Topological Polygons

Abstract

It is quite natural to marry geometry to topology, since the classical geometries over ℝ or ℂ carry special topologies, which are indispensable for characterizing these geometries (in Hilbert’s foundations of geometry [1899], the necessary topological assumptions are disguised in terms of orderings; in order to include ℍ and ${\mathbb O}$, Kolmogoroff [1932] states topological axioms: compactness, connectedness). In this chapter we give a survey of the theory of topological generalized polygons, focusing on the case of compact connected topologies. For projective planes (i.e., generalized triangles) there exists a rich theory which is expounded in the monograph by Salzmann, Betten, Grundhöfer, Hähl, Löwen & Stroppel [1995]. As a topological analogue of Theorem 1.7.1 of Feit & Higman [1964], Knarr [1990] and Kramer [1994a] have proved that compact connected n-gons of finite topological dimension exist only for n ∈ {3, 4, 6}; see Theorem 9.5.1 below. At present, no example of a non-classical compact connected hexagon is known, but many generalized quadrangles of this type have been constructed (this is analogous to the finite case; compare Section 3.8.2 on page 134).