Theo Grundhöfer

Topology in generalized quadrangles

Abstract 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.

Flag-Homogeneous Compact Connected Polygons

All flag-homogeneous compact connected polygons are classified explicitly. It turns out that these polygons are precisely the compact connected Moufang polygons.

Chapter 23 – Linear Topological Geometries

Publisher Summary This chapter discusses linear topological geometries and offers some observations concerning the role of topology in the foundations of geometry. Continuity certainly is an essential part of the geometric imagination. In fact, topological elements are indispensable for characterizations of the classical geometries. This was clear at the turn of the century, despite the fact that the abstract concepts and notions of topology had not yet emerged. Thus, Hilbert expressed the necessary continuity assumptions in terms of ordering properties. One direction of research in the first decades of this century was constituted by the control (and, to some extent, the elimination) of the influence of ordering axioms. The notion of a topological projective plane was introduced and studied systematically by Skornjakov, Salzmann and Freudenthal. Unlike Hilberts system of axioms, this topological approach includes complex and quaternion projective spaces, as well as the octonion plane; compare the lucid review in Freudenthal. There is a related approach, which eliminates all incidence axioms in flavor of assumptions on topological transformation groups characterizing the groups of motions.

A synthetic construction of the Figueroa planes

The constructions of the Figueroa planes by Figueroa, Hering-Schaeffer and Dempwolff make essential use of the collineation groups. Here we give a synthetic construction of these planes, which avoids coordinates and groups.

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hähl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hahl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).

Slanted symplectic quadrangles

By ‘slanting’ symplectic quadrangles W(F) over fieldsF, we obtain very simple examples of non-classical generalized quadrangles. We determine the collineation groups of these slanted quadrangles and their groups of projectivities. No slanted quadrangle is a topological quadrangle.

On Restrictions of Automorphism Groups of Compact Projective Planes to Subplanes

It is shown that the restriction of automorphisms of a compact projective plane to a closed Baer subplane or to an open subgeometry, respectively, is a quotient mapping (in contrast to restrictions to arbitrary subgeometries). In the proof, we investigate lineations in towers of Baer subplanes.

Compact disconnected planes, inverse limits and homomorphisms

Every compact disconnected projective plane can be written as an inverse limit of finite discrete incidence structures. Every finite projective plane is a continuous epimorphic image of some compact disconnected projective plane. There exist compact disconnected projective planes of Lenz type V which do not admit any continuous epimorphism onto a finite projective plane.

Non-Existence of isomorphisms between certain unitals

We show that the Ree-Tits unitals are neither classical nor isomorphic to the polar unitals found in the Coulter-Matthews planes. To this end, we determine the full automorphism groups of the finite Ree-Tits unitals.

Sharp homogeneity in affine planes, and in some affine generalized polygons

Let G be a collineation group of a generalized (2n + 1 )-gon Γ and let L be a line such that every symmetry σ of any ordinary (2n + 1 )-gon in Γ containing L with σ(L) = L extends uniquely to a collineation in G. We show that Γ is then a Desarguesian projective plane. We also describe the groups G that arise. As a corollary, we treat the analogous problem without the restriction σ(L) = L.