affine and projective) planes are the so-called shift planes and in addition to these, in the case of planar partition functions, the underlying (affine and projective) translation planes. We introduce a method that allows us to combine two continuous planar functions ℝ → ℝ into a continuous planar function ℝ2 → ℝ2. We prove various extension and embedding results for the associated affine and projective planes and their collineation groups. Furthermore, we construct topological ovals and various kinds of polarities in the associated topological projective planes.

Coordinate rings of topological Klingenberg planes I: The affine perspective

Although the coordinate ternary field of a topological affine plane is topological, the converse does not hold. However, an affine plane is topological precisely when its coordinate biternary fields are topological. We extend this result to topological biternary rings and their topological affine Klingenberg planes. Then we examine the locally compact situation. Finally, following the ideas of Knarr and Weigand, we show that in certain circumstances, the continuity of the ternary operators is sufficient to ensure that the biternary ring is topological. This facilitates the construction of locally compact, locally connected affine Klingenberg planes.

A note on Hilbert and Beltrami systems

Hilbert and Beltrami (line- ) systems were introduced by H. Mohrmann, Math. Ann. 85 (1922) p.177- 183. These systems give examples of non- desarguesian affine planes, in fact, the earliest known examples are of this type. We describe a construction for “generalized Beltrami systems”, and show that every such system defines a topological affine plane with point set ℝ2. Since our construction uses only the topological structure of ℝ2- planes, it is possible to iterate this process. As an application, we obtain an embeddability theorem for a class of two- dimensional stable planes, including Strambach’s exceptional SL2R- plane.

A Class of Topological Affine Translation Planes Having no Topological Projective Extension

A Class of Topological Affine Translation Planes Having no Topological Projective Extension

Konstruktion topologischer affiner Ebenen mit nichtstetigem Parallelismus durch Knicken von Geraden

We prove the existence of a class of topological affine planes having non-continuous parallelism by using [2, Satz 5.2]. For this, we introduce a new method of constructing affine Salzmann-planes with a ‘monotonically increasing’ slope (see 2.1) by bending lines on two special curves, which are not necessary lines. Furthermore, the limit inferior of a sequence of topological planes with fixed point space is defined. As application of our new method, we construct a sequence of affine Salzmann-planes such that the limit inferior of this sequence is again an affine Salzmann-plane and fulfils the assumptions of [2, Satz 5.2]. Applying this theorem repeatedly, we get a sequence of non-isomorphic topological affine subplanes with non-continuous parallelism.

Topologische Divisionsalgebren ohne zugehörige topologische affine Ebene

We prove that many (non-associative) topological division algebrasD of dimensionn ∈ N over the centreK do not yield topological affine or projective planes (of Lenz-Barlotti type V) in contrast to the results of SKORNJAKOV [20], SALZMANN [18] and [19], GRUNDHOFER [7], HARTMANN [11] and RINK [17] concerning projective planes coordinatized by compact or special topological ternary fields. In particular, this holds for every non-trivial and non-archimedian valuation topology ofK distinct from the order topology ifK is a real-closed field, and if the division algebraD =Kn carries the product topology.

Topologische Affine Ebenen Mit Nichtstetigem Parallelismus

In this paper, we present a general method of constructing topological affine planes having non-continuous parallelism. We prove that a topological affine plane E with point set Lk×Lk, and with a special ‘K-algebraic slope’ has a topological affine subplane with non-continuous parallelism (Satz 4.6). Here, K is a real-closed subfield of a real-closed field L. The crucial tools needed to make our method work are the notion of a slope and the notion of K-algebraicity, a concept which is introduced and intensively studied here. As an application of our general method, we obtain in Section 5 affine Salzmann planes with lines being bent countably infinitely often admitting a subplane with non-continuous parallelism. This provides a negative answer to a question posed by H. Salzmann [13, p. 52].

Cartesian groups not belonging to topological projective planes

A new class of topological Cartesian groups is introduced by bending the lines of the affine coordinate plane over an ordered skewfield countably infinitely often. These Cartesian groups belong to topological affine planes with continuous parallelism, but they need not yield a topological projective plane if the topology is not equal to the order topology of the skewfield, in contrast to the results of HARTMANN [6] for generalized Moulton planes.

Vierdimensionale ?*�SU2?-Ebenen

Examples of 4-dimensional topological affine planes with a 5-dimensional group of automorphisms containing C*·SU2C are given. They are of Lenz-Barlotti class I.2.

Topological affine spaces

Some characterizations of the topological affine spaces are already known [2,5,6]; they are given via the topologies on the sets of points and hyperplanes. According to the definition made by Sorensen in [6], a topological affine space is an affine space whose sets of points and hyperplanes are endowed with non-trivial topologies such that the joining of n independent points, the intersection of n independent hyperplanes and the construction of parallel hyperplanes are continuous operations. In this paper we give a new characterization of such spaces by means of the topologies on the sets of points and lines.

Uniform semigroups and fixed point properties

LetS be a uniform semigroup (this includes all topological groups and affine semigroups). We show that a certain space of uniformly continuous functions onS has a left invariant mean iffS has the fixed point property for uniformly continuous affine actions ofS on compact convex sets. This is closely related to but independent of the results of T. Mitchell in [13] and A. Lau in [10]. Interesting examples and consequences are given for the special cases of topological groups and affine convolution semigroups of probability measures on a locally compact semigroup or group.

Topological affine planes composed of two desarguesian halfplanes and projective planes with trivial collineation group

Topological affine planes composed of two desarguesian halfplanes and projective planes with trivial collineation group