Stable planes
type: Item
instance of: scholarly article
title: Stable planes
zbMATH Open document ID: 0802.51015
DOI: 10.1016/0012-365X(92)00512-P
published in: Discrete Mathematics
publication date: 1994-12-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 599971
zbMATH Keywords: topological linear space
zbMATH Keywords: open embedding
zbMATH Keywords: stable planes
author: Markus Stroppel
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
cites work: Topologische Geometrien auf dem Möbiusband
cites work: 4-dimensional compact projective planes with a 7-dimensional collineation group
cites work: Wirkungen und Geometrien der Gruppe \(L_2\times\mathbb{R}\)
cites work: Point homogeneous flat affine planes
cites work: \(R^2\)-planes with 2-dimensional point transitive automorphism group
cites work: Pasting of \(\mathbb{R}^ 2\)-planes
cites work: Isomorphism types of arc planes
cites work: Q3951152
cites work: Geometric lattices with topology
cites work: Embedding geometric lattices with topology
cites work: Topological geometric lattices
cites work: \(R^ 2\)-planes with 3-dimensional automorphism group fixing precisely a line
cites work: Topological projective spaces
cites work: Lie algebras of locally compact groups
cites work: Vierdimensionale stabile Ebenen
cites work: Locally compact connected groups acting on Euclidean space with Lie isotropy groups are Lie
cites work: Halbeinfache Automorphismengruppen von vierdimensionalen stabilen Ebenen sind quasi-einfach
cites work: Symmetric planes
cites work: Classification of 4-dimensional symmetric planes
cites work: Weakly flag homogeneous stable planes of low dimension
cites work: Central collineations and the parallel axiom in stable planes
cites work: Q3911029
cites work: Equivariant embeddings of low dimensional symmetric planes
cites work: Homogeneous compact projective planes.
cites work: Stable planes of low dimension admitting reflections at many lines
cites work: A local ''fundamental theorem'' for classical topological projective spaces
cites work: Two-dimensional stable planes with non-solvable automorphism group
cites work: Topology and dimension of stable planes: On a conjecture of H. Freudenthal.
cites work: Stable planes with isotropic points
cites work: Q3324552
cites work: Actions of SO(3) on 4-dimensional stable planes
cites work: A criterion for stability of planes
cites work: Actions of \(Spin_ 3\) on 4-dimensional stable planes
cites work: Stable planes admitting a classical motion group
cites work: Four-dimensional compact projective planes with a nonsolvable automorphism group
cites work: Topologische projektive Räume
cites work: Q3032917
cites work: Lokal desarguessche Salzmann-Ebenen
cites work: Lokal desarguessche Geometrien auf dem Möbiusband
cites work: Zweidimensionale topologische Geometrien, in denen lokal die dreifache Ausartung des Desarguesschen Satzes gilt
cites work: Characterization of the three classical plane geometries
cites work: Zur Klassifikation topologischer Ebenen. I, II
cites work: Zur Klassifikation topologischer Ebenen. III
cites work: Topological planes
cites work: Kollineationsgruppen ebener Geometrien
cites work: Geometries on surfaces
cites work: Kompakte, vier-dimensionale projektive Ebenen mit 8-dimensionaler Kollineationsgruppe
cites work: Homogene kompakte projektive Ebenen
cites work: Homogene affine Ebenen
cites work: Compact 16-Dimensional Projective Planes with Large Collineation Groups. IV
cites work: Compact 8-dimensional Projective Planes
cites work: Q3767936
cites work: Q3791788
cites work: Generalized symmetric planes
cites work: Connected 4-dimensional stable planes with many central collineations
cites work: Classification of 4-Dimensional Generalized Symmetric Planes
cites work: Q5828496
cites work: Q3249418
cites work: Vierdimensionale \({\mathbb{C}}^*\cdot SU_ 2{\mathbb{C}}\)-Ebenen. (Four- dimensional \({\mathbb{C}}^*\cdot SU_ 2{\mathbb{C}}\)-planes)
cites work: Salzmann-Ebenen mit hinreichend vielen Punkt- oder Geradenspiegelungen
cites work: Eine Charakterisierung der klassischen Geometrien
cites work: Zur Klassifikation von Salzmann-Ebenen mit dreidimensionaler Kollineationsgruppe
cites work: Zur Klassifikation von Salzmann-Ebenen mit dreidimensionaler Kollineationsgruppe. II
cites work: Q5603756

Stable planes (cites work)
cites work: Zentrale und axiale Kollineationen in Salzmann-Ebenen

Stable planes (cites work)
cites work: Gruppentheoretische Charakterisierungen klassischer desarguesscher und moultonscher Ebenen.

Stable planes (cites work)
cites work: A characterization of quaternion planes

Stable planes (cites work)
cites work: Reconstruction of incidence geometries from groups of automorphisms

Stable planes (cites work)
cites work: Planar groups of automorphisms of stable planes

Stable planes (cites work)
cites work: Quasi-perspectivities in stable planes

Stable planes (cites work)
cites work: Quaternion Hermitian planes

Stable planes (cites work)
cites work: Locally Compact Hughes Planes

Stable planes (cites work)
cites work: Q4032322

Stable planes (cites work)
cites work: Q3032916

Stable planes (cites work)
cites work: Topological affine spaces

Stable planes (cites work)
cites work: On topological projective spaces and their Grassmannians

Stable planes (cites work)
cites work: Q3211894

Stable planes (cites work)
cites work: Defining topological projective spaces and topological Grassmann spaces

Stable planes (full work available at URL)
full work available at URL: 0012-365x(92)00512-p

Stable planes (OpenAlex ID)
OpenAlex ID: W2912469652