Eine projektive Ebene heist topologisch, wenn der Punktraum IP und der Geradenraum 5~ so mit Topologien versehen sind, das Schneiden und Verbinden stetige Operationen werden. In einer Theorie solcher Ebenen wird man konkrete und umfassende Resultate nur unter topologischen Zusatzvoraussetzungen erwarten konnen. Ahnlich wie in der Theorie der topologischen Gruppen hat sich die Forderung, das die betrachteten Topologien lokalkompakt und zusammenhaingend sein sollen, als guter Ausgangspunkt erwiesen.

Zur Klassifikation von 8- und 16-dimensionalen lokalkompakten Translationsebenen nach ihren Kollineationsgruppen

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 Hilbert's 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.

Linear Topological Geometries

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On the kernel and the nuclei of 8-dimensional locally compact quasifields

Die vorliegende Arbeit untersucht lokal kompakte zusammenhangende topologische Translationsebenen, auf deren Translationsachse L∞ eine Gruppe von stetigen affinen Kollineationen eine Bahn hat, die zu einer Sphare relativ groser Dimension homoomorph ist. Das diese Situation naturlich auftritt und haufig vorkommt, zeigt das folgende.

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Lokalkompakte zusammenhängende Translationsebenen mit groβen Sphärenbahnen auf der Translationsachse

Topological pseudo-ovals, elation laguerre planes, and elation generalized quadrangles

This note deals with the transposition of translation planes in the topological context. We show that a topological congruenceC of the real vector space ℝ2n has the property that every hyperplane of ℝ2n contains a component ofC. This makes it possible to define the transposePτ of the topological translation planeP associated withC; it is proved that the translation planePτ is topological also. The relationship between collineation groups and the relationship between coordinatizing quasifields ofP andPτ are also discussed.

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Thomas Buchanan

Papers

On the kernel and the nuclei of 8-dimensional locally compact quasifields

The transposition of locally compact, connected translation planes