Stable planes
A stable plane is a linear space \(\mathbb{M} = (M, {\mathcal M})\) in which the point set \(M\) and the set of lines \({\mathcal M}\) carry locally compact, finite-dimensional, connected Hausdorff topologies such that the geometric operations of joining points and intersecting lines are continuous on their domains of definition and such that the set of pairs of intersecting lines is open in \({\mathcal M} \times {\mathcal M}\) (stability axiom). A fundamental result of \textit{R. Löwen} [J. Reine Angew. Math. 343, 108-122 (1983; Zbl 0524.57011)] says that \(M\) and \({\mathcal M}\) are of the same dimension 2, 4, 8, or 16. Classical examples of \(2l\)-dimensional stable planes are the \(2l\)-dimensional projective planes \(P_ 2\mathbb{F}\) over the \(l\)-dimensional real alternative division algebra \(\mathbb{F}\) (i.e. the reals, the complex numbers, Hamilton's quaternions and Cayley's octonious) and their open connected subsets with the induced geometry. Given two stable planes \(\mathbb{D} = (D, {\mathcal D})\) and \(\mathbb{E} = (E, {\mathcal E})\) the author deals with the problem whether or not there is an open embedding \((\pi, \lambda)\) of \(\mathbb{D}\) into \(\mathbb{E}\), that is, \(\pi : D \to E\) and \(\lambda : {\mathcal D} \to {\mathcal E}\) are injective continuous maps and \(\pi (D)\) is open in \(E\). The latter condition can be replaced by the assumption that \(\lambda\) maps each pencil of lines \({\mathcal D}_ p\) through \(p \in D\) onto the pencil \({\mathcal E}_{\pi (p)}\). From this fact a simple criterion that a given stable cannot be embedded as an open subplane of a compact projective plane is derived. This criterion applies to many stable planes as illustrated by a number of examples. A short report on the present state of the program of classifying stable planes according to the dimension of their automorphism groups and an extensive list of references is included in the paper.
scientific article; zbMATH DE number 599971
scientific article