A characterisation of the generalized quadrangle \(Q(5,q)\) using cohomology
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.
scientific article; zbMATH DE number 1747857
scientific article