Topological Projective Planes Research Articles

16-dimensional compact projective planes with 3 fixed points

Let 9 = (P, £) be a topological projective plane with a compact point set P of finite (covering) dimension d = dim P > 0. A systematic treatment of such planes can be found in the book Compact Projective Planes [15]. Each line L e £ is homotopy equivalent to a sphere §/ with / 1 8, and d = 2/, see [15] (54.1 1). In all known examples, L is in fact homeomorphic to §/. Taken with the compact-open topology, the automorphism group Σ = Aut^ (of all continuous collineations) is a locally compact transformation group of Ρ with a countable basis, the dimension dim Σ is finite [15] (44.3 and 83.2). The classical examples are the planes κ over the three locally compact, connected fields IK with ( — dim IK and the 16-dimensional Moufang plane G = έΡ® over the octonion algebra Θ. If 9 is a classical plane, then Aut^ is an almost simple Lie group of dimension Q, where C = 8, €2 = 16, €4 = 35, and Cg = 78. In all other cases, dim Σ < ^ Q + 1 < 5Λ Planes with a group of dimension sufficiently close to C( can be described explicitly. More precisely, the classification program seeks to determine all pair s (&, Δ), where Δ is a connected closed subgroup of Aut & and bf ^ dim Δ 5^ for a suitable bound be 4/ — 1 . This has been accomplished for t ^ 2 and also for 64 = 17. Here, the case ( = 8 will be studied; the value of bt varies with the configuration of the fixed elements ofA. Most theorems that have been obtained so far require additional assumptions on the structure of Δ. If dim Δ 27, then Δ is always a Lie group [12]. By the structure theory of Lie groups, there are 3 possibilities: (i) Δ is semi-simple, or (ii) Δ contains a central torus subgroup, or (iii) Δ has a minimal normal vector subgroup, cf. [15] (94.26). The first two cases are understood fairly well:

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.

Coordinate rings of topological Klingenberg planes II: the algebraic foundation for a projective theory

Ternary fields are the coordinate rings of affine and projective planes; however, the planes constructed over topological ternary fields are not necessarily topological. Surprisingly, the explanation of this phenomenon becomes evident in the more general theory of topological Klingenberg planes as we exhibited in [3] for the affine case. However, in the projective setting, we have a more formidable task. We must develop a new coordinate ring that admits a topological structure suitable for coordinatizing topological PK-planes. We accomplish this in two stages. In this paper, we revisit the standard coordinate rings [1, 11], discuss and resolve their deficiencies by developing a new coordinate ring as a unique extension of these refined standard rings. In a subsequent paper [4], we show that this new ring can be suitably topologized to coordinatize a topological PK-plane. This last result can then be used to explain why topological ternary fields do not necessarily coordinatize topological projective planes.

Existenzbedingungen für Dualitäten projektiver Ebenen

In a projective plane a permutation of the set of all points and lines is constructed, using only the operations join and meet. Under certain conditions (identities in a coordinatizing ternary field; special cases of Desargues- and Pappos-theorem) this permutation is a duality. For a topological projective plane this duality proves, that point space and line space of the plane are homeomorphic.

Oriented Rank Three Matroids and Projective Planes

Recently, Goodman et al. have proven two conjectures by Grünbaum right, showing that any arrangement of pseudolines in the plane can be embedded into a flat projective plane and that there exists a universal topological projective plane in which every arrangement of pseudolines is stretchable. By Folkman and Lawrence’s theorem , this plane contains every finite (simple) oriented rank three matroid. In this paper, we will also consider embeddings of oriented rank three matroids into topological projective planes, but we will take a quite different viewpoint: we shall show that there exists a projective plane Π that contains the combinatorial geometry of every finite, orientable rank three matroidMn , such that any choice of orientations χnof theMn , n∈N, extends to an orientation χ ofΠ . Furthermore, these orientations correspond to archimedean orderings of Π, hence the reorientation classes of every finite rank three matroid can be studied by the set of archimedian orderings of Π. Since, by a celebrated result of Prieß-Crampe , any archimedian projective plane can be completed and thus embedded into a flat projective plane, our results yield another proof of Grünbaum’s conjectures and a new proof of the rank three case of Folkman and Lawrence’s theorem.

A Note on the Place Topology of Projective Planes

It is shown that the place topology induced by a proper epimorphism of a projective plane Π, which is known to make Π a Lenz-topological plane, makes Π even a topological projective plane, if the extended radical of some underlying ternary field is bounded.

Convergence of automorphisms of compact projective planes

We show that a pointwise convergent sequence (σn)n∈N of continuous collineations of a compact projective plane converges uniformly if and only if the pointwise limitα of (σn)n∈N has a quadrangle in its image. Moreoverα is then a continuous collineation. Furthermore, we derive that a homomorphism between topological projective planes is continuous if and only if it is continuous at some point.

Extending Homomorphisms of Dense Projective Subplanes by Continuity

Let Ψ be a dense projective subplane of a topological projective plane Π. We show that a continuous homomorphism a of Ψ is extendable to a continuous homomorphism of Π if and only if there is a line Z of Ψ such that the restriction of α to the Ψ-points of Z is continuously extendable to some mapping defined on all Π-points of Z. In particular, each projective collineation of Ψ is extendable to a projective collineation of Π yielding the well-known result that (z, A)-transitivity of Ψ extends to (z, A)-transitivity of Π.

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.

There are uncountably many universal topological planes

We prove the existence of uncountably many nonisomorphic topological projective planes, each ‘universal’ in the sense that it contains an isomorphic copy of every pseudoline arrangement.

Topological projective planes and uniform valuations

The valuation topology of any uniformly valued ternary field (K, T, v) can be extended to the projective plane II over (K, T) making it a topological projective plane in the sense of Salzmann. Appealing to Pries-Crampe's celebrated fixed point theorem for ultrametric spaces, our result allows us to present a wide variety of new, totally disconnected, compact and non-compact topological projective planes.

On flags in a topological projective plane

In this paper we study some properties of the flag space ∑(π) of a topological projective plane π. The line set, sayL, of ∑(π), splits into two subfamiliesL0 andL1 which an non-trivial partitions of the point set of ∑(π). For any two distinct lines l′, l′' belonging to one of the two subfamilies, there is exactly one line l in the other subfamily meeting botl l′ and l′'; l turns out to be a continuous mapping of (-l′,l′'−. Further, the intersection o: two lines, having exactly one point in common, is a continuous and open mapping. Conversely, every irreducible semilinear space, satisfying the above properties, is isomor phic to the flag space of a topological projective plane.

Large automorphism groups of 8-dimensional projective planes are Lie groups

We prove the following theorem: LetP be an 8-dimensional compact topological projective plane. If the connected component Δ of its automorphism group has dimension at least 12, then Δ is a Lie group.

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.

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.

Topological ternary fields not belonging to a topological projective plane

Every topological projective plane E can be coordinatized by a topological ternary field. As an application of the papers [4] and [5] we exhibit examples of topological ternary fields which do not yield a topological projective plane in contrast to the results of SKORNJAKOV [13], PICKERT [7], SALZMANN [11] and [12], RINK [9], and GRUNDH/SFER [6]. Thus, the question whether every topological ternary field belongs to a topological projective plane (cp. [6] p.87, [7] p.265, or [8] p.193) is answered in the negative. In section 2 we show that if E is a topological projective plane and F any non-discrete affine subplane of E coordinatized by a ternary field T, then T belongs to a topological projective plane if and only if T is the restriction of a ternary field of E to Dmn T, that is, the projective extension of F is a projective subplane of E (Theorem 2.1 and Corollary 2.4). To prove the first result, in section 1 we consider an arbitrary topological projective plane E with a special affine subplane EA having non-continuous paralelism. In [3] and [5] we have shown the existence of such affine subplanes (see Example 1.5). Then we shall obtain a topological ternary field T by taking an affine line, namely the 'exceptional line', as new line at infinity. But T does not belong to a topological projective plane because EA has a non-continuous paralelism (Theorem 1.9).

Topology in generalized quadrangles

In analogy to topological projective planes, topological generalized quadrangles are investigated. The point set and all lines of every (locally) compact connected quadrangle are integral homology manifolds and ANRs.

Sechzehndimensionale lokalkompakte Translationsebenen, deren Kollineationsgruppe G2 enth�lt

New results of Salzmann and Hubig say that a 16-dimensional (locally) compact topological projective plane in which the group \(\mathbb{G}\) of continuous collineations has dimension ≥40 is a translation plane. It is therefore important to determine all 16-dimensional locally compact translation planes with dim \(\mathbb{G}\)≥40. From previous work of the author ([10]), it is known that such a plane is either the classical octonion plane, or dim \(\mathbb{G}\)=40 and \(\mathbb{G}\) contains a subgroup isomorphic to the compact exceptional group G2, but no larger compact simple subgroup. In the present paper, all planes satisfying the latter property more generally with dim \(\mathbb{G}\)≥38 are explicitly determined. Together with the classification of all 16-dimensional locally compact translation planes in which \(\mathbb{G}\) contains Spin(7) given by the author in [8], one thus knows all 16-dimensional locally compact translation planes with \(\mathbb{G}\) containing G2 and dim \(\mathbb{G}\)≥38. Via suitable Baer subplanes, the classification makes use of analogous results for 8-dimensional planes ([7]).

Minimal topological projective planes

The atoms of the lattice of compatible topologies on a given projective plane are examined. The notion of a field of type V is generalized to ternary fields of type V. These are always minimal, and they arise for example as coordinate structure of strictly uniformizable or of orderable topological planes. Hence, such planes are minimal.

Compact 16-Dimensional Projective Planes with Large Collineation Groups. IV

Let be a topological projective plane with compact point set P of finite (covering) dimension. In the compact-open topology (of uniform convergence), the group Σ of continuous collineations of is a locally compact transformation group of P.THEOREM. If dim Σ &gt; 40, thenis isomorphic to the Moufang plane 6 over the real octonions (and dim Σ = 78).By [3] the translation planes with dim Σ = 40 form a one-parameter family and have Lenz type V. Presumably, there are no other planes with dim Σ = 40, cp. [17].