Compact ovoids in quadrangles. I: Geometric constructions (Q1818719)

An ovoid in a generalized quadrangle \(Q\) is a set of points which meets every line of \(Q\) in a unique point. The authors consider closed ovoids in compact connected generalized quadrangles \(Q\) (of finite topological dimension), and they show that the existence of such an ovoid has strong consequences for the topological parameters of \(Q\) (these parameters are an analogue of the order of a finite generalized quadrangle). The study of ovals in connection with subquadrangles leads to a number of new results on subquadrangles of compact connected generalized quadrangles. Many of the deeper results depend on the fibre bundle interpretation of an ovoid, and on a dose of algebraic topology. Part II deals with classical generalized quadrangles, and part III with Clifford algebras and isoparametric hypersurfaces [both to appear in Geom. Dedicata].