Rainer Löwen

Four-dimensional compact projective planes with a nonsolvable automorphism group

We contribute to the enumeration of all four-dimensional compact projective planes with an at least seven-dimensional automorphism group (cf. Betten [8]) by treating the nonsolvable case. Moreover, we find that the only possible six-dimensional nonsolvable automorphism group is ℝ2 · GL+2ℝ.

Compact spreads and compact translation planes over locally compact fields

We prove that a spread S over a locally compact nondlscrete field F defines a topological translation plane if and only if the spread is compact. For F=R, this is implicit in Breunings thesis [Bre], cf. [B 2]. For the proof, we describe the point set of the projective translation plane as a quotient space of some projective space, with identifications taking place in one hyperplane. This is new even for F=R.

Ends of surface geometries, revisited

Using the Freudenthal compactification, we show that each stable plane whose lines are connected curves has for point space either the open disk, or the compact surface of genus 1, or the Möbius strip. This continues an investigation of Salzmann (Pacific J. Math.29 (1969), 397–402).

Compactness of the Automorphism Group of a Topological Parallelism on Real Projective 3-Space

We conjecture that the automorphism group of a topological parallelism on real projective 3-space is compact. We prove that at least the identity component of this group is, indeed, compact.

Self-Orthogonal Compact Spreads and Unitals in Topological Translation Planes

AbstractA spread of

Four-dimensional compact projective planes admitting an affine Hughes group

V = \mathbb{R}^{2l}

Parallelisms of \(\mathrm{PG}(3,\mathbb R)\) admitting a 3-dimensional group

is a set of l-dimensional subspaces L ⩽ V partitioning V ∖ {0}. We construct examples of compact spreads that are identical with their sets of orthogonal spaces L⊥. In the corresponding topological translation planes, every Euclidean sphere is a unital with the additional property that every point at infinity has flat feet.

Non-classical 4-dimensional Minkowski planes obtained as brothers of semiclassical 4-dimensional Laguerre planes

In 2000, Bodi and Immervoll considered compact, connected smooth incidence geometries with mutually transversal point rows and mutually transversal line pencils. They made the very natural assumptions that the flag space is a 31-dimensional closed smooth submanifold of the product of the point space and the line space (both of which are 21-manifolds) and that both associated projections are submersions. They showed that then the number of joining lines of two distinct points and the number of intersection points of two distinct lines are constant. Here we prove that both constants are equal to one. Thus, smooth projective planes are characterized using only compactness and connectedness plus the purely local (in fact, infinitesimal) conditions stated above.