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The point space of compact generalized quadrangles with parameter 1

The point space of compact generalized quadrangles with parameter 1 Abstract. Up to duality point space and line space of a compact generalized quadrangle with parameter 1 are homeomorphic to the point space and line space of the real orthogonal quadrangle. 1991 Mathematics Subject Classi®cation: 51E12, 51H15. 1 Introduction This paper settles the long pending question positively whether the point space of a compact quadrangle with parameter 1 is always as in the classical case (see [3, 3.18]). The basic idea for the proof is rather simple. Take four points pY qY rY s that form an ordinary quadrangle. Then q 4 p q q 4 q q q 4 r q q 4 s q R 3 and q 4 p q 4 q q 4 r q 4 s P, where q 4 v denotes the points not collinear with a point v. What remains to investigate is how the four copies of R 3 are put together to form P. It turns out that this depends only on the twisting number, a topological invariant introduced in [11]. The twisting number of a compact quadrangle with parameter 1 is either 0 or 1. For each of these numbers there is a classical example. The real orthogonal http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

The point space of compact generalized quadrangles with parameter 1

Schroth, Andreas E.
Forum Mathematicum , Volume 12 (4) – May 29, 2000
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References (6)

Publisher
de Gruyter
Copyright
Copyright © 2000 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2000.012
Publisher site
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Abstract

Abstract. Up to duality point space and line space of a compact generalized quadrangle with parameter 1 are homeomorphic to the point space and line space of the real orthogonal quadrangle. 1991 Mathematics Subject Classi®cation: 51E12, 51H15. 1 Introduction This paper settles the long pending question positively whether the point space of a compact quadrangle with parameter 1 is always as in the classical case (see [3, 3.18]). The basic idea for the proof is rather simple. Take four points pY qY rY s that form an ordinary quadrangle. Then q 4 p q q 4 q q q 4 r q q 4 s q R 3 and q 4 p q 4 q q 4 r q 4 s P, where q 4 v denotes the points not collinear with a point v. What remains to investigate is how the four copies of R 3 are put together to form P. It turns out that this depends only on the twisting number, a topological invariant introduced in [11]. The twisting number of a compact quadrangle with parameter 1 is either 0 or 1. For each of these numbers there is a classical example. The real orthogonal

Journal

Forum Mathematicum – de Gruyter

Published: May 29, 2000

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