Unstetige Kollineationen 4-dimensionaler Ebenen. (Discontinuous collineations of four-dimensional planes) (Q578585)

There exist a lost of topological projective planes which do admit a discontinuous collineation, cf. \textit{H. A. Keller} [Pac. J. Math. 121, 397-406 (1986; Zbl 0546.12014)]. Among them is \({\mathcal P}_ 2{\mathbb{C}}\)- the classical complex projective plane. It has been conjectured by D. Betten that this property however characterizes \({\mathcal P}_ 2{\mathbb{C}}\) inside the class of all locally compact 4-dimensional topological projective planes, i.e. the class of all those topological projective planes \({\mathcal P}\) which are homeomorphic to \({\mathcal P}_ 2{\mathbb{C}}\)- cf. \textit{H. Salzmann} [Arch. Math. 20, 551-555 (1969; Zbl 0189.208)]. The author confirms this conjecture if in addition \({\mathcal P}\) contains a closed desarguesian Baer-subplane. Furthermore, such a subplane is shown to exist if the translation group of some line of \({\mathcal P}\) has a dimension of at least 3.