Compact ovoids in quadrangles. III: Clifford algebras and isoparametric hypersurfaces (Q1567097)

[For Part I see \textit{L. Kramer} and \textit{H. Van Maldeghem}, ibid. 78, No. 3, 279-300 (1999; Zbl 0942.51007), and for Part II see \textit{L. Kramer}, ibid. 79, No. 2, 179-188 (2000; Zbl 0953.51004).] The author uses real Clifford algebras and their modules for the construction of generalized quadrangles; this is inspired by work of \textit{D. Ferus}, \textit{H. Karcher} and \textit{H.-F. Münzner} [Math. Z. 177, 479-502 (1981; Zbl 0452.53032)] and \textit{G. Thorbergsson} [Duke Math. J. 67, No. 3, 627-632 (1992; Zbl 0765.51015)]. In fact, the author gives a new algebraic proof for the fact that the geometries obtained in this fashion are generalized quadrangles. The paper contains a comprehensive introduction to these quadrangles from the point of view of (topological) incidence geometry. The author gives explicit geometric constructions for spreads, ovoids, and sections, and he uses algebraic topology to decide which of these quadrangles have closed ovoids or spreads. Furthermore, the author determines whether the normal sphere bundles of the corresponding focal manifolds admit sections, or whether they are topologically trivial.