A characterisation of the generalized quadrangle \(Q(5,q)\) using cohomology
type: Item
instance of: scholarly article
title: A characterisation of the generalized quadrangle \(Q(5,q)\) using cohomology
zbMATH Open document ID: 0998.51005
DOI: 10.1023/A:1013812619953
author: Matthew Brown
published in: Journal of Algebraic Combinatorics
publication date: 2002-05-30T00:00:00Z
review text: The author generalizes a theorem of \textit{J. A. Thas} and \textit{S. E. Payne} [Geom. Dedicata 52, 227-253 (1994; Zbl 0804.51007)]: Let \(S\) be a generalized quadrangle of order \((q,q^2)\), \(q\) even, which has a subquadrangle \(S'\) isomorphic to \(Q(4,q)\). If each ovoid \(O_X\) of \(S'\) consisting of those points collinear with some point \(X\in S\setminus S'\) is an elliptic quadric, then \(S\) is isomorphic to \(Q(5,q)\). The new proof allows for dropping the assumption on \(q\). The proof utilizes the first cohomology group over \({\mathbb Z}_2\) of a simplicial simplex \(\Gamma\) that is associated to the geometry \(S\) and is based on the duality relation \(H_n(\Gamma,{\mathbb Z}_2)\cong H^n(\Gamma,{\mathbb Z}_2)\). The manuscript is self-contained and covers all technical details very thoroughly.
Mathematics Subject Classification ID: 51E12
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1747857
zbMATH Keywords: generalized quadrangle
zbMATH Keywords: subquadrangle
zbMATH Keywords: cohomology
zbMATH Keywords: ovoid
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1747857