Flocks of topological circle planes

Abstract

We prove that every flock of a finite-dimensional locally compact connected circle plane is homeomorphic to R or S 1 and that every flock of a real Miquelian circle plane defines a compact 4-dimensional translation plane. Furthermore we investigate (topological) properties of regulizations. These properties are used to relate the automorphism group of a flock to the automorphism group of the corresponding translation plane.

A new family of locally compact 4-dimensional translation planes admitting a 7-dimensional collineation group

“…The new planes were also found by S. Blaschke [4]. He obtained them from flocks of the real Laguerre plane using a general method to construct translation planes from flocks of Miquelian circle planes described by N. Rosehr [11]. Nevertheless, as not all translation planes can be obtained in this way, it might still be interesting to see how the new planes can be constructed using information about their automorphism groups.…”

Compact Tits quadrangles as Lie geometries of topological Laguerre spaces

“…is a disjoint union of (n − 2)-spheres, a contradiction; see [22], Theorem 2.1(a). Hence for all co-lines x ∈ C ⊆ T x there is another tangent hyperplane of O incident with C.…”