Nonexistence of disjoint compact Baer subplanes
type: Item
instance of: scholarly article
title: Nonexistence of disjoint compact Baer subplanes
zbMATH Open document ID: 0941.51014
author: Rainer Löwen
published in: Archiv der Mathematik
publication date: 2000-07-30T00:00:00Z
review text: A closed subplane \({\mathcal Q}=(Q,{\mathcal M})\) in a topological compact connected finite-dimensional projective plane \({\mathcal P}=(P,{\mathcal L})\) is a Baer subplane if the point set \(Q\) of \({\mathcal Q}\) has half the dimension of the point set \(P\) of \({\mathcal P}\). Baer subplanes frequently arise as fixed point sets of automorphisms. In particular, if two Baer subplanes are the fixed point sets of Baer involutions, then it is well known that they have both a point and a line in common. The author shows that this remains true for any two compact Baer subplanes (not necessarily associated with Baer involutions). This result is in marked contrast to the finite case where the point set of every finite desarguesian projective plane may be partitioned into mutually disjoint Baer subplanes. The method of proof is to first show that for a Baer subplane \({\mathcal Q}=(Q,{\mathcal M})\) of a compact connected \(4k\)-dimensional projective plane \({\mathcal P}=(P,{\mathcal L})\), where \(k=1,2,4\), the complement \(P\setminus Q\) is a locally trivial fibre bundle over \({\mathcal M}\) with fibres homotopy equivalent to the sphere of dimension \(k-1\). The fibre map \(\alpha\) sends a point of \(P\setminus Q\) to the unique line to the unique line in \({\mathcal M}\) incident with it. In the next step, using various homology and cohomology groups, Lefschetz duality and the topology of compact Baer subplanes, the author then proves that the fibre bundle associated with a Baer subplane does not admit a cross section. The final step is to construct a cross section to \(\alpha\) from a second disjoint Baer subplane \({\mathcal Q'}=(Q',{\mathcal M'})\) and thus obtain a contradiction. By a result of Salzmann two Baer subplanes that have a line in common also have a point in common so that in an assumed counterexample both point sets \(Q\) and \(Q'\) and line sets \({\mathcal M}\) and \({\mathcal M'}\) are disjoint. By duality one obtains a fibre map \(\beta:{\ma…
Mathematics Subject Classification ID: 51H10
Mathematics Subject Classification ID: 51H20
zbMATH DE Number: 1375104
zbMATH Keywords: compact projective plane
zbMATH Keywords: Baer subplane
zbMATH Keywords: fibre bundle
zbMATH Keywords: cross section
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
full work available at URL: s000130050033
OpenAlex ID: W2055601618
DOI: 10.1007/S000130050033
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description: scientific article; zbMATH DE number 1375104